^/^.
\
A SYNOPSIS
ELEMENTAIIY RESULTS
PUKE MATHEMATICS
CONTAINTNO
PROPOSTTTONS.'FORMUL^, AND METHODS OF ANALYSIS.
WITH
ABRIDGED DEMONSTRATIONS.
SlTPLEMFNTFI) ItY AN InDEX TO THE PaI'ERS ON Pi HE JIaTHEMATU S -WHUJI AKE TO
BE FOVNn IN THE PHINCIPAL JoiBNALS AND TllANSACTlONS OF LEAUM I) Socill lEP,
poTH English and Foreion, of the present centuky.
G. S. CARR, M.A.
LONDON :
FRANCIS noi)(;soN, 80 farkin(tDon street, e.c
CAMBRIDGE: MACMILLAN & BOWES.
1886.
(AU rights reserved )
C3
neenng
Librai:'^
LONDON :
PRINTED BY C. F. HODGSON AND SON,
GOUGH SQUARE, FLEET STREET.
76. 6j6
N TVERSJT?
PREFACE TO PART I
Tin: work, of which tlio part now issued is a first instal-
ment, has been compiled from notes made at various periods
of the last fourteen years, and chiefly during* the engagements
of teaching. !Many of the abbreviated methods and mnemonic
rules are in the form in which I originally wrote them for my
pupils.
The general object of the compilation is, as the title
indicates, to present within a moderate compass the funda-
mental theorems, formulas, and processes in the chief branches
of pure and applied mathematics.
Tlie work is intended, in the first place, to follow and
supplement the use of tl,ic ordinary text-books, and it is
arranged witli tlie view of assisting tlie student in the task of
revision of book-w^ork. To this end I have, in many cases,
merely indicated the salient points of a demonstration, or
merely referred to the theorems by which the proposition is
proved. I am convinced that it is more beneficial to tlie
student to recall demonstrations with such aids, than to read
and re-read them. Let them be read once, but recalled often.
The difference in the effect upon the mind between reading a
mathematical demonstration, and originating cue wholly or
IV PEEFACE.
partly, is very great. It may be compared to tlic difference
between the pleasure experienced, and interest aroused, when
in the one case a traveller is passively conducted through the
roads of a novel and unexplored country, and in the other
case he discovers the roads for himself with the assistance of
a map.
In the second place, I venture to hope that the work,
when completed, may prove useful to advanced students as
an aide-memoire and book of reference. The boundary of
mathematical science forms, year by year, an ever widening
circle, and the advantage of having at hand some condensed
statement of results becomes more and more evident.
To the original investigator occupied with abstruse re-
searches in some one of the many branches of mathematics, a
work which gathers together synoptically the leading propo-
sitions in all, may not therefore prove unacceptable. Abler
hands than mine undoubtedly, might have undertaken the task
of making such a digest ; but abler hands might also, perhaps,
be more usefully emj^loycd, — and with this reflection I have the
less hesitation in commencing the work myself. The design
which I have indicated is somewhat comprehensive, and in
relation to it the present essay may be regarded as tentative.
The degree of success which it may meet with, and the
suggestions or criticisms which it may call forth, will doubt-
Ic: ■! have their effect on the subsequent portions of the work.
With respect to the abridgment of the demonstrations, I
may remark, that while some diffuseness of explanation is not
only allowable but very desirable in an initiatory treatise,
conciseness is one of the chief reciuiremcnts in a work intended
PREFACE. V
for tlio piii-i)OSos of revision and rcfiTeiico only. In order,
liowever, not to sacrifice clearness to conciseness, much moro
la])our has been expended upon this part of the subject-matter
of the book than will at first sip^ht be at all evident. The only
])alpal)le I'esult lK'in<^ a compression of the text, the result is
so far a neji^ative one. The amount of compression attained
is illustrated in the last section of the present part, in which
moro than the number of propositions usually given in
treatises on Geometrical Conies are contained, together with
the figures and demonstrations, in the s})ace of twenty-foui*
pages.
The foregoing remarks have a general application to the
work as a whole. With the view, however, of making the
earlier sections more acceptable to beginners, it will be found
tliat, in those sections, important principles have sometimes
been more fully elucidated and more illustrated by exam})les,
than the plan of the work would admit of in subsequent
di\isions.
A feature to which attention may be directed is the uni-
form system of reference adopted throughout all the sections.
AVithtlie object of facilitating such reference, the articles have
been numbered progressively from the commencement in
large Clarendon figures ; the breaks which will occasionally
be found in these numbers having been purposely made, in order
to leave room for the insertion of additional matter, if it should
be re(piired in a future edition, without distui-bing tlie oi'iginal
numbers and references. With the same object, demonstrations
and examples have been made subordinate to enunciations and
formidie, the former being [)rinted in small, the latter in bold
Vi rREFACE.
type. By tliese aids, tlic intordcpcndoncc of propositions is
more reudily sliown, and it becomes easy to trace the connexion
Ijetwecn theorems in different branches of mathematics, with-
out the loss of time which would be incurred in turning to
separate treatises on the subjects. The advantage thus gained
will, however, become more apparent as the work proceeds.
The Algebra section was printed some years ago, and does
not quite correspond mth the succeeding ones in some of
the particulars named above. Under the pressure of other
occupations, this section moreover was not properly revised
before f>'oing to press. On that account the table of errata
will be found to apply almost exclusively to errors in that
section ; but I trust that the hst is e:s.haustive. Great pains
liave been taken to secure the accm^acy of the rest of the
volume. Any intimation of errors will be gladly received.
I have now to acknowledge some of the sources from which
the present part has been compiled. In the Algebra, Theory
of Equations, and Trigonometry sections, I am largely in-
debted to Todhunter's well-known treatises, the accuracy and
completeness of which it would be superfluous in me to dwell
upon.
In the section entitled Elementary Geometry, I have added
to simpler propositions a selection of theorems from Town-
send's Modern Geometiy and Salmon's Conic Sections.
In Geometrical Conies, the line of demonstration followed
agrees, in the main, with that adopted in Drew's treatise on the
subject. I am inclined to think that the method of that
author cannot be much improved. It is true that some im-
portant properties of the ellipse, which are arrived at in
I'KKI'ACR. Vll
Drew's (\)nic Si^ctioiis tlirou^^h ccitnin iiitcnncdialc jn'oposi-
tions, can be dcMliicod at once from tlie circle ])y tlic metliod of
orthogonal projection. But the intcM-mediate propositions can-
not on that account be dispenscnl w itli, for they are of value in
th(Mns('lv(\^. ]\Ioreov(>r, tlie nietliod of projection applied to
Ili(> hyperbola is not so successful; because a pi-oj)erty which
lias first to bo proved true in the case of the equilateral
hyperbola, might as will be proved at once for the general case.
I have introduced the method of projection but spanngly,
alwavs giving prefei'ence to a demonstration which admits of
being n])])li(Ml in the same identical form to the ellipse and to
the hyperbola. The remarkable analogy subsisting between
the two curves is thus kept prominently before the reader.
The account of the C. G. S. system of units given in the
preliminary section, has been compiled from a valuable con-
tribution on the subject by Professor Everett, of Belfast,
published by the Physical Society of London.* This abstract,
and the tables of physical constants, might perhaps have found
a more appropriate place in an after part of the work. I have,
however, introduced them at the commencement, from a sense
of the great importance of the rcfonu in the selection of units
of measurement Avhich is embodied in the C G. S. system,
and from a belief that the student cannot be too early
familiarized with the same.
The Factor Table wliich folluAvs is, to its limited extent, a
rei)rint of Burckluirdt's " Tnhlr.^ ilrs diviscurs,'' published in
* "Illustrations of the Centimetrc-Grammc-Sccoiid System of Units."
London : Taylor and Francis. 1875.
Vlll riJEFACE.
1814-17, which give the least divisors of all numbers from 1
to 3,036,000. In a certain sense, it may be said that this is
the only sort of purely mathematical table which is absolutely
indispensable, because the information which it gives cannot
be supplied by any process of direct calculation. The loga-
rithm of a number, for instance, may be computed by a
formula. Not so its prime factors. These can only bo
arrived at through the tentative process of successive divisions
by the prime numbers, an operation of a most deterrent kind
when the subject of it is a liigh integer.
A table similar to and in continuation of Burckhardt's has
recently been constructed for the fourth million by J.^Y. L.
Glaisher, F.R.S., who I believe is also now engaged in com-
pleting the fifth and sixth millions. The factors for the seventh,
eighth, and ninth millions were calculated previously by Dase
and Rosenberg, and pubhshed in 1862-05, and the tenth
million is said to exist in manuscript. The history of the
formation of these tables is both instructive and interesting.*
As, however, such tables are necessarily expensive to pur-
chase, and not very accessible in any other way to the majority
of persons, it seemed to me that a small portion of them
would form a useful accompaniment to the present volume.
I have, accordingly, introduced the first eleven pages of Burckh-
ardt's tables, which give the least factors of tlie first 100,000
integers nearly. Each double page of the table here pi-iiUed is
* Sec " Factor Table for the Fourth JifiJltoii." By James Glaisher, F.R.S.
London: Taylor and Francis. 1880. Also Camh. J'hil. Soc. Proc, Vol. TIL,
Pt. IV., and Nature, No. 542, p. 402.
an exact rcpi'otluctiuii, iu all l)iit tlie tyi'e, of ;i Hiii^^e (|iiar(o
I)ai^n: of JJm-c'kluirdt's great work.
It may l)e noticed licro that Prof. Lebesquo constructed
ji tal)le to a))out tliis extent, on tlio ])lan of omlttinj^ tlio
n)ulti|)les of seven, and tlius re<lucins^^ tlie size of tlie tabic
))y about one-sixtli.* ]3ut a small calculation is re(|uired iu
using the table which counterbalances the advantage so gained.
The values of the Gamma-Function, pages 30 and 31 , have
been taken from Legendre's table in his ^'Excrciccs de Calcnl
Infnjral,'' Tome I. The table belongs to Part II. of tliis
Volume, but it is placed here for the convenience of having
all the numerical tables of Volume I. in the same section.
In addition to tlie autliors already named, tlie following
treatises have been consulted — Algebras, by AYood, Bourdon,
and Lefebure de Fourcy ; Snowball's Trigonometry ; Salmon's
Higher Algebra ; the Geometrical Exercises in Potts's Euclid ;
and Geometrical Conies by Taylor, Jackson, and Renshaw.
Articles 2G0, 431, o69, and very nearly all the examples,
are original. The latter have been framed with great care, in
order that they might illustrate the propositions as completely
as possible.
G. S. C.
Hadi.ry, ^Iiiii>i,i;.sKx ;
AIu)/ 23, 18ba.
* "Tables divcrscs pour la decomposition des nombrca en leura HietLurs
premiers." Par V. A. Lebesque. Paris. 18G-ii.
b
EIUiATA.
Art. 13,
„ 66,
Line 1,
„ 66,
„ 6,
,, 90,
„ 4,
„ 99,
„ 1,
„ 107,
„ 1,
„ 108,
„ 2,
,, 131,
„ 1,2
,, ,.
„ 5,
„ 133,
„ 3,0,
)) >>
„ 8,
M )>
„ 9,
„ „
„ 10,
„ 138,
„ 4,
„ 140,
„ 182,
», 6,
„ 191,
„ 4,
„ 220,
„ 6,
„ 221,
„ 4,
,,•237,
>, 11,
„ 238,
» 6,
„ 239,
„ 11,
„ 218,
„ 4»
„ 267,
„ 4,
„ 274,
„ 8,
,, 276,
„ 13,
,, ,,
„ 14,
,, 283,
„ 3,
„ 288,
„ 7,
„ 289,
„ 4,
„ 290,
9
„ 325,
p 17,
■ a^b^ read + a'b\
„ 333,
,, 361,
,, 481,
,, 614,
„ 617,
,, 614,
,, 651,
„ 704,
,, 729,
Article 112 should
numerators 1, 1, 1
denominator r— 1
taken
(190)
5
(-1)*
0.*;
4
204, 459
459
10.9.8
1.2.3
(-? + !)*
?(^^_j in nimiorator
(i'o3)
{^^ + !/ + ^Y
(1)
X- = 1
(a-- — 4x- + 8) on left side
(234)
(29)
(267)
in
p + 2
ip-l)
x= 1
n-\
U{r,n-\)
Jr{r+\, «-l)
^2
i'l /'.>/'•), last lino but one
C^)
3628
«-3
apiilying Descartes' rule
i'>
„ (11, 12)
„ (i'lO)
be as folbnv.s: —
1 i-_2y3 + y 2 ^ 1 + 2^3 + y 2 ^ n_
(i + Vai-'-a" 11 + va
1,^, a».
w-1.
taken m at a time.
(360).
6.
(-IP.
3.r.
34.
102, 306.
9)1.
7.8.9.10.
{q + lf^ Notation of (96).
2
(IG4).
2{x + i/+zy.
square of (1).
x"=-l.
(x-—ix + 8)\
Deic.
(28).
(266).
2\U.
P + \.
u = b.
n + \.
ll{>:,r-\).
H{„,r).
1\
(^),
10284.
H-\.
Ihle.
id/.
Trunsjiosi
/I.
a — K.
(9, I(», 1).
(960).
2y3 4-^/2^(ll -4^/3)
73
TABLE OF CONTEXTS.
PAUT I.
SECTION I.— MATHEMATICAL TABLES. Pap,
Introduction. The C. G. S. Systkm of Units —
Notation and Definitions of Units... ... ... ... 1
Physical Constants and Formulas ... ... ... ... 2
Table I. — English Measures and Equivalents in C. G. S. Units 4
II. — Pressure of Aqueous Vapour at different temperatures 4
III. — Wave lengths and Wave frequency for the principal'
lines of the Spectrum ... ... ... ... ... 4
lY. — The Principal Metals — Their Densities ; Coeffi-
cients of Elasticity, Rigidity, and Tenacity ; Expan-
sion by Heat ; Specific Heat ; Conductivity ; Rato
of conduction of Sound ; Electro-magnetic Specific
Resistance ... ... ... ... ... ... 5
V. — The Planets — Their Dimensions, ^Masses, Densities,
and Elements of Orbits ... ... ... ... 5
VI. — Powers and Logarithms of TT and e ... ... ... G
Vll.^Square and Cube Roots of the Integers 1 to .30 ... 6
VIII. — Common and Hyperbolic Logarithms of the
Prime numbers from 1 to 100... ... ... ... G
IX. — Factor Table —
Explanation of the Talilo .. ... ... ... 7
The Least Factors of all numbers from 1 to 00000... S
X. — Values OF THE Gamma-Function 30
SECTION IT.— ALriEBRA. x.^of
Ailiilo
Factors ... ... ... ... ... ... ... ... 1
Newton's Rule for expanding a Binniuial ... ... ... 12
Multiplication and Division ... ... ... ... ... 2S
Indices ... ... ... ... ... ... 20
Highest Common Factor 30
xii CONTENTS.
No. of
Artu-lo
Lowest Common Multiple 33
Evolution —
Square Root and Culic Root ... ... ... ... 3.5
Useful Transformations ... ... ... ... ... 38
Quadratic Equations 45
Theory OP Quadratic Expressions SO
Equations IN ONE Unknown Quantity.— Exam rr,i:s 54
Maxima and Minima by a Quadi-atic lv[uation 58
Simultaneous Equations and Exam it, i:a 59
Ratio AND Proportion ... ... ... ••• ••• ••• ^^
The /.• Theorem 70
Duplicate and Triplicate Ratios ... ... ... ... ^2
Compound Ratios ... ... ... ... ... •■• '^
Variation ... ... ... ... ••• ••• •■• ••• 7G
Arithmetical Progrkssion ... ... ... ... ... •-• 79
Geometrical Progression ... ... ... ... ... ... 83
Haumonical Progression 87
Permutations AND Combinations ... ... ... 94
Surds ... ...
108
Simplification of -/a + \/fe and Va-^\/h 121
Simplification of Va-^ V~h 124
Binomial Theorem 125
Multinomial Theorem 137
Logarithms ... ... ••• •.• 142
Exponential Theorem 149
Continued Fractions and Convergents ... ... ... ... 160
General Theory of same ... ... ... 167
To convert a Series into a Continued Fraction ... ... 182
A Continued Fraction with Recurring Quotients ... 186
Indeterminate Equations ... ... ... ... ... ... 188
To reduce A Quadratic Surd to a Contini-kd Fkaction 195
To form high Convergents rapidly ... ... ... 197
General Theory ... ... ... ... ... ... 199
Equations —
Special cases in the Solution of Simultaneous Equations, .. 211
Method by Indeterminate Multipliers ... ... ... 21.S
^liscellancous Equations and Solutions ... ... ... "J 11
On Symmetrical E.xpressions ... ... ... ... iM'.l
Imaginahy Expressions ... ... ... ... ... ... 2"J;?
^IeT1K)D OF InDETEKMINATI; CoEl'FiiTlN IS ... ... ... ... 'I'.Vl
Method OK Proof BY I NDiCTioN ... ... ... ... ... 233
Parfial FuACTiONS. — F(tuR Cases 235
CONVEHGENCY AND J')lVEK(iENCY OF SeKIES ... ... ... ... 2:5;»
General Theorem of ^ (a:) ... ... ... ... ... 2li>
CONTENTS. XI U
N... <.f
Expansion of a FKAnmN ... ... ... ... ... ... -1-H
Ui;cLKi:iN(i Skhiks -•'''I
Tho General Term ... ... ... ... ... ... -">7
Case of Quailratic Fuctiir with Iiiia^Miiary Koots... ... 258
Lagrann^e's Rule ... ... ... ... ... ... 203
SlMMATIOX OF SkRIKS BY THE ^[Ernon OF DlFFIMiKNCE.S ... ... 2r,t
Interpolation of a term ... ... ... ... ... 2t;7
DiRKCT Fact(M{ial Skrik.s ... ... ... ... ... ... 2(58
Invkusk Factorial Serifs 270
Su.MMATiON IJY Paktial Fractions ... ... ... ... ... 272
CoMi'OsiTE Factorial Series ... ... ... ... ••• 271
Miscellaneous Series —
Sams of tho Powers of the Natural Numbers 27G
Suraof (i+(a + J)r+(a + 2(Z);-Hctc 27'J
Sum of n'' — n (7i — l)''4-&c. ... ... ... ... ... 2H.5
POLYOONAL Nu.MI!ERS ... ... ... ... ... 287
FuiURATE Numbers 289
Hypergeometrical Series 21>1
Proof that c" is incommensurable ... ... ... ... 21*5
Interest ... ... ... ••■ ••. ••• ••• ••• 2'JG
Annuities '^^^2
PROr.AniLITIES... ••• ••• ■•• ^*^^
Inequalities ... ... ... ••• 330
Arithmetic ^Ican > Geometric 'Mean ... ... ... 332
Arithmetic Mean of ?h*'' powers > m^^ power of A. ^l. ... 334
Scales OF Notation ... ... ... -.. ••• •^•i2
Theorem concerning Sam or Difference of Digits ... 3i7
Theory OF NuMRERS 3^9
Highest Power of a Prime ji contained in \m^ ... ... 305
Format's Theorem ... ... ... .... ... ... 3r)9
Wilson's Theorem ... ... ... ... ... ••■ 371
Divi.sors of a Number ... ... ... ... ... 374
S, (livisil.lebv2» + l 380
SECTION TIT.^THHOKY OF EQX'ATTON.S.
Factors OF AN EyuATiox ... ... ... ... -.. ••• '^'"^
To compute /(a) numcricallv 4i»3
Di.scriminati(m of Roots ... ... •.. ... ... ^O'J
Descartes' Rule OF SicNs 410
The DEiiiVED Functions of/ (.}•) ... ... .. -.. ■•• 424
To remove an assigned term ... ... ... ... 428
To transform an equation ... ... ... .. •■• 430
Ei^TAL Roots OF an Equation ... ... ... •• ••• 4.32
Pra-tical Rule •^^''
XIV
CONTEXTS.
No. of
Article.
Limits OF THE Roots 448
Newton's Method 452
Rolle'fi Tlieorem 454
Newton's Method OF Divisors ... ... ... ... ... 459
RECiPROC-\Ti Equations 4GG
Binomial Equations 472
Solution of .^"±1 = bj Do Moivre's Tlicorcm 480
Cubic Equ.viions ... ... ... ... ... ... ... 483
Cardan s Method 484
Trigonometrical Method ... ... ... ... ... 489
Biquadratic Equ.ations —
Descartes' Solution ... ... ... ... ... ... 492
Ferrai'i's Solution ... ... ... ... ... ... 496
Euler's Solution 499
Commensurable Roots 502
Incommensurable Roots —
Sturm's Theorem ... ... ... ... ... ••• 506
Fourier's Theoi-em ... ... ... ... ... ... 518
Lagrange's Method of Approximation ... ... ... 525
Newton's Method of Approximation ... ... ... 527
Fourier's Limitation to the same ... ... ... ... 528
Newton's Rule for the Limits of the Roots 5.30
Sylvester's Theorem .. . ... ... ... 532
Horner's Method 533
Symmetrical Functions of the Roots of an Equation —
^ms of the Powei'S of the Roots ... ... ... ... 534
Symmetrical Functions not Powers of the Roots... ... 538
The Equation whose Roots are the Squares of the
Differences of the Roots of a given Equation
Sum of the m"' Powers of the Roots of a Quadratic
Equation
Approximation to the Root of an Equation through the
Sums of the Powers of the Roots
E-KPANsioN of an Implicit Function of a;
Determinants —
Definitions ...
General Tlieory
To raise the Order of a Dotonninant
Analysis of a Deterniinaut ...
Synthesis of a Detcrminiuit
Product of two Determinants of the 7i"' Onlei
Synimetrioal Determinants ..
Reciprocal Determinants ...
Partial and Ci>niplcnu'ntary DcierininnntH
545
548
551
55-i
556
564
568
569
570
574
575
576
CONTENTS. XV
\u. ..f
Aril. I.'.
Theorem of a Partial Ik-ciprocal Dclonuiiiunt ... ... .')77
Product of DiU'ercuce.s of /i Quantitius ... ... ... 578
Product of Squares of UiiTereiices of samo ... ... 579
Rational Algebraic Fraction expressed as a Ucleniiinant 5sl
Eli.mi.n.mio.n —
Solution of Linear Ecjuations ... ... ... ... r»s2
Orthogonal Transformation ... ... ... ... r)Sl.
Theorem of the ?i.—'2"' Power of a Deteniiinant... ... r).s5
Bezout's Method of Elimination ... ... ... ... 5SG
Sylvester's Dialytic Method ... ... ... ... 587
^lethod by Symmetrical Function.^ ... ... ... 5H8
Eliminatiox BY llu;iii:sT Cu.MMON Factou ... ... 5'J3
SECTION IV.— PLANE TRIGONOMETRY,
Angular Measuremext ... ... ... ... ... ... COO
Trigonometrical Ratios ... ... ... ... ... ... GOG
Formula) involving one Anglo ... ... ... ... G13
Formula; invoMng two Angles and Multii)lc Angles ... G27
Formula-" involving three Angles ... ... ... ... G7-1
Ratios OF 45°, G0=, 15°, 18°, <fcc GOO
Properties of THE Triangle 700
The s Formula) for sin ^^, &c. ... ... ... ... 70-i
The Triangle and Circle 709
Solution of Triangles —
Right-angled Triangles ... ... ... ... ... 718
Scalene Triangles. — Three cases ... ... ... ... 720
Examples on the same ... ... ... ... ... 859
Quadrilateral in a Circle 733
Bi.sector of the Side of a Triangle 738
Bisector of the Angle of a Triangle ... ... ... 71-2
Perpendicular on the Base of a Triangle ... ... ... 711i
Regular Polygon AND Cikcle 74G
Subsidiary Angles ... ... ... ... ... 749
Limits of Ratios 753
De Moivre's Theorem 75G
Expansion of cos ?i0, &c. in ])owcrs of sin and cosO ... 758
Expansion of sinO and cos in jiowers of ... ... 7G4
Expansion of cos" ami shi" U in cosines or sines of
multiples of (^ ... ... ... ... ... ... 772
Expansion of cos 7i0 and sin7j8 in powers of biuO ... 775
Expansion of cosn9 and sin «0 in powers of cos ... 779
Expansion of cos nO in descending powers of cos ... 780
Sin a -f- c sin (a + /3) + <tc., and similar series 783
XVI
CONTENTS.
No. of
Artiol.;.
Gregory's Scries for in powers of tauO .. ... ... 71>1
Formulas for tlio calculation of TT ... ... .. ... 792
Proof that TT is iucomracnsural)lc ... ... ... ... 795
Sina; = ?i sin (a; + a.). — Series for a; ... ... ... 790
Sum of sines or cosines of Angles in A. P. ... ... 800
Exi^ansion of the sine and cosine in Factors ... ... 807
Sin 1/0 and cosnf expanded in Factors ... ... ... 808
Sin(? and cos in Factors involving (^ ... ... ... 815
e' — 2cos6 + e"' expanded in Factors ... ... ... 817
De Moivre's Property of the Circle ... ... ... 819
Cotes's Properties ... ... ... ... ... ... 821
Additional Formulae ... ... ... ... ... ... 823
Properties of a Right-angled Triangle ... ... ... 832
Properties of any Triangle... ... ... ... ... 835
Area of a Triangle ... ... ... ... ... ... 838
Relations between a Triangle and the Inscribed,
Escribed, and Circumscribed Circles ... ... 841
Other Relations between the Sides and Angles of a Triangle 850
Examples of the Solution of Triangles ... ... ... 859
SECTION v.— SPHERICAL TRIGONOMETRY.
Introductort Theorems —
Definitions
Polar Triangle
Right-angled Triangles —
Napier's Rules
Oblique-angled Triangles.
Formula) for cos a and cos A
The (S Formula) for siniJ, sinJa, <tc. ...
sin^ — sin JB _ sin
sin a sin b sin c
cos 6 cos (7 = cot a sin i— cot ^1 sin
Napier's Formula) ...
Gauss's Formulas
Spherical Triangle and Circle —
Inscribed and Escribed Circles
Circumscril)cd Circles
Si'UERiCAL Areas —
Spherical Excess
Area of Sphei'ical Polygon ...
Cagnoli's Theorem ...
Lhuillior's Theorem
870
871
881
882
884
894
895
89G
897
898
900
902
903
904
905
CONTKNTS. XV 11
N... o<
ArtirUi.
PoLTiiEnnoNs ... ... ... ... ... ... ... i'06
The five RcgulaT Solids 1>07
Tlio Aiii^'lo hot woiMiAdjiuvnt Faces i'OO
Kadi i of Iii.siTil)L'd and Circiim.scriheil Spheres... ... 'JlO
SECTION VI.— ELEMENTARY GEOMETRY.
Miscellaneous Propositions —
Reflection of a point at a single surface ... ... ... 920
do. do. at Buccussive surfaces ... ... i.'2I
Relations between the sides of a triangle, the segments
of the base, and the line drawn from the vertex ... i'22
Equilateral triangle Yli'C; P.'P + Pi>" + i'0" 923
Sum of squares of sides of a quadrilateral ... ... 024
Locus of a point whose distances from given lines or
points are in a given ratio ... ... ... ... 92(3
To divide a triangle in a given ratio ... ... ... 930
Sides of triangle in given ratio. Locus of vertex ... 032
Harmonic division of base ... ... ... ...' 933
Triangle with Inscribed and Circum.scril}ud circles ... 935
TuE PRor.LE.Ms OF TUE Tangencies 037
Tangents and cliord of contact, fty =. u- ... . ... 0-48
To find any sub-multiple of a line ... ... ... 950
Triangle and three concurrent lines ; Three cases ... 951
Inscribed and inscribed circles ; /?, s — t/, &c. ... ... 953
Ni.N'E-PoixT Circle... ... ... ... ... ... ... 954)
CoNSTULCTiox OK Tkiano.les ... ... ... ... ... 900
Locus of a point from which the tangents to two circles
have a constant ratio .. . ... ... ... ... 003
CoLLiNEAR AND CoxcuKUENx Systems 007
Triangle of constant species circum.scribcd or inscribed
to a triangle ... ... ... ... ... ... \)77
Radical Axis —
Of two Circles l*8i
Of three Circles il'j7
Inveksiox —
Inversion of a point ... ... ... ... ... loOU
do. circle ... ... ... ... ... lii(,»0
do. right line ... ... ... ... ... Iiil2
Pole and Polar ... ... ... ... ... ... ... \u\C,
Coaxal Circles ... ... ... ... ... ... ... Iu21
Centres and Axes of Similitude —
Homologous and Anti-homuldgons pi )ini.s ... ... Iu37
do. do. chord.s ... ... Iu38
C
xvin
CONTENTS.
Constant product of anti-siniilitudc
Circle of similitude
Axes of similitude of three circles
Gergonne's Theorem
Anharmoxic Ratio and Pencil
HoMOGUAi'Hic Systems of Points
Involution ...
Projection ...
On Perspective Drawing ...
Orthogonal Projection
Projections of the Sphere
Additional Tueorems —
Squares of distances of P from equidistant points on a
circle ...
Squares of perpendiculars on radii, etc. ...
Polygon n-ith inscribed and circumscribed circle
of perpendiculars on sides, &c....
Sum
No. of
Article.
1U43
1045
1046
1049
1052
1058
1066
1075
1083
1087
1090
1094
1095
1099
" SECTION VII.— GEOMETRICAL CONICS.
Sections of the Cone —
Defining property of Conic PS = ePM
Fundamental Equation
Projection from Circle and Rectangular Hyperbola
Joint Properties of the Ellipse and Hypereola —
Definitions ...
CS:CA:CX
F8 ±PS'=AA'
CS" = AC ^ PC
SZ bisects ^ QSP
If PZ be a tangent PSZ is a right angle
Tangent makes equal angles with focal distances
Tangents of focal chord meet in directrix
CN.CT =AC'
as :PS =e
NG : NC =^ PC' : AC
Auxiliai-y Circle
J'N: QN = PC :AC
P^- : AN. NA' = L'C- : JO'-
Cn.a = PC'
sy.s'Y' = PC'
PP = A(!
To draw two tangents
Tangents subtend equal angles at the focus
1151
1156
1158
1160
1162
1103
1164
1166
1167
1168
1169
1170
1171
1172
117;i
1174
1 1 7t;
1177
1178
1179
1180
Ubl
TON TK NTS.
No. of
Arti.l.'.
To draw two tanj^ents
1201
Asymptotic PHOi'F.nTir.s ok Titr TTviM;i;itor,\ —
RN^-IW = IK" 11^:^
rn.p>=n(" ... n-i-
('!■:= AC ll-<'
J'l> i.s i-Mnillcl (o the A.svni|.l..t.- 1I>^7
Qh'=r- ' "^^
rrj = n an.i (>v = <iV n^i^
Qn.Qx= Plr= UV'--(iV- ll'-'l
Arn.TK = cs' ^ ••• ii-'"-i
Joint Propkrtiks of Ellipre and HYi-r.Kr.oLA rksi'mkh. Cox-
jniATK DiAMETKKS —
QV-.FV.VF'^CD'' :Cr^ 11^.'^
rF.cn = AC .no nin.
rF.ra = nc" ami ff-fo'^zAC ii '■'•'»
PG.FG' = Cr)' ll'-'7
Diameter bisects parallel cliord.s ... ... ... ... 11'''^
Supplemental chord.s
Diameters arc mutually conjugate
CV.CT=r!F^ l-"-5
CN=dR, CB = pN l--^0.-S
CN^ ± CK' = A C\ FF- ^F}r- = FC- 1 '2' i7
CP± (72)^ = ^1 dfc 7>C''' I'-^H
FS.PS' = cn' 1-^-5
OQ . Oq \ OQ' . O'i = CD- : CF- 121t
SR:QL = e 1-'1'»
Director Circle ... ... ••• ■•• ••• ■•• 1-''
Properties of Parabola deduced from the Ellipse ... 121'.»
The PAHAnoLA —
Defining property I',S' = P.!/- •.• 1'2"20
Latus Rectum = 4JN •• ^'-^'^
If FZ be a tangent, I'SZ is a right angle 1 -•^3
Tangent bisects Z ,ST.U and fe';^ .If l—'i
ST=SF = Sa l--'^
Tangents of a focal fliord intersect at right anglos in
directrix ... ... ... ... .•• •■• ■ ■ l'--*^
A^ = AT l--^27
NG = 2AS l^^*^
FN"- = iAS.AN l^^O
SA:SY:SF 1-'^^
SQ:SO:SQ' l--^-^
Z OSQ = OSQ' and QOQ' = { QSQ' 1 ■^■'-i-
DiAMF.TK.nS ... ... ... ... ... ■•• •■• l-'^'^
The diameter bi.sects itanilh'l chords ... ... ... I'-^^S
XX
CONTENTS.
QP=4P.9.Pr
0(2.0q :0(/ : Oq =rS iFS
Pai-abola two-thirds of circnmscribing parallelogram
Methods OF Drawing A Cuxic
To find the axes and centre
To construct a conic from the conjugate diameters
Circle OP Curvature
Chord of curvature = QV^ -^ PV ult
c • 1 1 . , CT)'' OD'- CL^
bemi-cliords ot curvature, — -— , rf^,=r, -77-r
C-P PJf AC
In Parabola, Focal chord of curvature = 4SP ...
do. Kadius of curvature = 2SP"' -^ ST
Common chords of a circle and conies are equally i
clined to the axis
To find the centre of curvature ...
MiSCKLLAM;OUS TlIKOKLMS
No. of
Article.
1239
1242
1244
1245
1252
1253
1254
1258
1259
12G0
12G1
1263
1265
1267
INDEX TO TROPOSITIONS OF EUCLID
REFERRED TO IX THIS WORK.
Tho references to Euclid are made in Koinan and ^Vrabic numerals ; e.g. (VI. 19).
BOOK T.
I. 4. — Triaui^'los arc equal and similar if two sides and the included
an<^le of each are equal each to each.
I. 5. — The angles at the base of an isosceles triangle are equal.
1. 0. — The converse of 5.
I. 8. — Triangles are equal and similar if tlie tliroe sides of eacli arc
ecjual each to each.
I. IT). — The exterior angle of a triangle is grojiter than the interior
and opposite.
I. 20. — Two sides of a triangle are greater than the third.
I. 26. — Triangles are equal and similar if two angles and one corres-
ponding side of each are equal each to each.
I. 27. — Two straight lines are parallel if tlicy make equal alternate
angles with a third line.
I. 29. — The converse of 27.
I. 32. — The exterior angle of a triangle is cqiial to tho two interior
and opposite; and tlic three angles of a triangle are equal
to two right angles.
C'riK. 1.— The interior angles of a ]-)olygon of n sides
= («-2)7r.
C'oK. 2. — The exterior angles = 27r.
I. 35 to 38. — Parallelograms or triangles upon tlie same or equal
bases and between tho same parallels are equal.
I. la.— The conq)lements of the parallelograms about the diameter
of a parallelogram are c([ual.
T. M . — Tlio square on the hypotenuse of a right-angled triangle is
equal to the scpiares <m the other sides.
I. 48. — The converse of 47.
XXll INDEX TO PTtOrOSITIOXS OF EUCLID
BOOK II.
II. 4.— If a, h arc the two parts of a riglit line, {a + iy = a" -\-1nh-\-h-.
If a right line be bisected, and also divided, internally or
externally, into two nnequal segments, then —
II. 5 and 6. — The rectangle of the unequal segments is eqnal to the
difference of the squares on half the line, and on the line
between the points of section; or (a + i) (a-h) = a? — l/.
II. 9 and 10. — The squares on the same unequal segments are together
double the squares on the other parts ; or
II. 11. — To divide a right line into two parts so that the rectangle
of the whole line and one part may be equal to the square on
the other part,
II. 12 and 13. — The square on the base of a triangle is equal to the
sum of the squares on the two sides lolus or mimis (as the
vertical angle is obtuse or acute), twice the rectangle under
either of those sides, and the projection of the other upon it ;
or a- = h- + c''-21jccosA (702).
BOOK III.
III. 3. — If a diameter of a circle bisects a chord, it is perpendicular to
it : and conversely.
III. 20. — The angle at the centre of a circle is twice the angle at the
circumference on the same arc.
III. 21. — Angles in the same segment of a circle are equal.
III. 22. — The opposite angles of a quadrilateral inscribed in a circle
are together equal to two right angles.
ITT. 31. — The angle in a semicircle is a right angle.
111. 32. — The angle betAveen a tangent and a chord from the pcint of
contact is equal to the angle in the alternate segment.
111. 33 and 34. — To describe or to cut of ti segment of a circle which
shall contain a given angle.
III. 35 and 30.— The rectangle of the segments of any chord of a
circle drawn through an inta'ual or external point is eiiual
to the square of the semi-chord perpendicular to the
diameter through the internal point, or to the square of the
tangent from the external point.
III. 37.— The converse of 3G. If the rectangle be equal to the .scpiare,
tlic lino which meets the circle touches it.
iJKKRI.'KKn T(i IN THIS WoKK
lUJUK IV.
IV. 2. — To insoi-ilif a trian^k' of yivcii form in a i-irclc
IV. 3. — To describu tlic same about a circle.
IV. 4. — To inscribe a circle in a triangle.
IV. 5. — To describe a circle about a triangle.
IV. 10. — To construct two-iiftlis of a right angle.
1\'. 11. — To construct a regular pentagon.
VI
VI. 4.
VI.
VI.
VI.
BOOK VI.
VI. 1. — Triangles and parallelograms of the same altitude arc
proportional to their bases.
VI. 2. — A right line parallel to the side of a triangle cuts the other
sides proportionally ; and conversely.
3 and A. — The bisector of the interior or exterior vortical angle of
a triangle divides the base into segments proportional to
the sides.
Eipiiangular triangles have their sides proportional honio-
logously.
5. — Tlie converse of -i.
0. — Two triangles are equiangular if they have two angles equal,
and the sides about them proportional.
7. — Two triangles are equiangular if they have two angles equal
and the sides about two other angles proportional, provided
that the third angles are both greater than, both less than,
or both equal to a right angle.
6. — A right-angled triangle is divided by the perpendicular from
the right angle upon the hypotenuse into triangles similar
to itself.
11 and l.'i. — Equal lyaralldoijrams, or trianjlcs which have two
angles equal, have the sides about those angles reciprocally
jiroportional ; and conversely, if the sides are in tliis i)ro-
jiDi-tion, the figures are eciual.
ll*. — Similar triangles are in the duplicate latio of their homo-
logous sides.
2". — Likewise similar jjiilyg'ons.
23. — E(|uiangular parallelogi-auis are in the ratio compoundetl of
the ratios of their sides.
B. — The rectangle of the sides of a (riangle is ccjual to the s(]uare
of the bisector of the vertical angle i>lus the rectangle of
the segments of the base.
VI
VI
VI.
VI.
VI.
VI.
XXIV INDEX TO PROPOSITIONS OF EUCLID.
VI. C. — The rectangle of the sides of a triangle is equal to tlie rect-
angle under the perpendicular from the vertex on the
base and the diameter of the circumscribing circle.
VI. D. — Ptolemy's Theorem. The rectangle of the diagonals of a
quadrilateral inscribed in a circle is equal to both the
rectangles under the opposite sides.
BOOK XI.
XI. 4. — A right line perpendicular to two others at their point of
intersection is perpendicular to their plane.
XI. 5. — The converse of 4. If the first line is also perpendicular to a
fourth at the same point, that fourth line and the other
two are in the same plane.
XI. 6. — Right lines perpendicular to tlie same plane are parallel.
XI. 8. — If one of two parallel lines is perpendicular to a plane, the
other is also.
XI. 20. — Any two of three plane angles containing a solid angle are
greater than the third.
XI. 21. — The plane angles of any solid angle are together less than
four r'nAit ano'les.
TABLE OF CONTENTS.
PART II.
SECTION VIIT.— DIFFERENTIAL CALCULUS. No. of
Article.
Introduction ... ... ... ... ••• ■■• •■• 1400
Successive differentiation ... ... ... ... 1405
Infinitesimals. Differentials ... ... ... ... 1407
Differentiation.
Methods 1411-21
SoccEssivR Differentiation' —
Leibnitz's theorem ... ... ... ... ... 14(30
Derivatives of the ?ith order (see Index) ... ... 14G1-71
Partial Differentiation 1480
Theory of Operations ... ... ... ... ... 1483
Distributive, Commutative, and Index laws... ... 1488
Expansion of Explicit Functions —
Taylor's and Maclaurin's theorems ... ... ...1500,1507
Symbolic forms of the same ... ... ... ... 1520-3
f(x + h,y + k),&c 1512-4
Methods of expansion by indeterminate coefficients.
Four rules... ... ... ... •■• ••• 1527-31-
Method by Maclaurin's theorem 1524
Arbogast's method of expanding ^ (2) ... ... 15:3i>
Bernoulli's numbers ... ... ... ... • . 1. ">:'.".'
Expansions of ^ (j; + /0 — ^ (•'')• Stirling and Boole 151G-7
Expansions of Lmplicit Functions —
Lagrange's, Laplace's, and Burmann's theorems, 1552, 1550-03
Cayley's series for --— ... ... ... •• 1555
Abel's series for if>{x-\-a) ... ... ... .•• 1-'"-
Indeterminate Forms 1580
Jacodians ... ... ... ... ••• •• • 1^*^^
Modulus of transformation ... ... ... ... lOUt
XXVI
CONTENTS.
No. of
Article.
QiAViics 1620
Euler's theorem .. ... ... ... ... 1621
Eliruinant, Discriminant, Iuvariant,Covariant, Hes.sian 1626-30
Theorems concerning discriminants ... ... ... 1635-45
Notation ^ = 6c-/, &c 1642
Invariants 1648-52
Cogredients and Emanents ... ... ... ... 1653-5
Implicit Functions —
One independent variable ... ... ... ... 1700
Two independent variables ... ... ... ... 1725
w independent variables ... ... ... ... 1737
Change of the Independent Variable ... ... ... 1760
Linear transformation ... ... ... ... ... 1794
Orthogonal transformation ... ... ... ... 1799
Contragredient and Contravariant ... 1813
Notation z„=p,&:,c.jq,r^s,t... ... ... ... 1815
Maxima and Minima —
One independent variable ... ... ... ... 1830
Two independent variables ... ... ... ... 1841
Three or more independent variables... .. ... 1852
Discriminating cubic ... ... ... ... ... 1849
Method of undetermined multipliers ... ... ... 1862
Continuous maxima and minima ... ... ... 1866
SECTION IX.— INTEGRAL CALCULUS.
Introduction 1900
Multiple Integrals 1905
Methods of Integration —
By Substitution, Parts, Division, Rationalization,
Partial fractions. Infinite sei'ies ... ... ... 1908-19
Standard Integrals ... ... ... ... ... ... 1921
Various Indefinite Integrals —
Circular functions ... ... ... ... ... 1954
Exponential and logarithmic functions ... ... 1998
Algebraic functions ... ... ... ... ... 2007
Integration by rationalization... ... ... ... 2110
Integrals reducible to Elliptic integrals 2121-47
Elliptic integrals approximated to ... ... ... 2127
Successive Integration ... ... ... ... ... 2148
Hyperbolic Functions cosh a-, sinh-i;, tanh.r ... ... 2180
Inverse relations ... ... ... ... ■• 2210
Geometrical meaning of tanh <S ... ... ... 2213
Logarithm of an imaginary quantity ... ... ... 2214
CONTENTS. XXVll
No. of
Article.
DkFINITK iNlKfiRALS
Summation of series ... ... ... ... ... 2'J3U
'I'liroKKMS KESrECTINU LiMITS OF iMECiliATION ... ... 2233
Methops ok evaluating Definite iNTEiiBAi-s {Eiyitt rules) ... 2245
Differentiation under the sign of Integration ... ... 2253
Integration by tliis method ... ... ... ... 2258
Change in the order of integration ... ... ... 2261
Approximate Integration hy Beunoulm's Series ... ... 2262
The Integrals j;(Z,m) and r(») 2280
logr(l + M) in a converging series ... ... ... 22'.'4
Numerical calculation of r (x) ... ... ... 2317
Integration of Algebraic Forms ... ... ... ... 2341
Integration of Logarithmic and Exponential F(>i;.m.s ... 23i>l
Integration of Circular Forms ... ... ... ... 2451
Integration of Circular Logarithmic and Exponential Forms 2571
Miscellaneous Theorems —
Frullani's, Poisson'.'^, Abel's, Kummer's, and Cauchy's
formula) 2700-13
Finite Variation of a Parameter ... ... ... ... 2714
Fourier's formula ... ... ... ... ... 2726
The Function \P{.c) 2743
Summation of series by the function «/' (.t) ... ... 2757
4/ (.«) as a definite integral independent of \p{l) ... 27»)<>
Nu.merical Calculation of log r{x) 2771
Change of the Variables in a Definite JMl'ltiple I.ntegral 2774
Multiple Integrals-
Expansions of Functions in Converging Seriks —
Derivatives of the nth order ... ... ... ... 2852
Miscellaneous expansions ... ... ... ... 2911
Legendre's function X„ ... ... ... ... 2936
Expansion of Functions in Trigonometrical Series ... 2955
Approximate Integration 2991
^lethods by Simp.son, Cotes, and Gauss ... ... 2992-7
SKCTION X.— CALCULUS OF VARIATIONS.
Functions of one Independent Variable ... ... ... 3028
Particular cases... ... ... ... ... ... 3033
Other exceptional cases ... •• •■• ■ 3045
Functions of two Dependent Variables ... ... ... 3051
Relative maxima and minima ... ... ... ... 3069
Geometrical applications ... ... ... ... 3070
XXVlll CONTENTS.
No. of
Article.
Functions of two Independent Variables ... ... ... 3075
Geometrical applications ... ... ... ... 3078
Appendix —
Oil the general object of the Calculus of Variations... 3084
Successive variation ... ... ... ... ... 3087
Immediate integrability ... ... ... ... 3090
SECTION XL— DIFFERENTIAL EQUATIONS.
Generation of Differential Equations ... ... .. 3150
Definitions and Rules... ... ... ... 3158
Singular Solutions ... ... ... ... ... ... 3168
First Order Linear Equations ... ... ... ... 3184
Integrating factor for il/dc+iVfZi/ = 3192
Riccati's Equation ... ... ... ... ... 3214
First Order Non-linear Equations 3221
Solution by factors ... ... ... ... ... 3222
Solution by difFei'entiation ... ... ... ... 3236
Higher Order Linear Equations 3237
Linear Equations with Constant Coefficients ... 3238
Higher Order Non-linear Equations ... ... ... 3251
Depression of Order by Unity... ... ... ... 3262
Exact Differential Equations 3270
Miscellaneous Methods ... ... ... ... ... 3276
Approximate solution of Differential Equations by
Taylor's theoi'em ... ... ... ... ... 3289
Singular Solutions OF Higher Order Equations 3.301
Equations with more than two Variables... ... ... 3320
Simultaneous Equations with one Independent Variable... 3340
Partial Differential Equations 3380
Linear first order P. D. Equations ... ... ... 3381
Non-linear first order P. D. Equations ... ... 3399
Non-linear first order P. D. Equations with more
than two independent variables ... ... ... 3409
Second Order P. D. Equations 3420
Law of Reciprocity ... ... ... ... ... ... 3446
Symbolic Methods ... ... ... ... ... ... 3470
Solution OF Linear Differential Equations BY Series ... 3604
Solution by Definite Integrals ... ... ... ... 3617
P. D. Equations with more than two Independent Variables 3629
Differential Resolvents of Algebraic Eqlaitons ... 3631
CONTENTS. XXIX
Xo. of
Article.
SECTION XII. — CALCULUS OF FINITE
DIFFERENCES.
F0KMri,.K KOR FlKST AND uth UlKKKKKNCF.S '^7(^6
Expansion by factorials ... ... ... ••• li/.^O
Gcnemting functions ... ... ... ... ■•• 3732
The operations 1/, A, and (/.r ... ... ... ••• 373-5
Herscbel's theorem ... ... ... ... ••• 3/.)7
A theorem conjugate to Machiuriu's ... ... ... 3759
Interpoi-ation
37G2
Lagrange's interpolation formula ... ... ... 370H
MhXHANlCAL QUADRATURK ^^^72
Cotes's and Gauss's formula3 ... ... ... ... 3777
Laplace's formula ... ... ... ••• ... o/lH
Summation of Series '^781
Approximate Summation 3820
SECTION XIII.— PLANE COORDINATE GEOMETRY.
Systems of Coordinates —
Cartesian, Polar, Trilinear, Areal, Tangential, and
Intercept Coordinates 4001-28
ANALYTICAL CONICS IN CARTESIAN COORDINATES.
Lengths and Areas ^032
Transformation of Coordinates 4048
The Right Line 4(>
Equations of two or more right lines
General Methods
Poles and Polars 4124
The Circle ^^^G
Co-axal circles 4U»1
The Parabola '^^^^
The Ellipse and Hyperbola 4250
Right line and ellipse 4310
Polar equations of the conic 433t.
Conjugate diameters 4o4b
Determination of various angles 4375
The Hyperrola referred to its Asymptotes 4387
The rectangular hyperbola 439-
4110
4114
XXX
CONTENTS.
The General Equation
The ellipse and hyperbola
Invariants of the conic
The parabola
Method without transformation of the axes . . .
Rules for the analysis of the general equation
Right line and conic with the general equation
Intercept equation of a conic ...
Similar Conics
Circle of Curvature —
Contact of Conics
CoNKOCAL Conics
No. of
Article.
4400
4402
4417
4430
4445
4464
4487
4498
4522
4527
4550
ANALYTICAL CONICS IN TRILINEAR COORDINATES.
The Right Line
Equations of particular lines and coordinate ratios
of particular points in the trigon
Anharmonic Ratio
The complete quadrilateral
The General Equation of a Conic
Director- Circle ...
Particular Conics
Conic circumscribing the trigon
Inscribed conic of the trigon ...
Inscribed circle of the trigon ...
General equation of the circle...
Nine-point circle
Triplicate-ratio circle ...
Seven-point circle
Conic and Self-conjugate Triangle...
On lines passing through imaginary points ...
Carnot's, Pascal's, and Brianchon's Theorems
The Conic referred to two Tangents and the Chord of
Contact —
Related conics ...
Anharmonic Pencils of Conics
Construction of Conics
Newton's method of generating a conic
Maclaurin's method of generating a conic ...
The Method or Reciprocal Polars...
Tangkntial Coordinates
Abridged notation
4601
4628
4648
4652
4656
4693
4697
4724
4739
4747
4751
4754
47546
4754e
4755
4761
4778-83
4803
4809
4822
4829
4830
4844
4870
4907
CONTKNTS. XXXI
No. of
Article.
51G7
r.l72
On Tin: 1m kksection ok two Conics—
Geonictricftl^mcanin^ of v/(- 1) ... ... 'I'-^l^'
The Methop of Pkojection '^•'-l
Invariants anp Covakiants 41K5G
To find the foci of the general conic ... .. ... 5008
THEORY OF PLANE CURVES.
Tangent and Normal -''l^'*^
Radius of Curvature and Evolute -"il-^^
Inverse Problem and Intrinsic Equation 51G0
Asymptotes
Asymptotic curves
Singularities of Curves —
Concavity and Convexity ... ... ... ... 5174'
Points of inflexion, multiple points, &c. ... ... 5176-87
Contact of Curves ... ... ... ... ... ••• 5188
Envelopes 5192
Integrals of Curves and Areas ... ... ... ... 519G
Inverse Curves 5212
Pedal Curves ... ... ... ... ... ... ••• 5220
Roulettes ... ... ... ••• ••• ••• ••• 5229
Area, length, and radius of cui'vature... ... ... 5230-5
The envelope of a carried curve ... ... ••• 5239
Instantaneous centre ... ... ... ... •■• o243
Holditch's theorem ... ... ... ... ••• 5244
Trajectories ^24G
Curves of pursuit ... ... ••• 5247
Caustics 52-*8
Quetelet's theorem ... ... ... ••• ••• 5-49
Transcendental and other Cuhves —
The cycloid 5250
The companion to the cycloid... ... ... ... 5258
Prolate and curtate cycloids ... ... ... ... 52G0
Epitrochoids and hypotrochoids ... ... ... 52G
Epicycloids and hypocycloids...
The Catenary ...
The Tractrix
The Syntractrix
The Logarithmic Curve 5284
The Equiangular Spiral 5288
The Spiral of Archimedes 5296
The Hyperbolic or Reciprocal Spiral 5302
GG
5273
5279
5282
XXXn CONTENTS.
No. of
Article.
The Involute of tlie Circle ... ... ... ... -5306
TheCissoid 5309
The Cassinian or Oval of Cassini ... ... ... 5313
The Lemniscate ... ... ... ... ... 6317
The Conchoid 5320
The Lima9on ... ... ... ... ... .. 5327
The Versiera (or Witch of Agnesi) ... ... ... 5335
The Quadratrix... ... ... ... ... ... 5338
The Cartesian Oval 5341
The semi- cubical parabola ... ... ... ... 5359
The folium of Descartes ... ... ... ... 5360
Linkages AND LiNKwoRK ... ... ... ... ... 5400
Kempe's five-bar linkage. Eight cases 5401-5417
Reversor, Multiplicator, and Translator ... ... 5407
Peaucellier's linkage ... ... ... ... ... 5410
The six-bar invertor ... ... ... ... ... 5419
The eight-bar double invertor ... ... ... 5420
The Quadruplane or Versor Invertor ... ... 5422
The Pentograph or Proportionator ... ... ... 5423
The Isoklinostat or Angle-divider ... ... ... 5425
A linkage for drawing an Ellipse ... ... ... 5426
A linkage for drawing a Lima9on, and also a bicir-
cular quartic ... ... ... ... ... 5427
A linkage for solving a cubic equation ... ... 5429
On three-bar motion in a plane ... ... ... 5430
The Mechanical Integrator ... ... ... ... 5450
The Plauimeter 5452
SECTION XIV.— SOLID COORDINATE GEOMETRY.
Systems of Coordinates ... ... ... ... ... 5501
The Right Line 5507
The Plane 5545
Transformation of Cooi;niNATES ... ... ... ... 5574
The Sphere ... ... ... ... ... ... ... 5582
The Radical Plane 5585
Poles of similitude ... ... ... ... ... 5587
Cymxdrical and Conical Surfaces ... ... ... ... 5590
Circular Sections ... ... ... ... ... 5596
Ellipsoid, HvriiRUOLOiD, and Paraboloid ... ... ...5590-5621
CONTENTS.
N... of
Article.
Centrai, Qcapiuc Surfack—
Tangent and diainotnil plaiu's... ... ... ... 5026
Eccentric values of the coordinates ... ... ... 5038
CoNKOCAL QuAi'urcs ... ... ... ... ... ... •''^•^0
Reciprocal and Enveloping Cones ... ... ... 56G4
Thk Genkral Equation of a Quadric 5073
Reciprocal Polars •'''"" !•
Theory of Tortuous Curves •'"^^•Jl
The Helix 575i;
General Theory of Surfaces —
General equation of a surface ... ... ... ... 5780
Tangent line and cone at a singular point ... ... 5783
The Indicatrix Conic ^^795
Eulei-'s and Meunier's theorems ... ... ... 5806-9
Curvature of a surface... ... ... -. ... 5826
Osculating plane of a line of curvatnri! 5835
Geodesics ... ... ... ... -.. ... ■.• 5837-48
Invariants ^856
Integrals for Volumes and Surfaces ... ... ... 5871
Guldin's rules ... ... ... ... ... ■•. 587i)
Centre of Mas.s 5884
Moments and Products of Inertia 5903
Momcntal ellipsoid
Momental ellipse
Integrals for moments of inertia ... ... ... 5978
Perimeters, Areas, Volumes, Centres of Mass, and Moments
of Inertia of various Figures —
Rectangular lamina and Right Solid... ... ... 6015
The Circle <'019
The Right Cone 0043
Frustum of Cylinder ... ... ... ... ... 0048
The Sphere
5925-40
5953
6050
The Parabola G067
The Ellipse
0083
Fagnani's, Grinitli's, and li;uid)ert's tlu'orenis ...0088-0114
The Hyperbola ''llS
The Paraboloid Ol"'^<3
The Ellipsoid <>11--
Prolate and Oblate Spheroiils 0152-05
rKKFACK TO VAUT Jl
ArOLOGlES for the noii-completioii of tliis voliniie ;it ;iii
earlier period are due to friends and enquirers. The hd)our
involved in its production, and the pressure of other duties,
must form the autlior's excuse.
In the compilation of Sections VIII. to XIV., the
following works have been made use of : —
Treatises on theDifFerential and Integral Calculus, by liertratid,
Hymer, Todhunter, Williamson, and Gregory's Examples
on the same subjects; Salmon's Lessons on Higher Algebra.
Treatises on the Calculus of Variations, by Jellett and Tod-
hunter ; Boole's Differeutial Equations and Supplement ;
Carmichtiers Calculus of Operations ; Boole's Calculus of
Finite Differences, edited by Moulton.
Salmon's Conic Sections; Ferrors's Trilinear Coordinates;
Kompo on Linkages {Fruc. of Roij. Soc, Vol. 23) ; Frost
aud Wolstenholme's Solid Geometry ; Salmon's Geometry
of Three Dimensions.
Wolstenholme's Problems.
The Index which concludes the work, and which, it is
hoped, will supply a felt want, deals with 890 volumes of
o2 serial publications : of tliese publications, thirteen belong
to Great Britain, one to Xew South Wales, two to America,
four to France, five to Germany, three to Italy, two to
Russia, and two to Sweden.
As the volumes only date from the year 180(j, the
XXXVl PREFACE.
important contributions of Euler to the " Transactions of
the St. Petersburg Academy," in the last centur}^ are
excluded. It was, however, unnecessary to include them,
because a ver^- complete classified index to Euler's papers,
as well as to those of David Bernoulli, Fuss, and others in
the same Transactions, already exists.
The titles of this Index, and of the works of Euler
therein referred to, are here appended, for the convenience
of those who may wish to refer to the volumes.
Tableau general des publications de I'Academie Imperiale de
St. Pek'i-sbourg depuis sa fondation. 1872. [B.M.C.:* 11. Ii.
2050, e.]
I. Commentarii Academias Scientiarum Imperialis Petropolitanae.
1726-1746; 14 vols. [B.M.C: 431,/.]
II. Novi Commentarii A. S. I. P. 1747-75, 1750-77; 21 vols.
[B. M. C. : 431,/. 15-17, g. 1-16, h. \, 2.]
III. Acta A. S. I. P. 1778-86; 12 vols. [B.M.C: 431, /i.;3-8; or
T.C. 8,a. 11.]
IV. Nova Acta A. S. I. P. 1787-1806; 15 vols. [B. M. C. : 431,
A. 9-15, LIS; or T.G. 8, a.23.]
V. Leonliardi Euler Opera minora coUecta, vel Commentationes Aritli-
meticte collectse ; 2 vols. 1849. [B. M. rj. : 853 J-, ee.]
VI. Opera posthuma mathematica et pliysica ; 2 vols. 1862. [B.M.C:
8534,/]
VII. Opuscula analytica ; 1783-5; 2 vols. [B.M.C: 50,/. 15.]
Analysis infinitorum. [B.M.C: 529,6.11.]
G. S. C.
Endslkigh Gakdkns,
London, N.W., 1886.
British Museum Ctitiilogui
i
[Correct tons nhich mi important are marknl with an imtirink.)
•Art.
1,
Lino 7, fu
• volume
read weight.
„ 10,
f^amnie-million
,,
gramme-six.
6,
,, 5.
1-407 1490
•4971499.
,, 6,
■(•.(•.7o:i")S
,,
M447'-'90.
123,
2
2 v/5
,,
2^/15.
259,
2,
a- + /3-
,,
(a' + 0-)».
276,
„ 6,
3;(- + w — 1
,,
3«- +'in — \.
291,
„ 1,
S
,,
7-
292,
„ 3&4, „
a
,,
a.
322,
,, 0, ,,
45 and 13
,,
35 and 10.
361,
„ 7,
3528
,,
8684.
459,
„ 3&0, „
-6
,,
-16.
470,
„ 1,
.r,„
,,
X .
489,
„ 0,
„
3
555,
M 1-i.
a number of rows
,,
two columns.
593,
„ 11&12, „
li and i?
,,
El and i?...
604,
•2,
one-sixtieth
,,
one-ninetieth.
713,
M 2, ,.
II.
,,
III.
897,
,, •'', >
cosic
,,
sin Jc.
922ii.,
/>H2c-
,,
2J2 + C2,
949,
„ last, ,
D
,,
C.
1076,
M 2,
(Me "The projections
. . . are p
■irallel."
1158,
„ last, ,
1201
,,
1217.
1178,
M 1.
rs
,,
PS'.
1241,
,, 1,
parallel
>>
conjugate.
1413,
„ 3,
-dxdv
,,
+ dndv.
1491,
„ 3,
-
M
=
1849,
„ 1.
+
,,
-
1903,
footnote, ,
Iw
,.
/W-
1925,
supply dx.
1954-6,
supply X.
2030-2,
erroneous, because / in (1427) is necessarily an
integer
2035,
„ 1,
, ax
,,
a.
2140,
,, 1,
y applies to the whole denominator.
2136,
„ last, ,
, 2294
2293.
2354,
, -
+
2392,
xi'
X -1.
2465,
P.
2
T
3237,
,. 1.
, (»-l).r
("-I).
3751,
s„ppl,, ,
ii-
4678,
ilele 2 in the second term.
4680,
supply the factor 4 on
the left.
4692,
;-),
dele 2 in the second term.
4903,
,, :J,
mpplij the factor 4 on
the loft.
5154,
M •*.
, 3155
,
51 -.5.
5330,
,, 2.
, m
-
/>.
and refer to Fig. 129, Art. 5332 on the cardioid is wanting.
MATHEMATICAL TAJiLKS.
INTRODUCTION.
The Ccnthnctrv-Grammc-Second system of units.
Notation. — The decimal measures of length are the kilo-
metre^ hectometre, decametre, metre, decimetre, centimetre,
miUimetre. The same prefixes are used with the litre and
gramme for measures of capacity aud v o lume; ^'^''f^
Also, 10' metres is deuouiinated a metre-><ei:en; 10"^ metres,
a seventh-metre ; lO^f grammes, a gramme-fy'tcen ; and so on.
A o;rainme-imttwH is also called a megagramme ; and a
milliontli-gramme, a microgramme ; and similarly "with other
measures.
Definitions. — The C. G. S. system of units refers idl pliy-
sical measurements to the Centimetre (cm.), tlie (Jramme (gm.),
and the Second (sec.) as the units of length, mass, and time.
Tlie quadrant of a meridian is approximately a metre-
seven. More exactly, one metre = 3-28U8Gyi feet = 89-370-i-j2
inches.
The Gramme is the Unit of mass, and the weight of a
gramme is the Uiiit qftveight, being approximately the wciglit
of a cubic centimetre of water; more exactly, 1 gm. =
15-432:U1) grs.
The jL/f;v' is a cubic decimetre: but one cubic centimetre
is the C. G. S. Unit of volume.
1 litre = -035317 cubic feet = '22000(37 gallons.
The l)i/ne (dn.) is the Unit of force, and is the force wliicli,
in one second generates in a gramme of matter a velocity of
one centimetre per second.
The 7'i/v/ is the Unit of worlc and energg, and is the work
done by a dyne in the distance of one centimetre.
The absolute Unit (f atmospltcric pressure is one meg:ulyno
})er square centimetre = 71«'0G1 cm., or 2!>"."j14 in. of mercurial
column at 0" at London, where ^ = y8ri7 dynes.
Elasticitij of Volume = l\ is the pressure per unit area
upon a body divided by the cubic dilatation.
15
MATHEMATICAL TABLES.
Rigidity = n, is the shearing stress divided hj the angle
of the shear.
Young's Modiihifi = M, is the longitudinal stress divided
by the elongation produced, = 9nh -h (3/j + ??).
Tenacity is the tensile strength of the substance in dynes
per square centimetre.
The Gramme-degree is the Unit of heat, and is the amount
of heat required to raise by 1° C. the temperature of 1 gramme
of water at or near 0°.
Thermal capacifi/ of a body is the increment of heat
divided by the increment of temperature. When the incrc-
raonts are small, this is the thermal capacity at the given
temperatu"e.
Specific heat is the thermal capacity of unit mass of the
body at the given temperature.
The Electrostatic iinit is the quantity of electricity which
repels an equal quantity at the distance of 1 centimetre mth
the force of 1 dyne.
The Electromagnetic unit of quantity = 3 X 1 0^'' electro-
static units approximately.
The Unit of potential is the potential of unit quantity at
unit distance.
The Ohm is the common electromagnetic unit of resistance,
and is approximately = \(f G. G. S. units.
The Volt is the unit of electromotive force, and is = 10^
C. G. S. units of potential.
The Weber is the unit- of current, being the current due to
an electromotive force of 1 Volt, with a resistance of 1 Ohm.
It is = -j^ C. G. S. unit.
Resistance of aWiTe = Specific resistance X Length -r- Section.
Physical constants and Formulce.
In the lutitudc of London, cj = 3:2-1908t' feet per second.
= l>!^ri7 centimetres per second.
In latitude X, at a liciglit h above the sea level,
g = (98O-0U56 — 2-.'')028 cos 2/\ — -OOOOO:]/;) centimetres per second.
Seconds ])endaliim = (iJ9-85G2 — -2536 cos 2\ — -0000003 h) centimetres.
THE 7';.17i"i7f.— Semi-polar axis, 20,854890 feet* = G-3;.4ll x lO^centims.
Mean semi-equatorial diameter, 20,9_'G202 „ * = 3782t x 10"
Quadrant oi" meridian, 39-377780 x 10' inches* = TOOOlOO x lO" metres.
Volume, r08279 cubic centimetre-nines.
JIass (with a density 5g) = Six gramme- twenty-sevens nearly.
* These dimensions nro liikcn frjiu C'larko'a "Geodesy," 1880.
MATIIEMA TICAL TA BLES.
Velocity in orbit = 2033000 ccntims per sec. Ohii.iuity, -2:f 27' lo".»
Aiii^ular vclucily of rotation = 1 -=- 13713.
Precession, t>0"'20.* Prounession of Apse, U"-2.'). I':cCentricity, e = -01079.
Centrifugal Ibrce of rotation at tlio equator, ;>-3'.)12 dynes per «^nunnio.
Force of attraction upon moon, -2701. Force of sun's attraction, -0839.
Katio of (/ to centrilu^ral force of rotation, g : rw* = 2H0.
Sun's horizontal parallax, h"7 to '.»'.* Aberrat i<.n, 20"-ll to 20"70.*
Semi-diameter at earth's mean distance, 1»>' l"b2.*
Approximate mean distance, 1>2,UUOUUO miles, or l"'i8 centimetre-tliirleens.t
Tropical year, 3Go2422l6 days, or 31,550927 seconds.
Sidereal year, 305-250374 „ 31,558150 „
Anomalistic year, 305-259544 days. Sidereal day, 8010 !• seconds,
TJllJ M0UN.—Uas8 = Earth's ma.'^s X -011304 = 0-98 10^ granimes.
Horizontal parallax. From 53' 50" to 01' 24".*
Sidereal revolution, 27d. 7h. 43m. 1 l-40s. Lunar month, 29d. 12h. l-lin. 2878.
Greatest distance from the earth, 251700 miles, or 4U5 centinieire-tens.
Least „ „ 225000 „ 303
Inclination of Orbit, 5° 9'. Annual regression of Nodes, 19° 20'.
Hulk. — {The yt'ar+l)-^19. The remainder is the Guhlen Number.
{Tlie Uulih'ti Number— 1) X 11-^30. The remainder is the Ejiact.
GRAVITATION.— Attraction between masses ) mm clvues
m, m' at a distanco / j ~ ;-' x l-54o x It/
The mass which at unit distance (1cm.) attracts an eijual mass with unit
force (1 dn.) is = v/(l-543x 10^; gm. = 3iV28 gm.
Tr.rr^/i!.— Density at 0°C., unity ; at 4°, 1 0000l3 (Kupffer).
Volume elasticity at 15°, 2-22 X iV".
Compression for 1 megadyue per sq. cm., 4-51x10-* (Amaury and
Descamps).
The heat required to raise tlie temperature of a mass of water from 0° to
i° is proportional to < + -00002/" + OiJ00003i* (Regnault).
G'yI-8'ii'iS'.— Expansion for 1° C, -003065 = 1-4-273.
Spccitic heat at constant pressure _ i.jAq
Specific heat at constant volume
Density of dry air at 0° with Bar. at 76 cm. = -0012932 gm. per cb. cm.
(Regnault).
At unit pres. (a megadyne) Density = -0012759.
Density at press, p = jtx 1-2759 X 10"'.
Density of saturated steam at t°, with j) taken) _ -7931 .0 9^;)
from Table 11., is approximately j (i -|- OUoOOO lO"*
SOUND. — Velocity = \/(elasticify of mediuvi -^ detusifij).
Velocity in dry air at t° = 3o2 10 ^(1 -+--00300/) centimetres per second.
Velocity in water at U' = 14;:U(J0 „ „
LIGHT. — A'elocity in a medium of absolute refrangibility /i
= 3004 X lO"'-^^ (Coruu).
If I' be the pressure in dynes per sq. cm., and / the temperature,
^i-l = 29(K; X lU-''i'-4-(l-|--OO30i;/) (Biot & Arago).
• These fliita iire from the "Nautical .Mmniiack" for 1S8:{.
t Inuisil ol' Vuuus, IbTt, " Aalruin. S c. XuI.lcs,'' Vols. 37, '<i8.
MATHEMATICAL TABLES.
Table I.
Various Measures and their Equivalents in C. G. S. units.
Dimensions.
1 inch = 2-5400 cm.
1 foot = 30-4797 „
1 mile = 160933 „
1 nautical do. = 185230 „
1 sq. inch = 6'451G sq. cm,
1 sq. foot = 929-01 „
] Pq. yard = 83G1-13 „
1 sq.mile = 2-59 X 10^°,,
1 cb. inch = 16387 cb. cm.
1 cb. foot = 28316
1 cb. yard = 761535 .,
1 gallon = 4541 „
= 277-274 cb. in. or the vo-
lume of 10 lbs. of water
at 62° Fall., Bar. 30 in.
1 grain
1 ounce
1 pound
1 ton
1 kilogramme
1 pound Avoir
1 pound Troy
Mas^.
= -06479895 gra.
= 28-3495
= 453-5926 „
= 1,016047 „
= 2-20462125 lbs.
= 7000 grains
= 5760 „
^'cIoc■lfl/.
1 mile per hour = 44704 cm. per sec.
1 kilometre „ = 27'777 „
Pressure.
1 gm.persq.cm.= 981 dynes per sq. cm,
1 lb. pcrsci.foot = 479 „
1 lb. per sq. in. = 68971
76 centimetres-)
of mercury [ = 1,014,000 „
at 0° C. )
^^^^- P^^ ^q- ^"- = 70-307 = ^
gms. per sq. cm. -014223
Force of Gravity.
upon 1 cramme = 981
1 grain = 6fi-^)Cj777
„ 1 oz. =2-7811x10*
„ 1 lb. = 4-4497 X 10-^
„ 1 cwt, =4-9837x10'
„ 1 ton = 9-9674 X 10»
WorJc (^ = 981),
1 gramme-centimetre = 981
dynes
erofs.
1 kilogram-metre
1 foot-grain
1 foot-pound
1 foot- ton
981 X 10-"^
1-937 xlO\,
1-356x10" „
= 3-04 X W „
1 'hor.se po-wer' p. sec. = 7-46x10®
ITeat.
1 gramme-degree C, = 42 X 10" ergs,
1 pound-degree =191x10- „
1 pound-degree Fah. = 106 x lU* „
Table II.
Pressure of Aqueous Vapour in
dynes per scquare centim.
Teinj).
Press me.
Temp.
Pressure.
-20°
1236
40°
73200
-15°
1866
50°
122t;00
-lu°
2 7; 10
60°
li)8500
- 5°
4150
80°
472900
0°
6133
100°
1014000
5°
8710
120°
1988000
10°
12220
140°
3626000
15°
16930
160°
6210000
■ 20°
23190
180°
10060000
25°
31400
200°
15600000
30°
42050
Table III.
Values for the principal Lines of
t he Spectrum in air at 1G0°C.
with Bar. 76 cm.
Wave-length
No.ot'vibriitions
iu centini.<s.
per second.
A
7-604x10-''
3-950 X 10"
n
6-867 „
4-373 „
('
6-56201 „
4-577 „
1) (mean)
5-89212 „
5-097 „
J'J
5-26913 „
5-700
F
4-86072 „
6 179 „
G
4-30725 „
6-973 „
^1
3-96801 „
7-569 „
//,
3-93300 „
7-636
M. I THEM A TFCA L TA BLES.
Elect. Magn.
Specific
Resistance
at 0= C.
9158
2081
00100
10850
1521
1015
9827
13300
5000
Rjite of
Coniluction
of Sound
in cm. per sec.
Ci -f ' CO -1 -* -O Ol -O Ol ' ' CO
-c IN. Ol '-C r^ 'O CO o 01 ';0
CN Ah Ah Ol CO CO -rf< O "'o -f
Ill-
coo l-ioooco 'co 'coco '
Specific
Heat be-
tween
& 100 c.
"O o r^ ~ X' i^ ~
6? I?? i:^?. 1 IS 1 l^:i:
o o o o -7- 9' r
Linear
Expansion
between
& lOOC.
-000875
•001483
•002801
00100
•00175
•00103
■001111
■001258
001200
•00227
•00204
•00081
§ 1 O 1 1^ O O -lO -^ -+ -? CO O tj. _:o X) .o
'I|l^;|iiii'i';!r5iii
1 I 1 00 1 ->? 00 1 '.o CO r- i^ 1
1 =
3
or:;:;;: ;
I 1 1 1 I x,^ I ! ^
■^ CO l^ XI 1
35 2
ir
E s :: = s r
I 1 |x|^^c5„ci| Ico
CJ CO l^ -f< -^ CO 'C
"0 01 CO ri — . ^
9> ^^^^^,
-0 CO — 1 lO t^ Ci 01
CO Ci 1 - -T^ I^ CO t^ -+ C5 Ci -t
01 ..0 CO -+ X rj< C'l X Ol ^
-^ ^ Ah 6 X 60 l'-~ l'^ I^ t^ I-* 61
Platinum
Gold
^Mercury
Lead
Silver
Copper
Brass, drawn
Iron, cast
Iron,wrou<^'lit
Steel
Tin, cast
Zinc, ca.st
Glas.s, flint
•s
c
C
025
2-01
007
1-00
072
024
0-13
015
027
i
2: T' ^ V '^ 9 9 '.^ f ■
Diameter
in
Miles.
i CO {: r: ^ ?i .0 8 :ri
X C-- l^ CO CO
X
II
X c; X -H '^ Ol -c w
01 OJ Ol Ol r-H
-^ 1 53 ?i ?i ?! ^' 2
a 6
.2 .-e -q
lis
£°-2
III
1 X ol -3 X ?: in X ?i
Ol CO o CO i^ •-= :^
,'ri5x:^^i-."2
1 CO t^ C- CO -- O Ol I^
Greatest distance
from Sun.
Earth's mean
distance = 1.
-v ;f, ,^ ,^ ,^ ?, Zi 5
, '3 X o ..0 CO CO - X
l^r^o-3l^'o'ooi
t^ - ^ 5 =
a X _ u r S -
MATHEMATICAL TABLES.
Table
YI. — Functions of it and e.
TT = 31 1.15;i2G
TT- = ;)-.s(;;t(;<»44
7r» = 31'<tin;2761
-/tt = 1-7724539
logjoTT = t-4971499
ic),o-.7r = &m7m^
TT-' = -3183099
7r-2=: -1013212
TT-' = -0322515
200''-=-7r = 63''-6619772
180°-^7r = 57°-2957795
= 2002r.4"-8
e = 2-7182Hm3
e* = 7 389n:.(;Jl
e-^ = 0-307s7!'4
e-2 = 01353353
lo2-„e = 0-43429448
low-, 10 = 2 -30258509
Table YII.
Table VIII,
No.
2
S([uaro root.
Cube root.
1-414213G
1-2599210
3
1-7320508
1-4422496
4
2-0000000
1-5874011
5
2-2360680
1-7099759
6
2-4494897
1-8171206
7
2-6457513
1-9129312
8
2-8284271
2-0000000
9
3-0000000
2-0800837
10
3-1622777
2-1544347
11
3-3166248
2-2239801
12
34641016
2-2894286
13
3-6055513
2-3513347
14
3-7416574
2-4101422
15
3-8729833
2-4662121
16
4-0000000
2-5198421
17
4-1231056
2-5712816
18
4-2426407
2-6207414
19
4-3588989
2-6684016
20
4-4721360
2-7144177
21
4-5825757
2-7589243
22
4-6904158
2-802o:;i»3
23
4-7958315
2-8438670
24
4-8989795
2-8841991
25
5-0000000
2-9240177
26
5-0990195
2 9624960
27
5-1961524
3-0000000
28
5-2915026
3-0365889
29
5-385164S
3-072316S
30
5-4772256
31072325
2
logio N.
log^.V.
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•69314718
3
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1-09861229
5
-6989700
1-60943791
7
-8450980
1-94591015
11
1-0413927
2-39789527
13
1-1139434
2-56494936
17
1-2304489
2-83321334
19
1-2787536
2-94443.^98
23
1-3617278
3-l;;540422
29
1-4623980
3-3672'.>5SH
31
1-4913617
3-4339S720
37
1-5682017
3-610111791
41
1-6127839
3-71357-207
43
1-6334685
3-761-2O01-J
47
1-6720979
3-8501 i-rt/.o
53
1-7242759
3 9 702; Ml 11
5i>
1-7708520
407753744
61
1-7853-298
4-110873S6
t)7
1-8260748
4-20460262
71
1-S5125S3
4-26267;t88
73
l-S(;;;3229
4-2:h)45:M.4
7'.^
1-8976271
4-36944785
83
1-9190781
4-41884061
89
1-9493900
4-48863637
97
1-9867717
4-57471098
101
2-0043214
4-61512052
103
2-01-28372
4-63472899
lor
2-0293838
4-67282883
Loi)
2-0374265
4-69134788
NoTi^. — The authorities for Table IV. are as follows: — Columns 2, 3, anil
4 (Mereury e.xcepted), Everett's experiments (Phil. Trans., 1867); (j is hero
taken = 981-4. The densities in these cases are those of the specimens
employed. Cols. 5 and 7, Kaid<;iiie. Col. 6, Watt's Diet, of Cliemistry,
Col. 8, Dulong and Petit. Col. 10, Wertheim. Col. 11, Matthiesseu.
'J'al^le V. is abridged from Loomis's Astronomy.
The values iu Tabic III. are An^r^itrom's.
BURCKTIARDT'S FACTOR TABLES.
For all M.'.Mr.KKS FJiOM 1 to 9'JOOO.
Explanation. — Tlic tables give the least divisor of cvciy
number from 1 up to 99000 : but numbers divisible by 2, 3,
or 6 are not printed. All tlie digits of the number whoso
divisor is sought, excepting the units and tens, will be found
in one of the three rows of larger figures. The two remaining
digits will be found in the left-haiid column. The least divisor
will then be found in the column of the first named digits, and
in the row of the units and tens.
If the number be prime, a cipher is printed in the place of
its least divisor.
The numbers in the first left-hand column are not conse-
cutive. Those are omitted which have 2, 3, or 5 for a divisor.
Since 2"-. 3. 5"^ = 300, it follows that this column of numl)er3
will re-appear in the same order after each multiple of 300 is
reached.
Mode of using TnE Tables. — If the number whose prime
factors are required is divisible by 2 or 5, the fact is evident
upon inspection, and the dinsion must be effected. The
quotient then becomes the number whose factors are required.
If this number, being within the range of the tables, is yet
not given, if is dirisihle by 3. Di\'iding by 3, we refer to the
tables again for the new quotient and its least factor, and so on.
Ex.\Mrr,ES. — Required the prime factors of 3101-55.
Dividinrr by 5, the quotient is G2031. This number is within the range
of the tables. But it is not found printed. Therefore 3 is a divisor of it.
Dividing by 3, the quotient is 20G77. The table gives 23 for the least factor
of 2ftr)77. Dividing by 23, the quotient is SW.
The table gives 2i» for tlie least factor of H'.tO. Dividing by 20, the quo-
tient is 31, a prime number. Therefore 31015-3 = 3.5.23.20.31.
Again, roipiired tiie divisors of 02881. The table gives 203 for the least
divisor. Dividing by it, the quotient is 317. Referring to the tables lor 31 7,
a cipher is found in tbe place of the least divisor, and this signifies that 317
is a prime nundjer.
Tlitrefore 02S81 = 203 X 317, the product of two primes.
It may be remarked that, to have resolved 02881 into these factors with-
out the aid of tiio tables by the method of Art. 3G0, would have iuvolved
fifty-nine fruitless trial divisions by prime uumbei-a.
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" - -« 2- 2- -=^^-
t-on ot^— ocoo — or^— no
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— O OOOf»—— MSlOlOl^O
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l^-r^t^M 00-1^0 ©on- t^OiOOM OOt^OlOO
— t^ooo) t»r^r^OM r-ot^oo 3>oo — 000 — r»o
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co'^iot^-nc^ — t^ -1^0
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«-r:i^ r:-oooco oooicot^
^ 2:: = 2z =•"2=5 '-2s = = = = '-s^ 2'"::s = =
_|t--c.p:o noot-o ot^oo- oowi^n o»oi-t-oo
°j;^n — m"^,-c« <o "e«
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1
S
-S = -2 *°°*S =«SS!2 ^ = 523 S^'^S^"
°^2 g^'^S^S «-»»2 = «
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r. - n t^ n - - '^ ^ 5 2°2°Z ° n ^ ° •" -
2:^f? S?2|j§«S S3555S5S
S?:=J^?: ???J?5S§g :?5SS5 S-S-2t-tf^ SSS?$?.2
30
Log T (n).
n
1
2 1 3
4
5
6
7
8
9
1 ("»('»
97497
95001 1 92512
90030 87655
85087
82627
80173
77727
1.01
9.9975287
72855
70430 68011
65600 63196
60799
58408
56025
53648
1.02
51279
48916
46561 44212
41870 139535
37207
34886
32572
30265
1.03
27964
25671
23384 21104
18831
16564
14305
12052
09806
07567
1.04
05334
03108
00889 ' 98677
96471
94273
92080
89895
87716
"5544
1.05
9.9883379
81220
79068 76922
74783
72651
70525
68406
66294
641 8S
LOG
62089
59996
57910 ' 55830
53757
51690
49630
47577
45530
43489
1.07
41469
39428
37407 35392
33384
31382
29387
27398
25415
23449
1.08
21469
19506
17549 1 15599
13655
11717
09785
07860
05941
'•4o29
1.09
02123
00223
98329 96442
94561
92686
90818
88956
87100
~525n
1.10
9.9783407
81570
79738
77914
76095
74283
72476
70676
68882
67095
1.11
65313
63538
61768 '60005
58248
56497
54753
53014
51281
49555
1.12
47834
46120
44411 142709
41013
39323
37638
35960
342y.s 1 32(;22
1.13
30962
29308
27659 26017
24381
22751
21126
19508
1789(1 1 iC.-JS'.t
1.14
14689
13094
11505 1 09922
08345
06774
05209
03650
02<j'.'t; H ■:,['.'
1.15
9.9699007
97471
95941194417
92898
91386
89879
88378
8(;s.-;; \ -:>:\:k\
1.16
83910
82432
.^0960 79493
78033
76578
75129
73686
7224s I 7< 'Sic
1.17
69390 1 67969
(;r,554 65145
63742
62344
60952
59566
5sls.-. !.-..;slu
1.18
55440
54076
52718 51366
50019
48677
47341
46011
44687
4336S
1.19
42054
40746
39444 38147
36856
35570
34290
33016
31747
30483
1.20
29225
27973
26725
25484
24248
23017
21792
20573
19358
18150
1.21
16946
15748
14556
13369
12188
11011
09841
08675
07515
06361
1.22
05212
04068
02930
01796
00669
99546
98430
97318
y6212
"5111
1.23
9.9594015
92925
91840
90760
89685
88616
87553
86494
85441
84393
1.24
83350
82313
81280
80253
79232
78215
77204
7619S
75197
74201
1.25
73211
72226
71246
70271
69301
68337
67377
66423
65474
64530
1.26
63592
62658
61730
60806
59888
58975
58067
57165
56267
55374
1.27
54487
53604
52727
51855
50988
50126
49268
48416
47570
4t;72s
1.28
45891
45059
44232
43410
42593
41782
40975
40173
39376
3^585
1.29
37798
37016
36239
35467
34700
33938
33181
32439
31682
30940
1.30
30203
29470
28743
28021
27303
26590
25883
25180
24482
23789
1.31
23100
22417
21739 21065
20396
19732
19073
18419
17770
17125
1.32
16485
15850
15220 14595
13975
13359
1274S
12142
11540
10944
1.33
10353
09766
09184 ' 08606
08034
07466
06903
06344
05791
05242
1.34
04698
04158
03624 ' 03094
02568
02048
01532
01021
00514
00012
1.35
9.9499515
990''>3
98535 ' 98052
97573
971 Of)
966)30
9(5166
95706
95251
1.36
94800
94355
9;5'.)13 93477
93044
9261 7
92194
;»1776
91362
',H);i53
1.37
90549
90149
S'J754 89363
KS977
Sh'5'.'5 SS218
87H46
87478
87115
1.38
86756
86402
86052 ' 85707
S5366
85030 S469S
S4371
84049
83731
1.39
83417
83108
82803 82503
82208
81916
81630
81348
81070
80797
1.40
80528
80263
80003
79748
79497
79250
79008
7S770
78537
78308
1.41
78084
77864
77648
77437
77230
77027
76829
766)36
7644i?
76261
1.42
76081
75:m)5
75733
75565
751.02
75243
75( »S9
74'.>39 74793
74.;52
1.43
74515
743K2
74254
74130
71(tlO
73Si»4 73783
736)76 73574
73476
1.44
73382
73292
73207
73125
73049
72d7() 7290S
72844 72784
72728
1.45
72G77
72630
72587
72549
72514
724H4 72459
72437 72419
72406
1.46
72397
72393
723',.2'
72396
72404
72416 72432
72452 72477
72506
1.47
72539
72576
7261 7
72662
72712
72766 72S24
72«8(') ' 72952
t:'-^ »22
1.48
73097
73175
7325H
73345
73436
73531 73630
73734 ! 73841
731 >53
1.49
74068
741 HK
74312
7444(.)
74572
74708 74848
74992 75141
75293
Note. — Tliis tabic is taken IVcmi Vol. 1 1, of Lcgendi-c's work, and nut
from Vol. I., as slati d ill <lio Vro'iirc : tin' Tinmlic rs p;v«n in A'ol. 1. l)«'inu;
innrcnrato in the srvfiiUi drciiniil (iI.k . In \',,] it. (hr values .arc ^ivon t<>
twelve places of dotinmls. 1 lie (ij,'uri' Ian juihttd in tlir s^viuth place is
L(
g r(/0.
31
n
1
2
3 4 5
6
7
8
i.r,o
0.9475440
75610
75774
75013
7(;ii6
76202
76473
76658
76847
77040
].iA
77237
77438
77642
77851
78ot;4
78281
78502
78727
78056
70189
i:>-2
70426
70667
70012
8(»l(;i
H()414
80671
8O032
81106
81465
81738
\.'>-A
82015
82205
8258(1
K-1H6S
X3161
83457
83758
84062
84370
84682
{.'A
84'.»0S
8.")318
85t;42
8507<»
86302
8(5638
8f>077
87321
87668
KHOlO
I :.:.
8S374
88733
8o<.»0(;
8li4t;3
8'.t,s;54
0(»2O8
00587
0O060
01355
01745
I ru;
021311
02537
02038
03344
03753
04166
04583
05OO4
05420
05857
1.57
06280
06725
071 1;5
07600
0805r)
085( 8
08063
00422
008H5
00351
loH
0.05OOH22
01206
01774
02255
02741
03230
03723
04220
04720
05225
1.511
05733
06245
06760
07280
07803
08330
08860
09305
09033
10475
I. GO
1102O
11560
12122
12670
13240
13804
14372
14043
15510
16008
l.Ol
it;6s(i
17267
17857
18451
10O48
10650
20254
2os(;-j
21475
2201*1
l.tV2
22710
23333
2306O
24501
25225
25863
26504
27140
27708
28451
1.G8
20107
20767
30430
31007
31767
32442
33120
33801
34486
35175
1.04
35867
36563
37263
37066
38.;73
303H3
40007
40815
41536
422(JO
1.65
42080
43721
44456
45105
45038
46684
47434
48ls7
48044
40704
1.(36
50468
51236
52007
52782
5356(J
54342
55127
55016
5670S
575(J4
1.67
58;-{o3
50106
50013
60723
61536
t;2353
63174
6300S
6482(;
65(;56
1.68
66401
67320
68170
60015
60864
70716
71571
72430
73203
74150
1.60
75028
750U1
76777
77657
78540
70427
80317
81211
82108
83008
1.70
83912
84820
85731
86645
87563
88484
89409
90337
01268
02203
1.71
03141
04083
05028
05977
06020
07884
08843
OOsd.".
00771
Hi 740
1.72
9.9602712
03688
04667
05650
06636
07625
0861s
oor.u
10613
11616
1.73
12622
13632
14645
15661
16681
17704
18730
107t;n
20703
21S30
1.74
22860
23012
24050
260O0
27062
28118
20178
30-J41
3130S
32377
1.75
33451
34527
35607
36600
37776
38866
30050
41055
42155
43258
1.76
44364
45473
46586
47702
48821
40044
51070
52200
53331
54467
1.77
55606
56740
57804
50043
60105
61350
62500
63671
64836
66004
1.78
6717 G
68351
69529
70710
71805
73082
74274
75468
7666b
77866
1.79
79070
80277
81488
82701
83018
85138
86361
87588
88818
00051
1.80
91287
02526
93768
05014
96263
07515
08770
ij0020
"1201
^2555
1.81
9.9703823
05005
06360
07646
08927
10211
11408
12788
14082
15378
1.82
16678
17081
10287
20506
21908
23224
24542
25864
27180
28517
1.83
29848
31182
32520
33860
35204
36551
37000
30254
40610
41060
184
43331
44607
46065
47437
48812
50100
51571
52055
54342
55733
1.85
57126
58522
50022
61325
62730
r>4140
65551
6t;066
68384
♦ ".0805
1.86
71230
72657
74087
75521
76057
783^37
70830
81285
H2734
84186
1.87
85640
87008
88550
00023
01400
02060
04433
05010
0738'.*
08871
1.88
9.9800356
01844
03335
04830
06327
07827
00331
108;i7
12346
13859
1.89
15374
16893
18414
19939
21466
22996
24530
26066
27606
20148
1.00
30603
32242
33703
35348
36005
38465
40028
41505
43164
44736
1.01
46311 17-'.'M l'.'l-71
51055
526.42
54232
55825
57421
50020
60622
1.02
62226 1 •■,:;>:; 1 r,.-,-l.45
67058
('.8675
70204
71017
73542
75170
76802
1.03
78436 son 7;; si 713
83356
85002
86651
88302
80057
01614
03275
1.04
04038 ''.».;.;. 15 I its-j 74
99946
01621
O3200
O4080
06()C>3
08350
To039
1.05
9.9911732 13427
15125
16826
18530
20237
21047
2365it
25375
27003
1.96
28815
30539
32266
33995
35728
37464
30202
40043
42(".88
4-4435
1.97
46185
47937
40693
51451
53213
54077
56744
58513
t;o28(".
«;2062
1.98
63840
65621
67405
60102
70082
72774
74570
763f.s
78160
70072
1.99
81779
83588
85401
87216
80034
00854
02678
045O4
06333
08it;5
tho one nearest to the true value whether in oxcosa or defect. This table, and
the table of Least Factors, have tai-h been subjected to two couiiilcte and ia-
dependont rovisivua before linuUy printing ofl".
ALGEBRA.
FACTORS.
1 a'-h'= (n-b) («+6).
2 <r-lr = (a-b) {(r-\-(tb-irb-).
3 a'-\-i/' = (a-\-b) {(r-ab-\-b').
And generally,
4 «"-.ft" = (a—b) («"-^ + «"--/>+ ... + //-') iilwiiys.
5 a" — b" = {(i + b) {a"-' — a"--b+...—b"-') if n be even.
6 a"-\-b" = {a-\-b) («"-^-a»- -6+...+6"-') if n be odd.
8 Gr+^O G^*+&) (^<' + c) = .^•^^+(^f + /> + r') .r-^
ft / I /\-' ■' I .1 / I 7' 'i-(bc-{-('a-\-(ih).i -\-(ihc.
9 (r^ + o)- = (r-\-'2(W-\-b-. ' V I I / .
10 {(i-b)- = ir-'lab-\-}r.
11 (^/ + 6)'' =: a'-^\\irb^?uib-^-b^ = a'^b'-^i\ab {a-\-b).
12 {a-bf = a'-[\(rb-{-{\alr-lf' = d'-li'-Wab {a-b).
Generally,
{a±by=a'±7(i'b-]-2\(vb-±'^'m'b'-\-:irui'b'±2](tW-^^
Newton's Rule ior forming- the coefficients : Miiltiphj (inn
coefficient by the index of the leading qnantifi/, and divide bij
the number of terma to that place to obtain the coefficient of the
term next following . Tims 21xr)-^3 gives 35, the following
coefficient in the example given above. See also (125).
To square a polynomial : Add to the square of each term
twice the 2)roduct of that term and every term that folloivs it.
Thus, {a-\-b-^r-]-f/y
= rt- + 2rt(ft + r-+f/) + //-+2/>(r+^/)+r- + 2rr/+^/-.
34 ALQEBUA.
13 a'-f a-6"+6' = {a--^ah-\-h-) {ir-ab-\-¥).
14 a'+b' = {a'+ab V2-\-I)') (a'-ab V2+6^).
15 („.+iy=,.Hi+2, (.+iy=.'+-L+3(,,.+i).
16 {a-\-b-^cY = (r-\-b--]-c'+2bc-\-2('a-\-2ab.
17 (a-^b + cf = a'-\-b'-[-c'-\-',^ {b'c^bc--\-c'a-\-ca'
■^a-b+a¥)-\-6abc.
Observe that in an algebraical equation the sign of any
letter may be changed throughout, and thus a new formula
obtained, it being borne in mind that an even power of a
negative quantity is positive. For example, by changing the
sign of c in (16), we obtain
{a-\-b-cf = a^ + h' + c'-2bc-2ca + 2ab.
18 a'+b'-c'+2(ib = {a-^b)--c' = («+6+c) {a+b-c)
^y (1).
19 (r-b'-r-\-2bc = a'-(b-cy = (a + b-c) {a-b-\-c).
20 a'-\-b'-\-c'-^abc = {a-^b+c) {a'-\-b'-\-c'-bc-ca-ab).
21 bc'+b'c-\-ca'+chi+ab'-\-(rb-{-a'-\-b'-\-e'
= {a-^b+c^((r+b'-^r).
22 bc'-\-b-c-\-c(r+c'a + ab''-^a-b + :\(ibc
= {a-\-b-\-c){bc-}-ca-\-ah).
23 bc'-\-b'c + m''-\-c'a-\-ab'+(rb + 2abr={b-{-c)(c-\-((){a-j-b)
24 b(r + b'c + cd' + c'n + ab- + (rb — 2a be — ({' — h' — r '
= {b^c-a) {c^(i-b) {(i^b-c).
25 bc^-b^c + ca'-c^a-irab'-irb = (b-c) (c-a) (a-b).
26 2b'c'-\-2c'a'-Jr2a'b~-a*-b'-c'
= {(i + b-]-c) ib-\-c-a) (c + a-b) (a-\-b-c).
27 .rH2.t%+2.r/ + // = (,,■+//) C^■-• + .^// + /^).
Generally for the division of {x + //)" — {x" -\- //") by .r- + xi/ -\- y-
see (545).
MULTIPLIOATIOS AXD J >I VISION.
35
MrLTirLTCATTON AXD DTVTSTOX,
15YTHE MKTIKil) OF DETACH HI) roKFFICI KNTS.
28 Ex. 1
(a*-SaV + 2ab' + b*) x (a'-2a6»-26').
1+0-3+2+1
l+U-2-2
1+0-8+2+1
-2-0+6-4-2
-2-0+6-4-2
1+0-5+0+7+2-6-2
Result a^-5aV- + 7a%* + 2o:'b'-Gab"-2b'
Ex. 2: (x^-bx' + 7x' + 2x'-6x-2)-r-(x*-Sx' + 2x + l).
1+0-3+2+1) 1+0-5+0+7+2-6-2(1+0-2-2
-1-0+3-2-1
0-2-2+6+2-6
+2+0-6+4+2
-2+0+6-4-2
+2+0-6+4+2
■ Result a;»-2.i— 2.
S(/ntlift}r Dici.sinn .
Ex. 3: EmployiTig the ]a.st example, the work stands thus,
1+0-5+0+7+2-6-2
0+0+0+0
+3+0-6-6
-2+0+4+4
-1+0+2+2
-0
+3
-1
1+0-2-2
Re.sult
[See also (248).
Note that, in all operations with detached coefficients, the result mn.st he
written out in successive powers of the quantity which stood in its successive
powers in the original cxpre.-^sion.
36 ALGEBRA.
INDICES.
29 Multiplication: a}x c(^ = a}'^^ = a^, or ^^a^;
a'
" X a" =
1 1 m + n
a'" "= a'"" ,
or
Va'-".
Division :
a'
4 . 4
L JL
' -i-a'" =
or
or
Involution :
(a*)i =
= a«'<4 = a*,
or
l/a.
Evolution :
ya»=:
a«^^=aA
or
Va\
a~
a
= 1
HIGHEST COMMON
FACTOR.
30 Rule. — To find the highest common factor of two ex-
pressions : Divide the one 'which is of the highest dimension hy
the other, rejecting first any factor of either exjrression which
is not also a factor of the other. Operate in the same manner
ujjon the remainder and the divisor , and continue the process
until there is no remainder. The last divisor ivill he the
highest common factor required.
31 Example.— To find the H. C. F. of
3.«^- lO.f^ + 15.1- + 8 and ^ - 2x' - 6.«» + k/' + 13,c + 6.
1- 2- 6+ 4 + 13+ 6 3 + 0-10+ + 15+ 8 3
3
3_ 6-18 + 12 + 39 + 18
-3- 4+ 6 + 12+ 5
2 ) -10-12 + 24 + 44 + 18
- 5- G + 12 + 22+ 9
3
-15-18 + 36 + 66 + 27
+ 15 + 20-.30-60-25
-3 + 6 + 18-
-12-
-39-
■18
2)6 +
8-
-12-
-24-
10
3 +
-3-
4-
9-
- 6-
- 9-
-12-
- 3
■ 5
+
5_15_15_
5 + 15 + 15 +
5
5
2)2-1- G+ 6+ 2 Result H. C. F. = .'c» + 3.r + 3.« + l.
1+ o+ 3+ 1
EVOLUTIOS. 37
32 Otherwise. — To form the H. C F. of two or more
algebraical expressions : Sfpanite the e.rprcHtiions into their
simplcd fdctorx. The 11. C. F. will be the product of the
factors co)nmoiL to all the exjyrGSsionSy taken in the loivest
powers that orcur.
LOWEST COMMON MULTIPLE.
33 The L. G. M. of two quantities is equal to their product
dicided hi/ the E. G. F.
34 Otherwise. — To form the L. C. M. of two or more
algebraical expressions : Separate them into their simplest
factors. The L. G. M. will he the product of all the factors
that occur, taken in the highest powers that occur.
Example.— The H. C. F. of a\h-xfchl and aXh-xfc'e is a=(6-.r)V
and the L. C. M. is a'{b — xY'c'de.
EVOLUTION.
To extract the Square Root of
., 3a \/a S\/a , 41a , ,
"'-—^ 2- + 16-+'-
Arranging accoi'ding to powers of a, and reducing to one denominator, the
16a2-24;a'-|-41a-24a5 +16
expression becomes
16
35 Detaching the coefficients, the work is as follows :-
16-24 + 41 -2-4 + 10 (4-3 + 4
16
8-3
-3
-24 + 41
24- 9
8-6 + 4 32-24 + 16
' -32 + 24-16
D li. 4a — 3o* + 4 T / , 1
Result — ' — =a — ^v/a+l
38 ALGEBRA.
To extract the Cube Root of
37 Sx' - 36a;'' ^ij + 66x'y - 63xhj ^y + 33,cy - 9^- ^y + y\
The terms here contain the successive powers of .r and \/y ; therefore,
detaching the coefficients, the work will be as follows: —
I. II. III.
6-3) 12 8-36 + 66-63+33-9 + 1(2-3 + 1
-6)
-18+ 9^
1
12-18+ 9
+ 9j
1
12-36 + 27
6-
-9 + 1
-s
6-9 + 1 12-18+ 9 f -36 + 66-63 + 33-9 + 1
+ 36-54 + 27
12-36 + 33-9 + 1
12-36 + 83-9 + 1
-12 + 36-33 + 9-1
Result 'la?—3x^y + y.
Explanation. — The cube root of 8 is 2, the first term of the result.
Place 3x2 = 6 in the first column I., 3x2^ = 12 in column II., and 2*= 8
in III., changing its sign for subtraction.
— 36-f-12 = — 3, the second term of the result.
Put -3 in I.; (6-3) X (-3) gives -18 + 9 for II.
(12 — 18 + 9) X 3 (changing sign) gives 36 — 54 + 27 for III. Then add.
Put twice ( — 3), the term last found, in I., and the square of it in II.
Add the two last rows in I., and the three last in II.
12-T-12 gives 1, the thu'd term of the result.
Put 1 in col. I., (6-9 + 1) xl gives 6-9 + 1 for col. IT.
(12 — 36 + 33- 9 + 1) X 1 gives the same for III. Change the signs, and
add, and the work is finished.
The foregoing process is bub a slight variation of Horner's
rule for solving an equation of any degree. See (533).
Transformations frequently required.
38 If^=^, then ;^ = ^^ [68.
39 If .'■+.'/ = «^_^,^^ S.r = \{u+b)
and cV — ?/ =^hy \a)=. \[a — 0)
40 i^^-^yr+i^v-^jY = 2 (.r+y^).
41 {,v-\-i,y-{.v-i/y = i.vij.
EQUATIONS. 39
42 0^'+.y)' = (.r-//)-'+4r//.
43 {^-f/Y = (.r+//)'-4f//.
44 Examples.
2 y g^ - b' + ^6' - x' _ 3 y/g^ - //- + y/r- - ^Z'^
v/c^-o^ y/c^-rf^
9(r-a;^) =4(r-tZ-),
,[38.
a^ = y^c'+w
To simplify a compound fraction, as
' .,+ 1
a^ — ab + h- a- + ah + li-
1 1
a*— a6 + 6* a* + ab + b''
multiply the numerator and denominator by the L. C. M. of all the smaller
denominators.
Result (a^ + ab + b') + (a'-ab + b')^a- + lr
(a- + ab + b')-(a--ab \-b') ab
QUADKATIC EQUATIONS.
'2(1
46 If «cr-+2^>( -fr = (I ; that is, if the coefficient of ,r be
an even number, .i' = .
47 Method of solution without the formula.
Ex.: 2.r— 7« + 3 = U.
7 3
Divide by 2, x'— -—x+ - = 0.
2 <j
40 ALGEBRA.
n w ^1 2 7 , /7\- 49 3 25
Complete the square, x^ x-\- { = = —- .
2 \ 4 / 16 2 16
Take square root, re— — = ± — ,
4 4
.r = ^^ = 3 or -
"4 2*
48 Rule for "completing the square" of an expression like
33^ — fci' : Add the square of half the coefficient of x.
49 The solution of the foregoing equation, employing formula (45), is
_fc±v/fe2_4^^ 7^y49_24 7±5 o 1
" = 2a = 4 = -^ = ^ "' 2-
THEORY OF QUADRATIC EXPRESSIONS.
If a, /3 be the roots of the equation ax--\-hx-\-c = 0, then
50 a.v'-]-kv-{-c = a {a—a){.v-S).
51 Sum of roots a+/8 = — -.
a
52 Product of roots a/3 = -.
a
Condition for the existence of equal roots —
53 b^—4<ac must vanish.
54 The solution of equations in one unknoAvn quantity may
sometimes be simplified by changing the quantity sought.
Ex.(l): 2.+ «»L-l+ l^Lte =14 (1).
Sx + 1 dx' + bx—l
6.^- + 5.g-l ^ 6(3j; + 1) ^ j^
3a; + l 6ar + bx — l
-^^^^ (^)-
EUi'ATloXS. 41
thus y + - = i^-
y
y having been dcUrininetl from this quadratic, x is afterwards found from (2).
55 Ex.2: a;H -, +X+-- =4.
(..!)%(.. l) = c.
Pat x-\ = ij, and solve the qnadi-atic in y.
X
56 l'^>i- '^ ■■ x' + x+^^2x' + x + 2 = -iy + 1
2x- + X + S v/2.«* + x + 2 = 2,
lx^ + x + 2 + 3y/2.x^ + x + 2 = 4.
Put v2x*-\-x + 2 = y, and solve the quadratic
57 Ex.^: ^^"+3|v.= ¥-'
'i , 2 ? Iti
'■' + 3 ■'■ = 3 ■
2.1
A quadratic in y =z x^ .
58 Tojind Md.vitnd (ind MininKt rahn'.s hi/ menus of a
Q 1 1 (I (h'd t ic Ju/ uatiou.
Ex.— Given ;/ = 3.r + G.c + 7,
to find what value of x will make y a maximum or mitiimnm.
Solve the quadratic equation
3.c' + 6a; + 7-y = 0.
Tl,>,s ^^ -3±y3y-12 ,.45
o
In order that .r may be a real quantity, we must have '^y not less than 12 ;
therefore 4 is a minimum value of //, and the value of x which makes y a
minimum is — 1.
O
oe
42 ALGEBBA.
SIMULTANEOUS EQUATIONS.
General solution icith tu-o unknotvn quantities.
Given
59 (ti^v-\-b,ij = Cil ^^. — c,b, — cA ^ ^ c,a,-c,a,
a.a-[-b,i/=eJ' ' a^h.—a,h^ h^a.—b^a^
General solution with three unknown quantities.
60 Griven chA^-{-b,y-^c,z = (U\
a..v-\-biy-^c.z = dj
_ d,(h,c,-hc.?i + d, {b,e,-b,Cs) + (h (brC,-b,e,) ^
«i {b-2Cz—bsCo)-{-ao {hc^—b,Cs)-\-a3 {biC. — b.eyY
niid symmetrical forms for y and z.
Methods of solving simultaneous equations bettceen two
unknown quantities x and y.
61 I. By substitution. — Find one unhioivn in terms of the
other from one of the tivo equations, and substitute this value
in the remaining equation. Then solve the resulting equation.
Ex.: .r + 52/ = 23 (1)]
77/ = 28 {-I)]'
From (2), y = 4-. Substitute in (1) ; thus
.i- + 20 = 23, .r=3.
62 IL By the method of Multipliers.
Ex.: '6x + 5y = 36 (1) \
2x-:hj= 5 (2)V
Eliniinitc .»■ l)y multiplyincr oq. (1) by 2, and (2) by 3; thus
6x + l0y = 72,
6x— % = 15,
I9y = 57, by subtraction,
.'/= 3;
,T = 7, by substitution in Lt[. (2).
I'Uil'ATJONS. 4'.\
63 Til. />// clKm^,in^: thr (/unnfitirs .sou;iIif.
Ek. 1: x-y= 2 (1))
.r-2/- + a- + v/ = :iO (-1))'
Let .(• + // = ", x — if = i\
Substitute tliese valiius in (1) and (2),
uv + u = 30 )
n = 10 ;
x-\-i/ = 10,
From which x = 6 and // = -i.
64 Ex.2: 2 -Ltl + 10 ^l^JL = 9 (1)
^* X — 1J x + y
z' + 7>r = 0i (2)
Substitute ;: for ^^^' in (1) ;
x-y
,. 2.-^1^ = 0;
2-^-92+10 = 0.
From which z = ~ or 2,
— !^ = 2 or — .
x-y I
7
From which x — '.iy or — »/.
Substitute in (2) ; tlius .'/ = 2 and x = 6,
or ^ ~ 77? ^"'^ '*^~ ~^'^'
65 Ex.3: 3.j; + 5^= a-y (1) )
2x + 7y = 3.vy (2))-
Divide each (juantity by xy ;
^+ ^ =1 (^))
y « f
- + ^=3 (I, •
V ■*' I
Multiply (o) by 2,- and (I) by 3, and by subtraction y i.s eliminated.
44 ALGEBRA.
66 IV. % substituting y = tx, tfhen the equations are
homogeneous in the terms tvhich contain a' and y.
Ex.1: 52x^ + 7.ry = 52/^ (1)7
^x-^ = n (2)5"
From (1), h^x' + ltx' = 6fx' (3) j
and, from (2), bx-Stx = 17 (4))'
(3) gives 52 + 7t = 6t\
a quadratic equation from whicli t must be found, and its value substituted
in (4).
X is thus determined ; and then y from y = tx.
67 Ex.2: 2x'' + xy + '3y' = l6 (1) |
3y-2x= 4 (2)3'
From (1), by putting y = tx,
x'(2 + t + 5t') = 16 (3)) .
from (2), a. (3^-2)= 4 (4)3 '
squaring, a;' (9^^-12^ + 4) = 16 ;
9t'-12t + 4^ = 2 + t + Sf,
a quadratic equation for t.
t beino- found from tliis, equat'on (4) will determine x ; and finally y — tx.
RATIO AND PROPORTION.
68 \i a\h v. e \ d\ then ad = be, and — = — ;
a-\-b __c-\-d ^ a — b _ e—d ^ a-^b _ e-\-d
~~b d ' b ~ d ' a — b c—d
69 " T = 17 = 7 = '^" ' """ T - i+</+/+&c.-
General theorem.
70 If ^ = 4 = 4 := kv. = k say, then
b d J
. ^ ^ pa'' + f/c"-]-re" + Szc. } I
lpb"-\-qd"-\-rf"-^&G.))
where /), q, r, Sic. arc any quantities ^vhatever. Proved as
in (71).
72/1770 AND ritOPOHTlON. 45
71 Rile. — To verify any equation between such proportional
quantities: Suhsfifufe for d, r, c, (Jv., their eqiiicalenta kh, Inly
kj\ ^'c. respect Ivcl ij, in the given equation.
Ex. — If a '.h '.: c : d, to show that
y/g — 6 _ \/a — \/b
^c — d Vc— -/d
Fni kb for a, and kd for c ; thus
^/a^> ^/kb-b s/by/k-i s/b
x/c-d -^kd-d Vds/k-l ^d
Va-K/ h ^ Vkh-Vb _ v/6 ( x/k-1) ^ v^
Vc-Vd s/kd-Vd ~VdWk-\) -/d'
Identical results being obtained, the proposed equation must be true.
72 li a : b : c I d '. e &c., forming a continued proportion,
then a : c :: cr : fr, the duplicate ratio of a I b,
a : di: a^ : b^, the triplicate ratio of a I b, and so on.
Also \^a : ^^h is the subduplicate ratio of a : 6,
a' : h^ is the sesquiplicate ratio oi a : h.
73 The fraction -^ is made to approach nearer to unity in
value, by adding the same quantity to the numerator and
denominator. Thus
-— !— IS nearer to 1 than — is.
6 + aj f)
74 Def. — The ratio compounded of the ratios a : b and c : d
is the ratio ac : Id.
75 li a : b :: c : d , and a' : b' :: c' : d' ; then, by compound-
ing ratios, aa : bl/ :: cc' : dd'.
VARIATION.
76 If rt oc c and Ij a c, then (a + b) cc c and \/ab a c.
77 If ^ Gc^ 7 i.u LI 1 "' '^
• • ^ - [ , then ac cc bd and — oc — .
and coed) c d
78 If <t cc^) ^ve may assume a = hib, where m is some constant.
46 ALGEBEA.
ARITHMETICAL PROGRESSION.
General form of a series in A. P.
79 a, a-\-(I, a + 2f/, « + ;W, a-\-{n — l)d.
a = first term,
d = common difference,
/ = last of n terms,
s = sum of n terms ; then
80 I =a-[-{n-l)d.
81 * = (« + /) I .
82 s={2a-\-{n^l)d}^.
Proof. — By writing (79) in reversed order, and adding both series
together.
GEOMETRICAL PROGRESSION.
General form of a series in G. P.
83 a, ar, ar, ar^, «r" "^
a = first term,
r = common ratio,
I = last of n terms,
s = sum of n terms ; then
84 l = ar"-\
85 s = a or a
r — 1 1 — r
If r be less than 1, and n be infinite,
86 s= -i^, since r" = 0.
I— r
Proof. — (85) is obtained by multi]>lyiiig (83) by r, and siil)(racting one
series from the other.
PEIiMUTATlONS AND COMIHSATIOXS. 47
HARMONICAL PROGRESSION.
, -, — , -y, &G. are in Aritb. Prog.,
87 cf, b, Cy d, &c. are in Harm. Prog, when the reciprocals
i_ 1_ 1 1
a b r a
88 Or when a : b :: a -b : b — c is tlie rehition subsisting
between any three consecutive terras.
89 «^'' term of the series = r^ ; -. [87, 80.
{n-l)a-{n-2)
90 Approximate sum of n terms of the Harm. Prog.
, &c., wlien d is small compared with a,
ft + rf' a + 2d' a-^Sd
_ {a-{-(l)"-a"
1 2
Proof. — By takintr instead the G.P. , + 7— —77-0 + ; — r^TK + ••• •
91 Arithmetic mean between a and h = — ^^.
92 Geometric do. = \/ab.
93 Harmonic do. = — —r.
<t-\-h
The three means are in continued proportion.
PERMUTATIONS AND COMBINATIONS.
94 Tlie nui'iber of permutations of v things taken all at a
thne = n{u-\){n^'>) ...\\.'lA = n\ or ;i"".
Proof hy IxnucriON. — Assume the foniiiila to ho true for n things.
Now take ?i + l things. After eaeh of those the remaining n things may bo
arranged in n ! ways, making in all nX n\ [that is (»t + l) !J permutations of
w + 1 things; therefore, &c. See also (23."^) for the mode of proof by
Induction.
4S ALGEBRA.
95 The number of permutations of n things taken r at a
time is denoted by P {n, r).
P {n, r) = n (w-1) {n-2) ... (w-r+l) = n(^>.
Proof. — By (94) ; for («—r) things are left out of each pei-mutation ;
therefore P (n, r) = nl -i- {n—r)l .
Observe that r = the number of factors.
96 The number of combinations of 7i things taken r at a
time is denoted by G (n, r).
r r,, r) - n{n-l)in-2) ...(n-r-j-l) _ n^^^
^ ' ^ ~ 1.2.3. ..r = 7T
= C {n, n — r).
r\ {n — r)
For every combination of r things admits of r ! permuta-
tions; therefore G {n, r) = P{ii, r) -^ r!
97 G {n, r) is greatest when r = ^u or i{n + \), according
as n is even or odd.
98 The number of homogeneous products of r dimensions
of n things is denoted by H(y, r).
^ ^ ' * ^ 1.2...r = V\ •
When r is > n, this reduces to
(>-+l)(>'+2)...(/^ + >— 1)
99
(V-I)!
PrOOK. — Jl{n, r) is equal to the number of terms in the pi-otluct of the
expansions by the Bin. Th. of the n expressions (1— a.i')~\ (1 — Z/.j)"\
(1 — cr)"', &c.
Pnt a=-h = c =■ &c. = 1. The number will be the coenTicIeut of x'^ in
(1-a:)-". (128, 129.)
ri'lUil UTA TIONS AND COM I'.LXA TIOXS. VJ
100 The niimbor of perimitations of n tliin<j:s tjiken all to-
gether, when a of them are alike, h of them alike, c alike, &c.
a ! ^1 c! ... &c.
For, if the a things were all different, they would form a!
permutations where there is now but one. So of b, c, &c.
101 The number of combinations of n things r at a time,
in which any^ of them will always be found, is
= C(n-p, r-p).
For, if the p things be set on one side, we have to add to them
r—j) things taken from the remaining n—2) things in every
possible way.
102 Theorem: C(n-\, /— 1) + C(;t-1, r) = C{u, >•)•
Peoof by Induction ; or as follows : Put one out of n
letters aside; there are G{u — l,r) combinations of the re-
maining 71 — 1 letters r at a time. To complete the total
C(n, /•), we must place with the excluded letter all the com-
binations of the remaining n—l letters /*— 1 at a time.
103 If there be one set of P things, another of Q things,
another of ii things, and so on ; the number of combinations
formed by taking one out of each set is = FQIl ... &;c., the
product of the numbers in the several sets.
For one of the P things will form Q combinations with
the Q, things. A second of the P things will form Q more
combinations ; and so on. In all, PQ combinations of two
things. Similarly there will be PQE combinations of three
things; and so on. This principle is very important.
104 On the same principle, if p, 7, r, &c. things bo taken
out of each set respectively, the number of combinations will
be the ])roduct of the iiuniberR of the separate combinations ;
that is, = C{rp) . ('{Qr/) . C{Rr) ... Sec.
60 ALGEBRA.
105 The number of combinations of n things taken m at a
time, when p of the n things are alike, q of them alike, r of
them alike, &c., will be the sum of all the combinations of
each possible form of m dimensions, and this is equal to the
coefficient of x'" in the expansion of
(l^-.^' + ,T2+•••+.^'')(l+c^' + cT'-h...+a?'')(l^-■T^-.^''+■•.+.^'•)••••
106 The total number of possible combinations under the
same circumstances, when the n things are taken in all ways,
1, 2, 3 ... 7i at a time,
= (p+l){g+l){r+l)...-l.
107 The number of permutations when they are taken m
at a time in all possible ways will be equal to the product of
m ! and the coefficient of x'" in the expansion of
&c.
SURDS.
108 To reduce >/2808. Decompose the number into its
prime factors by (360) ; thus,
V28iJ8 = y2\ 3M3 = 6 Vl3,
^a'" 6'" c^ = a'» b'^" c = fV' h' c' h c- = a' h" c' Vh6'
109 To briug 5^3 to an entire surd.
5y;3 = vo'. 3 = yi875,
a;» y^ z' = a;- y^ z^" = V^z^.
110 To rationnllse fractions hnvinf:; .surds in their
drnnminators.
j_^ y?. 1 ^ ^49 ^ y4o
SUIiVS. 61
111 .J3^o=^^r-'<"^^'^°''
since (9 - ^80) (9 + ^80) = 81 - 80, by ( 1 ) .
^^^ l+2y8-v/2 (l + 2y3)'--J 11 + 4/3
^ (1 + 2/3+^2) (11-4/3)
73
1^3 ?/3-v/2 3*--i*"
Put 3* = a-, 2' = ij, and take G the L.C.M. of the deriominafors 2 and 3, tlien
„ . 1 3' + 3«2' + 352* + 3'2» + 3«2« + 2*
thereiore — -= z
3i-2* 3^-2'
= 3 y9 + 3 y72 + 6 + 2 yG48 + 4 y3 + 4 v/2.
114. . Here the result will be the same as in the last exainplo
^^^ y3+/2
if the signs of the even terms be changed. [See 5.
115 A surd cannot bo partly rational ; that is, y/a cannot
be equal to >^''h + c. rrovcd by squaring.
116 'J he product of two unlike squares is irrational;
^7 X y/'^ = ^/2], an iri-atioual (piantity.
117 The sum or difference of two unlike surds cannot
produce a single surd; that is, \/a-\-x/h cannot be equal
to \/c. 15y S(jnaring.
118 If " -\- \/m = J'i-^'^n; then a = h and w = n.
Theorems (115) to (118) are i)roved indirectly.
119 If ^/a+ W>= ^.c-\- s^ij,
then
By squaring and by (HH).
52 ALGEBRA.
120 To express in two terms \/7 + 2V6.
Let v/7 + 2v/6= ^x+^tj;
then x + y = 7 by squaring and by (118),
and X-2J = ^7'-{2^6y = a/49-24 = 5, by (119) ;
.•. ic = 6 and y = 1.
Result ye+i.
General formula for the same —
121 \/a±^b=\/i{a-{-x/a'-b)±\/i{a-x/a'-b).
Observe that no simplification is effected unless a' — b is a
perfect square.
122 To simplify v/a+ Vb.
Assume \/a-\- Vb = x-{- \^y.
Let c = y/a^—b.
Then x must be found by trial from the cubic equation
4cr^— SccV = a,
and 7/ = cv'^—c.
No simplification is effected unless a^—h is a perfect cube.
Ex.1: V7 + 5^2 = x+y7j.
c= ^49- 50= -1.
4.(;* + 3.t; = 7 ; .-. x=\, y
Result 1 + v/2.
Ex. 2: y9v/a — 11n/2 = v/-T+ v/y, two different surds.
Cubing, 9 v/3 - 11 v/2 = a; v/.x' + 3.« y?/ + Sy ^x ^y^y,
.-. 9v/3 = (.T + %)ya;-) . .^^o^
liy2 = (:3x- + 2/)y2/) ' ^ ^
.-. .r = ."{ and ?/ = 2.
lUSOMlAL THEOREM. 53
123 To simplify v/(12 + 4y3 + '4yr) + 2yi5).
Assume v/(12 + 4v/3 + 4v/5 + 2yi5) = ^x^ ^y-\- ^z.
Square, and equate corresponding surds.
Result v/3+yi+-/5.
124 To express \/A + B in the form of two surds, wliere A
and B are one or both quadratic surds and n is odd. Take (/
such that q (A^—B-) may be a perfect n^^ power, say />", by
(361). Take s and t the nearest integers to V'y (^4 + /?)''
and Vq{A--B)\ then
2Vq
Example: To reduce y89y8 + lU9y2.
Here A =89^3, B = 109^/2,
A''-B' = l; .-. p=l and q = I.
vq (A + By = 9+f \ f being a proper fraction ;
^qiA-By=l-f\' .-.8=9,1=1.
Result i(^9 + l + 2±v/9 + l-2) = y3+v/2.
BINOMIAL THEOREM.
125 (n+by =
126 General or (/•4-l)^" term,
r!
127 or , ''[, , a"-n/
if n be a positive integer.
If b be negative, the signs of the even terms will be changed.
54 ALGEBRA.
If n be negative tlie expansion reduces to
128 {a+br^ =
129 General term,
v\
Elder's proof.— Let the expansion of (1 +.'«)", as in (125),
be called /(7i). Then it may be proved by Induction that the
equation f{'>n)Xf{n) =f{m + n) (1)
is true when m and n are integers, and therefore universally
true ; because the form of an algebraical product is not altered
by changing the letters involved into fractional or negative
quantities. Hence
/(m + ?i+j9 + &c.) =fim)Xf{n)Xf(p), &c.
Put 7n = 71= 2^ = &c. to Jc terms, each equal —, and the
theorem is proved for a fractional index.
Again, put —n for m in (1) ; thus, whatever n may be,
f{-^i)Xf{n)=f{0) = l,
which proves the theorem for a negative index.
130 For the greatest term in the expansion of (a-^-by, take
... -, ^ c {n^-l)b {n-l)b
r = the mtegral part of ^^ — -—f- or ^^ f— ,
° '■ a-{-b a — o
according as n is positive or negative.
But if b be greater than o, and n negative or fractional,
the terms increase without limit.
Required the 40th term (.f ( 1 —
Examples.
Hero r = 39 ; a = 1 ; b = - '- ; n= 12.
By (127), ilio term will he
_42! / _ 2..\-_ _ ^-^ ■ iU-i^. (2.,-y« (^(3^
y!3i)!\ :W . 1-2.3 \sl ^ '
lilNOMIM. TllbUiUKM.
Roqniied the Slst term of (a — .r)"*.
Here r = 30 ; h=-x\ 7i = — 4.
By (129), the term is
^ 4.5.6...80.:U.:V2.:i1 ,>. ^_ 31 .82. 33 «•'"
^~^^ 1.2. 3... 30 " ( -^^ - 1.2.3 -a" ''^ ^^
131 IleqairoJ the greatest term ia the expansion of — — when a
— = (l-|-.r)"'. Here n = ^^ a = 1, h = x in the formula
(n-l)fc _ 5x|> _ 231 .
a -6 1-H-
thert'foro r = 23, by (130), and the greatest term
_ , ,.o3 5.6.7...27 /14\'»^ 24 .25 .26 .27 lU^
~^ ^ 1.2.3... 23\17/ 1.2.3.4 \17/'
132 Find tlie fir.st negative term in the expansion of (2a + 3&)'*'.
We must take r the first integer which makes n — r-\-\ negative; there-
fore r>Jt + l = V +1 = 6| 5 therefore r = 7. The term will be
17 14 11 8 8 2. C 1"\ ,
(2(7)-»(36/ by (126;
17. 14.11 . 85^2^ 1 J/_
7! ■" (2«)5'
133 Required the ooeffioient of .?;" in the expansion of i- — -^ j .
g±M=(2.3..V(2-a.r'=C^)'(i-^)-'
the three terms last written being tliose which produce .r'*' after niultiplyii
by the factor (l-|-3a;-|- Jx*) ; for we have
33(|)%3„.x3^(^;:-)%lx35(|)^'
giving for the coeflicit nt of J'" iu the result
The coefficient of j" will in like jnanjicr be Ibi i !] /' ".
56 ALGEBRA.
134 To write the coeflBcient of x^'"*^ in the expansion of (x- ^)
The general term is
(2jt + l-r)!r! x"- (2«+l-r)!r!
Equate 4n—4r + 2 to 3m+l, thus
. _4n — Sm + l
Substitute this value of r in the general term; the required coefficient becomes
(2n+l)\
The value of r shows that there is no term in x^'"*^ unless — — "^."^ is an
[i(4n + Sm + 3)]\ [i{4n-Sm + iy
ae of r sho'
integer. **
135 An approximate value of (1+a?)", when x is small, is
l-\-7iXy by (125), neglecting x^ and higher powers of x.
136 Ex. — An approximation to \/y99 by Bin. Th. (125) is obtained from
the first two or three terms of the expansion of
(1000-1)* = 10-1 . 1000-5 = 10- 3^^ = mi- nearly.
MULTINOMIAL THEOREM.
The general term in the expansion of (a-{-hx-\-cx--\-&G.y is
j3, «(»-!) (»-2)...(p+l) ^^„ j,^,^^, ,^„«r..,..
ql rl si ...
where j;-f-r/ + r + 5+&c. = n,
and the number of terms p, q, r, &c. corresponds to the
number of terms in the given nndtinomial.
]> is integral, fractional, or negative, according as n is one
or tlie other.
If n be an integer, (137) may bo written
138 , ]'■ , , a' h" (••■ </, . . . .r''+2''+3.,
pi (/ . r\ .V I
I neduccd fi-om Mio Tliii. Thoor.
MUI/VISOMIA L TIIKOUEM.
57
Kx. 1. — To write f lie enofficient of a'fec" in tlieoxpanfiinii of {n +{> -f r + i/)"
Hero put )/= 1(». ,. = 1, i> = 'A, q=\, *- = r., s = () in (IMS).
Kesi.lt
10!
:^! 5
= 7.8.0.10.
Ex. 2. — To obtiiin till' CDcflSc-icnt of .r* in tlio cxiciusion ol
(l-2,« + 3.i'»-4a;')'.
Here, compariiii,' with (l;{7), we liave ii = \, h = —2, c = :i, </ = — 4,
q + '2r + ;is- = 8,
1
1
2
2
2
1
2
1
4
Tlie nuniljer.s 1, 0, 1, 2 are particular values of p, q, r, s respectively,
which satisfy the two equations given above.
0, 2, 0, 2 are another set of values which also satisfy those equations ;
and the four rows of numbers constitute all the solutions. In forming these
rows always try the highest possible numbers on the right first.
Now substitute each set of values of p, q, r, s in formula (138) succes-
sively, as under :
||l'(-2)",S'(-4)- = 57(3
2^1«(-2r3«(-4)^ = 384
4!
Z|i«(_o).3-.(_4^
8(54
ir.(_oy.;).(_4^o^ SI
Result 1!>(I.-,
Ex. 3.— Required the coefficient of v* in (H-2.-C— 4'c'-2a.-»)~*.
Hero (/ = 1, /i = 2, r = — 4, i1 = — 2, ?; = — ', ; inid the two ciinations arc
p + q + '■ + s = — \,
'J + 2/- + :rs- = 4,
— S.
1
1
2'
2
1 2
2
1
! -^
4
0|0|
58 ALGEBRA.
Employing formula (137), the remainder of the work stands as follows :
2M-^})(-l)'"'^"^-*''(-'>"= "
iT(-i)(-|-)(-|)l-^^=(-4)'(-)»= 15
Result 22f
139 The number of terms in the expansion of the multi-
nomial {a-\-h-{-r--\- to n terms)'' is the same as the number of
homogeneous products of n things of /' dimensions. See (97)
and (98).
The greatest coefficient in the expansion of {a-{-J>-\-<^-{-
to m terms)", n being an integer, is
Proof. — By making the denoniiuator in (138) as small as possible. The
notation is explained in (96).
LOGAEITHMS.
142 log,, ^^' = •^' signifies that a' = N, or
Def. — TJie logarithm of a nmiiber is the power to irliirJt tin
base must he raised to produce that number.
143 log.« = l, log 1 = 0.
144 log MN = log 3/4-log N.
log—- = log .1/ — log N.
log (3/)" = «log3/.
log:;/ j/ = J- log J/. [li'^
EXPONENT! A L Til IKlU EM. 59
145 '"f^"' ^ n;;77/
Tliat is — 'riir lixjiirilhiii of a innnhrr In miij Imsr is niiml fo
(he loijnvitJnn- of flu' nuMibcr dlrulitl hi/ fhc /ni/(irifhiii of the
hdur, the two last named logaritlnns being taken to any tlie
same l^ase at ])leasui'e.
Pi;(iOl'. — Let, log,. u = .)' and \i)<y,.l,= i/\ ihvn a = c-'', b = r". Eliiiiiiuitc <■.
c = a" = h"; .'. a = b", that is, \o<r,^a = -' ■
y y. e. d.
146 l<>K/.<^ = ,— ^- •'>'< '•=--" ii' (1 ^5)-
147 ,og,„.v = ;2gby(l4n).
is called the modulus of the common system of logarithms ;
that is, the factor which will convert loi>arithms of nnml^ers
calculated to the base e into the corros]~)ondino; loo-nrit1ims to
the base 10. See (154).
EXPONENTIAL THEOREM.
149 *' = 1 + r.r + '-^ + '^ + etc.,
where c = {a-])-\ ('(-\Y + \ (n-\y-Scc.
PiJOOF. n^ = fl + (a — l)j^ Ivxpand tliis Ijy JJinomial Thcoiviii, ami
collect the coedicients of .c ; thus c is obtained. A.ssnme r.,, c„ Arc, as the
coefficients of the succeeding poweis of ;r, and with this assumption write
ont the expansions of a"", a", and a'"*". Form the product of the first two
series, which product must be equivalent to the third. Therefore equate
the coefficient of .c in this product with that in the expansion of a'*". In
the identity so olitained, equate the coefficients of the successive powers of
y to determine Cj, f,, &c.
60 ALGEBRA.
Let e be that value of a which makes c = 1 , then
150 ,.' = i + .,.+ |l4.i_4.&c.
151 ''^^ + ^ + ^ + ^ + ^^-
= '2-718281828... [See (2;»5).
Proof. — B}^ making x = 1 in (1-^0).
152 By making a; = 1 in (149) and ,t = c in (150), we obtain
a = e"" ; that is, c = log^ a. Therefore by (149)
154 \og,n = {a-l)-i(a-iy-hi(a-iy-&G.
155 l«g(l+.r)= .,.-£- + :;^_±-+&c.
156 \og{l-.v) = -.v~:^-^-'^-&c. [154
— »» 4
157 .-. l..sl±£=2J,, + :^ + 4^+&c.^-.
Put for ,r in (157); tluis,
158 l..g», = ■> )^ +|(!^y'+i(^f+&c.( .
(m-\-l ,\\m-\-\' ;)\m + l' )
Put ^ ^ for ,r in (157); thus,
2/^ + 1 ^
159 lot,^ (// + !) -loi^M*
CUNTIS ri:i> FRA < "I'lOXS.
<;i
CONTINUED FRACTIONS AND CONVERGENT S.
160 'l^> liii'l foiiV('rt<( Ml ts to 3-M.i:)9 =
ill tlu" rule for II. ('. F.
;n4ir,<>
roceea hh
100000
99113
31415!)
300000
887
854
83
29
4
4
14159
887
5289
4435
854
66
15
25
194
165
28 !
The contiinKMl fraction is
3 + 1
7 + 1
15 + &C.
or, as it is more conve-
niently written,
1 1
3 +
7+ 15 +
cV-C.
The convergents are formed as follows : —
3 7 15 1 25 1 7 4
3 22 333 355 9208 95(53 76149 814159
1 ' 7' 106' 113' 2931' 3044' 24239' lOOOOO'
161 KuLE. — AVrite the ([uotients in a row, and the first two
convergents at sight (in the example 3 and 3+y). Multiply
the numerator of any convergent by the next quotient, and
add the previous numerator. The result is tlie numerator of
the next convergent. Proceed in the same wa} to determine
the denominator. The last convergent should be the original
fraction in its lowest terras.
162
Furtnuld fin' fhrniinii- tlir f(nirrr::rnf.s.
If -?^, ^^^!i^, ''" are any consecutivt' converu-ents, and
qn-2 7«-i V» . .
^»-2» <'«-!' ",> f^"' coi'r(^S|)on(liiiii: (|iiotienls: then
62 ALGEBBA.
The /i"' convergent is therefore
7» (lnqn-\-\rqn-l
The true value of the continued fraction will be expressed
by
163 p^anPn-r-^Pn^.^
(f„q»-i-\-qn->
in which ct'^ is the complete quotient or value of the continued
fraction commencing with a^.
164 Pnqn-i—Pn-iqn = ± 1 alternately, by (162).
The convergents are alternately greater and less than the
original fraction, and are always in their lowest terms.
165 The difference between F^ and the true value of the
continued fraction is
< and >
quqn+i qniqn+qn+i)
and this difference therefore diminishes as n increases.
Pkoof.— By taking tlie difference, ^« - Y"^'"^^" - (163)
Also F is nearer the true value than any otlier fraction
witli a less denominator.
166 l'\iFn+\ is greater or less than F'^ according as F„ is
greater or less than i^,,+i.
Grucral Theory of Vontlnucd Fnfrfion.s.
167 b'irst class of continued [ Second class of contimu d
fraction.
fraction.
1/ _ ^1 ^'2 ^h
<'i — ''■•— ''.{ — &c.
ill, hi, itc. are taken as positive (piaiitities.
CONTINI'JJJ) Fh'AcriONS. {]^
'. '■'' , Sec. ;ii"(' tcfiiKMl (•(nii/ioiKii Is of the coiit iiiiicd IViK'-
(ion. It' tlir compoiicnt s he iiitiiiilc in iiiiiiihrr, the coiit iiiiicil
IVactiou is said lo bo iuliniU-.
Let the successive conver<ients bo tloiiotod l)v
^'i . 1'-i _f'\ ^1 . Pa _ ^^i ^^2 ki .
; and so oi
168 1 lit' law of formation of the convergents is
I'^or /', I For r,
Pn = f. P„ 1 + '^, />« .' { Pn = ftn pn 1 " '>„ />« .'
•/« = <f„ Un - I + f'n 7« - -1 ( V" — ^^< qn-l—f^H Hn-l
[Proved by Induction.
The relation between the successive differences of the
converg-ents is, by (108),
169 l^_lK^ h„..U„U !K_1K^\
7»+i Un qn^\ \q. v.-i
Take the — sign for 7'', and th(> + ^ov I '.
171 The odd convergents for i^, ^, ^'•\ &c., continually
Vi V.t
decrease, and the oven convero-onts, '-, ^-, S:c., continually
'/•' qi
increase. i^^'O
Every odd convn'oHMit is greater, and ovorv even con-
vergent is less, than all following convergents. (lGi>)
172 l^i.i'. — If the diffei-onco between consecutive conver-
gents diminishes without limit, the infinite contiiuied fraction
is said to l)e dcjiiilfr. if the same difference tends to a fixed
value greater than zero, the infinite continued fraction is hi-
drfinifr ; the odd convergents tending to ..nc value, and the
even converu'ents to another.
64 ALGEBRA.
173 F is definite if the ratio of every (|uotioiit to tlie next
component is greater than a fixed quantity.
Proof. — Apply (1(39) successively.
174 F is incommensurable when tlie romponents arc* all
proper fractions and infinite in number.
Proof. — Indirectly, and by (168).
175 if a be never less than /> + l, the convero'ents of V are
all positive proper fractions, increasing in magnitude, Pn and
(/„ also increasing with 7^. By (167; and (168).
176 If, ill this case, V be infinite, it is also definite, being
= 1, if a always =h-\-\ while h is less than 1, (175); and
being less than 1, if a is ever greater than h-{-\. By (ISO).
177 V is incommensurable when it is less than 1 , and the
components are all proper fractions and infinite in number.
180 If in the continued fraction V (167), we have a„ = h„ -\- 1
always; then, by (168),
'p,^=^ hy-\-hih.2-\-h]^h.2,b-i-{- ... to n terms, and q^z= p^^-{-\.
181 Ifj iu the continued fraction F, a,^ and h,i are constant
and equal, say, to a and h respectively ; then ^,;„ and (/„ are
respectively e(pml to the coefiicients of x!'"'^ in the expansions
/. h T a-\-hv
of - , 2 and .,.
1 — ax — J? 1 — ^^c — hxr'
Proof. — p,i and q^ are the w*'' tei-uis of two recurring seiies. See (IGS)
and (251).
182 /''> v(»n'('i't (I Scries info a dnitiuucil Fractiini.
The series i + ^ + :!l + ... + —
is ('(jual to a continued fraction 1^ 0^'^'^), with ;/ -|- 1 com-
poneiits ; th(^ first, second, and //-t-l"' coinj)on('iits being
1 ir,v u'i ,.r
u iii-{-i(.r «,, + "" -i<^'
[Proved by Induction.
COSTISUFA) FUACTIOXS. 05
183 Tlio scries
1 r r'- .r"
-+ — + -^+...+
r rr, rr^r. f'''\i'- ... ''»
is o(|ual to ;i coiitiimod fraction T (1C»7), with // + 1 coiiipo-
ncnts, the first, second, and // + !"' components beini^-
J_ l\r !JI_1±, [Proved l)v In.lucfinn.
184 'Hie sio-n of w may be changed in eitlicr of the state-
ments in (182) or (18;3). '
185 Also, if any of these series are convergent and iidinite,
the continued fractions become infinite.
186 To find fhr rahir of a rnntinurd fraction with
rrrnrrinii: qnitfirnf.s.
Let tlie continued fraction be
where ?/ = -- -
so tliat there are m recurring quotients. Form the |/"' con-
vergent for X, and tlie m^^' for //. 'Vhvu, by substituting the
complete quotients a„-\-i/ for a„, and a,,,.,,, -f-// for */„,,„ in (lC),s),
two equations are obtaincMl of the forms
from which, by (eliminating //, a (iii;i(lr;it ic (miumI ion foi' (h--
terminimr ;'' is obtained.
1R7 Tf !^^ ^h—
i)i' a colli iinicd I nii't ion, ;ni<
El I^
7i' (/"
66 ALGEBRA.
tlie correspoiKliiii^ first vi convorgents ; then '" "\ developed
by (1<J8), ]iroduces tlie continued fraction
1 hn />„_! h, b.
(In + (ln-l-\- (ln--2-\- '" + fh + «1
tlie quotients being the same but in reversed order.
INDETERMINATE EQUATIONS.
188 Given aa-\-hii = c
free from fractions, and a, /3 integral values of x and // wiiich
satisfy the equation, the complete integral solution is given by
.r= a — ht
y = fi+at
where t is any integer.
Example. — Given Src + -h/ = 1 1 '2.
Then X = 20, y = 4< are valnrs ;
x = 20-:U\
y= .l. + r,M •
The v.alues of x and y may be exhiliited as niuler:
t = -2 -I I 2 .S -i 5 6 7
x= 26 28 20 17 14 11 8 5 2-1
7/=-G -1 -4 !) 11- 1!) 24 29 84 89
For solutions in positive integers / must lie between \" = 6j; and — t ;
tliat is, t must be 0, 1, 2, 8, 4, 5, or 6, giving 7 positive integral solutions.
189 If the equation be
(u—hjl = c
tlie solutions are given by
,v— a-\-ht
INDETEnMIXA TE FA^J'A TIONS.
EXAMPLK : 4c -8// = li>.
Here X = 10, // = 7 satisfy ilie equaticjii ;
' ~ fiinii.Nli :ill tilt' solutions.
,/ = 7+ U *
The simultiinoous vahu'S of /, .r, und // will he as follows : —
t=-l) -i -:i --2 -I 1 -J :;
., = -5 -2 1 1- 7 10 l:{ 10 I'.'
,j = -rA -9 -5-1 3 7 11 1.-) 19
The number of positive integral solutions is infinite, ami the least positive
integral values of x and ij are given by the limiting value of /, viz.,
t>-\- and t>-\-'
that is, t mast be —1, 0, 1, 2, 3, or greater.
190 It" two values, a and /3, cannot readily be found by
inspection, as, for example, in tlie equation
17.t' + 13// = 14900,
dlridr In/ f/ic huisf roi'ffirient, and equate the re iiiaiiiliKj frac-
tions to t, an intc/jer; thus
*+■■'+ if ="«+!;, '"■
4a— 2 = l-.it.
Repeat the process ; thus
4 4
Pat
^ + •2 = 4
H.
u
=
1,
t
=
2,
X
=
18/ + 2 _
4
7 =
f ;
and _y + .,; + /= 114(5, by (1),
7/ = 114(5-7-2 = ll;w
The general solution will be
.,' = 7-l.S^
II = 1137 + 17/,
Or, changing the sign of / for convenicncp,
.(• = 7 + 13/,
y = 1137-17/.
68 ALGfEBBA.
Here the number of solutions in positive integers is equal to the number of
^ ■ X. , 7 , 1137
integers lymg between — and -— - ;
or ~ Tq ^^^ ^^Tf ; t^^t is, 67.
191 Otherwise. — Two values of x and y may be found in
the following manner : —
17
33
Find the nearest converging fraction to y^. [By (160).
This is — . By (1G4) we have
17x3-13x4 = -1.
Multiply by 14900, and change the signs;
17 (-44700) + 13 (59600) = 14900
a = -44700
which shews that we may take , , ^^^
^ ( /5 = 59600
and the general solution may be written
x = -44700 + 13/,
y= 59600-17^.
This method has the disadvantage of producing high values of a and y8.
192 The values of x and //, in positive integers, which
satisfy the equation ax + bi/ = c, form two Arithmetic Pro-
gressions, of which h and a are respectively the common
differences. See examples (188) and (189).
193 Abbreviation of the method in (169).
Example : ll.i;— 18;/ = 63.
Put X = 92, and divide by 9 ; then proceed as before.
194 To ohld'ni iiifriinil s(thitioN.s' nf (H-\-f)t/-\-rz = (I.
Write the equation thus
ax -{-III/ = (J — cz.
Put successive integers for ;:, and solve for .r, // in encli cnse
ItEDUGTION OF A QtlADJiATfC srUD. GO
TO Iv'KDlTCF] A QUA1>HATI(^ SLIHI) TO A
CONTINUED FRACTION.
195 EXAMI'I-K :
^29= 5+v/29-r, = 5-h '^
,29 + 5'
y29 + 5_ ^, v/29-:5^ ^ 5
4 '^^ . 4 "^^ ,'29 + 3'
5 ~ ^ 5 ^^29 + 2'
v/29 + 2_ . , v/29-3_ , 4
" 5 ~ "^ 5 ^ ■^V^29 + :>'
/29 + 3_ g , v/29-5 _ 2, •
—4 - ^ + "—4—- ^ + v^29+.y
^/29 + 5 = 1U+ V 29 -5 = 10 +
v/29 + 5'
Tlio (iiiotients 5, 2, 1, 1,2, 10 arc the gTcatest integers
contained in the quantities in the first cohimu. The quotients
now recur, ami the surd \/29 is equivalent to tlie continued
fraction
1_ 1_ 1_ 1 J 1_ 1_ 1 ]
5+ 2+1+1+ 2+ 10+ 2+ 1+ 1+ 2 + c^c.
The convcrgents to v/29, formed as in (IGO), will be
5 11 10 27 70 727 1524 2251 3775 9801
T' 2' 3' 5' 13' 135' 283' 418' 701' 1820'
196 Note that the last quotient 10 is the greatest antl twice
the first, that the ><i'ron(l is the first of the recurring ones, and
that the recurring quotients, excluding the last, consist of
pairs of equal terms, <[iu)tients e(|ui-distant from the first and
last being ecjual. These properties are universal. (See 204
-210).
To for)) I liiii'h ro)H'C)'i»'r)ifs rnpUllji.
197 Suppose m. the number of recurring (piotients, or any
70 ALGEBRA.
multiple of that number, and let the m^^' convergent to \/Q be
represented by F„,; then the 2??^"' convergent is given by the
formula F„„ = i .Jf;,+ -^^- by (203) and (210).
198 i'or example, in approximating to \/29 above, there ai-e five recurring
quotients. Take m = 2x5 = 10 ; therefore, by
i^,y = ^-— — , the 10^'^ convergent.
1820
Therefore F,, = {||^^ + 29x^
1820 ) ^ 192119201
)801 ) 35675640^
the 20^'' convergent to \/29 ; and the labour of calculating the interveninf
convergents is saved.
GENERAL THEORY.
199 'J'he process of (174) may bo exhibited as follows :-
= a.
^±^ = ..+
''Q + r„,-
200 Tl
1 1 1
v/ g = f / , + a, -h a, + a^-irScv.
Tli(! (juotients '/,, rr,, a.^, iScc. are the integral parts oF the fi'ac-
tions on tlie left.
Ri'iDUCTios OF A (,n' A i>i:.\'r h' srii'it. 71
201 'I''"' •'•I'intions coinicct iii^- tlic rnniiiiiiii^- (|ii;iiit it ii'S iii-c
r, r= (I. )'.,—(■ >:5 —
r., = ^/,. ,r
,-.= ^=:^
Tlie ?/*'' r()iivorn:(Mit to \/(? will bo
202 ^ = ^iiLZ!zL-_Lii!±l2_ [By Tndnctioii.
The tnu> value of v'^^ i^ ^^''^^^t tliis becomes wIh-ii we
substitute for (/„ the complete quoticDt ^ ' ", of wliicli <i„
is only the integral part. This gives
By tlie relations (1 '.)<)) to (203) tlie following theorcius ai'*^
demonstrated : —
204 All the (juantities (/, r, and r ai'e positive integers.
205 'I'Ik' greatest c is r.,, and c, = a^.
206 No >i or /• can be greater than 2'/,.
207 n /■„ = 1, then r„ = a,.
208 I'^or all values of n groat(>r tlian 1, rt—r„ is < >•„.
209 'Hi'- number of (juotients cannot be greater than 2a'l
The last (luotient is 2(i,, and after that the terms repeat.
The first complete quoti(>nt that is repeated is ^ \ '\ and
(7o, 7-0, r., commence each cycle of re}»eated terms.
72 ALGEBRA.
210 I^et </,„, r,„, r,„ be the last terms of the first cycle ; then
<*»n-ij '''m-ii Cm-\ ^^^'c respcctivcly equal to rto, r.,, c-2', «,«-2j ''m-2j
c„,_2 are equal to a.^, r.^, r.^, and so on. rp^ (187).
EQUATIONS.
Special Cases in the Snlntion of Simidtaneous Equations.
211 First, witli two unknown quantities.
a^e-\-hyii = i\\ ^ _ cA—Co bi _ ^1^2 — ^2 ^1
If the denominators vanish, w^e have
^ = '\ and X = cc, ?/ = 00 ;
«2 b.,
unless at the same time the numerators vanish, for then
a._h,_c,, 0. 0.
a, ~ h ~ c./ ' '^ ~ '
and the equations are not Indepmdent, one being produced by
multiplying the other by some constant.
212 Next, with three unknown quantities. See (60) for
the ecpiations.
If (l^, (L, (I,; all vanisli, divide each equation by .v, and we
have three equations for finding the two ratios -'- and • , two
only of which equations are necessary, any one being dedu-
cible from the otlier two if the three be consistent.
213 ^« solve simultaneous equulions bt/ huleterntnuitr
Multipliers.
Ex. — Take the equations
,/' + 2// + ;lv + lip = 27,
'5.'c + r,//+ 7,v+ //• = -l-S,
bx + 8y-\- 10,v - 2/r = 05,
7x + 6y + 5,^ + iw = 53.
MISCELLANEOUS KQJWTIONS. 7:\
Multiply the first by A, the second by /?, the third l)y C,
leaviiiLi;- one C(inution nnnniltiplied ; and then add the results.
Thus (J+3//4-5r7-{-7)./' + (2.l-h'")/>'4-8^' + (;)//
+ (;5J +7/>' + l()C'-f-5) .v-[-(4J + /;-26'^-4) w
= 27-l + 18Zy + GoO + 5;].
To determine either of the unknowns, for instance .f,
equate the coefficients of the other three separately to zero,
and from the three equations find A, B, G. Then
^ 27J+48/y + G50 + o:]
* A + W-\-hG-\r7 '
MISCELLANEOUS EQUATIONS AND SOLUTIONS.
214 ^''±1 = 0.
Divide by x^, and throw into factors, by (2) or (o). See also
(480).
215 .r'-7d-i) = i).
X = —1 is a root, by inspection; therefore ''+1 is a factor.
Divide by x-\-l, and solve the resulting (quadratic.
216 aH n; I' = !'>''>.
x*-\-lC).>- = l-j.'),/' = (')<■) X 7,/',
•' + 2 -""^ 2'
.i;" = /i''
= 7.
Rrr-E. — Divide the absolute* terra (here 455) into two
factors, if possible, such that one of them, minus the scpian*
of the other, equals the coefficient of x. ^^ee (483) for i^cMicral
solution of a cubic equation.
I.
74 ALGEBRA,
217 .r*-i/ = 145(>0, .r-v = 8.
'*"t^^ .f = «+v and 2/ = z—v.
Eliujiiiate c, and obtain a cubic in 2, which solve as in (216).
218 .i^-/ = 3093, ci— 1/ = 3.
Divide the first equation by the second, and subtract from
tlie result the fourth power of x—y. Eliminate {x^-\-if)j and
obtain a quadratic in xij.
219 On forming Symmetrical Expressions.
Take, for example, the equation
(y-c){z-h) = aK
'Vo form the remaining equations symmetrical with this, write
the corresponding letters in vertical columns, obser\dng the
circular order in which a is followed by h, h by c, and c by a.
So with X, ;?/, and z. Thus the equations become
0/-rj {z-h) = a\
[z-a) Gr-e) = b\
{.v-h){y-a) = c\
To solve these equations, substitute
x = h + c,-\-x\ y = c-\-a + y', :: = a -{- b -\- ::' ;
and, ]nulti})lying out, and eliminating // and ;:, we obtain
^^ho{b + r)-a(lr-hr)
hc — ca — ab
niid tlicrefore, by symmetiy, the values of y and ;:, by the
niK> just given.
220 // + .^' + //- = ^r (1),
:^-^-.r-\-x.v = fr (2),
'♦■H/ + .*//-r^ (3);
••• :5(//.r + .v.' + ,o/)-=r :l/n--\-2r',r-\-2'rlr-a'-b'-c* (4).
iMAniXAUY i<jxi'i:j:ssi()xs. 75
Now add (1), (2), and (3), and av(> o])tain
From (4) and (5), (,r + // -|- ;:) is obtained, and then (1), (2),
and (o) are readily solved.
221 .,.-^^;/~ = «'^ (I),
jr-z^=fr (2),
z^-.n/=f- 05).
Mnltiply (2) by (:>), and subtract tlie square of (1).
Result X (3./'//^ - Jf -if- ::'') = h'<- -n\
X _ y ^
/A.2_ft^ (-V--/.* a'b''-r' ^ ^^^'
Obtain X" by proportion as a fraction witli numerator
= x^ — yz = a^.
222 .v=n,-\-bz (1),
,^ = az^c.i (2),
z = Lv-^(a/ (3).
Eliminate a between (2) and (3), and substitute tlie value of
X from equation (1).
XieSUlC -r-" — r^ '„ j: ;,-
IMAGINARY EXPRESSIONS.
223 'Hic following are conventions : —
That v/(-'f-) is equivalent to a^^{—li); that a y,/{ — \)
vanishes wlien a vanishes; that the symbol a, y(— 1) is sub-
ject to the ordinary rules of Algebra. \'(— 1) is denoted
l)y /.
76 ALOEBEA.
224 If a + ?'/3 = 7 -f- i^ ; then a = y and (i = B.
225 « + //3 and a — /'fS are conjugate expressions ; tlieir pro-
duct = a- 4-/3-.
226 The sum and ])roduct of two conjugate expressions are
both real, but their difference is imaginary.
227 The modulus is -\-x/a^+^^.
228 If the modulus vanishes, a and /3 must vanish.
229 If two imaginary expressions are equal, their moduli
are etiual, by (224).
230 The modulus of the product of two imaginary expres-
sions is equal to the product of their moduli.
231 Also the modulus of the quotient is equal to the
quotient of their moduli.
METHOD OF INDETERMINATE COEFFICIENTS.
232 If A + Re + a«- + . . . = .4' + B'x -\- G'x' + . -_. be an equa=.
tion which holds for all values of .^', the coefl&cients .1, B, &c.
not involving ;r, then A— A', B = B\ C = G', &c. ; that is,
the coefficients of like powers of x must be equal. Proved by
putting X = 0, and dividing by x alternately. See (234) for
an example.
233 METHOD OF PROOF BY INDUCTION.
Ex. — To prove that
5
Assume 1 + - + •» -\- ... +ir = - -"-.
iwirriAL FiLurnoxs.
= »(» + !) (2n + l) +6 (n + 1)- ^ (m + 1) {n (2n + l) -l-G r«-H)}
6 ' (5
^ (« + lU» + 2)(2u + 3) _ n (n'+\){ 'l n+l)
G 6 '
where n' is written for «+l ;
o
It is thns proved that i/ the formula he true for n it is also true for n + 1.
But the formula is true when n = 2 or 3, as may be shewn by actual
trial ; therefore it is true when >t = 4 ; therefore also when n = 5, and so on ;
therefore universally true.
234 Ex. — The same theorem proved by the method of In-
determiuate coefficients.
Assume
1^2^ + 3-+.. .+n^ =A + Bn +Cn^ +Du^ +&c.;
.-. 1 + 2- + 3-+. ..+»' + (« + !)- = .-l+5(/^ + l) + (7(« + l)- + D(n + l)'' + &c.;
therefore, by subtraction,
«H2n + l = B + C(2n + l) + D{3n' + Sn+l),
Avriting no terms in this equation which contain higher powers of n than the
highest which occurs on the left-hand side, for the coefficients of such terms
may be shewn to be separately equal to zero.
Now equate the coefficients of like powers of n ; thus
1
, and ^ = 0;
3jD= 1,
■•■ ^=i-
2C' + 3D = 2,
•■ ^ = 1'
i + C + D = 1,
therefore the sum of tlie Heries is equal to
n TT «=" _ «(» + !) (2n + l)
G "^ 2 "^ 3 " 6 ■
PARTIAL FRACTIONS.
In the resolution of a fraction into partial fractions four
cases present tlieniselves, \\]\\c\\ arc illustrated in tlie follow-
ing examples.
78 ALGEBRA.
235 First. — When there are no repeated factors in the de-
nominator of the given fraction.
3a;— 2
Ex. — To resolve :r—, -— — into partial fractions.
{x—l){x—2){x—o)
^''^""^ (a-l)(J"-2)(«-3) " ^:il "^ ^-2 "^ x-3 '
Sx-2 = A(x-2)(!c-S)-\-B(x-S)(x-l) + C{x-l)(x-2).
Since A, B, and C do not contain x, and this equation is true for all values
of X, put x = l ; then
3-2 = ^(1-2) (1-3), from whicli A = ^.
Similarly, if x be put = 2, we have
6-2 = i? (2-3) (2-1) ; .-. B = -4 ;
and, putting a; = 3,
9-2 = 0(3-1) (3-2); .-. G = \'
H 3a; -2 ^ __1 ^ 7
®°°® (a;-l) (a!-2) (a;-3) 2(a!-l) a;-2 2 (a;-3)'
236 Secondly. — When there is a repeated factor.
Ex. — Eesolve into partial fractions i- — -.,„' j! -
^ (a;— 1)^(33 + 2)
. lx^-\Ox'^^x A . B C ^ D
A^«"^« (-^=i?(.:t^ = (^^» "^ (^::ir "^ ^^ "^ ^rr2-
These forms are necessary and sufficient. Multiplying up, we have
7x'-lOx' + 6x = A ix-\-2) +B (x-1) ix + 2) + C (x-iy(x + 2) +D (x-iy
(I).
Makea; = l; .'. 7-10 + 6 = ^(1 + 2); .-. ^ = 1.
Substitute this valne of A in (1) ; thus
7x'-lOx' + 5x-2 ^ B (x-].){.v i2) + C (x-iy(x + 2)+D (x-iy.
Divide by a; — 1 ; thus
7x'-Sx + 2 = B(x + 2) + C(x-l)(x + 2)+D(x-iy (2).
Make X = 1 again, 7 -3 + 2 = J? (1 + 2) ; .-. B = 2.
Substitute this value of B in (2), and we have
7a;*-5a;-2 = G (x-l) (x + 2) +D (x-iy.
Divide by .T-l, 7a; + 2 = G (x + 2)+D (x-l) (3).
Put a; = 1 a thiwl time, 7 + 2= C (1+2); .-. C = 3.
I'AirriAL FRACTIONS. 79
Ijastl}^ make a; = —2 in (3),
-14 + 2 = X>(-2-l); .-. D = i.
1 2 3 4
Result 7 z—j. + 7 TTT H , H rii"
(a— 1)' (a;-l) «-l a; + 2
237 Thirdly.— When there is a quadratic factor of imaginary
roots not repeated.
Ex.— Resolve ,t— tw^2-. — , ix into partial fractions.
Here we must assume
Ax-{-B Cx + D
(a5»+l)(j!» + a! + l) a;^ + l x' + x+l'
x-i-l and X- + X + 1 have no real factors, and are therefore retained as
denominators. The requisite form of the numerators is seen by adding
too'ether two simple fractions, such as — —- ^ r~,-
° ^ x + b x + d
Multij)l}iiig up, we have the equation
1 = (Ax + B) (x' + x + l) + {Cx + D) (x' + l) (1).
Let a;- + l = 0; z. x^ = —I.
Substitute this value of x- in (1) repeatedly ; thus
1 = (Ax + B) X = Ax' + Bx = -A + Bx ;
or Bx-A-l = 0.
Equate coefficients to zero ; .'. 5 = 0,
^ = -1.
Again, let ar + .r + l=0;
.-. x-=-x-l.
Substitute this value of x^ repeatedly in (1) ; thus
1 = {Cx + D) i-x) = -Cx'-Dx = Cx + C-Dx-
or (G-D)x + C-l=0.
Equate coefficients to zero ; thus ^ ' = 1,
1)= 1.
XT 1 _ = ''+1 _. ^
^^'"'^^ (..^ + l)(x^ + * + l) .tHx + 1 a-Hl
238 Fourthly. — When there is a repeated quadratic factor
of imaginary roots.
Rv —"Resolve 40.^' — 103 ■ ^ i)artiiil fractions.
80 ALGEBRA.
Assume
40.7; -103 ^ Ax + B _Cx±B _ Ex + F
(x + iy {x'-4x + Sy (.r2_4x + 8y (.'?;--4a; + 8)- a;--'-4^ + 8
4- -^ + -^;
(.r+l)- a; + l'
40.t;-103 = {iAx + B) + {Cx + D)ix-—ix + 8) + iEx + F)(x--4:X + 8y} {x + l)-
+ {G + H(x + 1)} (x'-4x + Sy (1).
In the first place, to determine A and B, equate rt;-— 4a; + 8 to zero ; thus
a;2=4a;-8.
Substitute this value of x- repeatedly in (1), as in the previous example,
until the first power of x alone remains. The resulting equation is
40a; -103= (17.4 + 65) a? -48^ -75.
Equating coefficients, we obtain two equations
17^ + 65= 40 ) f .. , A = 2
48^ + 75 = 103)' ^^«--^^«^^ B = l.
Next, to determine and D, substitute these values of A and 5 in (1) ;
the equation will then be divisible by a;^— 4a; + 8. Divide, and the resulting
equation is
= 2x + l3+{Cx + B+(Ex + F)(x'-4x + 8)] (x + iy
+ {G + H(x + l)]{x'-4x + 8y (2).
Equate a;'- — 4a; + 8 again to zero, and proceed exactly as before, when
finding A and B.
Next, to determine E and F, substitute the values of (7 and D, last found
in equation (2) ; divide, and proceed as before.
Lastly, G and H are determined by equating a' + l to zero successively,
as in Example 2.
CONVERGENCY AND DIVERGENCY OF SERIES.
239 Let ai-{-a.^-\-a;i-\-&c. be a scries, and (7„, a„+^ auy two
consecutive terras. The foUowino- tests of convergency may
be applied. Tlie series will converge, if, after any fixed term —
(i.) The terms decrease and are alternately })0sitive and
negative.
(ii.) Or if "- is always (j renter than some (piantity
(' n 1-1
greater tlian unity.
SERIES. 81
(iii.) Or if — — i.s never less tluui tlic corrcspoiidiii^ I'atio
'''1 + 1
ill a known coiivei\u:ing series.
(iv.) Or if l-^—n) is always tjreafrr than some (juan-
tity greater than unit3^ [% - tl' and iii.
(v.) Or if l^-^—ii — l]\og)i is always i/rrdfcr tlian
V^'»j+i ^
some quantity greater than unity.
240 The conditions of divergency are obviously the converse
of rules (i.) to (v.).
241 The series ai-^a.,x-\-a.iX^-{-&c. converges, if ^^
always less than some quantity p, and x loss than
1
[By 239 (ii.)
242 To make the sum of the last series less than an assigned
(iiiantitv /s make ,v less than , , I' hvincr the o^reatest co-
efficient.
Grnrral Tltcnron.
243 If •/> ('") be positive for all positive intec^ral values of .r,
and continually diminish as <>■ increases, and if )n be any posi-
tive integer, then the two series
<^(l) + (^(2) + (^GJ) + ^(l) +
<li{l)-\-m(f>{m)-\-m-4>{m-)-\-m''(t>{m')-Y
arc either both coiivern-ent oi* diverofent.
244 Ajiplication of tliis theorem. To asc<'rtain whotlier the
is diverjT^ent or convero-init when p is «^i-eater than unitv.
41
82 ALGEBRA.
Taking m = 2, tte second series in (243) becomes
1.2,4,8,0
^ 2'^ ^ 4p ^ 8^
a geometrical progression whicli converges ; therefore the
245 I'lie series of which --:- ^,- is the general term is
?i (log ny'
convergent if j9 be greater than unity, and divergent if p be
not greater than unity. [By (243), (244).
246 The series of which the general term is
1
n\{ii)X'{n) V{n){r^'(n)}^'
where \ (n) signifies \ogn,X'^{n) signifies log {log ()i)}, and
so on, is convergent if ^ be greater than unity, and divergent
if j) be not greater than unity. [By Induction, and by (243).
247 The series ai + cu + SLC. is convergent if
ncu log (n) log2 {n) log''(7i) {log,^i {n)y
is always finite for a value of p greater than unity ; log'' (7;)
here signifying log (log iz), and so on.
[See Todhunter's Alr/ehra, or Boole's Finite Bijjcrences.
EXPANSION OF A FRACTION.
42.' — 10a3
248 A fractional expression such as :, — --'. — -- may
\ — bx-\-l\x^ — Q>x,
be expanded in ascending powers of x in three different ways.
First, by dividing the niiraerator by the denominator in
tlie ordinary way, or by Synthetic Division, as shewn in (28).
Secondly, l)v the metliod of Indeterminate Coefficients
(2:32).
Thirdly, by Partial Fractions and the Binomial Theorem.
SERIES. 83
To expand by tlie method of Indeteriiiiii:ite CoefficiLiits
proceed as follows : —
Assume , '^''^ ~\^'^' . , = -1 + ^'•'- + C.>- + J).c' + E.v' + & c.
4x-lUr = .1+ llx+ Cx--\- nx''+ Ex'+ I<\r''+...
— OAx— GBx-— GC'u;*- Gi*./;'— OiiV'-...
+ ll.lj;-4-ll/^a;' + liac*+llA/'+...
- G.-Lc'- 6Bx*- G^.V-...
Et|uate cocUicients of like powers of x, thus
.1 = U,
JJ- 6A = 4, .-. J! = I ;
C- 6B + IIA =-lO, .-. C= li;
D-6C+UB-GA= 0, .-. 1)= 40;
E—6D + IIC— OB = U, .-. E=UO;
F-6E + 11D-6C= 0, .-. i''=;30-i;
The formation of the same coefficients by synthetic division is now
exhibited, in order that the connexion between tlio two processes may be
clearly seen.
The division of 4a; — lO.r l)y 1— ('..i'-f U.r-G,/' is as follows:—
+ 4-10
+ 6 24 + 84 + 240 + GGO
-11 -44-154-440-1210
+ 6 + 24+ 84+ 240 + GGo
+ 4+14 + 40 + 110 + 304+
^l n C D E F
If wc> stop :it the term llO.r', then the undivided remainder will lie
;i04.j;''— 'JTO/^ + CtiOi/, and the complete result will be
4.r + 14x- + 40..H110. + ^_^^^,fZ:^-
249 Here the conchidiiig fraction may be regarded as the
sum to inlinity after four terms of the series, just as the
original expression is considered to be tlie sum to inlinity of
the whole series.
250 Tf the general term be reipiired, the method of ex-
pansion by partial fractions must be adopted. See (257),
wliere tlie Lrcneral term of the foregoing series is oljtained.
84 ALGEBRA.
RECURRING SERIES.
a^^-\-(ii.i'-]- Uod'- -\-ayv'^-\- &c. is a recurring series if the co-
efficients are connected by the relation
251 (In = Ih «« - 1 + 7>2 «« - 2 + • • . + Pm (in - m-
The Scale of Relation is
252 1 -PI^V -JhO^ —... —lhn^V''\
The sum of n terms of the series is equal to
253 [The first m terms
—piV (first tn — l terms + the last term)
—p^x^ (first m— 2 terms + the last 2 terms)
—IhJC^ (first m— 3 terms + the last 3 terms)
-~i>i«-i'^'""^ (first term + the last m — \ terms)
—p,nX"' (the last m terms)] -^ [l—p^.v—pocV^— ... — />,„cr"'].
254 If the series converges, and the sum to infinity is re-
quired, omit all " the last terms " from the formula.
255 Example. — Required the Scale of Relation, the general
term, and the apparent sum to infinity, of the series
4'c + 14r + 40,v^ + 110,ii'^ + 304^^-8o4/+ ... .
Observe that six arbitrary terms given are sufficient to determine a Scale
of Relation of the form l—px — qx' — rx^, involving three constants p, q, r,
for, by (251), we can write three equatious to determine these constants ;
namely, 110= 40p4- 14(2+ 4r\ The solution gives
304 = llOp + 402 + 14r k p = G, 7 = - 1 1, r = 6.
854 = 304;j+110g + 40rJ
Hence the Scale of Relation is 1 — 6.« + ll.r — 6.r^.
The sum of the series without limit will be found from (254), by putting
Pi = ^, Pi = — 11» P3 =6, m = 3.
The first th ree terms = 4,c + 1 4.r + 40.i-''
— 6xthe first two terms = —24^-— 84a;*
-I- 1 l.r X the first term = + 44a;'
4«-10a:-
RE CURBING SERIES.
^^ 4.r-10x'
1 - G.i; + 1 Ix* -<;.(•»'
tlie meaning i)f which is tliat, if this fmction bo expiunluil in asuentling
powers of x, the first six terms will bo those given in the question.
256 To obtain more terms of the series, we may use the Scale of Relation ;
tlius the 7th term will be
(6 X 854- 1 1 X 30i + 6 X 1 10) a:^ = 2440a;^
257 To find the general term, S must be decomposed into
l)artial fractions; thus, by the method of (2'35),
4.B-10a;- _ 1 , 2 8
l-6.c + ll.j;'-(3a;-' l-Sx 1— 2x 1-a;
By the Binomial Theorem (128),
, ^, = l+3.c + 3-.r + +S''x'\
1 — Sx
r-=-r = - + 2=.c + 2^r + + 2" * '.c",
l—zx
— =-3-3.c-3.r - -Sx".
Hence the general term involving x" is
(;3'> + 2''»'-3)x''.
And by this formula we can write the " last terms" required in (2.")3), and
so obtain the sum of any finite number of terms of the given series. Also,
by the same formula we can calculate the successive terms at the beginning
of the series. In the present case this mode will be more expeditious than
that of employing the Scale of Relation.
258 If 5 in decomposing -<S' into partial fractions for the sake
of obtaining the general term, a quadratic factor ^vitli ima-
ginary roots sliould occur as a denominator, tlie same method
must be pursued, for the imaginary quantities will disappear
in the final result. In this case, however, it is more con-
venient to employ a general formula. Sup[)ose the fraction
which gives rise to the imaginary roots to be
L-\-Mx _ L-\ -Mx
<i+b,r + x"- ~ {p—r){q-x)'
p and q being the imaginary roots of ri-{-hx-{-x' = 0.
Suppose j) = (i-\-i)^,
q = a—ij^y where i = v —1.
86 ALGEBRA.
If, uow, the above fraction be resolved into two partial
fractions in the ordinary way, and if these fractions be ex-
panded separately by the Binomial Theorem, and that part of
tlie general term furnislied by these two expansions written
out, still retaining j) and r/, and if the imaginary values of p
and q be then substituted, it will be found that the factor will
disappear, and that the result may be enunciated as follows.
259 The coeflQcient of a;" ^ in the expansion of
L + iU.r
will be
^g^-^[Ha«-'^-C(«,3)a"-^HC(«,o)a»-^'-...l
^^"+^^ +C(«-l,5)«"-«j8»-...}.
260 With the aid of the known expansion of sin nO in
Trigonometry, this formula for the ti^'* term may be reduced to
in wliich 6 = tan — , <^ = tan ^
L + 3/a
If n be not greater than 100, sin (viO — ^) may be obtained
from the tables correct to about six places of decimals, and
accordingly the «*'' term of the expansion may be found with
corresponding accuracy. As an example, the 100*'' term in
the expansion of - — ^ ., is readily found by this method
^ , 41824 „9
to be -^- x^.
To (/cfrrniinc whether a i>;ii'cn Scries i.s rernrri)i;ii- (tr not.
261 If certain tirst terms only of the series be given, a scale
of relation may be found which shall produce a recurring
BECUBRINQ Sl^RII'JS. 87
series whose first terms arc those given. The method is
exemplified in ('255). The innnber of niikiiowii coefficients
j), </, r, &c. to be assumed for the scale of relation must be
equal to half the number of the given terms of the series, if
that number be even. If tlie nund^er of given terras be odd,
it may he made even by })refixing zero for the first term of
the series.
262 Since this method may, however, produce zero values
for one or more of the last coefiicients in the scale of relation,
it may be advisable in practice to deteruiine a scale from the
first two terms of the series, and if that scale does not jn-oduce
the following terms, we may try a scale determined from the
first four terms, and so on until the true scale is arrived at.
If an indefinite nnmber of terms of the series be given,
we may find whether it is recurring or not by a rule of
Lagrange's.
263 Let the series be
S= A + ]Jx + G.r -h Dx' -f- &c.
Divide unity by S as far as two terms of the quotient, wliicli
will be of the form p-\-qx, and write the remainder in the form
/S'V, S' being another indefinite series of the same form as S.
Next, divide S by S' as far as two terms of the quotient,
and write the remainder in the form S"x-.
Again, divide /S" by S'", and proceed as before, and repeat
this process until there is no remainder after one of the
divisions. The series will then be proved to be a recurring
series, and the order of the series, that is, the degree of tlie
scale of relation, will be the same as the number of divisions
which have been effected in the process.
KxAMi'LK. — To determine whether the series 1, o, (i, 10, 15, '21, 28, .'U),
45, ... is recurring or not.
Introducing x, we may write
S = l+3.r + 6.r+10.rH15.):' + 21,/' + 2,V' + :](u''+45..:\.. .
Then we .shall have ' = 1 — ;lf4-... ^vith a rcniaimler
6x' + 8.r' + 15.1!* + 24.c'' -f- obx'' + itc.
TlH-iir..io S'= 3 + 8.j;+15.7;-4-2kr» + .'3.V + A-c.,
.s" ;; 9
88 ALGEBBA.
with a remainder -^ (.r + 8a;H ().)■' + 10.x-'^ + &c. ...)•
Therefore we may take S" = 1 +8.1- + 6.r + 10a* + &c.
Lastly -^„ = 3 — .13 without any remainder.
Consequently the series is a recurring series of the third order. It is, in
fact, the expansion of
SUMMATION OF SERIES BY THE METHOD OF
DIFFERENCES.
264 Rule. — Form successive series of differences until a series
of equal differences is obtained. Let a, b, c, d, &c. be the first
terms of the several series ; then the 7^^'' term of the given
series is
265 a+i.-m+ ("-^(r'^ e+ ("-^("-f-^ .l+
The sum of n terms
266 =„« + !^M4 + ^iIi=ii§^<- + &c.
Proved by Induction.
Example: a...l+ 5 + 15 + 35 + 70 + 126 +
h ...4 + 10 + 20 + 35 + 5G + ...
c ... 6 + 10 + 15 + 21+...
(Z...4+ 5+ 6 + ...
e ... 1+ 1 + ...
The lOOti' term of tlie first series
■ ^ 1.2 ^ 1.2.3 ^ 1.2.3.4
The sum of 100 terms
,,,.,^100.09 , .100. 99. 98«j^ 100. 99. 98.97 ,, 100.99.98.97.96
== 100+ -^^4+-^-2r3-^+ 1.2.3.4 ^^ 1.2.3.4T.r-
FAiToh'fAl, ,s7';/.7 /•;>'. ^ SO
267 '^'-^ interpolate a term bclwecn two terms of u series by
the motliod of ililTtTeiices.
Ex. — Given log 71, log 72, log 73, log 74, it is required to find I<>g 7"2o4.
Form tbo scries of diBerences from the given logarithms, as in ('itJG),
log 71 log 72 log 73 log 74
a... l-8ol2".83 l-8573:i2') l-8G:i3229 1-8G02:U7
6 ... -0060742 •00r)9904 -0059088
r ... --0000838 - -0000816
(Z ... —00000-22 coii.sidered to vimish.
Log 72'o4 mnst be regarded as an interpolated term, the number of its
place being 2-54.
Therelore put 2-.'')l. for n in formula (265).
Result log 72-54 = 1-8605777.
DIRECT FACTORIAL SERIES.
268 Ex.: 5.7.9 + 7.0.11 -I- <.i.n .1;] + 11 .1:3.15 + ..
d = common difference of factors,
m= miml)er of factors in each term,
n = number of terms,
a = first factor of first term —d.
>*♦'• term == (fi-\-n(/) {a + n + i</) (^/ + ;/ + //<- I r/).
269 To find the sum of u terms.
Rule. — Fnnii the last term with the ne.rt hitjlirsf fnrtor take
thefi'i\st ti'Dii ir'ilh flie next loiiwst factor, and dlrlde Ixj {m + 1 ) '/.
Proof. — By Induction.
Thus the sum of 4 terms of the above series will be, putting d = 2, vi — 3,
^, 11.13.1 5.17-3.5.7.9
n = 4, a = 3, h = ^j^^, .
Proved either by Induction, or by the ractliod of Indeterminate Coefficicnt.s.
90 ALGEBRA.
INVERSE FACTORIAL SERIES.
270 Ex.: 5^7^9 + 7^9_ii + 9.11,13 + 11.13.15+---
Defining d, in, oi, a as before, the
1
;}tii term =
(n-]-ud) {a + n-\-l(I) ... {a-^n-\-m — id)
271 To find the sum of n terms. Rule. — From the first
term wanting its last factor take the last term wanting its first
factor, and divide hy (m — 1) d.
Thus the sum of 4 terms of the above series will be, putting d = 2, m = 3,
1 1
5.7 13.15
^^ = ^'" = ^^ (3-1)2 •
Proof. — By Induction, or by decomposing the terms, as in the following
example.
272 Ex.: To sum the same series by decomposing the terms into partial
fractions. Take the general term in the simple form
(r-2)r(r + 2)
Resolve this into the three fractions
Substitute 7, 9, 11, &c. successively for r, and the given series has for
its equivalent the three series
Ij 1 + 1+ 1 +i + 1 +_1_|
8 1 5 7 9 11 13 2u + 3 )
2^ C _ 2 _ 2 _ 2 _ 2 _ 2 2_ )
8 I '7 9 11 13 2» + 3 2n + 5)
+ 1 j 1 + 1 +1_+ + _i_ + _l_ +^1-1,
8l 9 11 13 2»i + 3 2u + 5 2» + 7)*
and the sum of n terms is seen, Jiy inspoction, to he
1(1_1_ 1 +_l_l=Jil 1_ ]
8(5 7 2« + 5 2n + 7) 4(5.7 (2» + 5) (2« + 7} 3 '
a result ol)taiiicd at once by the rule in (271), taking -— - — for the first
•^ ^ '^ 5.7.9
term, and -t- .-- for the »i"' or last term.
(2n + 3)(2n + 5)(2n + 7)
FACTOh'IAL SERIES. 91
273 Analogous series may be reduced to tlie types in (268)
and (270), or else tlie terms may be decomposed in the manner
shewn in (272).
Ex.: -J_+_i_+__7__+_10L.+
1.2.3 2.3.4 3.4.5 -i. 5. <;
has for its general term
3»-2 __1 ,_5 4_ , .......
n(n + l)(n-\-2) ,i n + 1 n + 2 '^ '^""^^>''
and we may proceed as in (272) to Hnd the sum of n terms.
The metliod of (272) includes the method known as "Summation by-
Subtraction," but it has the advaut;igo uf being more general and easier of
application to complex series.
COMPOSITE FACTORIAL SERIES.
274 If the two series
M N-5 i^r _l5.G .,^5.6.7 3^5.6.7.8 ,^
/I ^-3 1,.. ,3.4 .. 3.4.5 3 , 3.4.5.0 t ,
be multiplied together, and the coefficient of .f* in the product
be equated to the coefficient of x*" in the expansion of (1 —x)'^^
we obtain as the result the sum of the composite series
5.6.7.8xl.2 + 4.5.C).7x2.3 + :K4.5.6x3.4
4! 2.11!
+ 2. 3. 4. 5X4. 5 + 1. 2. 3. 4X5. G
7! 4!
275 Generally, if the given series be
Aa + Aa.+ --+A,. ,(?„-! (I),
where (^)^ = ,. (r + 1) (/• + 2) ... (r + v- 1),
and r,.= (?i— r) (//-/• + 1) .. {n- ri-p-l) ;
the sum of n — l terms will bo
/>!7! (;,4-;, + r/-l)!
1)2 ALGEBRA.
MISCELLANEOUS SERIES.
276 Sum of the powers of the terms of an Arithmetical
Progression.
1+2+;)+...+// =
1 +2»+y +...+«'= 5 ^^^j' =s.
H o.+y+ ... + „. = "(" + 1) (2« + l)(3»-+3» -1) ^ ^, ,
[By the method of Indeterminate Coefficients (234).
A general formula for tlie sum of tlie 7-"' powers of
1.2.3 ... n, obtained iu the same way is
>• + !
wliere Ji, A.,, &g., are determined by i)uttiug j> = 1, 2, 3, &c.
successively in the equation
1
2 0^ + 1)!
~(;?+2)!"^r(/>)!'^r(r-l)(i>-l)!^"'"r(r-l)...(r-7>+l)
277 «"' + (^/ + ^/)"' + (^/ + 2^/)"'+... + (^/ + //</)"'
^ (>, + !) ^,'" + ,s^,,,^f'"- V/+.S,(' (m, 2i «'"-->/-
Proof. — By Binomial Theorem and (276).
278 Summation of a scries parthj Arithmetical and
pa rill/ Geometrical.
Ex.\m?lt;. — To hud the sum of the series 1 +3,i' + 5.r + to n
terms.
Let s = l+;?.r + 5.r' + 7.i:» +... + (2/1-1) x"-',
S.V = .i+3.r + oj;'+ ... + {2n-:i) j""' + (*Ju- 1) x",
.'. by Kiiblractioii,
6- ( 1 -.,•) = 1 + ±v + 2.r^ + 2./.» + . . . + 2.t" - ' - (2« - 1 ) x"
1 — r""'
= l+2a^^- -^--(2H-l).r",
1 — X
l_(2n-l).r" 2x(l-.r''-')
• '- 1-x + (I-../--
279 A general formula for tlio sum of n terms of
^- \^' + 0-ry ■
Obtained as in (278).
Rule. — Mi(Itij)Ji/ Inj the ratio and subtract the resulting
series.
280 r-'— = l+.r+.»"+.t'+...+cr"-'+-pi
281 Tj-^, = l+2.r+;ja-+lr''+...
, n-i , (n-{-\)r'*~)i.v"^'
(l-cf)-
282 (^<-l).r+(/i-2).rH('<-.'i).''^'+... + -V-H'i'""'
= (^^-i);;-"^;+->"" . B,(253).
283 i^,,^W//^--])^;/(»-])0/-2)_^^^.^.^^^,„^
Hv making 4^-=^ in (12r»). .etrrat*.
94 ALGEBRA.
284 The series
»-,-j , (n-4)(n-5) _ {n-6){n-r,){n-7) ,
^~"T""^ ;5! 4! ^*
^ , ( ^y., 0i-r-l)(n-r-2)...(n-2r-[-l)
consists of ^^ or ^^~ terms, and the sum is given by
/S' = — if 71 be of the form 6m-\-S,
n
S = if ?i be of the form 6/?i + l,
S = if ?i be of the form 6m,
n
>S^ = — if 7i be of the form 6m±_2.
n
Proof.— By (545), putting;) = x^-y,q = xy, and applying (546).
285 The series 7i^-n («-!)'•+ Hd^l^ {n-2y
o !
takes the values 0, n\, ^n{n + l)\
according as r is <n, =n, or =)i + l.
Proof. — By expanding (e^ — 1)", in two ways: first, by tlie Exponential
Theorem and Multinomial ; secondly, by the Bin. Th., and each term of
the expansion by the Exponential. Equate the coeflicients of a;** in the two
results.
Other results are obtained by putting r = n-{-2, n-\-o, &c.
The series (285), when divided by r\, is, in fact, equal to
the coefiBicient of aj'" in the expansion of
POLYGONAL NUMBERS.
05
286 By exactly the same process we may detlucc from the
I'unctiou {t"*' — f'*}" the result tliat tlie scries
n'-n (N-'2y+ "^"~^^ (,,_.l.)'-_&c.
takes the values or 2'*.?^!, according as r is < n or = n;
this scries, divided by r ! , being e<j[ual to the coefficient of x''
in the expansion of
f ^.3 ,-,.5 yi
POLYGONAL NUMBERS.
287 Tlie n^^' term of the r'^' order of polygonal numbers is
equal to the sum of n terms of an Aritli. Prog. Avhose first
term is unity and common difference r — 2 ; that is
= 1 {2+{n-l)(r-2)] = n + ln {n-l){r-'2).
288 I'hc sum of It terms
__ uiu-^l) u{u-l)(n-\-l)(r-2)
2 "^ «
By resolving into two scries.
Order.
«*•> term.
1
2
3
4
5
6
n
Inin + l)
n-
iH(3u-l)
(2»-l)»
11 11111
1 2 3 4 5 6 7
13 6 10 15 21 28
1 4 9 16 25 36 49
1 5 12 22 35 51 70
1 6 15 28 45 66 91
r
n+^i^(r-2)
1, r, 3 + 3 0—2), 4 + 6 (r-2), 5 + 10(;— 2),
6 + 15 (r- 2), (fee.
96
ALGEBRA.
la practice — to form, for instance, the 6*'' order of poly-
gonal numbers — write the first three terms by the formula,
and form the rest by the method of differences.
Ex.: 16 lo 28 45 66 91 120 ...
5 9 13 17 21 25 29 ...
[r-2 = 4] 4 i 4 4 4 4 ...
FIGURATE NUMBERS.
289 The n"' term of any order is the sum of n terms of the
preceding order.
The n^^ term of the r*^' order is
njn+l)_^in +r-2) = /j („ ^ ,. _ j ) . [By 98.
(r-1)
290 The sum of n terms is
n{n-\-l)...{n + r-l) _
H{n,r).
Order.
Figurate Numbers.
nth term.
1
1, 1, 1, 1,
1, 1
1
2
1, 2, 3, 4,
5, 6
n
3
1, 3, 6, 10,
15, 21
7?, (w+1)
1.2
4
1, 4, 10, 20,
35, 5G
1.2.3
5
1, 5, 15,_35,
70, 12G
1.2.3.4
G
1, G, 21, 5G,
126, 252
7i(n + l)(7i + 2)(7i + 3)(7z-l-4)
1.2.3.4.5
iiYi'h'iiCKoM irrincAi. si:riks. 97
IIYPEUGEOMETKICAL SERIES.
291 ,+^_^,,. + ^(^+lMim),,.=
l.y 1. -2.7(7+])
a(a+1)(a + 2)/3(/3+1)(^ + 2) ^, ^^,
"^ 1.2.;$.y(y+l)(7 + 2)
is convergent if .r is < 1 ,
and divergent if x is > 1 ; (-•5'* ''•)
and if x = 1, the series is
convergent if -y — a — /3 is positive,
divergent if y — a — /3 is negative, (239 iv.)
and divergent if 7 — a — /3 is zero. (239 v.)
Let the liypergeometrical series (291) be denoted by
F{a, ft, y) ; then, the series being convergent, it is shewn by
induction that
292 ria,ft-^\,y jyi} ] concl.ulin- ^vitl.
l-/r, I-/.-,. ,
1-c^c. ... \-k,,z,..
^vlH"^c /.',, I:,, Jr., Sec with .-:,,,, are given l)y tlic foniiiilie
. _ (a + r-1)(y+)— 1-y8).r
(y + 2>—2j(y + 2>— 1)
_ ()8 + r)(y+r-a).r
(y+2;-l)(y+2r)
f'(a-f-r, /8 + r, yH-2r)
Tlie continued fi-action may be conchuhMl at nny point
with k-ir^Ur' When r is infinite, r/o^ = 1 and tlic continurfl
fraction is infinite.
o
98 ALGEBRA.
293 Let
"l.y ' 1.2.y(y+l) ' 1 . 2.;{.y (y+1) (7+2)
f{y) ::^ 1 + _!^ + '^' + -^ /"^i 1,/ _1_.>X + '^^•
1.7 1.2.V v+1) 1.2.,{.v(v4-l (-/+2)
the result of substituting — for ic in (291), and making
/3 =:= « = X . Tlien, by last, or independently by induction,
/(y + 1) _ 1_^ p^_ P2_ 'Pj!]_
Ay) 1+1+1 + ... + I+&C.
with j),„ =
(y+m — 1) (y+/>«]
294 In this result put y = ^ and -^ for ,r, and we obtain by
Exp. Th. (150),
Or the continued fraction may be formed by ordinary division
of one series by the other.
295 ('"' is incommensurable, m and n being integers. From
the last and (17-1), by putting x = ' .
INTEREST.
If r be the Interest on £1 for 1 year,
11 the inimber of years,
/' the I'l-incipal,
A the auiouut in n. years. Then
296 At Siiuple Interest A = P{l-^)n').
297 At Compound Interest A = /*(t+r)". % (-^-i)-
i\Ti:i:i:sr AM) .l\.\ ///'//vS. 99
298 But if the payments of ") .„
Interest be made 7 C A = I' h + -j
times a year ) ^ ^^
If A be an amount due in 11 years' time, and /' the ])resent
worth of -1. Theu
299 At Simple Interest 7* = -j-^ .
By (-200).
300 At Compound Interest /' = , . By (297).
301 Discount = A- P.
ANNUITIES.
302 The amount of an Annu- 1 ,,(,,_])
ity of £1 in n years, [■ = nA~ .^ >'• ^y (82).
at Simple Interest .
303 1 'resent value of same = '^yr — :
,tA-hi{N-\)r n^(09,j).
304 Amount at Compound \ _ ( 1+r)" — 1 ^ ,^-.
Interest ) "(l+r)-l '
Present worth of same — ~ . ' , '' . 'b' (-J^'*^*)-
(l+r)-l
305 Amount when the pay- ^ ( j 1 _!1 Y"' _ 1
ments of Interest/ _ ___7_ iw (2:is).
are made q times -|)er C / i 1 '* V' 1 '
annum J \ (/ ' ~
1
Present value of same =
100 ALGEBRA.
306 Amount wlicn the pay- )
meuts of the Annuity f _ ( 1 -f >•)"— I
are made m times per [ ' i ~
annum J m \{l-{-r)'-—l}
Present value of same
^ l-(l+ r )-
m{(l-fr)i-l}
307 Amount when the In-
terest is paid q times
and the Annuity m
times per annum ... J "^ V\ qi'
Present vahie of same
m
(i+v)-
PROBABILITIES.
309 If ^11 tlie ways in which an event can happen be m
in number, all being equally likely to occur, and if in n of
these m ways the event would happen under certain restrictive
conditions ; then the probability of the restricted event hap-
pening is equal to n-^m.
Thus, if the letters of the alphabet be chosen at random,
any letter being equally likely to be taken, the probability of
a vowel being selected is equal to -i^q. The number of un-
restricted cases here is 26, and the number of restricted
ones 5.
310 Ifj however, all the m events are not equally probable,
they may be divided into grou{)s of ccpially probable cases.
The probability of the restricted event happening in each
group separately must be calculated, anel tlie siun of these
probabilities will be the total })robability of the restricted
event liappening at all.
I'UOHAIIILiriHS. lol
ExAMPLK. — Tlicro are three bags A, B, and G.
A contains 2 white and .'} black balls.
B „ 3 „ t
C „ 4 „ 5
A bapf is taken at random and a hull drawn from it. Required the pro-
bability of the ball being white.
Hero the probability of the bag A being chosen = J, and the 8ub.sc([nonfc
probability of a white ball being drawn = l-
Therefore the [jrobability of a white ball being drawn from .1
~ 3 5 15-
Similarly the probability of a white ball being drawn from B
- 1' X 3 - l'
~ 3 7 ~ 7
And the probability of a white ball being drawn from G
-1 J* - i
~ 3 ^ 9 ~ 27'
Therefore the total probability of a white ball being drawn
.j2_ 1 4 ^ 401
15 7 27 945'
If a be the number of ways in wliicli an event can liappen,
and J) tlie number of ways in wliieli it can fail, then the
311 rrobabilitv of the event lia])penin2r = r.
312 l'rol)al)ihty of the event failing =
Thus Certainty = 1.
If p, p' be the respective probabilities of two iudcpcndcnt
events, then
313 rrol)al)ility (^f both liappening = pp'.
314 )} of not /yo//i happening == i—pp'.
315 )) of one happening and one faiHng
316 „ of l)o(h failing = (!—/>) (1 —/>').
102 ALGEBlLi.
If the probability of an event happening in one trial be j>,
and the probability of its failing q, then
317 Probability of the event happening r times in n trials
= C{n, r)2fff-\
318 Probability of the event failing r times in n trials
= C {n, r) ^j" "''(/''. [By induction.
319 Probability of the event happening at lea><t r times in
n trials = the sum of t\iQ fivHt n—r-[- 1 terms in the expansion
of 0; + ry)".
320 Probability of the event failing at least r times in n
trials = the sum of the last n — r-[-l terms in the same ex-
pansion.
321 The number of trials in which the probability of the
same event happening amounts to j/
_ iog(i-;/)
log (!-;>)'
From the equation (1 — j^)*" = 1 —])' -
322 Dei'inition. — AVhen a sum of money is to be received if
a certain event happens, that sum multiplied into the proba-
bility of the event is termed the expectation.
Example. — If three coins be taken at random from a bag
containing one sovereign, four half-croAvns, and five shillings,
the expectation will be the sum of the expectations founded
upon each way of drawing three coins. But this is also equal
to the average value of three coins out of the ten ; that is,
-i^ths of 35 shillings, or 10s. Qd.
323 The probability that, after r chance selections of the
numbers 0, 1, 2, 3 ... 7i, the sum of the numbers drawn will
be 6', is equal to the coefficient of .t'* in the expansion of
ri;i)r.M:iLiiii:s. lo:
324 'I'lie probability of the existence of a certain cause of
an observed event out of several known causes, one of wliieli
vi^ist liave produced the event, is proportional to tlie a jn-iarl
probability of the cause existinu: multiplied by the probability
of tlie event happening from it if it does exist.
Thus, if the a priori probabilities of the causes be /',, /'.
... Sec. J and the corresponding probabilities of tlie event hap-
pening from those causes (^),, (/_, ... itc, then the probal)ility
of the ?•"' cause having produced the event is
X{1'Q)
325 If A'> P-i ••• &c. be the a jfrlori probabilities of a second
event hap])ening from the same causes respectively, then,
after the first event has happened, the probability of the
second happening is t {PQ)
POP'
For this is the sum of such probabilities as \^''//, (i which is
the probability of the r^'' cause existing multiplied by the
probability of the second event happening from it.
Ex. 1. — Suppose there are
4 vases containing each 5 wliite and (i l)l:ick ball.s,
2 vases containing each ^ white and 5 black balls,
and 1 vaso containing '2 white and 1 black ball.
A white ball has been drawn, and the probability that it came Irani tiie group
of 2 vases is required.
Here P, = ^ ]'.. = 'f , P, = !
Therefore, by ('S-l), the pn)l)ability i-c(iuii-fd is
4,:.^ 2.;^ L2 427
7.11 7.8 7.3
Ex. 2. — After the white ball has been drawn and ro])laced. a ball
drawn again; required the probability of the ball being lilack.
104 ALGEUltA.
He-o P; = A, p; = |, p. = |
The probability, by (325), will be
4. 5.6 2.3.5 1.2.1
7.11.11 7.8.8 7.3.3 58639
4.5 2.3 1.2 112728'
7.11 7.8 773
I£ the probability of the second ball being white is required, QiQ^Qi
must be employed instead of P{P'.P'z.
326 The probability of one event at least happening out of
a number of events whose respective probabilities are a, h, c,
&G., is P1-P2 + P3-P4+&C.
where P^ is the probability of 1 event happening,
and so on. For, by (316), the probability is
l-(l-a) (l-h) (l-c) ... = y.a-^ah + ^ahc-, ...
327 The probability of tlie occurrence of r assigned events
and no more out of 01 events is
where Q„. is the probability of the r assigned events ; Q.,.+i the
probability of r + l events including the r assigned events.
For ii a, h, c ... be the probabilities of the r events, and
a, 1/, c ... the probabilities of the excluded events, the re-
quired probability will be
ahc ... {l-a'){l-h'){l-c') ...
= ahc ... {l-1.a-{-'^aU-:ia'l/c'-\-...).
328 Tlu^ probability of ani/ r events hapjiening and no more
Note.— If a = h = c = &c., tlien ^Q, = C (n, r) Q,, cl'c.
INEQUALITIES. 105
ii\i<:QLiALrrib;s.
330 ^^'+^^-+- •"^'^" lies between the -Teatest uiul least of
/>i + '>-.>+ ••• +^'»
the fractions i^, ^, ... -^, the (leiiominators being all of
the same sign.
PuoOF. — Let k be the greatest of the fractions, and - any other; then
ar<kbr. Substitute in this way for each a. Similarly if k be the least
frapCtion.
331 ^ > V"0.
332 «.+«.+ ...+»„ > y„,7^~^,;
71
or, Arithmetic mean > Geometric mean.
Proof. — Substitute both for the greatest iind least factors their Arith-
metic mean. Tiie product is thus increased in value. Repeat tlie process
indefinitely. The limiting value of the G. M. is the A. ]\I. of the quantities.
333 q:^' > {^l+!i)'\
excepting when m is a positive pi'opei' IVaet ion.
PlJOOK
„'" + ?,'" =("t'')"'[(l+.r)'" + (l-..)"'},
diere .»■ = " -'. Kinploy Hin. Tli
a + b
334 ":'+""'+..+": > ^'.+".+ ■+".. y\
excepting ^vhen /// is a positive proper fraction.
I"
106 Ahdi'JiiUA.
Otherwise. — The ArUhmetic mean of the m"' poivers is
greater than the m"' power of the Arithmetic mean, excepting
when m is a positive proper fraction.
Pkoof.— Similar to (332). Substitute for the greatest and least on the
left side, employing (333).
336 If -'' and m, are positive, and x and mx less than unity ;
then (l + ciO-'"> l-mx. (125, 240)
337 K ,1^ m, and n are positive, and n greater than ra ; then,
by taking' x small enough, we can make
For X maybe diminished until l^nx is > {l—mx)'^, and this
is > (l-\-xy\ by last.
338 If ^ be positive, log {l-\-x) < os. (150)
If X be positive and > 1 , log (l+.r) > <r- ^. (155, 240)
If X be positive and < 1, ^ogj^^, > ■<'• (^''^^^
339 When n becomes infinite in the two expressions
1.3.5... (27Z-1) .^^^^^ o.-^.7...(2y^ + l)
2.-4.(3 ... 2yi ' 2. 4. 6. ..2m
the first vanishes, the second becomes infinite, and their
product Ues between -J and 1.
Sliewii by adding 1 to each factor (see 7o), and multi-
plying the result by the original fraction.
340 II' '" he > II, and ii > <(,
SrA LES OF NOT. I TloX. 1 ( i7
341 If ", f> 1)0 |)ositiv(> quantities,
„V/' is > ('i+I'f"'.
SinulaHy a'' !,'■.■'> {^' + ',' + ' f"*' ■
These and similar theorems may be proved hy takinf^ lou^a-
rithms of each side, and employing the Expon. Th (loH), Sec.
SCALES OF NOTATION.
342 It" iVbe a whole number of h-{- I digits, and /• the radix
of tlie scale, ^<' = JJ,,'" +I>u-xr"-^ +p„_,r"-'-+ ... -^j>ir-\-p,,,
where j^„, 2>„_i, ...2>„ are the digits.
343 Similarly a radix-fraction will be exju'essed by
where 2>i, p.,, cVe. are the digits.
ExAMi'LKS : o-12l! in the scale of 7 == 3 . 7'' + •!■ . 7- + - • 7 + (J ;
•104o in the same scale = - + ^., + -,- + '* .
7 r 7'^ 7'
344 Kx.— To transform :U2G8 from the scale of 5 to the
scale of 11.
RuLK. — JJii'ide sucrcssiiu-Jij hi/ tlw iinr radix.
11 3426S
11 1 1348 -<
111-40-3
1—9 Result \[)'3t, in which t stands fm- Id.
108 ALGEBRA.
345 Ex.— To transform -tOcl from the scale of 12 to that
of 7, e standing for 11, and f for 10.
Rule. — Multiply successively hy the neio radix.
•tOel
7
5-i657
1
6-1931
7
1-0497
7
0-2971 Result -5610
346 Ex. — In what scale docs 2f7 represent the number 475
in the scale of ten ?
Solve the equation 27-- + 10;- + 7 = 475. [178
Result >• = 13.
347 The sum of the digits of any number divided by ?' — 1
leaves the same remainder as the number itself divided by
r— 1 ; r being the radix of the scale. (401)
348 The difference between the sums of the digits in the
even and odd places divided by r-\-l leaves the same re-
mainder as the number itself when divided by r + 1.
THEORY OF NUMBERS.
349 If ft is prime to b, , is in its lowest terms.
b
Proof. — Let = — i, a fiaction in lower terms.
b 1)^
Divide a by «,, remaimler n., (juotieiit 5,,
h by 6,, remainder h., quotient 7, ;
and so on, as in liiidinii- the H. C. F. of a and a„ and of b and i, (see 30).
Let «„ and b„ be the highest eoninion factors thus determined.
THEORY OF NUMBERS. 100
Tlu'ii, hocaiise — = ' , .-. = ■'•'=', (70;
6 ^, /' '' — Q\''\ "i
and so on ; thus - = ' = • = txc ~ 7 •
T hero ft )!■(.' u and b arc C(|uimnlti|)k's of a„ and b„ ; that is, a is not prime to b
if any fraction exists in lower terms.
(t (I
h
('(jiiiimiltiples of a and />.
350 It" " is prime to A, and -^= '.-; then n' and //arc
Pkoof. — Let - reduced to its lowest terms be ^ . Then ^- = — , and,
b 1 1 '>
since p is now prime to 7, and a prime to /', it folhiws, by ;M-ft, that ^ is
neither greater nor less than ; that is, it is (.Hjual to it. Therefore, &c.
351 If '^^ is divisible by c, and a is not ; then h must be.
II T , (lb a
Pi;nOl'.— Let =7; •■• ' = ■
c c l>
Hut '( is prime to c ; therefore, by last, b is a multiple of c.
352 If <^f Jind h be each of them prime to r, ah is i)rime
to r. [By (351).
353 If abed... is divisible by a prime, one at least of the
factors a, h, c, &c. must also be divisible by it.
Or, if p be prime to all but one of the factors, that factor
is divisible by j>. (:ir.l)
354 Therefore, if a" is divi.sihle by ^^ i> cannot be jirime to
'/ ; and if j) be a prime it must divide a.
355 If " is prime to h, any power of <i is prime to any
} tower of It.
Also, if a, 0, c, &c. are prime to each otlier, the product
of any of their powers is prime to any other protluct of their
powers.
110 ALGEBRA.
356 No expression with integral coefficients, siicli as
A + B,e + Cx' -h . . . , can represent primes only.
Proof. — For it is divisible by x if ^1 = ; and if not, it is divisible by A,
when x=: A.
357 '^^^G nnmber of primes is infinite.
Pkoof. — Suppose if possible p to be the greatest prime. Then the pro-
duct of all primes up to p, plus unity, is either a prime, in which case it
would be a gieater prime than p, or it must be divisible by a prime ; but
no prime up to p divides it, because there is a remainder 1 in each case.
Therefore, if divisible at all, it must be by a prime greater than p. In
either case, then, a prime greater thanp exists.
358 If (I be prime to h, and the quantities a, 2a, oa, ...
(b—i) a be divided by h, the remainders will be different.
Proof. — Assume ma — nb = ma — )ib, iii and n being less than b,
"" - "~"' Then by (350).
b m — in
359 A number can be resolved into prime factors in one
way only. [By (353).
360 To resolve 6040 into its prime factors.
Rule. — Divide hij the prime numbers snccessiveli/.
2x51 5040
2 1 504
21 252
2 1 126
7 1 63
3L9_
3 Thus 5040 = 2'. 3-.. 5. 7.
361 Required the least multiplier of 4704 wliicli will make
the product a perfect fourth power.
By (19G), 4704 = 2'. 3. 7-.
Then 2'. 3'. 7- x 2'. 3». 7- = 2». 8'. 7' = 84',
the indices 8, 4, 4 being the least multipli's i)f I which :ire not less than
5, 1,2 I'espcctively.
Tlius 2". 3'. 7- = 3584 is the multiplier reciuired.
TllEUliY UF MJMUl'JIiS. Ill
362 All mmilHTS :uv ..f one of tin- forms 2// <.|- 2//-f 1
•2n(n'2n — \
„ ,, ',\n ov:\n±\
,, ,, 1// or l//il oi- I// + 2
,, ,, ly/ or ly/±l or 4/<— 2
,, ,, r)/M)i- ."iz/il or r>//i-
aiul so oil.
363 AH square numbers are of tlie form .w< or .u/il-
l'i;00K. — By squarino; tlii" lonns o/t, .">» ± 1, I'tuzt-. wliii-h ciniiitrelieiHl
all numbers whatever.
364 All cube numbers are of the form "Jn or 7^'dzl-
And similarly for other powers.
365 The highest power of a prime jh which is contained in
tlu> i)roduct III ! , is the sum of the integral parts of
m m m p
1> F V
For there are ' factors in //^ ! which p will divides '., which
it will divide a second time ; and so on. The successive
divisions are eejuivalent to dividing b}'
ExAMi'LK. — The hitrliesfc power of 8 which will divide 29!. Heio the
factors 3, <3, l», 12, 15, 18, 21, 24, 27 can be divided by 'A. Their ntinilier is
'" I
"■ = !> (the inte,t,M-al pai-t).
Till' factors it, 18, 27 can be divided a second time. Their nmnlx'r is
"'_ = ;5 (the integral part).
One factor, 27, is divisil)le a third time. "7;^ = 1 (intei^'ral i):irt).
9-f 3+ 1 = 13; that is, 3" is the highest power of 3 which will divide 29!.
366 'I'hc [)roduct of :mv /• consecutive integers is divisiljlc
byr!.
PkooK: «("-!) ••• (»-'•+!) is necessarily an inteu'cr, by ('.'<;).
112 ALCIEBEA.
367 If ^^ be a prime, every coefficient in the expansion of
{a-\-hy\ except tlie first and last, is divisible by n. By last.
368 If " l)t> a prime, the coefficient of every term in the ex-
pansion of {(i-\-h-\-c ...)", except a", &", &c., is divisible by n.
Proof.— By (367). Put /3 for (6 + c+ ...).
369 Frrmaf.s Theorem.— U p be a prime, and N prime to
p ; then iV^"^ — 1 is divisible by p.
Proof : W={l + \ + ...y = N^Mp. By (368).
370 If V be any number, and if 1, r/, ^>, c, ... (p — 1) be all
the numbers less than, and prime to p ; and if n be their
number, and x any one of them ; then ,if — 1 is divisible by p.
Proof. — If x, ax, hx ... (p — l)x be divided by p, the remainders will be
all different and prime to^ [as in (358)] ; therefore the remainders will be
1, a, b, c ... (p — l) ; therefore the product
x"ahc... ip—l) = ahc ... {p — \)+Mp.
371 Wilson's Theorem.— If p be a prime, and only then,
l + (^— 1) ! is divisible by p.
Put j) — 1 for r and n in (285), and apply Fermat's Theorem
to each term.
372 If i' be a prime = 2y« + l,then {vlf + {-iy is divisible
hy p.
Proof. — By multiplying? together equi-di.'^tant factors of (j» — 1) ! in
Wilson's Theorem, and putting 2n + l for /).
373 \,Qt N =(('' !>''(''' ■'• in prime factors ; the number of in-
tegers, including 1 , which are less than u and prime to it, is
Proof. — The number of intogcrs piimo to N contained in ri" is n"-
Similiirly in //", /•'•, &c. Take the ])r()duct of those.
TUEORT OF NUMBERS.
113
Also tlio miinhcr of intcfjfors less tliaii mikI ])i-imo to
(Xx ^fxScc.) is the ])roduct of the coiTcspoiuliiig miiuhcrs
for X, ^[, &c. separately.
374 The number of divisors of N, incliidiiif^ 1 and ^V itself,
is = (y + l) (v 4-1) (/' + !) ...• For it is equal to the number
of terms in the product
(l+./ + ...+r7'')(l+/.-h...+^")(l+r+...+'")---<-'tc.
375 The number of ways of resolving N into two factors is
half the number of its divisors (374). If the number be a
S(juare the two equal factors must, in this case, be reckoned
as two divisors.
376 If the factors of each pair are to be prime to each other,
put j;, 7, r, &c. each equal to one.
377 The sum of the divisors of ^V is
a^^-'-l h'^^'-^ c^-^'-l
a-l ' h-1 ' c-1 '"
Proof. — Bj the product in (-374), and by (85).
378 If 7^ be a prime, then the j? — 1*** power of any number
is of the form mj^ or )fij)-\-l. By Fermat's Theorem (3(30).
Ex. — The 12"* power of any number is of the form 13fM. or 12m-\-l.
379 To find all the divisors of a number ; for instance, of 50-1.
I.
II.
1
504
2
2
252
2
-i
126
2
8
63
3
3
6
12
2J,
21
3
9
18
3G
72
7
7
7
14
28
56 21 42
84
168
63
126 252 504
ExprANATiON. — Kesolvc 504 into its priino factors, placing thi-m in
column 11.
114 ALGEBRA.
The divisors of 504 are now formed from tlie numbers iu column II., and
placed to the right of that column in the following manner : —
Place the divisor 1 to the right of column II., and follow this rule —
Multiply in order all the divisors which are written down by the next number
in column II., which has not already been used as a multiplier : place the first
netv divisor so obtained and all the folloiving products in order to the right of
column II.
380 >^r tlie sum of the r^^' powers of tlio first n natural
numbers is divisible by 2}i-\-l.
Proof : x {x"- V) (x" - 2") . . . (»' - »0
constitutes 2» + ] factors divisible by 2w+l, by (36G). Multiply out, re-
jecting X, which is to be less than 2ji-|-l. Thus, using (372),
x'"-S,x"'-' + S2X-"-'- ... S,,.,x-' + (-iy([ny = M(2n + l).
Put 1, 2, 3 ... (« — 1) in succession for x, and the solution of the (n— 1)
equations is of the form Sr = M{2n + l).
THEORY OF EQUATIONS.
FACTOKS OF AN EQUATION.
iicncral form of a rational integral equation of the w^'' degree.
The left side will be designated /(,/•) in the following
summary.
401 If /('■) bt-' divided by « — rt, tbe remainder will be /(")•
By assuming /(,(;) = P {x — a)-\-lL
402 If « be a root of the equation /(.i;) = 0, tlien/(rt) = 0.
403 To compute /(rt) numerically; divide f{.r) by x — a,
and the remainder will bef{a). [101
404 Ex^^MPLE.— To find the value of 4x''-2x' + l2x'-x'' + 10 when x = 2.
4-3 + 12 +0 -1 +0 +10
8 + 10 + 4-i + 88 + 174 + 34.8
4 + o + 22 + -lrl + 87 + 174 + 358 Thus /(2) = 358.
If a,h,c...h be the roots of the eciuation f {(r) = ;
then, by (401) and ( 1U2),
405 /Or) =7>o (.t-^/) G^-'>) O'-O ... ('^-/O.
By multiplying out the last cijuation, and equating coefficients with
cquatiou (100), cousidcriiig j'u = 1, the following results lu-o obtained :—
116 THEORY OF EQUATIONS.
406 —ih = the sum of all tlie roots of /(a?).
_ ( the sum of the products of the roots taken
^'" ~ \ two at a time.
_ ( the sum of the products of the roots taken
~~i^ ~ \ three at a time.
X 1 y _ ( the sum of the products of the roots taken
^ /ir — "^ rata time.
( — l)"j?,j = product of all the roots.
407 The number of roots of /(.^') is equal to the degree of
the equation.
408 Imaginary roots must occur in pairs of the form
a + /3v/^, a-i3\/^.
The quadratic factor corresponding to these roots will
then have real coefficients ; for it will be
■x'-2ax + a-+ii\ [405, 226
409 If /(■'«) be of an odd degree, it has at least one real root
of the opposite sign to p,j.
Thus a;^ — 1 = lias at least one positive root.
410 If /('O ^Q of an even degi^oe, and jhi negative, there is
at least one positive and one negative root.
Thus a;'*— 1 has +1 and —I for roots.
411 If several terms at the beginning of the equation are of
one sign, and all the rest of another, there is one, and only
one, positive root.
Thus x^ + 2x*-\-Sx^ + x'^ — 5x—4< = has only one positive root.
412 If all the terms are positive there is no positive root.
413 If all the terms of an even order are of one sign, and
all the rest are of another sign, there is no negative root,
414 Thus **— a;" + «'—;« + 1 = has no negative root.
DISCRIMINATION OF ROOTS. 1 1 7
415 If :>11 tlio indices are even, aiul all tlie terms of the same
sijjfii, there is no real root; and if all the indices are odd, and
all the terms of the same sign, there is no real root but zero.
Thus x* + x^ + l = has no real root, and x'^ + x^ + x = has no real rout
but zero. In this last equation there is no absolute term, because such a
terra would involve the zero power o( x, which is even, and by hypothesis is
wanting.
DESCARTES' RULE OF SIGNS.
416 In the following theorems every two adjacent terms in
/(.r), which have the same signs, count as one " continuation
of sign"; and every two adjacent terms, with different signs,
count as one chanw of siefu.
417 /(<'■)' multiplied by (■/.■ — a), has an odd number of
changes of sign thereby introduced, and one at least.
418 ./* (c) cannot have more positive roots than changes of
sign, or more negative roots than continuations of sign.
419 Wlien all the roots of f{.r) are real, the number of
positive roots is equal to the number of changes of sign in
f{.r) ; and the number of negative roots is equal to the number
of changes of sign in/(— .r).
420 Thus, it being known that the roots of the equation
a;*-10.t'' + 3o.c2-50.c + 24 =
are all real ; the number of positive roots will be equal to the number of
changes of sign, which is four. Also f(—x) =x*+lOx'^ + 3bx- + 50x + 2i! = 0,
and since there is no change of sign, there is consequently, by the rule, no
negative root.
421 If the degree of /(./•) exceeds the number of changes of
sign in f{x) and /(—a;) together, by /t, there are at least /j.
imaginary roots.
422 If, between two terms in /(,/■) of the same sign, there
be an odd number of consecutive terms wanting, then there
must be at least one more than that number of imaginary
roots ; and if the missing terms lie between terms of different
118 TEEORY OF EQUATIONS.
sign, there is at least one less than the same number of
imaginary roots.
Thus, in the cubic equation x^ + 4x — 7 = 0, there must be two imaginary
roots.
And in the equation x°—l =■ there arc, for certain, four imaginary roots.
423 If an even number of consecutive terms be wanting in
f{x), there is at least the same number of imaginary roots.
Thus the equation x^ + 1 — has four terms absent ; and therefore four
imaginary roots at least.
THE DERIVED FUNCTIONS OF f{.v).
Rule for forming the derived functions.
424 Multiply each term hy the index of x, and reduce the
index by one; that ^s, differentiate the function with respect
to X.
Example. — Take
/ («) = x^+ a;*+ a;'- x'-x-l
f (x) = 5x'+ 4x^ + dx"" - 2a; - 1
f(x) = 20x' + 12x' + 6x-2
f(x) = 60a;2 + 24« +6
f(x) = 120x+24.
f (x) = 120
/' (*)j /' ('''')y <^c. are called the first, second, &c. derived functions of/ (a;).
425 To form the equation whose roots differ from those of
f{x) by a quantity a.
Put x = y-\-a infix), and expand each term hy the Binomial
TJieorem, arranging the results in vertical columns in the foU
lowing manner : —
/(a + 2/) = (« + 2/)'+(a + 2/)H(a + 7/)»-(a + y)=-(a + 7/)-l
a"
+ a* +
a" -
a? -
- a -1
+ ( 5a*
+ 4a'' +
3a^ -
2a -
• l)i/
+ (10a»
+ Go? +
3a -
1)^^
+ (10a»
+ 4a +
1)/
+ ( 5a
+ 1)/
+ f
TBANSFOJIMATION OF AN EQUATION.
119
('om paring this result with that seen in (421), it is seen that
426 /•("+.'/) =/(")+/'(")//
\A lA LI L±
80 tliat tlie coefficient generally of ?/'' in tlie transformed
equation is ' , ^ ^
r
427 To form tlie equation most expeditiously when a has a
miinerical value, diride f{,i') continuously hi/ x — a, and the
succcssli-e remainders ivlU furnish the coefficients.
ExAiirLE. — To expand f(y + 2) when, as in (425),
f(x) = Z' + x' + x'-x'-x-l.
Divide repeatedly by x — 2, as follows : —
1 + 1 + 1 - 1 - 1-1
+ 2 + + 14 + 2G +50
1 + 3 -f 7 +
-f 2 +10 +
13 +
31 +
1 + 5 +17 + 47
+ 2 +14 + 02
+ 119
+ 49=/(2)
= rC2)
1 + 7+31
+ 2 +18
+ 109 =
(i)
11
1 + 9 I +49 =
+ 2 I
14
_/'(2)
|3
+ 11
1 - f(^\
|5
That these remainders
are the required eoeflicients
is seen by inspecting the
form of the equation (420) ;
for if that equation bo di-
vided by x — a = II repeat-
edly, these remainders aro
obviously produced when
a = 2.
Thus the equation, whoso roots are each less by 2 than the i-oots of the
proposed equation, is ?/+ll?/' + 49(/"'+109//'+ 119// + 49 = 0.
428 To make any assigned term vanish in the transformed
equation, a must be so determined that the coeffieient of tliat
term shall vanish.
Example. — In order that there may be no term involving if in equation
(420), we musi have /*(a) = U.
Find /'(a) as in (424);
thus 120« + 24 = 0; .-. a = -\.
The equation in (424) must now be divided repeatedly by a; + | after the
manner of (427), and the resulting equation will be minus its seeoud term.
120 THEORY OF EQUATIONS.
429 Note, tliat to remove the second term of tlie equation
y(-t') = 0, tlic requisite value of a is = — ^ ^ ; tliat is, the
coefficient of the second term, with the sign changed, divided hy
the coefficient of the first term, and hy the numher expressing
the degree of the equation.
430 To transform /'(''') "^^o an equation in y so that // -■ <p {.<■),
a given function of x, 2^ut x = (j)~'^{y), the inverse function of y.
Example. — To obtain an equation whose roots are respectively three times
the roots of the equation x^ — Gx + 1 = 0. Here 2/ = 3a; ; therefore x = —,
and the equation becomes ^ — -^ + 1 = 0, or ?/— 54^ + 27 = 0.
431 To transform /(,v) = into an equation in which the
coefficient of the first term shall be unity, and the other
coefficients the least possible integers.
Example. — Take the equation
288x^ + 24>0x''-176x- 21 = 0.
Divide by the coeflacient of the first term, and reduce the fractions ; the
5 11 7
equation becomes x^ + —• x^— — x — KB — ^'
Substitute -^ for x, and multiply by h^ ; we get
Next resolve the denominators into their prime factors,
3 57^ , life' _ yic" ^Q
'^ "^2.3^ 2.3'^^ 2^3
The smallest value must now be assigned to h, which will suffice to make
each coefficient an integer. This is easily seen by inspection to be 2'-. 3 = 12,
and the resulting equation is i/ + 10if — 88y — l26 = 0,
the roots of which are connected with the roots of the original equation by
the relation y = I2x.
EQUAL ROOTS OF AN EQUATIOli^.
By ox})audiiin^ 7X''^ + ^') i^^ powers of ;■• by (i05), and also
by (1'2()), and c(iiiating the coefficients of z in the two ex-
EQUAL nOOTS. 121
pansions, it is provctl tliat
tVoiii wliicli result it appears tliat, if tlie roots a, b, r, &c. arc
all uTUMiual, /'(.*') and /'(,»■) can liave no common measure in-
volving ,*•. If, however, there are r roots each equal t<^ d,
s roots e(|ual to h, t roots equal to r, &c., so that
f{^v) = iK{,'-ay {.v-hy {.v-i'Y ...
then
433 /(..■) = ^ + ;^;! + (-ia^ + *..;
and the greatest common measure <'f/((') and /'(,<■) will bo
444 {.v-(iy-' {.v-hy-' {.v-cY-\..
When .v, = a, /(..), /'{,'), .•■/'-'Oi') all vanish. Similarly
when .v = b, &c.
Prdctiral mctlnnl offfiH/in^' the (-([Udl nKits.
445 Lft / (.0 = A', X: a1 X\ Xl . . . X;::, where
A', = product of all the ftictors like {x — <^^,
A1= „ „ {x-a)\
Xl= „ „ (.-«)».
Find the greatest comraon measure of /(j-) and /'(f) = I'\ (.') say,
I'\(x)aud'F;Cv) = F,(x),
F^(x) And F:(x) = F3(x),
Lastly, the greatest common measure of F,„.i(x) and /''„,-i(*) = I'\,0') = 1-
Next perform the divisions
f(x) -i- F,(.r) = <p,(.r) .say,
F,(x)-^F,(') ='P,{.r),
And, iinully, 0, (.,') -4- <Pi{x) = A',,
i''«.-.C'-)=V'..C'-) = A',,.. [T. 82.
1:
122 THEORY OF EQUATIONS.
The solution of the equations Xj = 0, X, = 0, &c. will furnish all the
roots of/ (a;) ; those which occur twice being found from X.^; those which
occur three times each, from X^ ; and so on.
446 If /('^) lias all its coefficients commensurable, X^^X^^X^,
&c. have likewise their coefficients commensurable.
Hence, if only one root be repeated r times, that root must
be commensurable.
447 III all the following theorems, unless othermse stated,
/(-/•) is understood to have unity for the coefficient of its first
term.
LIMITS OF THE KOOTS.
448 If the greatest negative coefficients in /(,/') and /( — <^')
be j) and q respectively, thenp + 1 and —(^ + 1) are limits of
the roots.
449 If x''-'' and x''-' are the highest negative terms in /(<i')
and /( — aO respectively, {l + \/p) and —{l-\-^q) are limits
of the roots.
450 If /-^ be a superior hmit to the positive roots of /( — j »
then — will be an inferior limit to the positive roots of /(-f)-
451 If each negative coefficient be divided by the sum of all
the preceding positive coefficients, the greatest of the fractions
so formed + unity will be a superior Hmit to the positive
roots.
452 Neivtons tnethod.—Viit x=h-\-i/ in /(./•) ; then, by (420),
Take // so that /•(/')' /(^O^ fW yf'W ^^^ ^11 positive;
then // is a superior limit to the positive roots.
453 According as /{a) andf{b) have the same or different
signs, the number of roots intermediate between a and b is
even or odd.
INTEGRAL BOOTS. 123
454 Rollrs Thcorrm.—Ono real root of the eriuation f (r)
lios iH'lwoeii every two adjacent real roots of /(./')•
455 ('OH. I.— /(•'•) cannot have more than one root gi-eatcr
than the greatest root in /(./■); or more than one less than
llic least root in/'(,r).
456 CoR. 2.— If f(.r) has m real roots, /'•(') has at least
III — /■ real roots.
457 CoH. 3.— If f(u') lias /t imaginary roots, f{.r) has also
/i at least.
458 CoK. -1.— If a, /3, y ... K be the roots of /(.<) ; then the
11 umber of changes of si^-n in the series of terms
f{^), /W, /(«, /(7).-/(-^)
is equal to tlie number of roots oif{.c).
NEWTON'S METHOD OF DIVISORS.
459 To discover the integral roots of an equation.
ExAMTLE. — To ascertain if 5 be a i-oot of 5 ) 105
«*-6x» + 80x*-i;6.i- + 10-. = 0. 21
— 1/b
If 5 be a root it will divide 105. Add the quotient to the ^ ) —\55
next coefficient. Result, —155. —'Si
If 5 bo a root it will divide -155. Add the quotient to _86
the next coefficient ; and so on. 5 ) 55
U
— 6
If the number tried be a root, tlic divisions will be effectiblo
to the end, and the last quotient will bo -1, or —}>o, if/u he >- n _ e
not unity. 2_Z_.
-1
460 In employing this method, limits of the roots may first
be found, and divisors chosen between those limits.
461 Also, to lessen the number of trial divisors, take any
integer m ; then any divisor a of the last term can be rejected
if a — m does not divide /(///).
In practice take /// = -f-1 and — 1 .
To find wliether any of the roots determiiK'd as above are
repeated, divide f{x) by the factors correspijndiTig to them,
and then applv tlie method of divisors to the resulting ecpiation.
124 THEORY OF EQUATIONS.
Example. — Take the equation
Putting X = 1, we find/(l) = —24. The divisors of 144 are
1, 2, 3, 4, G, 8, 9, 12, IG, 24, &c.
The values of n — m (since rii = l) are therefore
0, 1, 2, 3, 5, 7, 8, 11, 15, 23, &c.
Of these last numbers only 1, 2, 3, and 8 will divide 24. Hence 2, 3, 4, and
9 are the only divisors of 144 wliich it is of use to try. The only integral
roots of the equation will be found to be ±2 and ± 3.
462 If /('^') and F{X) have common roots, they are con-
tained in the greatest common measure of /(a:) and F{X).
463 If /(■'') l^as for its roots a, (p {a), h, (jt {b) amongst others ;
then the equations /(.*) = and/[(^(A')j = have the common
roots a and h.
464 But, if all the roots occur in pairs in this ^vay, these
equations coincide.
For example, suppose that each pair of roots, a and b, satisfies the equation
a + h = 2r. We may then assume a — b = 2z. Therefore/ (2 + r) = 0. This
equation involves only even powers of z, and may be solved for z'.
465 Otherwise: Let a& = z ; then /(;«) is divisible by {x — a){x — l)
= x^ — 2rx + z. Perform the division until a remainder is obtained of the
form Px + Q, where P and Q only involve z.
The equations P = 0, Q = determine z, by (462) ; and a and I arc found
from a + b = 2r, ab = z.
EECIPROCAL EQUATIONS.
466 A reciprocal equation has its roots in pairs of the form
a, — ; also the relation between the coefficients is
Pr =Vu-r^ OY else p, = —Pn-r-
467 A reciprocal equation of an even degree, with its last
term jwsit ire, may be made to depend upon the solution of an
equation of half the same degree.
BINOMIAL EQUATIONS. 125
468 KxAMiM.K : •l-,/''^-24,r''-f57r'-7o./-''-f :)7./--2-1../ + t =
is a rocii)rtic;il ('(lualioii of :iii cncii dcLi-i'd', with its last term
positive.
Any reciprocal equation w liicli is 7iot of this form may bo
reduced to it hi/ diridiiuj hi/ ,/ -f I //" the la.sf term l>e posit Ire ;
(UkI, 1/ till' lust term he neijafirc, l>i/ diriduuj hi/ ,r— 1 or r' — ],
so us to liriiiij thi' t'tjiiiition tn mi < rm. tlrr/ree. Then proceed
in tlie foHowinn' manner : -
469 First brin<^ toovther e(|nidistant ti'rins, and diviih' the
equation by ,/'"*; tlius
By })utting ,v -\ = //, ;ind by making- repeated use of tlie
relation ,/'- -| =. i,r -\ ) "~ -> ^^'<^' eipration is reduced to
a cubic in ?/, the degree being one-half that of the original
equation.
]"*ut ;) for .r -| , and p„, for x„^-\ .
470 Tlie relation between the successive factors of the form
j)„, may be exi)ressed by the e(juatioii
471 'I'he equation for ji,,,, in terms of ^>, is
P,u = p'-nip'" -+ \ ,, p'" '- ...
_L f_i V ^" f»/ — >•— 1) ... (in — '2r-\-\) „,_..^,
I>y (^')i")), putting 7 = 1.
BINOMIAL EQUATIONS.
472 If a be a root of .r" — 1 = 0, then a"* is likewise a root
where m is any positive or negative integer.
473 If " be a root of ./" + 1 = 0, then a-'"^' is likewise a root.
126 THEORY OF EQUATIONS.
474 If ^''' ai^d ^2- be prime to each other, x'"' — l and x^—1
have no common root but unity.
Take iim — qn = 1 for an indirect proof.
475 If n be a prime number, and if a be a root of ct;" — 1 = 0,
the other roots are a, a^, a^ ... a'\
These are all roots, by (472). Prove, by (474), that no two can be equal.
476 If ^i be not a prime number, other roots besides these
may exist. The successive powers, however, of some root
will furnish all the rest.
477 If r*/'— 1 = has the index n = m2)q; m, ]}, q being
prime factors ; then the roots are the terms of the product
(l+a + a^+ ... +a-^)(H-/3 + /3'--[- ... +/3''-^)
X(l + 7 + 7'+ - +7'"')>
where a is a root of «'"— 1,
/3 „ x^-l,
7 » »''-!'
but neither a, /3, nor 7 = 1. Proof as in (475).
478 If n = m^ and
a be a root of x""—! = 0,
(i „ x^>^-a = 0,
7 » r.--|3=0;
then the roots of x''—l = will be the terms of the product
(l+« + a^+ ... +„-^)(l+/3 + /3-^+...+r-^)
X(l + 7 + 7^"+... +7""')-
479 a^" + 1 = may be treated as a reciprocal equation, and
depressed in degree after the manner of (468).
480 The complete solution of the equation
.1 - -1 =
is obtained by De Moi\Te's Theorem. (757)
The 71 different roots are given by the formula
0? = cos ± V — 1 sill
71 n
in which r must have the successive values 0, 1, 2, 3, &c.,
concluding with ^ , if n be even ; and with -~ , if // be odd.
CUBIC EQ UA TIONS. 1 2 7
481 Similarly the n roots of the ofiuatiuu
.r" + 1 =
are given by the formula
n u
r taking the successive values 0, 1, 2, 3, &c., up to ' ~^ , if
n be even ; and up to ' , if )i be odd.
482 'I'he number of different values of the product
is equal to the least common multiple of m and n, when m and
7/ are integers.
CUBIC EQUATIONS.
483 To solve the general cubic equation
a;^ + jj.r + qx -f /• = 0.
Remove the term j^at^ by the method of (429). Let the trans-
formed equation be .v'^-\-q,r-\-r = 0.
484 Cardan s mcfhoiL — The complete theoretical solution
of this equation by Cardan's method is as follows : —
Put x = i/-\-:i (i.)
yH,v^ + (3v.v + 7)(y + ,v) + r = 0.
Put Si/:: + q = 0; .'. ^ = - 3^
Substitute this value of //, and solve the resulting quadratic
in //^. The roots are equal to 1/ and .r* respectively ; and we
have, by (i.),
485 r
{-iWf+j^r+i-^-vj+f;}'
128 THEOBY OF EQUATIONS.
Tbe cubic must have one real root at least, lij'- (400).
Let »i be one of the three values of j ^ "^ \/ TT "*" '^ ( ^' ^°^ " "°® I
of the three values of j ^ \/ X "^ 9" [ '
486 Let 1, n, a- be the three cube roots of unity, so that
a=-l +1- y^, and ci' = - 1- - L yZs. [472
487 Then, since Viu^ = my I, the roots of the cubic will be
m + ii, am-ta'-n, u'in-\-n)i.
Now, if in the expansion of
I 2 ^V 4 ^ 273
by the Binomial Theorem, we put
fx = the sum of the odd terms, and
V = the sum of the even terms ;
then we shall have m = /u. + y, and « = ^ — v;
or else m = /u + v/ — 1, and n = fji — y v— 1 ;
according ^^ \/ 'T '^ §^ i^ ^^^^ ^^ imaginaiy.
By substituting these expressions for in and n in (487), it appears that —
488 (i-) If V" + ^ ^® positive, the roots of the cubic will be
2/^, —/i + >'%/— 3, —fi — yv—o.
r- cj^
(ii.) If -r "^ 97 ^^ negative, the roots will be
2/1, —fx + y^S, — /J — vn/S.
,2 3
(iii.) If + t^ = 0, the roots are
4 27
2ot, —:?/!, -m;
since m is now equal to fi.
489 '/^/'^' Trigonometrical method. — The equation
.1'^ + r/.r + r =
may be solved in tlie following manner, by Trigonometry,
when -p + 77= is negative.
4 27
Assume <6' = ?; cos a. Divide the equation by n^; thus
cos'' a + -2- cos a -\ ^ = ^^
But cos« a - -? cos a - -^ = 0. By ((357)
4 4
TilQVADRATW EQUATIONS. 129
Equato coefficients in tlic two (M|n;itions ; the result is
n must now be found ^villl the aid of tlie Trigonometrical
tabh's.
490 The roots of the cubic will l)e
n cos a, n cos (jTr+a), n cos (^tt— a).
.2 3
491 Observe that, according as -- + ^- is positive or nega-
tive. Cardan's method or the Trigonometrical wall be practi-
cable. In the former case, there will be one real ami two
hnaginary roots ; in the latter case, three real roots.
BIQUADRATIC EQUATIONS.
492 Descartes' Solution. — To solve the equation
.v' + qA' + r.v -\- s = (i.)
the term in .r' having been removed by the method of ('t29).
Assume (.t.-+(u'+/) {.i--i\v-\-^-) = (ii.)
Multiply out, and equate coefficients witli (i.) ; and t1ie fol-
lowing equations for determining /", g, and e are obtained
ir+/='/ + ^'% .ir- /=A Kf=''*' i'''')
493 ^.«4.2rye'+(r/-4v)e2-j- = (iv.)
494 The cubic in t? is reducible by Cardans method, when the biquadratic
hits two real and two imaginary roots. For proof, take « ± //5 and — a ± y as
the roots of (i.), since their sum mu.st bo zero. Form the sum of eucb pair
for tlio values of e [see (ii.)]. and "PP'Y t^° ^^^^^^ ^" ('i88) to the cubic in e*.
1/ the biquadratic has all its roots real, or all imaginary, the cubic will liave
all its roots real. Take n ± //3 and —a ± iy for four imaginary roots of (i.),
and form the values of e as before.
495 If' «'> f^f y' ^e the roots of tfie cubic in e', tlie roots of tlie biquadratic
will he _i(„+/5 + y), l(a+/3-y), 4(/9 + y-n), ^ (y + a-/3).
130 THEORY OF EQUATIONS.
For proof, take w, x, y, z for the roots of the biquadratic; then, by (ii.), the
sum of each pair must give a value of e. Heuce, we have only to solve the
symmetrical equations
1/ + 2 = CI, w + ,7;=— a,
Z -I- iC = /3, ?o + ;/ = — /j,
a- + ?/ = y, w-\-z = —y-
496 Ferraris solution. — To the left member of the equation
x^+lKv^-\-qx^-\-ra:-\-s = 0,
add the quantity ax^ + bx + —, and assume the result
= (,.+|,+,„y.
497 Expanding and equating coefficients, the following
cubic equation for determining m is obtained
8m^—4iqm^-\-{2j)r—8s)m-\-4!qs—2^^s — r = i).
Then x is given by the two quadratics
2 , » , , 2(Lv-{-b
^ + 2 '*' + '*' = ± -TvTT
498 The cubic in m is reducible by Cardan'' s method ivhen the biquadratic
has two real and tioo imaginary roots. Assume a, /3, y, B for the roots of the
biquadratic ; then aft and yB are the respective products of roots of the two
quadratics above. From this find m in terms of aftyS.
499 Elders solution. — Remove the term in x^; then we
have .V'*' + q.v"^ -\-r.v-\-s = i).
500 Assume x = i/-\-z-\-u, and it may be shewn that i/, z^^
and u^ are the roots of the equation
fA.±f-4-'^"'~^'t-^=0
^ 2 ^ ^ 10 ^ (>i
501 The six values of y, z^ and //, thence obtained, are
restricted by the relation yzii. =
Thus X = //-!-;.■ + /< will take four different values.
COMMENSURABLE ROOTS. 131
COMMENSURABLE ROOTS.
502 '1'" t'""l tlio commonsuraUlo roots of an ofiiiation.
First transform it by pultinjr •'' = 'ir "^^^ ^^^^ equation of
tlic form .r"+/>i.r"-'+;>.>.r"--+ ... +/>„ = 0,
liaving j>o = 1» iiii^l tlio remaining coefHcients integers. (l-Gl)
503 This ecpiation cannot have a rational fractional root,
and tlie inte<]:ral roots may be found Ijy Newton's method of
Divisors (451)).
These roots, divided each by h, will furnish the commen-
surable roots of the original equation.
504 Example. — To find the comrnensurablo roots of tlie equation
8 1 /' - 20 7x' - 9x» + 89.r + 2,r - 8 = 0.
Dividing bj 81, and proceeding as in (431), \vc find the requisite substitu-
tion to bo a; = -^.
The transformed equation is
,/_23/-V + 801)/- + lG2//-5832 = 0.
The roots all lie between 24 and —34, .by (451).
The method of divisors gives the integral roots
G, —4. and 3.
Therefore, dividing each by 9, we find the commensurable roots of the original
equation to be 3, — f , and |.
505 To obtain the remaining roots ; diminish the transformed equation by
the roots G, —4, and 3, in the following manner (see 427) : —
1_23- 9 + 801 + 1G2- 5832
6— 102— GGG + 810 + 5832
6
-4
1-17-111 + 135 + 972
- 4+ 84 + 108-972
-21- 27 + 243
3_ 54-243
1_18- 81
The depressed equation is therefore
if - 18(/ - 81 = 0.
The roots of which arc (1+ \/2) and 9 (1— v/2) ; and, consequently, the
incommensuiable roots of the proposed equation are 1+ V'l and 1— v/2.
132 THEORY OF EQUATIONS.
INCOMMENSURABLE ROOTS.
506 Sturm's Theorem. — lff(x), freed from equal roots, bo
divided by /('>')> and the last divisor by the last remainder,
changing the sign of each remainder before dividing by it,
until a remainder independent of x is obtained, or else a re-
mainder which cannot change its sign; then /(a^), /'('<-')' ^^^
the successive remainders constitute Sturm's functions, and
are denoted by f{a-) , / (<r) , f^ {.v) , &c /« Gv) •
The operation may be exhibited as follows : —
M^r) = q,f,{.v)-f,{.v),
507 Note. — Any constant factor of a remainder may.be
rejected, and the quotient may be set down for the corres-
ponding function.
508 An inspection of the foregoing equations shews —
(1) That /„, (ft') cannot be zero; for, if it were, /'('^O ^^^
/i (x) would have a common factor, and therefore /(<^') would
have equal roots, by (432).
(2) Two consecutive functions, after the first, cannot
vanish together ; for this would make/*„ (x) zero.
(;3) AVhen any function, after the first, vanishes, the two
adjacent ones have contrary signs.
509 If, as X increases, f(x) passes through the value zero,
StnrrrC s Junctions lose one change of sign.
For, before ./'(-O tf^kes the value zero, /(a-) and/, {x) have contrary signs,
and afterwards tliey have the same sign; as may be shewn by making h
small, and changing its sign in the expansion off{x + li), by (420).
510 If d-nij other of Sturm's functions vanishes, there is
neither loss nor gain in the number of changes of sign.
This will appear on inspecting the equations.
511 Ri:sui;r. — The nnmhvr of roots off(,v) hrtu'cen a and h is
equal to the difference in the number of cluoiges of sign in
Sturm's functions, when x = a and when x = b.
INCOMMEN suit ABLE ROOTS.
133
512 <'"";• — Tlio total iiiiin])ci- ui" roots of /■(,*■) will he foiiiid
by taking a = -\- cc mid h = — cc ; tho aU^n of each fuiictioii
will then bo the same as that of its first term.
\VTien tho number of functions exceeds the degree of f{x)
by unity, the two following theorems hold : —
513 If thr jir^t trrins hi all flii'fdnrtionii, aj'trr t lie first ^ are
/losifirc ; all flu' roots off{j) arc real.
514 If the first terms are not all positive ; then, for every
r/iiiiK/c of si(j)i, there will be a pair of imaginart/ roots.
For the proof put .r = + co and — oo, and examino the number of
changes of sign in each case, applying Descartes' rule. (-tl6).
515 If ^ (•'') ^^^ no factor in common with /(<>'), and if <p (x)
and _/"(,/') take the same sign when /(,<■) = 0; then the rest of
Sturm's functions may be found from f{.f) and <p (</;), instead
of /'(./■). For the reasoning in (509) and (510) will ajjply to
the new functions.
516 If Sturm's functions be formed without first removing
equal roots from /(,/'), the theorem aWII still give the number
of distinct roots, without repetitions, between assigned limits.
For if/(.r) and /, (x) be divided by their highest common factor (see 444),
and if the quotients be used instead of /(./■) and/, (.r) to form Sturm's func-
tions ; then, by (olo), the theorem will ajjply to tho new set of functions,
which will dilfer only from those formed froin/(,c) and/, (./.) by the absence
of the same factor in every term of the series.
517 Example. — To find the position of the roots of the equation
x'-4x^ + x- + Gx + 2 = 0.
Sturai's functions, formed according to f(x)=x*—4a^+ x^+ Gx+ 2
the rule given above, are here calculated.
The first terms of the functions are all
positive ; therefore there is no imaginary
root.
The changes of
sign in the func-
tions, as X passes
through integral
values, are exhi-
bited in the adjoin-
ing table. There
are two changes of
sign lost while x
passes from — 1 to
0, and two more
lost while X passes
from 2 to 3. There
A(^') =
2x»— G.c*+ x+ 3
A(^) =
5x--l(Jx- 7
Ai-^) =
X- 1
Ai^) =
12
x =
-2
-1
1
2
3
4
f(x) =
/,(.«) =
A(^) =
A(^) =
1
+
+
+
+
+
+
+
+
+
+
+
+
+
4-
+
+
+
+
+
+
+
No. of changes )
of sign )
4
4
2
2
2
134 THEOEY OF EQUATIONS.
are therefore two roots lying between and — \ ; and two roots also between
2 and 3.
These roots are all incommensurable, by (503).
518 Fourier's Theorem. — Fourier's functions are the fol-
lowing quantities f{x), f'{.v), f'i^v) /"(cv).
519 Properties of Fourier's functions. — As x increases,
Fourier's functions lose one change of sign for each root of
the equation /(.f) = 0, through which x passes, and r changes
of sign for r repeated roots.
520 If any of the other functions vanish, an even number
of changes of sign is lost.
521 Results. — The 7iumher of real roots of f{x) hetween a
and j3 cannot he more than the difference hetween the numher
of changes of sign in Fourier's functions when x = a, and the
numher of changes ivhen x = j3.
522 When that difference is odd, the number of intermediate
roots is odd, and therefore one at least.
523 When the same difference is even, the number of inter-
mediate roots is either even or zero.
524 Descartes' rule of signs follows from the above for the
signs of Fourier's functions, when x = are the signs of the
terms in /(«); and when a3 = oo, Fourier's functions are all
positive.
525 Lagrange's method of approA'wiating to the incom-
mensiirahle roots of an equation.
Let a be tlie greatest integer less than an incommen-
surable root oif{x). Diminish the roots of /(,r) by a. Take
the reciprocal of the resulting equation. Let h be the greatest
integer less than a positive root of this equation. Diminish
the roots of this equation by h, and proceed as before.
526 Let a, h, e, &c. be the quantities thus determined ; then,
an approximation to the incommensurable root oif{x) will be
the continued fraction x = a + -— -,-
h-\- f-l-
INCOMMENSURABLE ROOTS.
]3i
527 AV/c/o/<'.v mcfluK/ <tf(ii>pr(uinnfti<ni.- If r, hv a ((uaiitity
a little less than one o£ the roots of the eciuatioii fU) = 0, so
that /(r, + //) = 0; tlieu c^ is a first approximation to the
value of tlie root. Also because
/(,-, + /,) =/(,.,) + /,/■{,-,) +^'/"(.-,) + &c (KG),
and h is but sniall, a second approximation to the root will be
In tlie same way a third ap]:»roximation may be obtained from
Co, and so on.
528 Fourier's limitation of Newton s method. — To ensure
that Ti, ^2, Cg, &c. shall successively increase up to the value
Ti + h without passing beyond it, it is necessary for all values
of X between c^ and Ci-\-h.
(i.) Thatf{;ii) andf'{,v) should have contrary signs.
(ii.) Thatflx) and f" {x) should have the same sign.
Fia. L
Fia. 2.
A proof may be obtained from the figure. Draw the curve
y=f(^x). Let OX be a root of the equation, and ON = Ci;
draw the successive ordinates and tangents NPy PQ, QBy &c.
Then OQ = c^, OS = Cg, and so on.
Fig. (2) represents Tg > OX, and the subsequent ai)proxi-
mations decreasinof towards the root.
530 Newton s Rule for Limits of the Roots.— hot the co-
ethcients of /(./) be respectively 'divided by the Binomial
coefficients, and let ^o, (T,, a, ... o„ be the quotients, so that
f{x) = ao-'j" + uai,/;"-
+ ''t^"''
+ ... + » ",.-!»;+«
13G TUEOUY OF EQUATIONS.
Lot ^Ij, ^lo, ^Ig . . . An be formed by tlic law A^ = a^— ri^_i(X^+i.
Write the first series of quantities over the second, in the fol-
loA\dng manner : —
«0, rti, ('2, ((3 (^n-l, ««,
ylo, Ai, ^2, ylg .-l,,_i, A,,.
"Whenever two adjacent terms in the first series have the
same sign, and the two corresponding terms below them in
the second series also the same sign; let this be called a
double 2Jermanence. AVhen two adjacent terms above have
different signs, and the two below the same sign, let this be
known as a variation-permanence.
531 Rule. — The number of double 'permanences in the asso-
ciated series is a superior limit to the number of negative roots
The number of variation-permanences is a superior limit to
the number of positive roots.
The number of imaginary roots cannot be less than the
number of variations of sign in the second series.-
532 Sylvesters Theorem. — Let /(f + X) be expanded by
(426) in powers of x, and let the two series be formed as in
Newton's Rule (530).
Let P (X) denote the number of double permanences.
Then P (X) ~ P [fi) is either equal to the number of roots
of /(ft'), or surpasses that number by an even integer.
Note. — The first series may be multiplied by [n_, and will
then stand thus,
/«(X), /-^X), [2/«-2(X), [l/"-^(^)-h/W-
The second series may be reduced to
On{^), r/„_,(x), r/„_,(x)...rr(x),
where G, (X) = {/'• (X)}^ - '^y^^ f-' (A) f^' (X).
533 Horner's Method. — To find the numerical values of the
roots of an equation. Take, for example, the ecpiation
x'-4t^ + x^-\-6x-\-2 = 0,
and find limits of the roots by Sturm's Method or otherwise.
INCOMMENSUnAIiLi: HOOTS.
137
It has been shewn
in (ol7) that this eqnation has two
incominensiirablo roots
between 2 and
:?. Th(«
|)rocess of
calciil;
itiiig the least of these roots is here
exhibited.
-4
+ 1
+ 6
+ 2(2-414213
2
-4
—6
_2
-\i
A
20000
2
-6
-19584
-3
i?, — 60U0
A,
4160000
2
4
6'/ 100
1104
-4896
^U
-2955839
o
1204161UUU0
2
176
276
1872
^4
-11437245184
P, 40
B^ — 30240U0
604364816
4
192
68161
-566003348
44
468
-2955839
A
38361468
4
208
68723
-28285470
48
C'j 676UO
n, -2887116000
A
10075998
4
561
27804704
- 8485368
52
6bl(;i
-2859311296
A
1590630
4
562
27895072
7)., oGO
68723
B, -2831416224
1
563
139948
282843)1590630(562372
6G1
Cg 6i>2H600
— 283U01674
1414215
1
22576
139970
28284)176415
562
6951176
22592
B^ -282861704
700
2828)"
169706
1
6709
5G3
6973768
- 28285470
5657
1
22608
700
282) 1052
D3 5040
C, 0996376
B, -28284770
848
4
11
21
28) 204
5044
69974
-2828456
197
4
11
21
2) 7
5648
69985
B, -2828435
5
4
11
5652
C, 69996
4
CL 7
P, 5,656
Root =
: 2-414213562372.
METnOD.— Diniinisli tlie roots by 2 in tlic manner of (427).
The resulting coefficients are indicated by .4,, J?„ C\, 7>,.
By Newton's rule (527), - {^r'A 5 ^^^^^ ^^' ~ V.' ** "" approximation to
the romaininfT pait of llic root. Tliis gives '3 for the next figure ; '4 will bo
found to be the eorrect one. The highest figure must bo taken which will
not change the sign of ^1.
Diminish the roots by -4. This is accomplished most easily by affixing
ciphers to J„ i^„ C'„ I\, in the manner shewn, and then employing 4 instead
of -4.
Having obtained ^1.,, and ob.scrving that its sign is +, retrace the steps,
T
138 THEORY OF EQUATIONS.
ti'ying 5 instead of 4. This gives A., with a minus sign, thereby pi-oving the
existence of a root between 24 and 2-5. The new coetficieuts are A.^, B.^, C.,, D.^.
— Y? gives 1 for the next figure of the root.
Affix ciphers as before, and diminish the roots by 1, distinguishing the
new coefficients as A^, B^, C^, D^.
Note that at every stage of the work A and B must preserve their signs
unchanged. If a change of sign takes place it shews that too hirge a figure
has been tried.
To abridge the calculation pi'ocecd thus : — After a certain number of
figures of the root have been obtained (in this example four), instead of
adding ciphers cut ott" one digit from B^, two from C'^, and three from D^.
This amounts to the same thing as adding the ciphers, and then dividing
each number by 10000.
Continue the work with the numbers so reduced, and cut off digits in like
manner at each stage until the D and C columns have disappeared.
Aj and Bj now alone remain, and six additional figures of the root are
determined correctly by the division of A^ by By.
To find the other root which lies between 2 and 3, we proceed as follows : —
After diminishing the roots by 2, try G for the next figure. This gives Jj
negative; 7 does the same, but 8 makes J., positive. That is to say, f{2'7)
is negative, and/ (2'8) positive. Therefore a root exists between 2*7 and
2-8, and its value may be approximated to, in the manner shewn.
Throughout this last calculation A will preserve the negative sign.
Observe also that the trial number for the next figure of the root given at
f(c)
each stagfe of the process by the formula — , {, will in this case be always
^ I y ^ /(c)' ^
too great, as in the former case it was always too small.
SYMMETEICAL FUNCTIONS OF THE HOOTS OF
AN EQUATION.
Notation. — Let a, h, c ... bo the roots of tlie equation
/(••'•) = 0.
Let .s,„ denote a'" + //" + . . . , tlic sum of tlie 7/^^'' powers of
the roots.
Let .s,„,p denote a"'h^' + lr(iP-\-a'"<'''-{- ... througli all the
permutations of the roots, two at a time.
Similarly let .9,„,p ,, denote (rh'\'''-\-a'"h^\V'-\- ... , taking all
the permutations of the roots three at a time ; and so ou.
SYM}n:rRICAL FUNCTTONS of TIIF h'OOTS. 13!)
534 SUM>^ OF nil': /vmi7;a'.s' of tuf norrrs.
wliere m is less tliau n, the degree oiJ'{.i).
Obtained by expanding by division each term in the vahio of/'(.i) given
at (432), arranging tlie whole in powers of .r, and equating coeiricieiits in llie
result and in the value ofy^^r), found by differentiation as in (1-21).
535 If "i ^0 greater than //, tlie forimihi will be
Obtained by multii)lying /(./•) = by .'""", substituting for .i; the roots
a, h, c, &c. in succession, and adding the results.
By these formula? .<„ s.,, .«„ &c. may be calculated successively.
536 To find the sum of the negative powers of the roots, put
m equal to n—1, )i — 2, ?^ — 3, &c. successively in (535), in
order to obtain s_i, s_o, s_3, &c.
537 To calculate .'?,. independently.
Rule : s,. = — r X rorjjicieut of x~'' in the e:q)amlo)i of
^'^"J n i'i '^''•^''''ii'^''>t'J I^OlV^^iS of .V.
Proved by taking f(x) = {x~ a) (x-h)(x — c) ... , dividing by x", and
expanding the logarithm of the right side of the equation by (loO).
538 SYMMETniCAL FUNCTIONS WTUGU AJ^E
NOT POWERS OF THE BOOTS.
These are expressed in terms of the sums of ])o\vers of
the roots as under, and thence, l)y (531), in terms of the routs
explicitly,
539 'V„,,^,.,, = .V,,,*,,*,/ — .V„, + /, A-,, — *•„, + ,,*,, — A', ,+ ,.?,„ +2.V„, + ^ + , ,.
The last equation may be ]iroved by mnlti})lying .<„,.,, by
."f, ; and expansions of other symmetiical functions may be
obtained in a similar way.
540 If </> ('•) be a rational integi'al function of ,r, then the
symmetrical function of the roots of ./'('), denoted by
140 TEE OUT OF EQUATIONS.
^ (a)-{-(j> (h) -{-<!> ((■) + &c., is equal to tlie coefficient of x''''^ in
the remainder obtained by dividing <{> 0*') /'('') ^7 /('*') •
Proved by multiplying the equation (432) by yji, and by tlieorera (401).
541 To find tlie equation wliose roots are tlie squares of
tlie differences of tlie roots of a given equation.
Let F{^) be the given equation, and S,. the sum of the r*''
powers of its roots. Let/(/c) and s^ have the same meaning
with regard to the required equation.
The coefficients of the required equation can be calculated
from those of the given one as follows : —
The coefficients of each equation may he connected tvith the
sums of the ])oivers of its roots hy (534) ; and the sums of the
poivers of the roots of the tivo equations are connected hy the
formula
542 2^, = nS,,,-2rSA._,-\- ^''('^''-^) S,S,,_,- ... +«S,,
Rule. — 2s,. is equal to the formal expansion of {S—Sf'' hy
the Binomial Theorem, with the first and last terms each mul-
tiplied hy n, and the indices all changed to siffixes. As the
equi-distant terms are equal we can divide by 2, and take half
the series.
Demonstration. — Let a, h, c ... be tbe roots of i^(a;).
Let <l>(x) = (x-ay-' + (x-hy+ (i-)
Expand each term on the right by the Bin. Theor., and add, substituting
Si, S2, &c. In the result change x into a, b, c ... successively, and add the n
equations to obtain the formula, observing tliat, by (i.),
(a) +0 (&) + ... =2.v
If r? be tlic degree of 7^ (,*•), then 'k)i(ii — l) is the degree
of /(.'). % (UG).
543 The last term of the equation 7" ('') = is equal to
n^T{a)FmF{y)...
where a, ft, y, ... are the roots of F' {•>'). Proved by shewing
that F'{a)F'{h) ... = n"F{a)F(ft) ...
544 If -^(''O ^^^s negative or imaginary roots, f(.i) must
have imaginary roots.
SYMMETRICAL FUNCTIONS OF THE ROOTS. 141
545 The sum of tlic v«"' powers of tlie roots of the (|u:i(l-
ratic ecjuation .*-— />.r+7 = 0.
,„ ., , in {ni — .*») ,„_t -
.v„, = ;>'" — ni]>"' -(/ + -j p (/-...
By (io") expaudiug the logarithm by (15G).
546 'I'he sum of tlie m^^' powers of the roots of </y* — 1 =
is n if ni be a multiple of )i, and zero if it be not.
By (537) ; expanding the logarithm by (15G).
547 If <!>{.,•) = a,-\-a,x-\-n,x- + &c (i.):
then the sum of the selected terms
^-illbe .v= — ;a"-"'<^(a.r)+y8"-"'<^()8./0+y"""'*(r'O + ^^'C.}
where a, /3, y, &c. are the n^^' roots of unity.
For proof, multiply (i.) by a""'", and change x into a.v; so with /?, y, &c.
and add the resulting equations.
548 To approximate to the root of an equation by means of
the sums of the powers of the roots.
By taking m large enough, the fraction tuill ^vill approx-
imate to the value of the numerically greatest root, unless
there be a modulus of imaginary roots greater than any real
root, in which case the fraction has no Umiting value.
549 Similarly tlie fraction •^''"•^'"^•-~'^'"":.' api)roximates, as m
increases, to the grrnfr.^f product of any pair of roots, real or
imnginary ; excepting in the case in which the product of tho
pair of imaginary roots, though less tlian the product of tho
two real roots, is greater than the scpiare of the least of tliem,
for then the fraction has no limiting value.
142 THEORY OF EQUATIONS.
550 Similarly tlie fraction ' '" '"'^^ '"'^y '""^^ approximates,
^'m'^'m + 2 ^m + 1
as m increases, to tlie sum of tlie two numerically greatest
roots, or to the sum of the two imaginary roots with the
greatest modulus.
EXPANSION OF AN IMPLICIT FUNCTION OF .r.
Let r{A,v''+)+!f{By-]-)^...-\-fiS.r^+) = (1)
be an equation arranged in descending powers of y, the co-
efficients being functions of x, the highest powers only of x
in each coefficient being written.
It is required to obtain y in a series of descending powers
of X.
First form the fractions
a — b a — c a — d n — s ,c)\
a — y8' a — y' a — S a — a
Let — — - t be the greatest of these algebraicalh^, or
a — n
if several are equal and greater than the rest, let it be the
last of such. Then, with tlie letters corresponding to these
equal and greatest fractions, form the equation
Au''-^ -^Ku'^i) (3).
Each value of ?/ in this equation corresponds to a value of v/,
commencing with ux^.
Next select the greatest of the fractions
/.• — / Ix—m I'—s /jx
r» — v^V •
K — A K — /x K — cr
/. _ ^,,
Let — -— ' := t' be the last of the greatest ones. Form
K — V
the corresponding equation Kh"-\- ...-\-Nh'' = (I (5).
Tlien each value of u in this equation gives a cori'cspondiiig
value of y, commencing with »./.
EXPANSION OF AN IMI'LICIT Fi'M'TIDN. 143
Proceed in tliis way until llie last tVactiun of tlio series (2)
is reached.
To obtain the second term in the exjjansion of //, put
,/ = .,.'(//+//,) Ill (1) 03),
cmplojnng tlie dilTerent vahies of n, and again of /' and //, /"
and ti, &c. in succession; and in each case this substitution
will produce an equation in // and x similar to the original
equation in //.
Eepeat the foregoing process ^vith the new eqnation in y,
observing the following additional rule : —
Wien all the values of t, t\ t", ^c. have heeu ohtained, the
negative ones only must be employed informing the equations
in n. (7).
552 To obtain y in a series of ascending powers of ■/■.
Arrange equation (1) so that a, /3, y, &c. may be in as-
ccndliKj order of magnitude, and a, h, c, &c. the lowest powers
of .t' in the respective coefficients.
Select f, the greatest of the flections in (2), and proceed
exactly as before, with the one exception of substituting the
vford jmsitive for negative in (7).
553 Example. — Take the equation
{x^ + x') + ( 3.^^ - hx^) 2/ + ( - 4.7; + 7x- + a;") >f - >/ = 0.
It is required to expand y in ascending powers of x.
The fractions (2) are - -'^. "qZo' ~ ^E ^ ' ^^' ^' ^' ^"^^ ^'
The first two being equal and gi-catest, we have t = 1.
The fractions (4) reduce to — - — ^ = i = ' •
Eqnation (3) is \+3n- 4ir = 0,
which gives ?/ = 1 and — ^, with / = 1.
Equation (5) is —4ir—u'' = 0,
and from this u = and —4*, with t' = ^.
"NVe have now to sub.stitute for ij, according to (G), cither
^(1 + y,), a:(-i + J/,), .tV, or a;*(-4^ + i/,).
Tut y = x (1 + 7/,), the tir.st of tlieso values, in the originul equation, and
arrange in ascending powers of y, thus
-4x* + (-rjx'-{-)y,+ (-4.c' + )yl - 10x'y\ - bx'y\ - .c'y] = 0.
The lowest power only of x in each cocflicient is here written.
144
THEORY OF EQUATIONS.
4-5
0-4'
5'
The fractions (2) now become
4-3 4-3 _4-5
0-1' 0-2' 0-3'
1, h -h -h -h
From tlieso /= 1, and equation (3) becomes
— 4— 5?t = 0; .•. « = — 4..
Hence one of the values of y■^ is, as in (G), yi = x (— f + 2/2)-
Therefore y = x {1 + «; (-f + 2/2)} = aJ-f'»'+ ...
Thus the first two terms of one of the expansions have been obtained.
DETERMINANTS.
554 Definitions. — The determinant
b, hi
is equivalent
to a^h-i—dtbi-, ^Tifl is called a determinant of the second order.
A determinant of the third order is
a.^ a^ = rti {Ihc^ — 63^2) + a. %c^ — b^c^) + a.^ {b,c., - b.,c^) .
b, b^ 63
Ci C.2 C;
Another notation is 2 ± (lyh.^c^, or simply {a^h.yC-^.
The letters are named constituents, and the terms are
called elements. The determinant is composed of all the
elements obtained by permutations of the suffixes 1, 2, 3.
The coefficients of the constituents are determinants of
the next lower order, and are termed minors of the original
determinant. Thus, the first determinant above is the minor
of ^3 in the second determinant. It is denoted by C^. So the
minor of % is denoted by vli, and so on.
555 A determinant of the 7^*^ order may be wintten in either
of the forms below
b, b.
,. «,. ... «„
.. 6, ... b.
... «„,
... a.,.
In the latter, or double suffix notation, the first suffix indicates
the row, aud the second the column. The former notation
will be adopted in these pages.
DETERMTXAXTS. ^^■
ft I a., «!
h, h., b.
A Compoiiift' ilotrrminant is one in which the
Tuimber of cohimns exceeds the innnber of rows,
and it is wntten as in the annexed example.
Its vahie is the sum of all the determinants obtained \)\ takiiif^
a number of rows in every possible way.
A Simple ih'tormlnnnt lias single terms for its constituents.
A Compound drtrrmuinut has more than one term in somo
or all of its constituents. See (')70) for an exam[)le.
For th(^ definitions of Si/mmetrical, Rcripronil, Parfial,
and Complcmciitari/ determinants; see (574), (575), and (576).
General Theory.
556 The number of constituents is n^.
Tlie number of elements in the complete determinant is [?^^.
557 The first or leading element is aih.,<\ ... /„. Any
element may be derived from the first by permutation of the
suffixes.
The sign of an element is + or — according as it has
been obtained from the diagonal element by an even or odd
number of permutations of the suffixes.
Hence the following rule for determining the sign of an
element.
RuT.E. — Take the suffixes in order, and put them hach to
their places in the first element. Let m he the whole number
of places passed over ; then ( — 1)"* "'^7/ give the sign required.
Ex. — To find the sign of the element n^h^CrJ\c.^ of tlio detei-minanfc
«4 h ^s 'h '^i
Move the .suffix 1, tlircc pliiccs 1 4 3 ''> 2
2, three places 1 2 4 3 ')
„ „ 3, one phwje 1 2 3 4 5
In all, seven places; therefore (—1)' = —1 gives the sign required.
558 n two suffixes in any element be transi)o.^ed, the sign
of the element is changed.
Half of the elements are plus, and half are minus.
559 The elements are not altered ])y changing the rows into
columns.
If two rows or columns are transposed, the sign of tho
U
146 TUEOUY OF EQUATIONS.
determinant is clianged. Because each element changes its
sign.
If two rows or columns are identical, the determinant
vanishes.
560 If all the constituents but one in a row or column
vanish, the determinant becomes the product of that con-
stituent and a determinant of the next lower order.
561 A cyclical interchange is effected by n — \ successive
transpositions of adjacent rows or columns, until the top row
has been brought to the bottom, or the left column to the
right side. Hence
A cyclical interchange changes the sign of a determinant
of an even order only.
The r*^ row may be brought to the top by r— 1 cyclical
interchanges.
562 If each constituent in a row or column be multiplied
by the same factor, the determinant becomes multiphed by it.
If each constituent of a row or column is the sum of m
terms, the compound determinant becomes the sum of m
simple determinants of the same order.
Also, if every constituent of the determinant consists of
m terms, the compound determinant is resolvable into the sum
of m^ simple determinants.
563 To express the minor of the r"' row and ¥^ column as
a determinant of the n — V^ order.
I*ut all the constituents in the r^^ row and /.-^^ column equal
to 0, and then make r— 1 cyclical interchanges in the rows
and L — 1 in the columns, and multiply by (_iy'-+^)(«-i).
r._. _ f_Y\(r-\*k-\){n-l)^
564 To express a determinant as a deter-
minant of a higher order.
Continue the diagonal with constituents
of " ones," and fill up with zeros on one side,
and with any quantities whatever (o, /3, y, &c.)
on the other.
1
a 1 {)
fi e a h <>'
V C h h f
h r) - /• c
vi:TEnMTy.iNTS.
U7
565 Tlio sura of tbe products of each constituent of a
column by tlio corresponding minor in another given cohimn
is zero. And the same is true if we read ' row' instead of
* cohimn.' Thus, referring to the determinant in (555),
Taking tlie /)''' and r/^' cohimns, Taking tho a and r rows,
For in each case we have a dotcnuinant with two columns identical.
566 I" i^ny row^ or column tlie sum of the pnxhicts of each
constituent by its minor is the determinant itself. That is,
Taking the j)^^ cohimn, Or taking the c row.
567 The hist equation may be expressed by
Also, if i'lpCq) express the determinant
' ^- r = A.
p p
a„ a
then
'2{apr,^) will represent the sum of all tlu^ determinants of the
second order wdiich can be formed by taking any two columns
out of the a and r rows. The minor of {dp, (\) may be wi'itten
(Ap, Cy, and signifies the determinant obtained by suppress-
ing the two rows and two columns of Op and c,. Thus
A = S {dp, Cg) {Ap, Cq). And a similar notation when three or
more rows and columns are selected. ^
568 Analyji'is of a determinant.
Rule. — To resolve into its elements a determinant
n*^ order. Express it as the sum of n determinants
(n—iy'' order by (5G0), and repeat the process with e
the new determinants.
EXAMl'LE :
«l «J «3 "i
c, q r, c^
«/, (/j d^ ll^
Again,
of thn
of the
ach oj
= a, b, b, b,
-a.
63 b^ 6,
+ ^.
b, b, 6,
-«4
b, b, b.
<'i <^i ''a
c, c, c,
r^ c, c,
d, d, d.
d, d, d,
dt (/, d.
./, c/, d.
= b, I c, r, I + b,
+ b.
i d, d, I
^ b, b, . .
c, C, C, I l/j t/j I 1 c/j ./,
di rfj d^
and so on. In the first scries the determinants have alternately ji
minus signs, by the rule for cyclical interchanges (501), tlie order bei
lus and
ngeven.
148 TUEOUY OF EQUATIONS.
569 Si/nthesh of a determinant.
The process is facilitated by making use of two evident
rules. Those constituents which belong to the row and
column of a given constituent a, will be designated *' a's con-
stituents." Also, two pairs of constituents such as a^, c, and
dg, c^, forming the corners of a rectangle, will be said to be
*' conjugate" to each other.
Rule I. — No constituent loill he found in the same term
with one of its oion constituents.
Rule II. — The conjugates of any two constituents a and b
will he cuinnion to a's and Us constituents.
Ex. — To write the following terms in the form of a determinant :
uhcd + Ifyl +fh^ + ledf+ cghp + 1 ahr + elpr
—fhpr—ahlr—ach^ — Ifhg — bdf^ — efhl — cedp.
The determinant will be of the fourth order ; and since every term must
contain four constituents, the constituent 1 is supplied to make up the
number in some of the terms. Select any term, as ahcd, for the leading
diagonal.
Kow apply Rule I.,
a ia not found with e,f,f, g,p,0...(l)- c is not found with /,/, I, r, 1, 0... (^).
1) is not found with e, h, h,p,l,0 ..(2). dis not found with g, h,h,l,r,0...(-i).
Each constituent has 2 (»— 1), that is, 6 constituents belonging to it, since
n =?'4. Assuming, therefore, that the above letters are the constituents of
a, b, c, and d, and that there are no more, we supply a sixth zero constituent
in each case.
Now apply Rule II. — The constituents common
to a and b are e, p ; to a and c — -/, /; to b and c — 1, ;
to a and d — g, ; to h and d — h, li, ; to c and d-
The determinant may now be formed. Tlie diagonal
being abed,; place e, p, the eoiijugatus of a and b, either as
in the diagram or transposed.
Then /and/, the conjugates of a and c, may be written.
1 and 0, the conjugates of b and c, must be placed as indicated, becau.so
1 is one of y/s constituents, sini-e it is nt)b found in any term with /i, and
must therefore be in the second row.
Similarly tlie places of y and 0, and of Z and r, are assigned.
In the c-ase of b and d we have h, h, from wliieh to clioose the two
conjugates, but, we see that is not one of them because that would assigu
two zero constituents to b, whereas b has but one, which is already placed.
By similar reasoning the ambiguity in selecting the conjugates I, r is
removed.
The foregoing method is rigid in the case of a complete determinant
/, /•, 0.
a e
f
7
l> b
1
h
/' U
(■
r
U It
/
d
DETEKMJXAXTS.
1 to
having different constituents. It becomes uncertain when the zero con-
Ktituonts increase in number, and when several coustitueuts are identical.
Ihit even then, in tlie majority of cases, it will soon afford a cluo to the
required arrangement.
570 I'RODUCr OF TWO DETERMINANTS OF
Tin: „"• (lUDEU.
C)
</, (I., ... a,
h, h, ... h,
L /..
{Q)
a, a> ... a„
X, X,
.U A, ... J„
/;, li ii,,
L, L, ... L,
. Ai = ay ai-\-(L,a.,-\- .
Ly r= aiXi4-^^X.+ ... +(?,.X,
Tlie values of A^, By ... L, in
tUo first column of S arc an-
nexed. For the second column
write //s in the place of as.
For the tliird columu write f's,
and so on.
For proof substitute the values of Ay, By, &c. in the determinant S, and
then resolve ,b' into the sum of a number of determinants by (oC>2), and note
the determinants which vanish through having identical columns.
Rule. — To form the determinant S, which is the product of
two determinants P and Q. First connect hij plus siijus the
constituents in the ro2VS of both the determinants F and Q.
Nou) place the first row of F upon each row of Q in turn,
and let each two constituents as thnj touch become jn-oducts.
Tliis is the first column of S.
Perform the same operation upon Q with the second rou- if
P to obtain the second column of S ; and again with the third
row of F to obtain the third column of S, and so on.
571 If the number of columns, both in F and Q., be n, and
the number of rows r, and if n be > r, then the determinant
S, found in tlie snme way from F and Q, is equal to the sum
of the C{ii, r) products of prirs of determinants obtained by
taking any r columns out of F, and the corresponding r
columns out of Q.
But if n be < r the determinant S vanishes.
For in that case, in every one of th
be two columns idcutical.
component determinants, there will
150 THEORY OF EQUATIONS.
572 The product of tlie determinants P and Q may be
formed in four ways by changing the rows into columns in
either or both P and Q.
573 Let the following system of n equations in XicV,.^ ... x^
be transformed by substituting the accompanying values of
the variables,
The ehminant of the resulting equations in ^^ ^., • • • ^» is
the determinant S in (570), and is therefore equal to the
product of the determinants P and Q. The determinant Q is
then termed the modulus of transformation.
574 A Symmetrical determinant is symmetrical about the
leading diagonal. If the E's form the r^'' row, and the K's
the k^^ row ; then B;, = K^ throughout a symmetrical deter-
minant.
The square of a determinant is a symmetrical determinant.
575 ^ Reciprocal determinant has for its constituents the
fii'st minors of the original determinant, and is equal to its
?i — l'^ power; that is,
A, ... A,
h ... 4
Proof. — ^Multiply both sides
of the equation by the original
determinant (o55j. The con-
stituents on the left side all
vanish excepting the diagonal
of A's.
576 Partial and Complementary determinants.
If r rows and the same number of columns be selected
fi'om a determinant, and if the rows be brouglit to the top,
and the columns to the left side, without changing their order,
then the elements common to the selected rows and columns
form a Partial determinant of the order r, and the elements
7wt found in any of those rows and columns form the Com-
plementary determinant, its order being n — r.
DETERMINANTS.
151
Ex.— Let the selected rows from the dctcrmiiiaTit («,''3r,(/^^J be tlio
Becond, third, and fifth ; and the selected columns bo the third, fourth, aud
fifth. The orijjinal and the transformed determinants will be
(J,
«3
"8
"4
^
h.
K
^
C^
C-l
fs
^4
d,
d.
d.
d.
C\
Ct
Cs
^4
d.
The partial determinant of the third order is (/'.
mentary of the second order is {'i^d.^.
The complete altered determinant is plus or minus, according as the
permutations of the rows and columns are of the same or of ditierent class.
Jn the example they are of the same class, for there have been four trans-
positions of rows, and six of columns. Thus ( — 1)'" = + ! gives the sign
of the altered determinant.
K
h K . K K
^8
Ci c, r, c.
^8
e^ Cs e, e.
flj
a^ Oj : a, «3
^^8
d, d, d, d.
del
is 0'3''i''i), find its comple-
577 TuEOREii. — A partial reciprocal determinant of the r^^
order is equal to the product of the r— 1"' power of the
original determinant, and the complementary of its corres-
ponding partial determinant.
Take the last determinant for an example. Here n=5, r=3 ; and by tho
theorem,
^8 B, B„
= A=
«i
a,
where B, C, E are the
G, C, C,
d.
d.
respective minors.
Es E, E,
Proof. — Raise the Partial Reciprocal to the original order five without
altering its value, by (5G4) ; and multiply it by A, with the rows aud
columns changed to correspond as in Ex. (57G) ; thus, by (570), we have
J?3 B, B, B, B,
c, C, C, C, C,
E^ E^ E^ ..^>....^»
;""i 0"
1
h K h
C, C4 O5
«. e, e.
A ti 60
A c, c,
A e, c,
0a, Oj
</, (7,
= A» a, a,
d,d.
(h «4 05
d, d, d.
a, a,
./, d.
578 The product of the differences between every pair of 11
quantities fli, a., ... "„,
{a, — (i.;){a,—(t,){a,-a,) ... (r/,-^/„)
X {(li — ^l■,^)(^(., — <l^) ... ((i.. — (i„)
X(an-X-(in)
Proof. — The determinant vanishes when any two of the quantities are
1
1
1 ...
1
(t^
(I.,
a-, ...
(1,
<
(tl
< '"
(I-
152
THEORY OF EQUATIONS.
equal. Therefore it is divisible by eacli of the factors on the left ; therefore
by their product. And the quotient is seen to be unity, for both sides of the
equation are of the same degree ; viz., ^n {n — 1). ' '
579 The product of the squares of the| _
differences of the same n quantities j ~
... S,;
Proof. — Square the determinant in (578), and
write Sf for the sum of the r"' powers of the roots.
580 With the same meaning for s-^,s.2..., the same deter-
minant taken of an order r, less than n, is equal to the sum
of the products of the squares of the differences of r of the n
quantities taken in every possible way ; that is, in C {n, r)
ways.
Ex.:
^1
§1
S-2
^0
^1
■'2
•''l
*2
-''3
^'2
•''3
*4
(«! - fl2)H (a, - a^y + &c. = 2 (flfi - «2)^
The next determinant in order
= S (a^-a^y (ffli— «3)^ (% — «i)^ («a-«s)^ ((^■i—ci'iy (as — a^y.
And so on until the equation (579) is reached.
Proved by substituting the values of s^, s.2 ... &c., and resolving the deter-
minant into its partial determinants by (571).
581 The quotient of
is given by the formula
qo^i"'-'' + f/i.r*"-'^-' + ... + </,.i^'"-"-'' + ...
where
/>o ... «o
/>2 fh K ••• ffi
h, h,
b,a.
Proved by Induction.
ELIMINATION. 153
ELIMINATION.
582 Solution of 11 li)i('(ir rquntious in u rnriahlrs.
The equations and the values of the varialjles are arranged
below :
«,.r, + f/,.r,+ ... + r/„.r„ = ^„ .r,A = J^^, + /;, f,+ ... + /., ^„
where A is the determinant annexed, and J^, B^, «, ... a
&c. are its first minors.
/i ... /.
To find the value of one of the unknowns x^.
HuLE. — Multiply the equations respectively by the minors
of the r''' column, and add the results, x^ will he equal to the
fraction whose numerator is the determinant A, with its ■?•'*
column replaced by Hj, ^2 ••• s„, and whose denominator is A
itself.
583 If SI, ^2 ••• ^« ^^''tl A all vanish, then x^, Xo ... x^ are in
the ratios of the minors of any row of the determinant A.
For example, in the ratios C^: Cj : C^ : ... : €„.
The eliminant of the given equations is now A = 0.
584 Orthoi^onal Transformation.
If the two sets of variables in the n equations (5y2) be
connected by the relation
.»•, + 4 + ... +.<•;= ?; + f: + ... + f;, (D.
then the changing from one set of variables to the other, by
substituting the values of the Ts in terms of the xs, in any
function of the former, or rice versa, is called orthogonal
transformation.
When equation (1) is satisfied, two results follow.
I. The determinant A = ± 1.
X
154
THEORY OF EQUATIONS.
II. Each of the constituents of A is equal to the corre-
sponding minorj or else to minus that minor according as A is
or
Proof. — Substitute the values of 4i, l^ ... 4„ in terras of x^, x^ ... a;„ in
equation (1), and equate coefficients of the squares and products of the new
variables. We get the n^ equations
a\^-V + = 1
fliCj + h^h^ + =
a.,ai + hjb^ + =
a\ + hl + = 1
a- + hl + = 1
a-,an + hA),, -f
Also A
Form the square of the determinant A by the
rule (570), and these equations show that the
product is a determinant in which the only con-
stituents that do not vanish constitute a diagonal
of ' ones.' Therefore
A'^ = 1 and A = ±l.
Again, solving the first set of equations
for ai (writing a, as a-^^a^, &c.), the second
set for ao, the third for a.^, and so on, we
have, by (582), the results annexed ; which,
proves the second proposition.
aiA = ^i +^,04-/ls0+ = .4i
a^\ = AS) + AS}-\-A^ +=^^3
&c. &c.
585 Theorem. — The 7i — 2*'' power of a determinant of the
n^^ order multiphed by any constituent is equal to the corre-
sponding minor of the reciprocal determinant.
Proof. — Let p be the reciprocal determinant of A, and /3,. the minor of i?^
in p. "Write the transformed equations (582) for the xs in terms of the 1%
and solve them for l^. Then equate the coefficient of x^ in the result with its
coefficient in the original value of .^0.
Thusp^3= A (/3ia;i+...+ /3,.a;,. + ...), and i, = &>i+ ... +Mr+ ••• ;
.-. A/3, = p&, = A"-'&, by (575); .-. /3, = A'-'^fc,.
586 To eliminate x from the two equations
«.r'" + 6.r"'-' + ca?'^-' + ... = (1),
aV+/>V-^+cV--'H-... =0 (2).
If it is desired that the equation should be homogeneous
in X and y ; put — instead of «, and clear of fractions.
following methods will still be applicable.
The
ELIMINATION.
155
1. Bezant's Method. — Suppose m > n.
Rule. — Bring the (ujuatiom to the same degree by multi-
plying (2) by «"•-". Then multiply (1) by a\ and (2) by a, and
subtract.
Again, multiply (1) by a\r-\-b\ and (2) by {'i,r-\-b), and
subtract.
Again, multiply (1) by ax--\-b'x-\-c\ and (2) by {a£'-\-bx-\-c),
and subtract, and so on until n equations have been obtained.
Each null be of the degree m—l.
Write under these the m—n equations obtained by multi-
plying (2) successively by x. The eliminant of the m equations
is the result required.
Ex.— Let the equations be ( a a;' + h ai' + c re" + d x' + ex +f = 0,
- =0.
( a a;"' + 6 a;' + c re" + fZ a
( a'x^ + h'x' + cx +d'
The five equations obtained by the method, and their eliminant, by (583),
are, writing capital letters for the functions of a, b, c, d, e, f,
A^ B, C, D, E,
A., Bn C, D, E,
A, Z?3 G, A E,
a' b' c d'
A,x* + B^a^+G^x^ + D,x + E, = '
A^* + B.^+C^' + D^ + Ej -
A^x' + B,x' + (7,a:^ + I)^^E, =
a x' + h' x^-\-c' x' + d' X =0
a x^^-b' x^-\-c x^-d' =
and
b'
d'
Should the cqaations be of the same degree, the eliminant will be a sym-
metrical determinant.
587
II. Silvester's Diahjtic Method.
Rule. — Multiply equation (1) successively by x, n — l times;
and equation (2) m — l times ; and eliminate x from the m-\-n
resulting equations.
Ex. — To eliminate x from a.i " + 6x' + c.r + tZ = I
px + qx + r =0 I
The m + n equations and their eliminant arc?
I'X- ->t qx ■\- r =
'px^-\-qx--'rrx =
px^^-qx'^ + ra? =
aa?-\-b3?-\-cx + d=
oa;* + 6«' + car+c/.c =0
and
2^ q r
<» i> q r
p q r
a b c d
a b c d
156 THEORY OF EQUATIONS.
588 III Method of elimination by Symmetrical Functions.
Divide the two equations in (586) respectively by the
coefficients of their first terms, thus reducing them to the
forms /Gr) =.r- + /)i^^-^+ ... -\-p^= 0,
<l> (cv) = .v'' + </i cr'*"^ + . . . + 7n = 0.
Rule. — Let a, b, c ... represent the roots of f{x). Form
the equation <p{a) (p (b) <{> (c) . . . = 0. This will contain sym-
metrical functions only of the roots a,b,c....
Express these functions in terms of pi, p2 "-by (538), ^'c,
and the equation becomes the eliminant.
Reason of the rule. — The eliminant is the condition for a common root of
the two equations. That root must make one of the factors (a), ^ (6) ...
vanish, and therefore it makes their product vanish.
589 The ehminant expressed in terms of the roots a, b,c ...
of /(,7'), and the roots o, )3, y ... of ^ {x), will be
(,,_«) (a-^) (a-y) ... (/.-«) (6-/3) {h-y) ... &c.,
being the product of all possible differences between a root of
one equation and a root of another.
590 The eliminant is a homogeneous function of the co-
efficients of either equation, being of the n^^ degree in the
coefficients of f{x), and of the m^^^ degree in the coefficients
of .^ (a').
591 The sum of the suffixes of p and q in each term of the
eliminant = ran. Also, if p, q contain z ; if p.y, q., contain z' ;
if 2?s, qs contain :^, and so on, the eliminant will contain z"'".
Proved by the fact that p^ is a homogeneous function of r dimensions of
the roots a, b, c ..., hy (40G).
592 If the two equations involve x and y, the elimination
may be conducted with respect to x ; and y will be contained
in the coefficients p^, j^z •••i ^Iv Q2 ••• -
593 Elimination by the Method of Highest Common Faetor.
Let two algebraical equations in x and y be represented by
^ = and i? = 0.
ELIMINATION.
157
It is required to eliminate x.
Arrange J and Ji according to descending powers of .i\
and, having rejected any factor which is a function of // only,
proceed to find the Highest Common Factor of J and B.
The process may be exhibited as follows :
c^A = (jxli +'"i^''i'l ''i» ^'2J Ci^ *'i a^6 the multipliers re-
c B = (hR +ro7?.> quired at each stage in order to avoid
^ _ T^ , " -({ \ fractional quotients ; and these must
<^3^i — (l^^h -r r^li^ ^^ constants or functions of y only.
c^Bo = QiRs + ri j q^^ q^^ q^^ q^ are the successive quo-
tients.
r^^Bi, r.jB^, rji^, r.^ are the successive remainders ; i\, rg, rg, i\
being functions of y only.
The process terminates as soon as a remainder is obtained
which is a function of y only ; i\ is here supposed to be such
a remainder.
Now, the simi)lest factors having been taken for Ci, c-,, c^
The values of x and v/,
which satisfy simulta-
neously the equations
A = {) and B = Q, are those
obtained by the four pairs
of simultaneous equations
following :
The final equation in y,
which gives all admissible
values, is
1^
I f it should happen that
tlio remainder i\ is zero,
th(> simultaneous equa-
we see that
1 iE
, theH.C
.F
of
Ci
and /•/
d.
3>
>j
^1
and r.2
^
J>
jj
(\ Co
and Vg
d.
»>
55
c
1^2 Cg
i,d.
and 7-.t
n
= and
B
=
.... (!)•
^2
d.
= and
n.
=
....(2)
d.
= and
B,
=
.... (:l)
= nnd
B,
=
.... (4)
tions (1), (2), {'■)), and (4) reduce to
7?,
B
r,= and „ = ; -/- = ^ and ^^ = ;
R
B,
^- = and f - = 0.
158 THEORY OF EQUATIONS.
594 To find infinite values of x or y whicli satisfy the given
equations.
Put X = ~. Clear of fractions, and make 2 = 0.
z
If the two resulting equations in y have any common
roots, such roots, together with .13 = oo, satisfy simultaneously
the equations proposed.
Similarly we may put y = —.
PLANE I^IMTJOXOMETRY.
ANGULAR MKASrHK.MKXT
600 'I'Jx' ii'iit of Circular mcasui'o is a liadiari, and is tlic
aiiulr at (lie centre of a circle which subtends an arc equal to
the radius. Ileiiee
601 Cireulai- measure of an anole = ^^ — .
radius
602 Circular measure of two right angles = H'T II 50 . . . =7r.
603 'I'he unit of Centesimal measure is a Grade, and is the
oni'-liundredth part of a right angle.
604 'i'he unit of Sexagesimal measure is a Degree, and is
the one-ninetieth part of a right angle.
To change degrees into grades, or circular measure, or
vice vprni'i, employ OTie of the ihrvQ equations included in
^"^ 90~100""7r'
where D, G, and G are respectively the nundjers of degrees,
grades, and radians in the angle considered.
THIGONOMETRICAL RATIOS
606 Let OA be fixed, and let the
revolving line OP describe a circle
round 0. Draw FN always perpen-
dicular to AA'. Then, in all ])osi-
tions of OP,
PN
- - = the sine of the angle AOI',
ON
z= the cosine of the angle AOP,
PN
= the tangent of the angle JO/'.
160 PLANE TL'IG GNOME TBY.
607 If P be above the line AA\ sin AOP is positive.
If P be helo2o the line AA', sin ylOP is negative.
608 It" y lies to the right of J5j5', cos AOP is positive.
If P lies to the /t// of BB\ cos ylOP is negative.
609 Note, that by the angle AOP is meant the angle through whicli
OP has revolved from OA, it-,.-; initial position ; and this angle of revolution
may have any magnitude. If the revolution takes place in the oppos'te
direction, the angle described is reckoned negative.
610 The secant of an angle is the reciprocal of its cosine,
or cos A sec A = 1.
611 The cosecant of an angle is the reciprocal of its sine,
or siu^ cosec^ = 1.
612 The cotangent of an angle is the reciprocal of its tangent,
or tan A cot ^ = 1.
Relations hctirrcn the trii>onomet7'ical functions of the
same un^le.
613 siuM+cosM = l. [1.47
614 sec'^ ^ l+tmiM.
615 cosecM = l+cotM.
616 tan A = ^i^. [606
cos^
If tail^ rrz "
h
617 sin A =
s/(r-\-/r
cosy4 = . [606
/?io • I tally! . 1 ^,_
618 sin.1 = , ('()Syl=: -. [617
vlH-iair/l x/1+taii-i
TIUOOSOMETliICA L HATIOS.
IGl
619 The Coniploment of .1 is = ^0^-A.
620 Tlie Supplement of A is = 180'-yl.
621 sin(00 -y1) = (OSy|,
taii(lM) -.l) = eot.l,
sec (1)0 —^) = c'osec A.
622 sill (18(r~^)= siiiyl,
cos (180°-^) = -cos ^,
tan (180 -.4) = -tan .4.
Ill the figure
Z QOX. = 1 80' — -•/. [607, 608
623 sin(— ^) = ~sm^.
624 cos( — ^) = cos^.
By Fig., and (607), (608).
The secant, cosecant, and cotangent of 180° — ^, and of
— y/, will follow the same rule as their reciprocals, the cosine,
sine, and tangent. [610-612
625 To reduce any ratio of an angle greater than 90° to the
ratio of an angle less than 90°.
Rule. — Determine the sign of the ratio by the rules (007),
and then substitute for the given angle the acute angle formed
by its two bounding lines, produced if necessary.
Ex.— To find all the ratios of 600°.
Measuring 300° {= 660°— 360°) round
the circle from ./ to P, we find the
acute angle AOP to be 60°, and F lies
helow AA\ and to the rUjki of BB' .
Therefore
sin 660° =r- sin 60'
2 '
cos 660° = cos 60° = J ,
and from the sine and cosine all the remaining ratios mav l)e
found by (610-616).
Inverse Notation. — The angle whose sine is x isdenoted
by sin"' ,/•.
Y
162 PLANE TRIGONOMETRY.
626 All tlie angles which have a given sine, cosine, or tan-
gent, are given by the formula
sin-^^= nir+^-iye (1),
cos-Vr =: 2mT±e (2),
U\\\\r= mr-\-6 (3).
In these formula} 6 is any angle wliich has x for its sine, cosine, or
tangent respectively, and n is any integer.
Cosec"'.T, sec"' a;, cot"' a; have similar genei'al values, by (610-612).
These formulae are verified by taking A, in Fig. 622, for 0, and making
n an odd or even integer successively.
FORMULAE INVOLVING TWO ANGLES, AND
MULTIPLE ANGLES.
627 sin {A-\-B) = sin A cos 5 + cos ^ sin B,
628 sill (A—B) — sin A cos^— cos^ sin B,
629 voii(A-\-B) = QosA cos ^— sin A sin i^,
630 coh{A — I}) = cos ^ cos J5 + sin A sin B.
Proof. — By (700) and (701), we have
sin C = sin A cos B + cos A siu B,
and sin G = sin (A+B), by (622).
To obtain .sin {A — B) change the sign of B in (627), and employ (623),
(624), cos(^ + B) = sin {(00"-^)-^j, by (621).
Expand by (628), and use (621), (628). (624). For cos (-1-5) change the
sign of B in (629).
631 tan (.4+/?)^ ^^'"^+^^'"-;^ .
632 tan (/!-/>') =
633 cot(.l + /i) =
634:
1-1
\i\ii A Ian B
tan
i - ian /,*
J+1
\i\uA <au/>*'
col
1 (M)< />'-!
cot
A + cot />'
col
.1 co< n-\-\
rol /> — col .1
Ohtaincul tVom (()27-6;>0).
MI'I/ni'LE AXOLES.
163
635
sin 2 A = 2 sin .1 cos^.
[C>27.
Put 7? = ^1
636
('()s2.1 = cos- J — siii-yl,
637
= 2 cos'^l — 1,
638
= 1 -2siiiM.
[029, Gl:{
639
2cos-^ = 1 +cos2 1.
[0:i7
640
2sin-.i = 1- cos 2.1.
[038
641
sin^-v'-r'.
[640
642
.o.-i-j-'+'-r^.
[639
643 Um^
646 cos A =
1 — cos A 1 — cos A
sin A
1 1 — cos .1
^ 1 + COS A sin .1 1 + cos A'
[641, G42, 613
1-tau^-:^
^ , , .A'
l + tan —
•2^ 2tan^
sin yl = _^. [643, 613
1 + taU" -rr-
648
•os^ =
1+tan.l iaii-^
, .1\ /.- A\ /l-Tsiiwt
, , yl\ . / ,- A\ j\ -^\\\A
+ y)""''"V ~~2)^ ^ 2
l + sin I 1+sin A _ cos A
649 ^i" (45
650 cosri')
651 ^^'•V'>+i7J-Vr3:-^i--T;;;rT"-r;^
[CU
[642
652
653
tan 2.1 =
cot 2.1 =
2 tan ^
1 -tan-M
col .1 - 1
2 col A
[6 51. ruti?=.'l
164
PLANE TRIGONOMETRY.
654
655
656
657
658
659
660
^ ^ 1 — tan A
X /A-o A\ 1— tau^
[631, 632
siu 3A = 3sm A—4< sinM,
cos 3^ = 4 cosM — 3 cos ^
tan 3^ =
3tan J— tanM
l-3tanM
By putting B = 24 in (627), (629), and (G31).
sin (A + B) sin {A — B) = sin" 4 — sin" B
= co^-B — cosM.
cos (^ + IJ) cos (^ - i^) = cos^ ^ - siu^ B
= cos"-B — sin" A.
From (627), &c.
661 sin Y + c<^® -9- = ± v/l + sin ^.
"^ [Proved by squaring.
662 sin — — cos — = ± \/l — sinyl.
663 sin 4 = 4 {\/l + sin^->/l-sin^}.
664 ^osA^-^ { v/1 + sin ^ + yi - siu ^} ,
when -- lies between —45° and +45°.
665 In the accompanying diagram the
signs exhibited in each quadrant are the
signs to be prefixed to the two surds in
the value of sin^y according to the quad-
rant in which — lies.
For cos - change the second sign.
L
Proof.— By examining ibe cbangosof sign in (UtH) and (662) by (607).
MULTIPLE ANGLES. 105
666 2 sin A cos B = sin (A + /i) + sin (.1 - /?).
667 2 cos^l sin /i = sin (A + 7^) - sin {A - li).
668 2 cos .1 cos /i = cos (.1 + U) + cos (J - B).
669 lAwA sin y^=z cos(.l-/;)-cos(.l + 7y).
[G27-G30
l-/y
670 sin .1 + sin B = l sin li+Z^ cos -i_
671 sni yl — sni B = 2 cos — ;^ — sin — - — .
672 cos ^ + cos jK = 2 cos — - — cos — - — .
01 o cos B — cos A = 2 sm — - — sni — - — .
Obtained by cbaugiug A into — .- — , and B into - ~ , in (G66-669).
It is advantageous to commit tbe foregoing fonnula; to memory, in words,
thus — 2 sin cos = sin sum + sin difference,
2 cos sin = sin sura — sin difference,
2 cos cos = cos sum + cos difference,
2 sin sin = cos difference — cos sum.
sin first + sin second = 2 sin half sum cos half difference,
sin first — sin second = 2 cos half sum sin half difference,
cos first + cos second = 2 cos half sum cos half difference,
cos second — cos first = 2 sin half sum sin half difference.
674 sin (J + 6+ C)
= sin A cos B cos C + sin B cos C cos A
-\- sin C cos A cos B — sin A sin B sin C.
675 cos(.l + 7J+C)
= cos A cos B cos C — cos A sin B sin C
— cos B sin C sin A — cos C sin A sin B.
676 Um{A + n-\-C)
_ tan A + tan B -f tan C — tan A tan B tan C
1 — tan 7^ tan C — tan C tan A — tan ^ tan li'
Prouf.— Put B + C fur L' in (G27), (02l>), and (Gol).
166 PLANE TRIGONOMETRY.
If A + B + C = 180°,
677 sill A + sill B + sill C = 4 cos— cos— cos—.
sm A + sill B — sill C = 4 siii — sin — cos—.
ABC
678 fos^ + cos7iH-cosC = 4 sill — sin — siii -+ 1.
£t Jd ^
cosyl + COS 1^ — COS C = 4 cos— cos— siii — — 1.
679 tan A + tan B + tan C = tan A tan ^ tan C\
680 cot 4 + cot I + cot ^ = cot 4 cot ^ cot ^.
681 sin 2A + sin 2i? + sin 2C = 4 sin ^ sin B sin C.
682 cos2^H-cos2ii + cos2C = — 4cos^ cos^ cosC— 1.
General formulae, including the foregoing, obtained by
applying (666-673).
If A + Bi-G = TT, and n be any integer,
-rto 4 ' nA . nB . nC
683 4 sm -rp sm -;j- sm -^
Ji ^ ^
/»o>i ^ ^^A nB nC
684 4 COS — COS— cos-^
cos
(!^-n^)+cos(^-.ii) + cos(^-..c) + cos^|r.
liA-\-B + C = 0,
635 4 sin '-^ sin '-^ sin ^ = - sin uA - sin yi j5 - sin nC.
2i 2i 2i
686 4 cos ^^4 cos ^ cos ^ = cos;iy/ + cos?iZ)'+cosyiO+l.
2t 2i 2t
Rule.— 7/; informulce (683) ^o (686), two factors on the left
be cliaiujed hy ivriting sin for cos, or cos for sin, then, on the
right side, change the signs of those ter)ns n-hirJi <1o not run tain
the angirs of the (dtpveit factors.
COMMON JxSUlJ-h^. I<i7
Thus, from (083), wc obtain
687 J sill - cos -— COS -
^ z ^
= _sm(^'^'^-„.l) + sm(^"2"-„y;) + sin('^''-«c) + siM".
A Formula for the construction of Tables of sines, co-
sines, &c. —
688 sin (?i + l)a — sin uu = sin «rt— sin (ii—l)u —A- sin 7ia,
where a = 10", and A = 2 (1 -cos a) = -0000000023504.
689 Formulae for verifying the tables —
sin,l + sin(72° + ^)-8in(72"-yl) = sin (3G^ + ^)-sin (36"-^!),
cos yl+cos (72° + yl) +COS (72^ + ^1) = cos (36^ + J) +cos (36^-^1),
sin (60° + ^) -sin (60°-^) = sin^.
RATIOS OF CERTAIN ANGLES.
690 sin 45° = cos 45° = -i^, tan 45° = 1.
691 sin 60^ = ^^, cos 60' = I , tan 60° = ^'6.
Li '-t
692 -15' = ^, -1-^^ = ^'
tanl5° = 2-y3^
cotI5' = 2 + v/'>3'
693 sml8^ = ^^, cosl8"=^'^ + 2'^^
tan]8^ = \/^-^|^.
^ 5
«/Nj • r^- \/5-|-l r 4'? vS— \/5
694 sin54=^^^~-, cos 54 = 2^2 '
tan54^ = V'^^^-
695 By taking- tlie complements of these ani^Hcs, llie same
table i>-ives tlie ratios ol ^Jn", 75', 72", and :'<> .
168
PLANE TRIGONOMETRY.
696 Proofs.— sin 15° is obtained from sin (45°-30°), expanded by (628).
697 sin 18'' from the equation sin 2x — cos 3.i-, where x = 18°.
698 sin 54^ from sin 3.7; = 3 sin a; -4 sin^r, where x = 18°.
699 And the ratios of various angles may be obtained by taking the sum,
difference, or some multiple of the angles in the table, and making use of
known formulae. Thus
12^ = 30°-18°, 7^° = ^, &c., &c.
PROPERTIES OF THE TRIANGLE.
700 c = aQOsB-{-bcosA.
a h c
701
sin A sin B sin C
702 «' = 6'+c'— 26c cos A.
703
Proof.— By Euc. 11. 12 and 13, a? = b' + c'-2c.AD.
cos A =
26c
If g = ^H-6 + g ^ ^^^ ^ denote the area ABC,
704 .u4=^^I^^i-^, ^o4 = ^^^.
[641,042, 703.0, 10, 1.
A I{k' — L\ (M — r
tan
705
706
707
708
A_J {s-b){s-c)
7-V .v(.y_„) •
sin A = ^ V .V (.y— a) (s—b) {s — c).
be
[635, 704
A = ^ sin A = Vs (s-a) (s-b) (.v-c), [707, 706
= 1 \^'2b-i'''-^2(-a--^'2irb--(i'-b'-r\
rifOl'EUTlh'S nr riilASULKS.
iC'.t
Thr Trianii/r tun I ('irrlt
Let
r = radius of inscribed circle,
r^= radius of escril)ed circle
touching tlie side a,
B = radius of circuinscribiu'
circle
709
l b roiii h i«^., A = ^ -1- ^ + -^ .
710 r =
. n . c
cos
It
[By a = r cot - + r cot - .
711
712
>
.y
— a
a cos
n
-^ cos
_
7? c
[Fioiu u = r„ tan +?•„ tan .
713 n = ,^J!—=%
1 sin yl 1 A
[Hy (III. 20) iiti.l (7()(;)
715
= iv/((/'+'-r^^^"e^^+(/^-''reosec^^| \
[702
Distance between the centres of inscribed and circum
scribed circles
716
= y/lV-'2i(i
[1)36
Radius of circle toucliini^: h, r and the inscribed circle
717 r = r tair' \ (/i-f T). | Hy sin 4 = ''^..
170
PLANE TRIGONOMETRY.
SOLUTION OF TRIANGLES.
Right-angled triangles are solved
by the formula?
718 e^=rt2+6-^;
la^e siu A ,
719 h, = ecos^,
\a z=: h tau A ,
&c.
Scalene Triangles,
720 Case I. — The equation
a h
sin A siu B
[701
will determine any one of the four
quantities A, B, a, h Avhen the re-
maining three are known.
721 The Ambiguous Case.
When, in Case I., two
sides and an acute angle
opposite to one of them
are given, we have, from
the figure,
. ^ e sin A
sin C = .
'C
Then C and 180 -C arc the values of C and C, by (622).
Also h = cvo» A ± v'a- — c- siu- A ,
because = A 1) + DC.
722 Wlicn an angle 1> is to be determined from the equation
... /, . .
.sill /; =: sin .1,
a
and '' is u small fracLiou ; tlic fiirular measure of B may be appi-oximatcd
a
\o by putting sin (U^C) i'or sin .1, and using theorem (rOC)).
SOL UTIOX OF TU I A M i L ES. 1 7 1
723 ('\^i' TF. — AVluMi two sides />, r and the inclndcMl aiigl(!
./ are known, tlie tliird side a is priven by tlic formula
(r= h- + (--'2hrvosA, [702
when logarithms are not used.
Otlierwise, eni])loy tlie followin<r formula with lo-^^ai-ithms
/j_f; i^-c , A
724 ^^»-^V-^-,TX7/*'^^>
Obtained from ('~'' = ^!°^^---^'A^, (701). and then applying
h-{-r sm/Z + smG
(670) and (07 1).
havinir l)oen found from the above eiiuation, and
2
— "trl being equal to 90'—-''-, we have
i? and G having been determined, a can be found by Case I.
726 If the logarithms of b and c are known, the trouble of taking out
log{b—c) and log(/> + c) may be avoided by employing the subsidiary angle
$ = tan"'—, and the formula
c
727 tan X(B-C) = tan (« - ^ ) cot ^ , [C55
Or else the subsidiary angle = cos"' ''' , and the formula
728 tan i (B- G) = tan' ^ cot ^J . [04:3
If a be ri(|uirod without ealcuhit iug the aiiglis />' mid / ', we may use the
formula
,1 [From the figure in 9(i0, by
(^< + (jMii ^ drawing a |)eijteiidiculnr
729 ** ~ cos ^ {u-c) ^^''^™ ^ '^ ^'^' pi'^'^i^^'ed-
730 If a be required in terms of ?», c, and A alone, and in a form adapted
to logarithmic eomputatiun, employ the subsidiary aiiglo
= sin"' ( ,' " .COS- ' ),
and the f..i-muhi a = (/- + .) cus«. [702, G37
172 PLANE TRIGONOMETRY.
Case III.^ — AVhen the three sides are known, tlie angles
may be found without employing logarithms, from the formula
731 co^A=!^±StzlL\ [703
'2oc
732 If logarithms are to be used, take the formulae for
sin^, cos^, or tan—-; (704) and (705).
QUADRILATERAL INSCRIBED IN A CIRCLE.
733 ,o.li = "'+'>'-/ -f . ^
2{ab-\-cd)
From AC- = a» + b''-2ab cos B = c- + (P +
2cdcosB, by (702}, and i^ + D = 180°.
734 ^\nB = -^S_
ab-\-cd
735 Q = x/{s-a)(s^b){s-c){s-d)
= area of A BCD,
and s =l{a-{-h + c-]-d).
Area = lah sin B + ^cd sin B ; substitute sin B from last.
736 AC^= (ae+hd)i„,l+he) _ ^,,,_ ,33
' " {ab-\-cd)
lladius of circumscribed circle
737 = -L ^\ab-\-cd) {ac-^bd) {ad-\-bc). [713, 734, 736
4^)
If AD bisect the side of the triangle ABC in D,
4A
b'-c''
733 tan 51)^ =
739 cot7?^I> = 2cot^ + cotJ?.
740 AD' = i (b' + c' + 2bc cos A) = hAb' + c'-ia').
If ylD bisect the angle yl of a triangle ABC,
B^C^b + c,__ A
2 b-
742 tan /;;>. 1 = cot ^^^- = ^^ tan
743 JD=^cos^
isriisn>iAuy angles.
173
If AD III- ]H>i{H'n.licular to BC
744
745
AD
hi' sin A li' sin G + c' sin 7?
//'' — r'^
tan J}-tnnC
tan 7) -h tan (/
REGULAR rOLYGUN AND CIRCLE.
Radius of circumscribing circle = R.
Radius of inscribed circle = r.
Side of polyo:on = a.
Number of sides = n.
a , TT
r = — cot — .
746 n = ^ cosec -
2 n
Area of Polygon
748 = \na- cot — = ItilV sill ^^ — nr- tan —
USE OF SUBSIDIARY ANGLES.
749 To adapt a±_h to logarithmic computation.
Take = tan"' / ; then a^h = a sec' 6.
750 i^'or a-h take ^ = tan"
h
til us
a — h
av/2 cos (0 + 45°)
COS0
751 To adapt a cos C±h sin C to logarithmic computation.
Take = tan"' ^ ; then
b
a cos C ± 6 sin = v/(a^ + i-) sin (9 ± C). [By 617
For similar instauces of the use of a subsidiary angle, see (72G) to (730).
752 To solve a quadratic equation by employing a subsidiary
angle.
If x-—22)X + i2 = l)e the equation,
[ Hy lo
174 PLANE TRIGONOMETRY.
Case I. — If a be < ^r, put ^ = sin'' B ; then
P
x = 2pcos'^. and 2^5 siV f-. [639,640
Case II. — If q be >»", put '^., = sec-0; then
X = p (Izki tan 0), imaginary roots. [614
Case III. — If q be negative, put ^ = tan* ; then
x = Vq cot and — y^ tan — . [644, 645
2 2
LIMITS OF RATIOS.
753 -g-=-r = '
when 9 vanislies.
AP AP
For ultimately ^=4i = l- [601,606 q
. e
754 n sin— = ^ when n is infinite, gy putting -- for in last.
755 (co^~) — 1 when n is infinite..
Proof. — Put ( l-siu"— ] ^, and expand the logarithm by (156).
DE MOIVRE'S THEOREM.
756 (t'os a+/ sin a) cos /8+/ sinyS) ... &c.
= cos (a+)8+7+ ...) + «■ ^iu (a+^+7+ •••).
where i = V — 1. [Proved by Induction.
757 (cos 6-\-i sin ^)" = cos n6+} sin >i^.
Proof. — By Induction, or by putting a, /5, &c. each = ^ in (756).
Expansion of cosnO, tj-c, in iwwcrs sinO and cosB.
758 c'osM^ = cos"^-C^(/J, 2) cos"-'^^ sin-^
4-C(»,4)cos"-^^sin^^-ctc.
759 sin n0 = n cos""' ^ sin 6—C{n, Ji) cos"'^ sin*^+&c.
I'liOOF. — Expand (757) by Bin. Th., and o<iuatc real and imaginary
parts.
TliiaOXOM ETHICAL ShJlilES. 170
760 Unu,e= >Hang-(-(»,:i)lM.r-g+Ac.
In series (758, 7.59), stop ut, aiul cwiliulc, all fmns willi indices grciiter
than n. Note, n is here an integer.
Let s^ = sum of the G{n, r) products of tana, tau/3, tany,
c^c. to n terms.
761 sill (a + ^+y+c^'C.) = cosa cosyS ... (.v,-.v, + .v,-.^c.).
762 i'Os(a-{-fi+y-\-Scc.) = cosa cos/S ... (1 -6',+.s-.U.).
Pkooi'. — By e(|u:itinf; real and imaginary jiarts in (7o6j.
Exjmnsions of the sine and cosine in powers of the angle
764 sill^=^--^+|:^-&C. (.OS^=l-^ + -[^+el'C.
(9
Proof. — Put - for 6 in (757) and n = x , employing (7'>ir) and (755).
766 e'<' = eos^+f siii^. e'' = cos 6—i sin 0. By (150)
768 c''-\-r-'" = 2 cos ^. t'''-e-" = 2/ sin ^.
770 itaii^-^— ^ l + »'tau(9 _ ,^
Expansion of ro.s-" av^(Z sin,'' 6 in cosines or sines of
midtiplcs of 0.
772 2" ' cos"^ = cos n9-\-n cos(//-2; l9
+ C'(/',2) cos (;/-!) ^4- ('(//,;{) cos (/< -(I) ^ + .Vc.
773 ^\'lleu // is even,
2"-i (-1)*" .sill" ^ =: cos n0-n cos (/<-2) ^
+ (;(//, 2) cos (n—l) e-C{n, ;{) COS {u-(\)e-\-kQ.,
774 And wlicn 7/ is odd,
2" '(_1)"2 si,i«^ = siu«^— « .sin (/I— 2)^
^^(///i) sin(//- l-)^~r' (;/,;{) sin (//-(I) ^ + ,<;.'c.
17(3 PLANE TRIGONOMETRY.
Observe that in these series the coefficients are those of the Binomial
Theorem, with this exception : If nbe even, the last term mud he divided by 2.
The series are obtained by expanding (e" ± e'")" by the Binomial Theorem,
collecting the equidistant terms in pairs, and employing (768) and (769).
Exjpansion of cosnO and sinnO in ijowers of sin 9.
775 AYheii n is even,
n 1 n' • o /J , n- (n- — 2") . 4 a
cos 116 = 1 — — — sm- 6 -{ ^-— ^ sm*
b !
776 When n is odd.
«^-l i„.fl^(»^-l)(«-3^)=:.»,
cosng = C08g n -!t^s\n'0 + y" ^>^\'' "' sia'e
b !
r j^2 2''
sin nO = n cos ) siu 6 — /^ siu ' 6-\-
777 When n is even,
i Ll sm^ 0— ^ '-^ — — — ^-^ ^ sin^ 0-\- &c. [ .
! / ! )
778 When n is odd,
^mnv = n sin^— ^ — ^ siir6/ + — ^ -i siii^
♦5 ! o I
- "("-!) oy in (>r-y) ,i„, g ^ ^tc.
Proof. — By (758), we may assume, when n is an even integer,
cos«^ — l+A^sin-6 + yl^siu*e+...+^^sin""^+..,.
Put d-\-x for 6, and in cos nd cos ?j.c — sin nO sin n.v substitute for cos nx and
sinn.T; their values in powers of iix from (764). Each tei-m on the right is of
the type ^^.^ (sin 6 cos a; + cos ^ sin. t)"''. Make similar substitutions for cos.«
and sina; in powers of x. Collect the two coeHicients of .r'^ in each term by
the multinomial theorem (137) and equate tlium all to the coeUicient of .t"
on the left. In this equation write cos"^ for 1 — sin" 6^ everywhere, sind then
equate the coefficients of sin'-''^ to obtain the relation between the successive
quantities A.,^ and A.^^^^ foi" the series (775).
To obtain the series (777) equate the coefficients of .i' instead of those of .c''.
When n is an odd integer, begin by assuming, by (7o9),
sin »/y = /I, sin ^ + yl, sin^G + itc.
TniriOXOMhJIlilCAL SERIES. 177
779 The expansions of cos nO and sin nB in powers of cos
are obtained by chanr^ing 6 into ivr— in (775) to (778).
780 Expansion of cos vO in drsrrvdlng poirrrs of cos B.
2 COS nO = (2 cos ey- n (2 cos ^)"- ■^ + !!i^^ (2 cos $)" " '-
,..(.-. -l)0^-r-2)...(.-2r+l)^.,_p,„ ..._^
r\
up to the last positive power of 2 cos B.
Pkoof. — By expanding each term of the identity
log(l-z.r)+log(l-^) = log{l-.-(.r + l-z)}
by (156), equating coefficients of -", and substituting from (768).
783 sin a-\-c sin {a-\-^)-^(r sin (a+2)3) + &c. to n terms
_ sina-rsin(a-ff)-c "sin(a+/<y 8)4-^"^'si» {a-\-n-l^} ^
~ l-2ccos;8+6-'
If c be < 1 and n infinite, this becomes
_ sill g-c sin (g—ffl
'°* l-2ccos)8+c^ *
785 cos a+c cos (a+^) + c- cos (a4-2i8) + &c. to n terms
= a similar result, changing sin into cos in the numerator.
786 Similarly when c is < 1 and n infinite.
787 Method of summation. — Substitute for tlie sinrs or
cosinrs their exponential valiirs (768). Sum the two resulting
geometrical series, and substitute the sines or cosines again for
the exponential values bij (760).
788 csin(a + iS) + ^sin(a-h2/8)+^sin(a4-.'^/3) + &c. to
infinity = c'"'""^ sin (a+c sin /3) — sin a.
789 (' <•<)> ( a + /3) + ^ cos (a + 2y8) + ^ cos (a + ;?)Sl + cVc. to
infinity — c' ^'"'^ cos (a-j- r sin ^) — cos a.
Obtained by tlie rnle in (7J?7).
2 A
178 PLANE THIGOXOMETRY.
790 If, in the series (783) to ( 789), /3 be changed into /> f n, the signs of
the alternate teims will thereby be changed.
Expansion of 6 in powers of tan B (Gregon/s series).
791 e = tan e-^j}^+ ^-^ -&c.
The series converges if tan 6 be not > 1.
Proof. — By expanding the logarithm of the valne of e"'* in (771) by (158).
Formulje lor the calculation of the value of w by Gregory*s
series.
792 ^ = tau-^i + tau-^i - 4tan-^i-taii-^;i^ [791
794 = 4taii^l-taii-^-iT + ^''^^^ 't^-
o /U 99
Proof. — By employing the formula for tan (A±B), (631).
To prove that ir is incommensurable.
795 Convert the value of tan 6 in terms of B from (764) and (765) into
a continued fraction, thas tanO = — -77 -z- -^ . ; or this result may
1— 3— 5— 7— (EC.
may be obtained by putting id for y in (294), and by (770). Hence
6__ _ «! «: p":
tany ~ 3— 5— 7— (tc.
Put "" for 6, and ussumc that tt, and therefore - -, is commensurable. Let
2 4
= , VI and n being integers. J lien we shall have i = .— - — -^ ,
4 7i dn — 5)i — tn— &c.
The continued fraction is incommensurable, by (177). But unity cannot be
equal to an incommen.^niable quantity. Thcrelore t is not commensurable.
796 ^^ sin j; = nsin (a; + a), iU = 7) sin 0+ - sin 2a + -^ sin3a + tfec.
797 ^ f ^''^" ^' — " *'^" 2/' ^ — .'/ — '" sin 2// + -|j sin 4?/— ^- sin 6^ + ic,
,1-7/.
•where vx = .
1 + u
Proof. — % suhs'itnt'ing the ea-pmential values of the sine or tangent (769)
and (770), and then eliminating x.
798 ('oeniciont of a;" in the cxi.ansion of c"-^ cos 6.1; = ^ cos n^,
■where a = r cos 6 and Z< = rsinO.
For proof, substitute for cos /^.r from (768); expand by (150); put
o = r cos^, h = T h'n\6 in the coellicicnt of x", and employ {1'>1).
rindoxoMirnnt -al ,s7;a7/';.s. 1 71)
799 W lion e is < 1 , ,^^-^^ = 1 + '2h cos + '2'r cos 20 + 2// cos :iO+...
'^^ I— e COB 6
where l>
l+v/l_e*
For proof, put c = " ., ami 2 cos ^ = .r + - -, expiind the fraction in two
feries of powers of x by the mcbhoJ of (257), and substitute from (768).
800 siiia+siiMa4-/8) + sin(a+-i)8) + ... + siii;a+(;/-l)/3j
sin(a+^^)si.4^
sin -^
801 (•()sa + ('()s(a + /8) + ('()s(a + 2^) + ...+C'()s;a + (/J-l,)^;
802 If the terms in tliese series liuve tlie signs + and —
alternately, change \i into /3-f-7r in tlie resnlts.
Proof. — Multiply the series by 2 sin ^ , and apply (669) and {QQQ).
803 If /3 = — in (800) and (801), each series vanishes.
804 Generally, If /3 = ^'^, and if r be an integer not a
niulti[)lc of n, the sum of tlie r^'' powers of the sines or C(^sines
in (800) or (801) is zero if r he odd; and if /• he even it is
General Theorem. — Dcnotinp^ the sniu of the scries
805 c + r,x + c,r + ...+ r,..." by F (.r) ;
then ccosa + c,eos^« + />j + ...+r„cos(a + H/3) = ^ {e''F(c'')+e-*'F(>r"')],
and
806 fsina + r,sin(«+/5) f ... +c-„sln(a \- n, I) = }-.{,■'' F {e'')-c-'' Fie-'')] .
Proved bv substituting for the sines and cosines their exponential values
(7Gt"'), Ac.
180 PLANE TRIGONOMETRY.
Expansion of the sine and cosine in factors.
807 'f ^" — 2ci?" ?/" cos n ^ + y-""
= 1^-- 2.r//cos^+/] \x'--2.vijQ0^{e^^-Vf
o
to n factors, adding — to the angle successively.
j^
Proof. — By solving theqaadraticon the left, wegefcic=i/(cos?i^ + isinn0)".
The n values of .r are found by (757) and (626), and thence the factors. For
the factors of a'"±y" see (480).
808 sin ?i<^ = 2^^-^ sill <^ siu (^<^ + ^) sin (^^ + -^ j . . .
as far as n factors of sines.
Pboof.— By putting x = ij = \ and ^ = 2</> in the last.
809 If '^ be even,
sin nij> = 2""^ sini^ cos</»(sin'— — sin^<^j (^sin' — — sin-</)j &c.
810 If ^^ be odd, omit cos (/> and make up n factors, reckoning
two factors for each pair of terms in brackets.
Proof. — From (808), by collecting equidistant factors in pairs, and
a pplying (659).
811 COS n<i> = 2"-' sin U + ^J sin U-Jr*^)... to n factors.
Proof.— Put d) + -^ for ^ in (808),
zn
812 Also, if n be odd,
cos n<l> = 2"-' cos <^ /siii'^ — sin'c^j l*^^^^'^ ~ sin'c^j ...
813 If 'i be even, omit cos (^.
PiiOOF.— As in (809).
814 n = 2-^ sin^ sin^ sin '^... sin (^i=ll^.
Proof. — Divide (8o9) by sin (p, and make ^ vanish ; then apply (754).
815 -^ = ''Sl-(-^)lS-(.01S>-(0
PnooF.-Put ^ = ^ in (SOD) and (812) ; divide by (8M-) and make n
intinite.
AJ'hITinXAL FOKMUL.r..
181
817 e'-2cos^ + e'-'
Proved bv sulistitutiiiu' ..•=! + -^-, 1/ = 1 - --f"' *^"^^ " ''o'" ^ •" (^<J").
making n intinitc, aud reducing one series of factors to 4 hiu'^ by putting
z=0. ^
De Moirrrs Proprrfi/ of the ^
Circle. — Take 7' any point, and jr
FOB = d any angle,
JWC = COD = &c. = ^"^ ;
n
OP = ,T ; OB = r.
819 J '" — 2a "r" cos w^ 4- r-"
= Plf PC- PD- ...ion factors.
By (807) and (702), since Pi?" = a- - 2.n- cos 5 + r, &c.
820 H .r = r, IV siiii|^ = PB.PC.PD ... &c.
821 Coles's properties. — If 6 = — — ,
822
v«~r' = PB.PC.PD ... &c.
.r^+r" = Pa.Ph.Pc ... &c.
ADDITIONAL FORMUL/IJ.
823 cot yl+tnii A = 2 eosot- 2 1 = sec A cosec A.
824 c<)secLM+cot2J = cot.4. sec J = 1+taii J taii^i.
826 co=*-^ = cos'— - — sill*-!-.
827
828
829
iaii.l + sec.l = taiirJ-V+^V
tan .1 +tMiiyi . 1 . ,,
= tan .1 tail B.
cot .l + cot Ji
sec"^ cosec/1 = sec. 1-f cosec* ^1.
182 PLANE TRIGONOMETRY.
830 n ^+1^+6 = -|,
tan B tan C+tan C Urn ^4 H-tau A tan B = l.
831 If A-{-B-\-C = Tr,
cot i^ cot C+ cot cot ^4 + cot yl cot B = l.
832 «J»"y + ^i» T=|- tan-i- + tan-i = -J
In a right-angled triangle ABG, G being the right angle,
833 eos2Z^ = g;. tan2B=-^.
834 tani^ =V(^6)- ii+»^ = i («+«»)•
In any triangle,
835 sini(^-J^) = 'i^cosiC.
cosi(^-i?) = ^siniC.
QQa ^inA-B _ a'-b' tan^^+tanj^ ,. c
^^^ siu^ + /i c' ' UmiA-UmiB a-b'
837 2 («' + ^' + ^') = ^^ ^os ^ + ^" ^^^ ^ + "^ ^^^ ^•
838 Area of triangle ABC = ^bc sin A
1 sinS sinC i / o ;o^ sin A sin iJ
839 = _2abc_ ^^g i^ ^^g iy^ ^.^^ 1 C\
840 = i {a+b-^cY tan i^ tan \B tan ^C.
With the notation of (709),
841 r = i{a + b + c) Um^A tan \n tan \C.
842 i^/^»' = ,''('''. . A = v/rr,nr,.
843 ^/cosl+^y cos/Z + r (M)sr:= l/»* sin.l sin/isinC.
ADDITIONAL FORMUL/E. 183
844 R-irr = \ {a cot A-\-h cot li-\-p cot C) = sum of per-
pendiculars ou the sides from centre of circumscribing circle.
Tliis may also bo shown by applyinf:^ Enc. VI. D. to the circle described
on R as diainetcr and the quadrilateral so formed.
845 >*a n >\ = fi^^t^ <*os I A cos }yB cos 1 C.
846 r = v/(r, r..) + ^ (r. r„) + v/(n. n)^_
847 1=1 + - +—. taii.M==V— •
>• >'a >*6 >'c *'6''c
849 If be the centre of inscribed circle,
0^ = — , ■ , cos I A.
a-\-o-{-c
850 rt (^> cos C—c QosB) = ¥—c'.
851 /> cos7i+c cos C = c cos {B-C).
852 'f cos .1 -\-b cos 7i+c cos C = 2a sin 2^ sin C.
'1(1 sin /i sinC
853 cos ^1 + cos B + cos C =l-\r
a + 6+c
854 If s = \{a + h + c),
1 —COS" a — COS" h — COS" c-\-'l cos a cos h cos c
= 4 sin.v sin {s — a) sin {s — h) sin (-v — c).
855 — 14-('Os-r/ + cos-/>4-<'os"^*+2 cosa cos 6 cose
= 4 cos A- COS {s—a) cos {s—h) cos (.v— c).
856 4 COS — COS — cos —
= COS .y 4- COS (.V — ^/) + cos (.v — />) + cos {s — c).
. . (I . h . (•
4 sni — sm — sni —
= — sin* + sin (*—«) + sin (*— 6)+sin (*--c).
858
.. = .(i+^+^ + ...).s(i-,.!^U + ...).
Proof. — Equate cocfScients of 0^ in the expansion of ' - by ("Oi) and
(81")) or of cos^ by (7t".'>) and (SlG).
184 PLANE TRIGONOMETRY.
859 Examples of the Solutions of Triangles.
Ex. 1 : Cask II. (724).— Two sides of a triangle b, c, being 900 and 700
feet, and the included angle 47° 25', to find tlie remaining angles.
tan ^^=^ = — "^ cot 4=1 cot 23° 42' 30" ;
2 b+c 2 8
therefore log tan ^ (J5 — C) = log cot — —log 8 ;
therefore i tan^ (J5-0) = i cot 23° 42' 30" -3 log 2,
10 being added to each side of the equation.
.-. L cot 23° 42' 30" = 10-3573942* / .-. ^ (5- 0) = 15° 53' 19-55"*
3 log 2 = -9030900 j and |(5 + C) = 66° 17' 30"
.-. itan|(B-(7) = 9-4543042 ( .-. 5 = 82° 10' 49-55"
And, by subtraction, G = 50° 24' 10-45".
Ex. 2: Case III. (732).— Given the sides a, b, c = 7, 8, 9 respectively,
to fiud the angles. . .
A /( s-b)(s-c) _ /4.3 _ /2 .
*""-2=V sis-a) -Vl2:5-Vl0'
therefore Ltau^ = 1<^+^ (log 2-1) = 96505 15;
2
therefore U = 24° 5' 41-43".*
\B is found in a similar manner, and G = 180°— J.— ^.
Ex. 3. — In a rigbt-angled triangle, given the hypotenuse c = 6953 and
a side 6 = 3, to fiud the remaining angles.
Here cos A = — ^ . But, since A is nearly a right angle, it cannot be
6953
determined accurately from log cos A. Therefore take
. A ll-cosA _ /3475
''"^^V"""^ -V6953'
therefore L sin ^ = 10 + ^ (log 3475 -log 6953) = 9-8493913;
therefore ^ = W 'o9' ro-o2"*
therefore ^ = 89° 58' 31 04" and i; = 0° 1' 28-96".
* See Chambers's Mathematical Tables for a concise explanation of the
method of obtaining these figures.
SPHERICAL TRIGONOMETRY.
INTRODUCTORY THEOREMS.
870 Definitions. — Planes through the centre of a sphere
intersect the surface in ^yr^f circles; other planes intersect
it in small circh's. Unless otherwise stated, all arcs are
measured on great circles.
The poles of a great circle are the extremities of the
diameter perpendicular to its plane.
The sides a, 6, c of a spherical triangle are the arcs of
great circles BG, CA, AB on a sphere of radius unity; and
the angles .1, B, C are the angles between the tangents to the
sides at the vertices, or the angles between the planes of the
great circles. The centre of the sphere will be denoted by 0.
The ])olar triangle of a spherical triangle ABC has for its
angvdar points A\ B\ C\ the poles of the sides /?C, CA, AB
of the primitive triangle in the directions of A, B, C respec-
tively (since each great circle has two i)oles). The sides of
A'B'C are denoted by a\ h\ c.
871 The sides and angles of the
polai- triangle are respectively the
supplements of the angles and
sides of the primitive triangle ;
that is,
n'-\-A = //+/; = r'-\-(' ^ n,
Ltt EC pi'oduc-cd cut the sidi'S A'H', \,
C'A' in a, 11. 11 is tJif pole of A'C\
therefore 1!II = y. Siniilarly C'li = ^,
therefore, by adilition, a -|- CJ[=ir and GII=.A\ because A' is the pole of BC.
The polar diai.'r;im of a spherical i»nIytron is formed in the same way, and
tlie same relations subsist between the sides and au'^les of the two Hy^ure.*'.
2 u
186 SPHERICAL TRIGONOMETRY.
Rule. — Hence, any equation between the slides and angles
of a spherical triangle jyroduces a siqjplementary equation by
changing a into tt — A and A into tt— a, ^c.
872 The centre of the inscribed circle, radius r, is also the
centre of the circumscribed circle, radius R\ of the polar
triangle, and v-^ll' = ^tt.
Pkoof. — In the last figure, let be the centre of the inscribed circle of
ABC; tlun 01), the perpendicular on BC, passes through A\ the pole of BG.
Also, OD = r; therefore OA'=h'r—r. Similarly 0B'= 0G'=Itt—7-; there-
fore is the centre of the circumscribed circle of A'B'C, and r-\-B'= ^tt.
873 The sine of the arc joining a point on the circumference
of a small circle with the pole of a parallel great circle, is equal
to the ratio of the circumferences or corresponding arcs of the
two circles.
For it is equal to the radius of the small circle divided by the radius of
the sphere ; that is, by the radius of the great circle.
874 Two sides of a triangle are greater than the third.
[By XI. 20.
875 The sides of a triangle are together less than the cir-
cumference of a great circle. [By XI. 21.
876 The angles of a triangle are together greater than two
right angles.
For ir—A + TT — B + TT—C is < 27r, by (875) and the polar triangle.
877 if two sides of ajriangle are equal, the opposite angles
are equal. [By the geomctric;il proof in (89-i).
878 If two angles of a triangle are equal, the opposite sides
are equal. [By the polar triangle and (877).
879 'I'he greater angle of a triangle has the greater side
opposite to it.
PnooF. — If J? be > ^, di-aw tie arc ItD mooting AC in P, and make
Z A lil> = .1, therefore BB = AD ; but BD + nG>BC, therefore AC>BC.
880 The greater side of a triangle has the greater angle
opposite to it. [By tho jx.lar triangle and (879).
OBL IQ i'E-A S UL ED Tit I A NO L ES.
187
RIGHT-ANGLED TlUAN(i LKS.
881 Napier s Rules. — Tn the triangle J BO let C be ;i li^Hit
angle, then «, QTr — 7?), (W — r), (Jtt — ./), and//, are called
the five circular jxn-/^. Taking any part for middle part,
Napier's rules are —
I. sineofun'diUrjiarf = jn-odiirt of faiKjfiifs ifaJjdcciif jvtrfs.
II. sine of middle part = product of cosines of opposite parts.
In applying the rules we can take J, B, c instead of their
complements, and change sine into cos, or vice vers/i, for those
parts at once. Thus, taking b for the middle part,
sin b = tan a cot .7 = sin B sine.
by the
Ten equations in all are given
rules.
Proof. — From any point P in OA, draw
PR perpendicular to OC, and EQ to OB;
therefore I'liQ is a right aiigle ; tlierefore OB
is perpendicular to PR and Q1i\ and therefore
to PQ. Then prove ai.y (oruiuhi by proportion
from the triangles of the tetrahedron OPQR,
which are all right-angled. Other\vise, prove
by the formulas for oblique-angled triangles.
OBLIQUE-ANGLED TRIANGLES.
882
cos a = cos b cos c-fsin h sin c cos .1.
Pkook. — Draw tangents at A
to the sides c, b to meet OH, OG
in D and E. Express VE'^ by
(7<i2) applied to each of tho
triangles DAE and 1)0E, and
subtiact.
If .1/? and. I'' are both > J,
]trodiicc them fo meet in ,1', the
pole of A, and employ the Iri-
augle A'BC.
If AB alone be > ^. pro-
duce 7>.l to meet BC.
The sup})leineutary formula, by (871), is
883 <*<>^l =r —cos /; cos r-f sill />' >in r <•<)>//.
188 8PHEEICAL TRIG0XOMETR7.
A /sill (.s — //) sill (.V — c
884 ^m^=yjt
8ill b 8111 c
885 cos4 = J ^"if-^i"(/-^^) .
^ ^ sill /> sill r*
oo/? i A /sill (.V — />) sill (5— r) T 1/ I / I \
886 tan— -=\/ ^^ A— — ^ — —-L whevQ fi = i{a-{-h-\-c).
'I ^ sill A- Sill ys — a)
Proof. — sin^— = \ (1 — cos^l). Substitute for cos /I from (872), and
throw the nnmerator of the whole expression into factors by (673). Similarly
for cos -.
The supplementary formulae are obtained in a similar way,
or by the rule in (871). They are
887 cos4 = J<^<^^{s-'iU'<^HS-C) _
2 Y sm B t*iu C
ooo . a /— COS .S cos (^' — ^
888 sill -n- = V • /> • /. —
2 ^ siu B sin C
ooft I t^ / — t'os 8 COS (^' — ^)
889 tan = d — — — — ^— - — '—
2 ^ Qos{S — B) cos(.S— C)
where S = i (A-[-B+C).
890 Let (T = \/siu.s* sin (* — //) sin (s — b) sin (a' — c)
= ^ \/l + ^ cos a cos 6 cos c— cos' a — cos- /> — cos' c.
Then the supplementary form, by (871), is
891 S = a/— cos 8' cos [S—A) cos [S — B) cos (,N — C'j
= 1^ \/ 1—2 cos^ cosii cusC— cos I — cos- Z^— cos-C.
2(r *^S
892 sin ^ — . . — . sinrt= . H • /> •
sin /> sin c sm fi sm C
[By sin^ = 2 sin fy cos ^ and (884, 885), &c.
893 The following rules will produce the ten formula^
(884 to 892)—
I. Write sin before each factor In the s cahics o/sin— ,
OULIQUE-AKCLEI) TRIASCLES. 189
cos \^ , tan ^ , sin A, and A, /// Phinr Trii/->iioinetri/ (701—
707), fo chta'ni flif cor I'cf^pouding f annul tn in Spherical Trijo-
iionirfri/.
11. To ohfain the svpplementarij forms of the five resultn^
transpose lanje and small letters everyivhere^ and transpose
sin and cos everywhere hut in the denominators, and write
minus before cos S.
n(\A sill A _ sin B _ sin C
sin a sin fj sin c
PuooF.— By (8S-2). Otliei-wise, in the figure of 882, draw PN perpendi-
cular to HOC, and NR, XS to UB, 00. Prove PRO and FSO right angles
hy T. 47, and therefore PN = OP sin c sin L' = OP sin ^ sin (J.
895 COS 6 cos (7 = cot a sin 6— cot A sin C
To remember tins formnla, take any four consecutive angles
and sides (as a, C, h, J), and, calling the first and fourth the
extremes, and the second and third the middle parts, employ
the following rule : —
Rule. — Product of cosines of middle parts = cot extreme
side X sin middle side — cot extreme angle X sin middle angle.
Pkoof. — In the formula for cos a (882) substitute a similar value for
cos c, and for sin c put sin C — — - .
smA
896 NAPIER'S FonmrL.E.
(1)
tan i (A -B) = ^-^-) — -^ cot — .
sni-o-(r< + 6) 2
/o\ i 1 ' < I in i'O^l ((t — h) . C
(-^ '""^^-' + ^^) = .....i (»+/,) *•"* 2-
(.3) (,.„i("-/')=jii^Tpqr7]y*""i7-
/ «\ A 1 / I /\ <*os i- (A — li) , r
(4) tan.U'' + '') = — r|:i:p7j^t»»^.
Rule. — /// ///c rahic of tan J- (A — 13) change sin to cos /o
o&^«m tan^(A + B). To obtain (3) «?«(/ (-4)/rc)??t (1) and (2),
transpose sides and angles, and change cot to tan.
Proof. — In the values of cos.l and cos 7^, by (883), put msina and
m sin b for sin ^ and sin B, and add the two equations. Then put
sinj4±sinZ> , . r i /.>nA r»i-T.i\
m = -^ . — , and transform by (()70-d72).
sill a ± sill 6
190
SrUERICAL TRiaOKOMETR Y.
897
(1)
(2)
(3)
GAUSS'S FORMULAE.
smi(.4 + 7J) _ (io^\{a-h)
cos ^6^ C0S-2C
cos^C sin^c
cosi(^ + i^) _ cosi(rt+&)
sin^C cosl^c
cosi(^-/J) _sini((< + ^)
sill ^ C sin ^c
From any of these formulae the others may be obtained
by the following rule : —
Rule. — Change, the sign of the letter B {large or small) on
one side of the equaJion, and ivrite sin for cos and cos for sin
on the other side.
Proof. — Take sin-^- (^ + 7?) = sin-^x4 cos }jB + coslA sin iZ>,
substitute the s values by (88-i, 885), and reduce.
SPHERICAL TRIANGLE AND CIRCLE.
898 Let r be the radius of the in-
scribed circle of ABG ; r« the radius of
the escribed circle touching the side a,
and B, Ba the radii of the circumscribed
circles ; then
(1) tan r = tan ^A sin (v— «) = ^ —
(3)
(4)
sni a .
SI
n^
sin?r^l siiioTi siiioC
2 cos ^A cos ^B cos I ^
cusS+cos (^^ — yl) + cos (S — B)-\-liic.
Pkook. — Tlio first value is found from tlie
ri^lit-auglcd triangle OAF, in which AF = s — a.
The otliei- vahies I)y (881-892).
spjiEuicAL 'nnASiii.i-: asi> ciikli:.
l'.»l
899 (1) i;ni r„ = tail ,\/l sin.v =
111 (.V — </)
(3) ^ -f'''[' sin M cos IB cos \C
sin A
(■>■) = T
i.s \A sini/i sin ^(7
oV
_(.os.S-c()s(N-.l) + c<)s(N-yi)+t'<>'^i''>'-^-')
Proof.— From tlie right-angled triangle O'AF', in which AF'= s.
NoiK.— The first two values of tan r„ may be obtained from those of
tan r by interchanging s and s—a.
900 (1) tiiii« =
(3)
(4)
tan Tift —cos S
c()s(.S-.l) S
_ sill \<i
sin A cos yj cos ^c
_ 2 sin ^^< sin ^j sin ^c
.ill .v + sin (,v — (i) + sin (.v — A) + c^'C.
Proof. — The fir.st value from the right-angled
triangle OBD, in which Z OTID = S — A. ^Tlie other
values by the formulie (887-892).
901 (1) h.nR„ =
tan^n _oos(S— ^)
— cds.S
05) =.-
(1.) =
(5) =
Sin tU<
sin A sin lb sin U'
2 sin hn <*os ^b cos Ic
;in (.V — rO + sin (.v — />) + .^in (.v — c)
Proof.— From the right-angled triangle o7>7^ in which z O'/;/^ = tt-.S.
192
SPHERICAL TRiaONOMETBY.
SPHERICAL AREAS.
902 area of ABC = (A-^B+C-tt) r- = Er
wliere E = ^+7i + C— tt, the spherical excess.
Proof. — By adding the three lunes
ABDG, BGEA, GAFB,
and observing that ABF = CDE,
get (
A+]l+^] 27rr' = 27rr + 2ABC.
TT IT IT I
903 AREA OF SPHERICAL POLYGON,
n being the number of sides,
Area = {interior Angles — (» — 2) tt] r'
= {277— Exterior Angles} r^
= {27r— sides of Polar Diagram} r.
The last value holds for a curvilinear area in the limit.
Proof. — By joining the vertices with an interior point, and adding the
areas of the spherical triangles so formed.
904 GagnolVs Theorem.
• 1 L^ _ \/ {si" ^ ^^^ { s — u) ^\n(s — h) sin(.y — r)y
sin c> tj — -T ■\ r-j \
2 cos -^a cos ^b cos ^c
Proof. -Expand sin \_\{A + B)~},{Tr-G)'] by (628), and transform by
Gauss's equations (897 i., iii.) and (669, 890).
905 LlhulUier's Theorem.
UuiE = y [tan k tan i (.s— ^0 ian I {.s^-h) tan i {s-c)}.
Proof. — Multiply numerator and denominator of the left side by
2 cos 1 (A + B-G + n) and reduce by (6G7, 668), then eliminate i {A+B)
by Gau8.s'.s foniiulio (S!i7 i., iii.) Tnuisfonii by (()72, 673), and substitute
from (886).
roLYIIHIili-ONS.
\\y.\
Vi)\M\VA)\{()NH.
Let tlic iiuiiilx'i- of fiiccs, solid angles, and cdircs, of any
|)(.lylicdr<.n hr /•', N, and /■>' ; tlu-n
906
//+.S = E+'2.
Pi;,),,i.-. — Project till' |)olvhc(lron upon Jin internal splicre. Let vi =
number of sides", and s = sum of anj^les of one of tlie spherieal poIy-,'on.s so
formed. Then its area = [s — {»i — 2) tt] r. by fOn.'i). .Sum this for all the
polygons, and equate to 4n-?-*.
THE FIVE REGULAR SOLIDS.
TiOt VI he the niimher of sides in each face, ii the nninher
of ]ilane angles in each solid angle ; therefore
907 ))iF= nS = '2E.
Fi-om these equations and (OOd), Hnd I\ S, and I'J in terms of ni ajid
«, thus,
1 ^ w / 1 1_ 1\ .1 ^ » ( 1 ^_ 1 _1\ 1 = 1 + ^ _ 1
F '2 V nt n 2 / ' N 2 V vi n 2 / ' E m h -1 '
In order that F, S, and E may be positive, we must havi- + >
a rehition which admits of five solutions in whole numbers, corresponding to
the five regular solids. The values of hi, », F, S, and E for the five regular
solids are exhibited in the following table : —
m
3
n
3
F
4
8
4
E
Tetrahedron
6
Hexahedron
4
3
6
8
12 i
Octahedron
3
4
8
n
12
Dodecahedron
5
3
12
20
oU ,
Icosahedron
i
3
5
20
12
30 i
908 'I'he siini of all the i)lane angles of any polyhedron
= '27r{S-'2);
Or, Four rliflit mnihn for crt ri/ n-rtcx less ciijJit riyht aiKjles,
2c
194
SPHEBIGAL TRIGONOMETB Y.
909 If I be the angle between two adjacent faces of a
regular polyhedron,
smi/ = cos •- sill — .
n m
Proof.— Let J*(^ = a be the edi^e, and *S'
the centre of a face, T the middle point of
PQ, the centre of the inscribed and circum-
scribed spheres, ABC the projection of PST
upon a concentiic sphere. In this splierical
triangle,
C =
and B
I.
2 n rii
Also STO
Now, by (881, ii.),
cos A = sin B cos BG ;
that IS, cos — = sui — sin ^1.
n m
= PST.
Q. e. d.
If r, B be the radii of the inscribed and circumscribed
spheres of a regular polyhedron,
910 r = 4 tan ^I cot — , « = -^ tan ^I tan ^.
Proof. — In the above figure, OS = r, OP = B., PT = -[^ ; and
OS = PT cot -'^ tan J J. Also OP = PT cosec AG, and by (881, i.),
5in AG = tan BG cot A = cot }J cot ; therefc)re, &c.
n
ELEMENTARY GEOMETRY.
MISCELLANEOUS PROPOSITIONS.
920 To find the point in a given line QY, the sum of whose
distances from two fixed points <S', S' is a minimum.
Draw SYB at right angles to QY,
making r7i' = TS. Join BS\ cntting
QY in P. Then P will be the required
point.
Proof. — For, if D be any other point
on the line, SD = Dli and SP = PR.
But BD + US' is > BS'; therefore, (to.
B is called the reflection of the point i9,
and SP.S" is the path of a ray of light
reflected at the line QY.
If ^, S' and QY are not in the same plane, make SY, YB equal perpen-
diculars as before, but the last in the plane of S' and QY.
Similarly, the point Q in the given line, the diSerence of whose distances
from the fixed points 8 and B' is a maximum, is found by a like construction.
The minimum sum of distances from 8, S' is given by
(^7^ + ^-7^)-= SS"'+4^SY.S'Y'.
And the maximum difference from S and R' is given by
{SQ-R'QY= {SHy-4^SY.IVY'.
Proved by VJ. D., since SBB'S' can be inscribed in a circle.
921 Hence, to find tlic
shortest distance from P
to Q en route of the hncs
AB, BC, CD', in other
words, the path of the ray
reflected at the successive
surfaces AB, BG, CD.
Find P, , the reflection of P at
the first surface; then Pj, the
reflection of 2', at the second sur-
face ; next Pj. the reflection of P,
at the third surface ; and so on if
196
ELEMENTARY GEOMETRY.
there be more surfaces. Lastly, join Q with P,, the last reflection, cutting
CD in a. Join aPj, cutting BG in b. Join hP^, cutting AB in c. Join cP.
PcbaQ is the path required.
The same construction will give the path when the surfaces are not, as
in the case considered, all perpendicular to the same plane.
922 If the straight line d from the vertex of a triangle
divide the base into segments _p, q, and if h be the distance
from the point of section to the foot of the perpendicular from
the vertex on the base, then
The following cases are important : —
(i.) When p = q, b'+c' = 2q'-^2d' ;
i.e., the sum of the squares of tiuo sides of a
triangle is equal to twice the square of half
the base, together ivith tivice the square of the
bisecting line drawn from the vertex.
(ii.) When p = 2q, 2b'-]-c' = 6q'+M\
(iii.) When the triangle is isosceles,
b'= c' = 2)q + (P.
[II. 12, 13.
(II. 12 or 13)
923 If be the centre of an equilateral triangle ABC and
P any point in space. Then
FA'+PB'-^PC = 3 {PO'-\-OA').
Proof.— PB' + PC = 2PD' + 2BD\ (922, i.)
Also PA' + 2PD' = 60D" + SPO\ (922, ii.)
and B0 = 20D;
therefore, &c.
CoR. — Hence, if P be any point on the
surface of a sphere, centre 0, the sum of the squares of
its distances from J, B, G is constant. And if r, the radius
of tlie sphere, be equal to OA, the sum of the same squares is
equal to Or".
MISCELLANEOUS PROPOSITIONS.
lo:
924 The sum of \\\c squares of
the sides of a (]uadrihitcral is equal
to the sum of the squares of the
diagonals plus four times the square
of the line joiuinpr the middle points
of the diagonals. (9-J2, i.)
925 Cor. — The sum of the squares
of the sides of a parallelogram is
equal to the sum of the squares of the diagonals.
926 In a given line AG, to find a point X whose distance
from a point P shall have a given
ratio to its distance in a given
direction from a line AB.
Through P draw BPC parallel to the
given direction. Produce AP, and make
CE in the given ratio to CB. Draw PX
parallel to EC, and XY to CB. There are
two solutions when CE cuts AP in two
points. [Proof.— By (VI. 2).
927 To find a point X in AC,
whose distance XY from AB parallel
to BC shall have a given ratio to its
distance XZ from BG parallel to AD.
Draw AE parallel to BC, and having to
AD the given ratio. Join BE cutting AG \n
X, the point required. [Proved by (VI. 2).
928 To find a point X on any
line, straight or curved, whose
distances XY, XZ, in given direc-
tions from two given lines AP, AB,
shall be in a given ratio.
Take P any point in the first line.
Draw PB parallel to the direction of XY,
and BC parallel to that of XZ, making
PB have to BC the given ratio. Join PC,
cutting AB in I). Draw DE parallel to
CB. Then AE produced cuts tlie line in
X, the point required, and is the locus of
such points. [Proof. — By (VI. 2).
198
ELEMENTARY GEOMETRY.
929 To draw a line XY through a given point P so that
the sogmeuts XP, FY, intercepted by a given circle, shall be
in a given ratio.
Divide the radius of the circle in that ratio,
and, with the parts for sides, construct a triangle
PDC upon PC as base. Produce GD to cut the
circle in X Draw XPY and GY.
Then PD + DC = radius ;
therefore PD = DX ■
But CY=CX;
thereforePDisparallelto(7r(I.5, 28) ; therefore
&c., by (VI. 2).
930 From a given point P in the
side of a triangle, to draw a line PX
which shall divide the area of the tri-
angle in a given ratio.
Divide EG in D in the given ratio, and
draw AX parallel to PD. PX will be the line
required.
ABD : ADG = the given ratio (VI. 1), and
API) = XPI) (I. 37) ; therefore, &c.
931 To divide the triangle ABG in a given ratio by a line
XY drawn parallel to any given line AE.
Make BD to BG in the given ratio. Then
make PY a mean proportional to BE and BB,
and draw YX parallel to EA.
Proof. — AB divides ABG in the given ratio
(VI. 1). Now
ABE : XBY :: BE : BD, (VI. 19)
or :: ABE: ABD;
therefore XBY = ABD.
932 If the interior and exterior vertical angles at P of the
triangle APB be bisected by straight lines which cut the base
in G and D, then the circle circumscribing GPD gives the
locus of the vertices of all triangles on the base AB whose
sides AT\ PP m-o in a constant ratio.
MISCELLANEOUS PROPOSITIONS.
199
Proof. —
The Z CPD = i(APB + BPE)
= a right angle ;
therefore P lies on the circumference of
the circle, diameter CD (III. 31). Also
AP : PB :: AC : GB :: AD : DB
(VI. .S, and A.), a fixed ratio.
933 AD is divided harmonically in B and G ; i.e., AD : DB :: AG I GB ;
or, the ivhole line is to one extreme part as the other extreme part is to the middle
part. If we put n, b, c for tlie lengths AD, BD, OD, the proportion is
expressed algebraically hy a : h :: a — c : c — b, which is equivalent to
'+! = -■
a c
934 Also AP : BP = 0A: OG = OG : OB
and AP' : BP' = OA : OB, (VI. 19)
AP'-AO' : GP" : BP'-BC\ (VI. 3, <k B.)
935 If Q be the centre of the inscribed circle of the triangle
ABG, and if AQ produced meet the circumscribed circle,
radius E/\nF; and if FOG bo a diameter, and AD perpendi-
cular to BG ; then
(i.) FC=FQ = FB='lR^\n:^.
ill) Z.FAI) = FAO=},{B-C),
and /_CAG = \{B^C).
Proof of (i.) —
Z.FQG= QGA-\-QAC.
But QAG =z QAB = BGF ; {\\\. 2\)
.-. FQG = FGQ; .-. FG = FQ.
Similarly FB = FQ.
Also Z GCF is a right angle, and
FOG = FAG = \A; (III. 21)
.-. FC = 2Esin4.
936 If -K, r be the radii of the circumscribed and inscribed
circles of the triangle AUG (see last figure), and 0, Q, the
centres; then ()Q^=IV—2Hr.
Proof. — Draw QH perpendicular to ^C ; then QII = r. By the isosceles
triangle AOF, OQ' = li'-AQ.QF (922, iii.), and QF = FG (935, i.), and
by similar triangles QFC, AQH, AQ : QII :: OF : FG ;
therefore AQ.FC = OF.QU = 2 Rr,
200
ELEMENTARY GEOMETRY.
The problems known as the Tangencies.
937 Given in position any three of the following nine data —
viz., three points, three straight lines, and three circles, — it is
required to describe a circle passing through the given points
and touching the given lines or circles. The following five
principal cases occur.
938 I. Given two points A^ J9, and the straight line CD.
Analysis. — Let ABX be the required
circle, touching GD in X. Therefore
GX'=^GA.GB. (III. 36)
Hence the point X can be found, and the
centre of the circle defined by the inter-
section of the perpendicular to (yD through
X and the perpendicular bisector of AB.
There are two solutions.
Otherwise, by (926), making the ratio
one of equality, and DO the given line.
Cor. — The point X thus determined is the point in GD at
which the distance AB subtends the greatest angle. In the
solution of (941) Q is a similar point in the circumference GD.
(III. 21, & I. 16)
939 II- Given one point A and two straight lines DG, DE.
In the last figure draw AOG perpendicular to DO, the bisector of the
angle D, and make OB = OA, and this case is solved by Case I.
940 III- Given the point P, the straight line DE, and the
circle AGF.
Analysis. — Let PEF be the required
circle touching the given line in E and the
circle in F.
Through IF, the centre of the given
circle, draw AHGD perpendicular to DE.
Let K be the centre of the other circle.
Join 11 K, passing through Z'', the point of
contact. Join AF, EF, and AF, cutting
the required circle in X. Then
^DHF = LKF; (1.27)
therefore UFA = KFE (the halves of equal
angles) ; therefore AF, FE are in the same
straight line. Then, because AX.AP = AF.AE, (III. 36)
and AF.AF :c= AG .AD by similar triangles, therefore ^X can bo found.
A circle must then be described through P and X to touch the given line,
THE PROnLEMS OF THE TANOE^X'rES.
201
by Caso I. There are two solutions with exterior contact, as appears from
Case 1. These are indicated in the diagram. There are two more in wliich
the circle J C lies witliin the described circle. The construction is quite
analogous, C taking the place of A.
941 IV. Given two points Ay B and
the circle CD.
Draw any circle through A, B, cutting tho
required circle in C, D. Draw AB and DC,
anil lot them meet in F. Draw FQ to touch
the given circle. Then, because
FC.FD = FA . FB = FQ\ (TIL 30)
and the required circle is to pass through
A, B ; therefore a circle drawn through A, B, (^
must touch FQ. and therefore the circle CJ),
in Q (III. 37), and it can be described by Caso
I. There are two solutions corresponding to
the two tangents from F to the circle CD.
942 V. Given one point P, and two circles, centres A and B.
Analysis. — Let FFO be tho required circle touching the given ones in F
and 0. Join the centres QA, QB. Join FG, and produce it to cut tho
circles in E and //, and the lino of centres in 0. Then, by the isosceles
triangles, the four angles at F, F, G, II are all equal ; therefore AE, BG are
parallel, and so arc .1/'', BII\ therefore AO : BO :: AF : Hit, and is a
centre of similitude for tho two circles. Again, Z IIJiK = 211 IjK, and
FAM = 2FNM (HI. 20) ; therefore FNM = IILK= IIGK (III. 21) ; there-
fore the triangles OFN, OKG are similar; therefore OF . OG = OK . OM ;
therefore, if OF cut the required circle in X, OX . OF = OK. ON. Thus
the point A' can be found, and tho problem is reduced to Case IV.
Two circles can be drawn througli F and X to touch tho given circles.
One is the circle FFX. The centre of the other is at tlie ]K)int where EA
and JIB meet if produced, and this circle touches the given ones in F and //.
943 An analogous construction, employing the internal centre of simili-
tude 0', determines the circle which passes through F, and touches one given
circle externally and the other internally. See (1017-1)).
The centres of similitude are tho two points which divide tho di.stanco
between the centres in the ratio of the radii. See (1037).
2d _
202
ELEMENTAUY C'EOMETRY.
944 Cor. — The tangents from to all circles which touch
the given circles, either both externally or both internally,
are equal.
For the square of tbe tangent is always equal to OK. ON" or OL . OM.
945 The solutions for the cases of three given straight lines
or three given points are to be found in Euc. IV., Props. 4, 5.
946 In the remaining cases of the tangencies, straight lines
and circles alone are given. By drawing a circle concentric
with the required one through the centre of the least given
circle, the problem can always be made to depend upon one
of the preceding cases ; the centre of the least circle becoming
one of the given points.
947 Definition. — A cenb-e of similarity of firo jjlane curves
v\s a point such that, any straight line being drawn through it
to cut the ciirccs, the segments of the line intercejited between
the ]Joint and the curves are in a constant ratio.
948 If AB, AG touch a circle at B and
C, then any straight line AEDF, cutting
the circle, is divided harmonically by
the circumference and the chord of con-
tact BG.
Proof from AE . AF = AB'. (III. 3G)
AB' = BD.DG+AD\ (923)
and BB.BC=EB. BE. (III. 35)
949 If " 5 /^j 7> in the same figure, be the
perpendiculars to the sides of ABG from
any point E on the circumference of the
circle, then /3y = a".
Prooi'. — Draw the diameter BII=d ; then EB'' = ft(l, because BEH is a
right angle. Similarly EG^ = yd. But EB . EC=ad (Yl. D.), therefore etc.
950 If EE be drawn parallel to the base
BG of a triangle, and if EB, FG intersect
in 0, then
AE : AG :: EG : OB :: FO : OG.
By VI. 2. Since each ratio = FE : BG.
Cou.— If AC = n . AE, then
BE = (n-j-l)OE.
MISCELLANEOUS riiOrOSlTIOKS.
203
951 T1h> tliroc lines drawn from the ann^les of a tii;in<^^l(' to
the niiihUe jioints of tlie opposite sides, intersect in tl»e same
point, and divide each other in the ratio of two to one.
For, l>y tlio last theorem, any ono of these lines is divided by each of tho
others in tho ratio of two to ono, measuring from tho same cxlrcniity, and
nuist therefore bo intersected by them in tho same point.
This point will bo referred to as the ccntruid of the triangle.
952 The perpendiculars frora the angles upon tho opposite
Bides of a triangle intersect in the same point.
Draw BE, CF perpendicular to the sides, and let A
them intersect in O. Let AO meet IW in J>. Circles
will circumscribe AEOF and BFEC, by (III. 31) ;
therefore Z FAO = FEO = FCB ; (III. 21)
therefore Z BDA = BFC = a right angle ;
i.e., AO \h perpendicular to BC, and therefore the
perpendicular from A on BC passes through 0.
is called the orthocentre of the triangle ABC.
CoTi. — The perpendiculars on the sides bisect the angles
of the triangle DEF, and the point is therefore the centre
of the inscribed circle of that triangle.
Proof. — From (III. 21), and the circles circumscribing OEAF and OECD.
953 If the inscrilied circle of a triangle ABC touches the
sides a, h, c in the points D, E, F ; and if tlie^ escribed circle
to the side a touches a and h, c produced in D', TJ', F' ; and if
then
BF' = nD'= CD = s-c,
and AE' = A F' = s ;
and similarly with respect to
the other segments.
Proof. — The two tangents from
any vertex tocithercirclfbeingoqual,
it folif)\vs that ll) + r= half tho
perimeter ol' A Bt\ whicli is made up
of three pairs of equal segments ;
therefore CD = s — c.
Also
AE'+ A F'= A r + Cl> + , 1 7? + Bjy
= 2-- ;
therefore AE'
AF' = s.
204
ELEMENTARY QEOMETBY.
The Nine-Point Circle.
954 The Nine-point circle is the circle described through
I), E, F, the feet of the perpendiculars on the sides of the
triangle ABG. It also passes through the middle points of
the sides of ABG and the middle points of OA, OB, OG ; in
all, through nine points.
Proof. — Let the circle
cut the sides of ABG
again in G, H, K; and
OA, OB, OG in L, M, N.
/.EMF=EDF (III. 21)
= 20DF (952, Cor.);
therefore, since OB is the
diameter of the circle cir-
cumscribing OFBD (III.
31), ill is the centre of
that circle (III. 20), and
therefore bisects OB.
Similarly OG and OA
are bisected at N and L.
Again, Z MGB = MED (III. 22) = OGD, (HI. 21), by the circle circum-
scribing OEGD. Therefore MG is parallel to OG, and therefore bisects BG.
Similarly H and K bisect GA and AB.
955 The centre of the
nine-point circle is the
middle point of OQ, the
line joining the ortho-
centre and the centre
of the circumscribing
circle of the triangle
ABG.
For the centre of the N. P.
circle is the intersection of
the perpendicular bisectors
of the chords DG, EH, FK,
and these perpendiculars
bisect OQ in the same point
N, by (VI. 2).
956 The centroid of
the triangle jUiG also lies on the line OQ and divides it in
B so that OB = 2BQ.
Pkook.— The triangles QUG, OAB are similar, and AB =: 2nG ; there-
fore A = 2GQ ; tluM-eforo Oli = 2h'(2 ; and Ali = 2ii'c; ; therefore B is the
centroid, and it divides 0(2 as stated (051).
CONSTIiUCTION OF Till ANGLES.
"201
957 Hence the line joining the centres of the circumscribed
and nine-point circles is divided harmonically in the I'atio of
2 : 1 by the centroid and the orthocentre of the ti-ian<^le.
These two points are therefore centres of similitude of the
circiimscri])ed and nine-point circles ; and any line drawn
through either of the points is divided by the circumferencca
in the^-atio of 2 : 1. See (1037.)
958 The lines BE, EF, FD intersect the sides of AJiC in
the radical axis of the two circles.
For, if EF meets BC in F, tlieu by the circle circumscribing FCEF,
FE . FF = FC . FB ; therefore (III. 3G) the tangents from F to the circles
are equal (985).
959 The nine-point circle touches the inscribed and escribed
circles of the triangle.
Proof. — Let be the orthocentre, and 7, Q
the centres of the inscribed and circumscribed
circles. Produce AI to bisect the arc 1>G in T.
Bisect AG in L, and join GL, cutting AT in .S'.
The N. P. circle passes through G, V, and
L (9o-i), and !» is a right angle. Therefore
GL is a diameter, and is therefore = R= QA
(957). Therefore GL and QA are parallel.
But QA = QT, therefore
A o., . ,A
GS = GT = CT sin
2Ii sin-
(935, i.)
Also ST =2GS cos 6
(e being the angle GST = GTS).
N being the centre of the N. P. circle, its
radius = NG = ^R; and r being the radius of
the inscribed circle, it is required to shew that
NI = NG-r.
Now NP = SN'-{-SP-2SN. SI cos 0. (702)
Substitute SN=IR-GS;
SI = TI-ST = 21i sin 4 -2GS cos ;
and GS = 2F sin'' lA, to prove the proposition.
If J be the centre of the escribed circle touching BC, and r„ its radius, it
is shewn in a similar way that NJ = NG + r^.
To construct a triatif^lefrom certain data.
960 When amongst Wxg data we have the sum or difference
of the two sides AB, AC; or the sum of the segments of the
base made by Ad, the bisector of the exterior vertical angle;
or the difference of the segments made by AF, the bisector of
206
ELEMENT A li Y GEOMETRY.
tlie interior vertical
lead to the solution.
Make AE = AD = AC. Draw
DII parallel to AF, and suppose
EK drawn parallel to AG to meet
the base produced in K; and
complete the figure. Then BE
is the sum, and IJD is the differ-
euce of the sides.
EK is the sum of the exterior
segments of the base, and Till is
the difference of the interior seg-
ments. Z BDH = BEG = iA,
ZADG = EAG =i(B+C),
L DCB = \BFB = i (0-7)).
le ; tlie followinf]f construction will
961 When the base and the vertical angle are given ; the
locus of the vertex is the circle ABC in figure (935) ; and the
locus of the centre of the inscribed circle is tlie circle, centre
F and radius FB, When the ratio of the sides is given,
se e (932).
962 To construct a triangle when its form and the distances
of its vertices from a point A' are given.
Analysis. — Let ABG be the required triangle. Oa
A'B make the triangle A'BG' similar to ABG, so that
AB : A'B :: GB : G'B. The angles ABA', GBG' will
also be equal; therefore AB : BG :: AA' : GC, which
gives GG', since the ratio AB : BG is known. Hence
the point G is found by constructing the triangle
A'CG'. Thus BG is determined, and thence the tri-
angle ABG from the known angles.
963 To find the locus
of a point P, the tan-
gent from which to a
given circle, centre A,
has a constant ratio to
its distance from a given
point B.
Let AK be the radius of
the circle, and jj : q the
given ratio. On A JJ take
AC, a third propoitional to
AB and AK, and make
AD:DB=]r : .j\
With ccnti-e D, aud a radius
TANGEXTS TO TWO ClRCLEFi.
207
Cfjunl to a mean proportional betwcon />/> and DC, describe a circle. It will
bo the required locus.
Prook. — Suppose r to bo a point on the rcMpiirod locus. Join 7' witli
A, J?, C, and 7>.
Describe a circle about PBC cutting Al' in l'\ and anotlicr about AliF
cutting P77 in (1, and join AH and UF. Tlien
TK- = AF'-A A'- = J7^--7U . AC (by constr.) = A V'-TA . A F (III. Z^\)
= Ai'.vF (II. '1) = ur.rjj (HI. :3G).
Therefore, by h^'pothe.sis,
2>- : f/ = (VP . rn : PP'^ = GP : rn = A D : T)B (by constr.) ;
therefore Z Z)7V? = PGA (VL 2) = PZ''7>' (III. 22) = PC// (III. 21).
Therefore the triangles DPP, BGP are simihir; therefore DF is a mean pro-
portional to DB and DC. Hence the construction.
964 CoE. — If l'> = q tlie locus becomes the pcrpciulicular
bisector of BC, as is otlicr\Ndse shown in (1003).
965 To find the locus of a point P, the tangents from wliich
to two given circles shall have a given ratio. (See also 10;3(3.)
Let A, B be the centres, a, h the radii
(rt > fe), and p : q the given ratio. Take c, so
that c : h = p : q, and describe a circle with
centre -1 and radius ^l.V = va' — c'. Find the
locus of P by the last proposition, so that
the tangent from P to this circle may have
the given ratio to FB. It will be the re-
quired locus.
Proof. — By hypothesis and construction
q' FT-
h- FT' + U'
ir'-o^ + ,^- _ AF'-AX'
BF' BP"
Cor. — Hence the point can be found on any curve from
which the tangents to two circles shall have a given ratio.
966 To find the locus of the point from which the tangents
to two given circles are ecpial.
Since, in (965). wo have now p = q, and therefoi-e c = h, the construction
simplifies to the following :
Take AN= y(a--6'-), find in ^IP take AB : AN : AC. The perpen-
dicular bisector of PC is the required locus. But, if the circles inter.sect,
then their common chord is at once the line required. See Radical Axis
(985).
208
ELEMENTARY GEOMETRY.
CoU'niear and Concurrent systems nfj)oints and lines.
967 Definitions. — Points lying in the same straight line are
colUnear. Straight lines passing through the same point are
concurrent^ and the point is called the focus of the pencil of
lines.
Theorem. — If the sides of the triangle ABG, or the sides
produced, be cut bj any straight line in the points a, 6, o
respectively, the line is called a transversal, and the segments
of the sides are connected by the equation
968 {Ah : hC) {Ca : aB) (Be : cA) = 1.
Conversely, if this relation holds, the points a, h, c will be
collinear.
Proof. — Througli any vertex A draw AD
parallel to the opposite side BG, to meet the
transversal in D, then
Ab : bG = AD : Ca and Be, : cA = aB : AD
(VI. 4), which proves the theorem.
Note. — In the formula the segments of the
sides are estimated positive, independently of
direction, the sequence of the letters being pre-
served the better to assist the memory. A point may be supposed to travel
from A over the segments Ab, bC, &c. continuously, until it reaches A again.
969 By the aid of (701) the above relation may be put in
the form
(sin ABb : sin bBC) (sin C^a : sin aAB) (sin BCc : sin cCA) = l
970 If be any focus in the plane of the triangle ABC, and
if AG, BO, CO meet the sides in a,b,c; then, as before,
{Ab : bC) {Ca : aB) {Be :cA) = l.
Conversely, if this relation holds, the lines Aa, Bh, Cc will
be concurrent.
Proof. — By the trans-
versal Bb to the triangle
AaG, we have (9G8)
{Ab : bG) {CB : Ba)
x{aO : 0.1) = 1.
And, by the transversal
Cc to the triangle AaB,
(Bc:cA)(AO: Oa)
x(aG: CB) = \.
Multiply these equations
together.
COLLINEAR AND COX('rnili:XT SYSTIJ^fS. 200
971 Tf J>r, ca, ah, in tlic last figure, be produced to meet the
sides of .l//Oin 1\ Q, R, then eac^h of the nine lines in the
tiu-nro will be divided liannoiiically, and tlie points J\ (,', R
^vill be collinear.
Proof. — (i.) Take hP a transversal to ABC; therefore, by (^08),
{Cr : PB) (Be : cyl) (Ab : hC) = 1 ;
tlu-rcforo, by (1170), CP : PB = Ca : uB.
(ii.) Take CP a transversal to Abe, therefore
(AB : Be) (cP : Pb) (bC : CA) = 1.
But, by (070), taking for focus to Abe,
(AB : Be) (ep : pb) (bC : CA) = 1 ;
therefore cP : Pb = ep : pb.
(iii.) Take PC a transversal to AOe, and b a focus to AOc\ therefore, by
(0G8 & 070), (.la : aO) (OG : Cc) (cB : BA) = 1,
and (Ap : pO) (OC : Cc) (eB : I?-l) = 1 ;
therefore Aa : aO = ylj) : i'O.
Thus all tlio lines are divided harmonically.
(iv.) In the equation of (070) put Ab : bC = AQ : QC the harmonic
ratio, and similarly for each ratio, and the result proves that P, Q, R aro
collinear, by (008).
Cor. — If in the same figure qr, rj), pq be joined, the three
lines will pass through P, (^, li respectively.
Proof. — Take as a focus to the triangle abc, and employ (070) and the
harmonic division of be to show that the transversal rq cuts be in P.
972 If a transversal intersects the sides AB, lUl, CD, &c.
of any polygon in the points a, h, c, &c. in order, tlien
{Aa : aB) {Bb : bC) {Cc ; cD) {Dd : (IE) ... kc = 1.
Pkoof. — Divide the polygon into triangles by lines drawn from one of the
angles, and, applying (908) to each triangle, combine the results.
973 Let any transversal cut the sides of a triangle and
tlieir three intersectors AO, HO, CO (see figure of V70) in tbe
points A', B', C, a, h' , c , respectively; then, as before,
(J7/ : IjC) {Ca : a B) {/)',■' : c'A') = 1.
Phoof.— Each .'^ide forms a triangle with its intorsector and the trans-
versal. Take the four remaining linos in smvissioi for transversals to each
trianMe, applying CJOS) symmctricallv, and ciMidiinr the twelve equations.
2 E
210 ELEMEXTAllY GEOMETRY
974 If the lines joining corresponding vertices of two tri-
angles ABC, abc are concurrent, the points of intersection
of the pairs of corresponding
sides are collinear, and con-
versely.
Proof. — Let tho concurront lines
Aa, Bh, Cc meet in 0. Take be,
ca, ah transversals respectively to
the triangles OBG, OCA, OAB, ap-
plying (9G8), and tlio product of the
three equations shows that P, E, Q
lie on a transversal to ABC. p
975 Hence it follows that, if the lines joining each pair of
corresponding vertices of any two rectilineal figures are con-
current, the pairs of corresponding sides intersect in points
which are collinear.
The figures in this case are said to be in pprspectlce^ or in,
homology, with each other. The point of concuiTcnce and
the hne of collinearity are called respectively the centre and
axis of perspective or homology. See (1083).
976 Theorem. — When three perpendiculars to the sides of a
triangle ABC, intersecting them in the points a, b, c respec-
tively, are concurrent, the following relation is satisfied ; and
converse^, if the relation be satisfied, the perpendiculars are
concurrent.
Afr-bC'+C(r-aB--\-Bc'-cA'' = 0.
Proof. — If the perpendiculars meet in 0, then Ab'' — bC-= AO'-—OC-,
&c. (I. 47).
Examples. — By the application of this theorem, the concurrence of the
three perpendiculars is readily established in the following cases: —
(1) When the perpendiculars bisect the sides of the triangle
(2) When they pass through the vertices. (By employing I. 47.)
(3) The three i-adiioflhe esci-ibed circles of a triangle at the points of
contact between the vertices are concui-rent. So also arc; the radius of the
inscribed circle at the point of contact wilh one side, and the radii of tho two
escribed circles of the remaining sides at tho points of contact beyond the
included angle.
In these cases employ the values of the segments criven in (953).
(4) The pei'j>endiculars equidistant from the vertices with three con-
current perpendiculars are also concun-cnt.
(5) When the three perpendiculars from the vertices of one triangle upon
the sides of the other are concurrent, then tho perpendiculars from the
vertices of the second triangle upon the sides of the tirst are also concurrent.
Proof. — If A, B, G and A\ If, C are corresponding vertices of the tri-
angles, join AB\ AC\ BC, BA\ CA, CR, and apply the theorem in conjuuc-
tiou witlj (I. 47).
TRiAXiiLES rinrvMscjiunxt} a rniAXi;rr:.
211
Trianii'lr.s' of rnnstant species ptrctimsrribrd to a trian<j;le.
977 Let J//C' be any triano-l(^, and ^ any point; and lot
circles circumscribe AOli, BOO, C(K\. The circumferences
will be the loci of the vertices of a triangle of constant form
whose sides pass through the
points ^1, //, C.
PkOOF. — Draw any line hAr fi-oni circle
to circle, and produce 1>C, cli to meet in n.
The angles AOD, CO A arc supplements
of tlie angles c and b (III. 22) ; therefore
BOC is the supplement of a (I. 32) ; there-
for© a lies on the circle OUC. Also, the
angles at being constant, the angles a, b, c
are constant.
978 Tlie triano-le ahe is a maximum wlien its sides are per-
pendicular to OA, on, 00.
PuooF. — The triangle is greatest when its sides are greatest. But the
sides vary as Oa, Ob, Oc, which are greatest when they are diameters of the
circles; therefore &c., by (111.31).
979 To construct a triangle of given S])ccies and of given
limited magnitude which shall liave its sides passing through
three given points .1, B, 0.
Determine by describing circles on the sides of ABC! to contain angles
equal to the supplements of the angles of the speciticd triangle. Construct
the figure nb'O independently from the known sides of ahe, and the now
known angles ObC = 0A(\ OnC = OBC\ &c. Thus the leJigths Oa, Ob, Oc
are found, and therefore the points a, b, o, on the circles, can bo determined.
The demonstrations of the following propositions will now be obvious.
Triant^les of constant species inscribed to a trian<;^le.
980 Let ahr, in the last figure, be a fixed triangle, and O
any point. Take any point .1 on l>r, and h^t tlie circles cir-
cumscribing OAr, OAb cut the oilur sides in /?, O. Then
.47:?(J will be a triangle of constant form, and its angles will
have the values A =■. Oha + Ora, &c. (III. -Ji.)
981 The triangle ABO will evidently be a minimum wht^n
OA, 01), 00 are drawn perpendicular to the sides of ah''.
982 To construct a triangle of given form and of given
limited magnitude having its vertices u})on three fixed lines
6r, ca, ah.
212
ELEMENTARY OEOMETIiY.
Construct the fitjure ABCO independently fi-om the known sides of ABG
and the angles at 0, which are equal to the sui)i)lenients of the given angles
a, b, c. Thus the angles OAG, &c are found, and therefore the angles ObC,
&c., equal to them (III. 21), are known. From these last angles the point
can be determined, and the lengths OA, OB, 00 being known from the inde-
pendent figure, the points A, B, C can be found.
Observe that, wherever the point may be taken, the angles AOB, BOO
COA are in all cases either the supplements of, or equal to, the angles c, a, b
respectively; while the angles aOb, bOc, cOa are in all cases equal to C zh c,
A±a, B±b.
983 Note. — In general problems, like the foregoing, wliicli
admit of different cases, it is advisable to clioose for reference
a standard figure which has all its elements of the same affec-
tion or sign. In adapting the figure to other cases, all that
is necessary is to follow the same construction, letter for
letter, observing the convention respecting positive and
negative, which applies both to the lengths of lines and to
the magnitudes of angles, as explained in (607 — 609).
Radical Ad is.
984 Definition. — The radical axis of two circles is that
perpendicular to the line of centres which divides tlie dis-
tance between the centres into segments, the difference of
whose squares is equal to the difference of the squares of the
radii.
Thus, A, B being the centres, a, h the radii, and IP the
the radical axis
985 It follows that, if tlie circles intersect, the radical axis
is their common chord ; and that, if they do not intersect, the
radical axis cuts the line of centres in a point the tangents
from which to the circles are equal (I. 47).
To draw the axis in this case, see (960).
nADfCAL AXIS.
21:^
Otliorwisc: let the two circles cut tlie line of centres in C, P anrl C, 7>'
Tcsptctivcly. I)i'sciil>c any ciri-lo tliroii^Mi (' and /', and another throii},di
r' iind //, intersectiiif^ the former in IJ and /''. Tlieir coniniou chord Ij F
will cut the central axis in the required \>o\ut f.
Proof. — IC. ID = IE. 1F= IC. 11/ (111. 'M) ; therefore the tangents
from I to the circles are equal.
986 T}ii-<n'inn. — The dift'orencc of the squares of tangents
from any })oint /' to two circles is equal to twice the rectangle
under the distance between their centres and the distance of
the point from their radical axis, or
Proof.
PK' -FT' = (Ar--Br-)-(a--V) = (AC/'-mr) - (AP-FP),
by (I. 47) & (t»84). Bisect AB in C, and substitute for each ditference of
squares, by (II. 12).
987 Cor. 1.— If r be on the circle whose centre is 7?, then
PK' = 2AB.PN.
988 Coif. 2.— If two chords be drawn through P to cut the
circles in A', A", Y, >" ; tlien, by (III. 30),
7^X . PX- P Y. P Y =2AB. PN.
989 If Ji variable circle intersect two given eireU^s at con-
stant angles a and ft, it will intersect tlieir radical axis at a
constant angle ; and its radius will bear a constant ratio to
the distance of its centre from the radical axis. Or
PN : PX = rt cos a — 6 cos fi : AB.
214 ELEMENTARY GEOMETRY.
Proof. — In the same figure, if P be tlie centre of the variable circle, and
if PX=PY be its radius; then, by (088),
FX (XX'- YY') = 2An . TN.
But XX' = 2a cos (I and YY' = 2h cos ft ;
therefore PX : PX = a cos a — & cos ft : AB,
which is a constant ratio if the angles (i, ft are constant.
990 Also FX : PN = the cosine of the angle at which the
circle of radius PX cuts the radical axis. This angle is
therefore constant.
991 Cor, — A circle which touches two fixed circles has its
radius in a constant ratio to the distance of its centre from
their radical axis.
This follows from the proposition by making a = ft = or Ott.
If P be on the radical axis ; then (see Figs. 1 and 2 of 984)
992 (i.) The tangents from P to the two circles are equal,
or FK = FT. (986)
993 (ii-) The rectangles under the segments of chords
through P are equal, or FX . FX' = FY . FY'. (988)
994 (iii-) Therefore the four points X, X', T, Y' are con-
cychc (III. 36); and, conversely, if they are concychc, the
chords XX' J YY' intersect in the radical axis.
995 Definition. — Points which lie on the circumference of
a circle are termed coney die.
996 (iv.) If P be the centre, and if PX = PY be the radius
of a circle intersecting the two circles in the figure at angles
a and /3; then, by (993), XX'=YY', or a cos a = Z> cos pJ ;
that is, The cosines of the angles of intersection are inversclj/
as the radii of the fixed circles.
997 The radical axes of three circles (Fig. 1046), taken two
and two together, intersect at a point called their radical centre.
PimOF. — Letyl, B, (7 b/ t'lc centres, a, h, c the radii, and X, Y, Z the points
in which the radical axes cut JJC, CA, AB. Wrire tlie equation of the defini-
tion ('J84) for each pair of circles. Add the results, and apply {iUO).
998 A circle whose centre is the radical centre of three
other circles intersects them in angles whose cosines are
inversely as their radii (996).
i.\\i:iiSit>.\'.
Henco., if this fourtli circle cuts one of the others or-
thogonally, it cuts them all orthogonally.
999 'I'he circle whicli intersects at angles a, ft, y three fixed
circles, whose centres are .1, li, C and radii a, A, r, has its
centre at distances from the radical axes uf the iixed circles
proportional to
/> cos ft — C cos y C cos y — (( COS a (/ COS a — h COS /3
BO ' CJi ' AB
And therefore the locus of its centre will be a straight line
passing through the radical centre and inchned to the three
radical axes at angles whose sines are projiortional to these
fractions.
Proof. — The result is obtained immediately by writing out equation ('J89)
for each p-.iir of fixed circles.
situated on a
The Method of Inversion.
1000 Definitions. — Any two points F, V
diameter of a fixed circle
whose centre is and radius
A-, so that 01\0r'= /r, are
called immerse itoints with re-
spect to the circle, and either
point is said to be the inverse
of the other. The circle and
its centre are called the circle
and centre of incersion, and.
k the constant of inversion.
1001 If every point of a plane figure be inverted Avith
respect to a circle, or every point of a figure in space witii
respect to a sphere, the resulting figure is called the inverse
or image of the original one.
Since OB : k : 0B\ therefore
1002 OP : OP' = OP' : fr = A^ : 0P'\
1003 Let D, jy, in the same figure, be a pair of inverse
points on the diameter 00'. In the perpendicular bisector of
VD\ take any point Q as the centre of a circle passing through
1), I)\ cutting the circle of inversion in R, and any straight
line through in the points P, B. Then, by (III. 3r.),
OB . OB = on . OU = OR- (1 000). Hence
216
ELEMENTARY GEOMETRY.
1004 (i-) i^i ^ '"ii'e inverse points; and, conversely, any
two pairs of inverse points lie on a circle.
1005 (ii-) The circle cuts orthogonally the circle of inver-
sion (III. 87) ; and, conversely, every circle cutting anotlier
orthogonally intersects each of its diameters in a pair of
inverse points.
1006 (iii-) The line IQ is the locus of a point the tangent
from which to a given circle is equal to its distance fi^om a
given point D.
1007 Def. — The line IQ, is called the axis of reflexion for
the two inverse points D, D', because there is another circle
of inversion, the reflexion of the former, to the right of 1(^,
having also D, I)' for inverse points.
1008 The straight hnes drawn, from any point P, within
or without a circle (Figs. 1 and 2), to the extremities of any
chord AB passing through the inverse point Q, make equal
angles with the diameter through FQ. Also, the four points
0, A, B, P are concyclic, and QA . QB = QO . QP.
Pkoof. — In eitlicr figure OR : OA : OQ and OR : OB : OQ (1000),
therefore, by similar triantrles, Z OR A = OAR and ORB = OR A in figure
(1) and the supplement of it in figure (2). But OAB = OBA (I. 5), there-
fore, &c.
Also, because Z OR A = OR A, the four points 0. A, B, P lie on a circle in
each case (III. 21), and therefore (^.4 . QR = QO . QR (III. 35, 3G).
1009 The inverse of a circle is a circle, and the centre of
inversion is the centre of simihtude of the two figures. See
also (1087).
PnoOF. — In the figure of (l<»lo), let be the point where the common
tangent RT of the two circles, centres A and R, cuts the central axis, and let
any other line through cut the circles in P, Q R\ Q'. Then, in the demon-
stration of (942), it is sliown that 01' ■ OQ,' = OQ . OP = h\ a constant
quantity. Tliercfore either circle is the inverse of the other, k being the
j'adius of the circle of inversion.
INVERSION.
217
1010 'r"o luaki' tlio inversions of two o-iveii cii-clos 0(iu:il
circles.
Rule. — Take the centre of inversion so that the squares of
the taiKjeiits from it to the given circles may he irroportioual to
their radii (965).
PKOOK.-(Fig. 1013) AT : 7?7i' = OT : 07?, = 07" : P, since OT : /.: : OR.
Therefore OT^ : AT = Jc^ : lili, therefore L' A' remains constant if OT^ <x AT.
1011 lleiiec three circles may be inverted into equal circles,
for the reciuired centre of inversion is the intersection of two
circles that can be drawn by (965).
1012 The inverse of a straight line is a circle passing
through the centre of inversion.
Proof. — Draw OQ perpendicular to the
line, and take P any otlicr point on it. Let
Q\ F be the inverse points. Then OP . 0F=
OQ-OQ'; therefore, by similar triangles,
Z OFQ' = OQr, a right angle ; and OQ is
constant, therefore the locus of F is tho
circle whose diameter is OC^.
1013 Example. — The inversion of a poly-
gon produces a figui'e bounded by circular
arcs which intersect in angles equal to tho
corresponding angles of the polygon, the
complete circles intersecting in the centre
of inversion.
l//>'
1014 If tlie extremities of a straight line V'Q' in the last
figure are the inversions of the extremities of l'(,^, tlien
pq : pq = v/(op . oq) : ^^{0P . oq').
Proof.— By similar triangles, FQ : FQ = OF : OQ' and FQ : F(2' =
OQ : OF. Compound these ratios.
1015 From the above it follows that any homogeneous
equation between the lengths of lines joining pairs of points
in space, such as ]H} . RS . TU = PR .QT . SfJ, the same
points appearing on both sides of the equation, will l)o
true for the figure obtained by joining the corresponding
pairs of inverse points.
For the ratio of each side of the equation to the corresponding side of tho
equation for the inverted points will bo the same, namely,
y'iOF.OQ.OU ...) : ^/{()F .OQ'.Oh'- ...).
218 ELEMENTARY GEOMETRY.
Pole and Polar.
1016 Defixitiox. — The iMar of any point P with respect
to a circle is the perpendicular to the diameter OF (Fig. 1012)
drawn through the inverse point F .
1017 It follows that the polar of a point exterior to the
circle is the cliord of contact of the tangents fron the point;
that is, the hne joining their points of contact.
1018 Also, FQ is the polar of F with respect to the circle,
centre 0, and FQ is the polar of Q. In other words, 0iy
point P hfhig on the polar of a point Q', has its oivn polar
alivays passing through Q'.
1019 The line joining any two points P, p is the polar of
Q', the point of intersection of their polars.
Proof. — The point Q' lies on both the lines P'Q', i^'Q'y and therefore has
its polar passing through the pole of each line, by the last theorem.
1020 The polars of any two points F,p, and the line joining
the points form a self-reciprocal triangle witli respect to
the circle, the three vertices being the poles of the opposite
sides. The centre of the circle is evidently the orthocentre
of the triangle (952). The circle and its centre are called the
polar circle and j^olar centre of the triangle.
If the radii of the polar and circumscribed circles of a
triangle ABC be r and P, then
r^ = 4iIV cos A cos B cos C
Proof. — In Fig. (052), is the centre of the polar circle, and the circles
described round AB(\BOC, COA, A0J3 are all equal; because the angle
BOO is the supplement of vl ; &c. Tlierofore 27i' . OD = OB . 00 (VI. C)
and r^ = OA . OD = OA . OB . 00 -^ 2/i'. Also, OA = 2Zi' cos.l by a diameter
through B, and (III. 21).
Coa.val Circles.
1021 Definition. — A system of circles having a counnon
line of centres called the central axis, and a coninion railical
axis, is termed a coaxal system.
1022 If be tlie variable centre of one of the circles, and
COAXAL CinCLES.
210
07v its rjuliuP, the whole system is included in tlie equation
()l--(>K-= ±8",
where S is a constant length.
1023 111 the first species (Fig. 1),
OP-OK' = S\
and S is the length of the tangent from I to any circle of the
system (985). Let a circle, centre I and radius B, cut the
Central axis in D, D'. When is atZ) or //, tlie circle whose
radius is OK vanishes. "When is at an infinite distance,
the circle developes into the radical axis itself and into a line
at infinity.
The points D, D' are called tlie J imiting points.
1024 In the second species (Fig. 2),
on-- 01- = 8-,
and S is half the chord R]i common to all the circles of
the system. Tliese circles vary between the circle with
centre I and radius S, and the circle with its centre at infinity
as described above. The points 7?, R' are the common points
of all circles of this system. The two systems are therefore
distinguished as the timiting jwints sjjccies and the common
2>oints sjxjcics of coaxal cii'cles.
220 ELEMENTARY GEOMETRY.
1025 There is a conjugate system of circles having R, B! for
limiting points, and D, 1)' for common points, and the circles
of one species intersect all the circles of the conjugate system
of the other species orthogonally (1005).
Thus, in figures (1) and (2), Q is the centre of a circle
of the opposite species intersecting the other circles or-
thogonally.
1026 In the first species of coaxal circles, the limiting
points D, D' are inverse points for every circle of the system,
the radical axis being the axis of reflexion for the system.
Proof.— (Fig. 1) OP-P = 0K\
therefore 01) . OB' = OIP, (II. 13)
therefore D, D' are inverse points (1000).
1027 Also, the points in which any circle of the system
cuts the central axis are inverse points for the circle whose
centre is I and radius S. [Proof.— Similar to the last.
1028 Problem. — Given two circles of a coaxal system, to
describe a circle of the same system — (i.) to pass through a
given point; or (ii.) to touch a given circle ; or (iii.) to cut a
given circle orthogonally.
1029 I. If the system be of the common points species, then, since the
required circle always passes through two known points, the first and second
cases fall under the Tangencies. See (91-1).
1030 To solve the third case, describe a circle through the given common
points, and through the inverse of either of them with respect to the given
circle, which will then be cut orthogonally, by (1005).
1031 II. If the system be of the limiting points species, the problem is
solved in each case by the aid of a circle of the conjugate system. Such a
circle always passes through the known limiting points, and may be called a
conjugate circle of the limiting points system. Thus,
1032 To solve case (i.) — Draw a conjugate circle through the given point,
and the tangent to it at that point will be the radius of the required circle.
1033 To solve case (ii.) — Draw a conjugate circle through the inverse
of either limiting point with respect to the given circle, which will thus be
cut orthogonally, and the tangent to the cutting circle at either point of
intersection will be the radius of the required circle.
1034 To solve case (iii.) — Draw a conjugate circle to touch the given one,
and the common tangent of the two will be the radius of the required circle.
1035 Thus, according as we wish to make a circle of the system loncli, or
cut <iii/in</(i)i(iUi/, the given circle, wo must draw a conjugate cii'cle to ctd
orlhiHjvnalli/, ur iuuch it.
CENTRES AND AXES OF SIMILITUDE. 221
1036 If three circles be coaxal, the squares of the tanp^cnts
drawn to any two of tlieni from a point on tlie tliii-d are in
the ratio of tlie distances of the centre of the third circle from
the centres of the other two.
PuoOF. — Let A, I), C bo tho centres of the circles ; PK, FT the tanr^onta
from a point P on tho circle, centre (,', to tho other two ; PN tho perpeu-
dicular on the radical axis. By (080),
PA"- = 2AG . PN and PT' = 2nG . PN,
thereforo PK' : PT' = AC : BG.
Centres and (Lies of similitude.
1037 Definitions. — Let 00' be the centres of sirailitudo
(Def . 1* 1-7) of the two circles in the figure below, and let any
line tlirough cut the circles in r, Q, P\ Q. Then the
constant ratio OP : OP' = OQ : OQ' is called the ratio of
similitude of the two figures ; and the constant product
OP .OQ' = OQ . OF is called the i)roduct of anti-si inilitade.
See (942), (1009), and (1043).
The corresponding points P, V or Q, Q' on the samo
straight line through are termed liomoltxjoas, and P, (/ or
Q, P are termed anti-homologous.
1038 Lt^'t any other line Opqp'q be drawn through 0.
Tlien, if any two points i', p on the one figure be joined, and
if P', j/, homologous to P,}) on the other figure, be also joined,
the lines so formed are termed liomolotjous. But if the points
Avhich are joined on the second figure are anti-liomologous to
those on the first, the two lines are termed anti-homulojous.
Thus, Pq, l/p' are anti-homologous lines.
1039 Tlie circle Avhose centre is 0, and wliose radius is
e(iual to the square root of the product of anti-similitude, is
called the circle of anti-similitude.
1040 The four pairs of homologous chords Pp and Fp'y
Qq and Q'q\ Pq and P'q, Qp and (/p of the two circles in the
figure are parallel. And in all similar and similarly situated
figures homologous lines are parallel.
Proof.— By (VI. '2) aud the dciiuitiou (917).
222
ELEMENT A E Y GEOME TB Y
1041 The four pairs of anti-liomologous cliords, Pj) and
^/'Z ' Q'l ^^^ Pp'i JPq ^^^ QP) Qp ^^^ ^^I'i of the two circles
meet on their radical axis.
Proof.— OP . OQ' = Op . Oq = ]c\
wliere k is the constant of inversion ; therefore P, -p, Q', q' are concyclic ;
therefore Pp and Q'q' meet ou the radical axis. Similarly for any other pair
of anti-homologous chords.
104:2 Cor. — From this and the preceding proposition it
follows that the tangents at homologous points are parallel ;
and that the tangents at anti-homologous points meet on the
radical axis. For these tangents are the limiting positions of
homologous or anti-homologous chords. (IIGO)
1043 Let 0, D be the inverse points of ^^^th respect to
two circles, centres A and JJ ; then the constant product of
anti-similitude
OF . OQ or OQ . OF = OA . OD or OB . OC.
JH^
CEXTHES AXJ) AXES OF SlMU.lTfhi:.
223
Proof. — By similar right-angled triangles,
0-1 : OT : OC and OB : OH : OD;
therefore OA . OD = Oli . OC
and also 0.1 . OG = OT = OP . 0(J,
and OB . OD = Olf' = or .OQ';
therefore OA . OB . OC . OD = OF . OQ . OL*' . OQ',
(1),
(III.:;.;)
therefore &c., by (1).
1044 The foregoing definitions and properties (10.S7 to
1U4;5), wliicli have respect to the external centre of simihtudo
0, hold good for the internal centre of simihtiide 0\ with the
usual convention of positive and negative for disitances
measured from 0' upon lines passing through it.
1045 Two circles will subtend equal angles at any point on
the circumference of the circle whose diameter is 00\ where
0, O are the centres of simihtude (Fig. 1043). This circle is
also coaxal with the given circles, and has been called the
circle of shnllitude.
Proof. — Let A, B be the centres, a, b the radii, and K any point on the
circle, diameter 00'. Then, by (D3"2),
KA : KB = AO : BO = AO' : BO' = a : h,
by the definition (O-io) ;
therefore a : KA = h : KTI ;
that is, the sines of the halves of the angles in question are equal, which
pioves the first part. Also, because the tangents from K are iu the ctnstant
ratio of the radii a, h, this circle is coaxal with the given ones, by (lUoG, 1)34).
1046 The six centres of similitude P,p, Q, q, R, r of three
circles lie three and three on four straight lines I'ijli, I'qr,
Qpr, Rpq, called axes
of simUitude.
Proof. — Taking any
three of the sets of points
named, say P, q, r, they are
shewn at once to be col-
linear by the transversal
theorem (*JGs) applied to
the triangle ABC.
For the segments of its
sides made by the points
P, 'i, r are in the ratios of
the radii of the circles.
224 ELEMENTARY GEOMETRY.
1047 From tlie investigation in (0 12), it appears that one
circle touches two others in a pair of anti-homologous points,
and that the following rule obtains : —
Rule. — The right line joining the points of contact imsses
through the external or internal centre of similitude of the tivo
circles according as the contacts are of the same or of different
kinds.
1048 Definition. — Contact of curves is either internal or external ac-
cordiug as the curvatures at the point of contact are in the same or opposite
directions.
1049 Gergo7ine''s method of descrihing the circles ichlch touch
three given circles.
Take Pqr, one of the four axes of similitude, and find its poles o, /3, y
with respect to the given circles, centres A, B, G (lOlG). From 0, the
radical centre, draw lines through a, /3, y, cutting the circles in a, a', b, b',
c, c. Then a, b, c and a', b', c will be the points of contact of two of the
requii'ed circles.
PjiOOP. — Analysis. — Let the circles
E, F touch the circles A, B, G in
a, b, c, a', b', c. Let be, b'c meet in
P ; ca, c'a' in ci ; and ab, a'b' in r.
Regarding E and F as touched by
A, B, G in turn, Rule (1047) shews
that Art', bb', cc' meet in 0, the centre
of similitude of jE/ and F ; and (1041)
shews that P, q, and r lie on the
radical axis of E and F.
Regarding B and G, or G and A,
or A and B, as touched hy E and F
in turn, Rule (1047) shews that P, q, r
are the centres of similitude of ]> and
G, G and A, A and B respectively;
and (1041) shews that is on the radical axis of each pair, and is therefore
the radical centre of A, B, and G.
Again, becansc the tangents to E and F, at the anti-homologous points
a, a', meet on Bqr, the radical axis of E and F (1042) ; therefore the point
of meeting is the pole of <(«,' with respect to the circle .1 (1017). Therefore
aa' ])as.ses through the pole of the line Pifr (1018). Similarly, bb' and cc'
pass through the poles of the same line J'qr with respect to J> and G. Hence
the construction.
1050 In the given configuration of the circles A, B, C, the
(leinoiistration shews that each of the three internal axes of
similitude P(/r, (,>rp, Jlju/ (Fig. 104()) is a radical axis and
connnon chord of tAvo of the eight osculating circles which
can be drawn. The external axis of similitude Vi^li is the
AXirAIDlOXIC JiATIO.
radical axis of the two remaining circles which touch J, 7/,
and C eitlier all externally or all internally.
1051 Tlie radical centre of the three p^iven circles is also
tlie common internal centre of similitude of the four paii-s of
osculating circles. Therefore the central axis of each pair
passes through 0, and is peri)endicular to the radical axis.
Thus, in the figure, EF passes thi-ough 0, and is })erpen-
dicular to Pqr.
Anharmonic Ratio.
1052 Dkfixitiox. — Let a pencil of
four lines through a point be cut
by a transversal in the points A, B,
(', D. The anharmonic ratio of the
])encil is any one of the three frac-
tions
A B . CD
AD.BC
or
AB.rn
AC.BI)
AC. liD
1053 The relation between these three different ratios is
obtained from the equation
AB . CD-\^AD . BC = AC . BJ).
Denoting the terms on the left side by 2' and </, the three anharmonic
i-atios may be expressed by
p : q, p :p + q, q :p + q.
The ratios are therefore mutually dependent. Hence, if the identity
merely of the anharmonic ratio in any two sj'stems is to be established, it is
immaterial which of the three ratios is selected.
1054 In future, when the ratio of an anharmonic pencil {0, AHl^'D] is
mentioned, the form All. CI) : AD. I!C will be the on<? intended, wliatever
the actual order of the points ,1, 7), <\ D may be. For, it should be observed
that, by making tlie line 01> revolve about 0, the ratio takes in turn eacii of
the forms given above. This ratio is shortly expressed by the notation
{0,AI!('1J}, or simply {AUCD}.
1055 If the transversal be drawn parallel to one of the lines, for instance
(>D, the two factors containing 7) become infinite, and their ratio becomes
unity. They may therefore bo omitted. The anharmonic ratio then reduces
to AB : liC. Thus, when IJ is at infinity, we may write
{0, ^17)'rx} = .47? : BC.
1056 The anharmonic ratio
A n . CD _ sin Aon sinrO/>
AD.BC siuAOJ) y^'niBifC
2g
226 ELEMENTARY GEOMETRY.
and its value is therefore tlie same for all transversals of the
pencil.
Proof. — Draw 07? parallel to the transversal, and let p be the perpendi-
cular from A upon Oli. Multiply eacli foctor in the fraction by p. Then
substitute i> . AB = OA.OB sin AOB, &c. (707).
1057 The anharmonic ratio (105(3) becomes harmonic when its value is
unity. See (933). The harmonic relation there defined may also be stated
thus : four points divide a line harmonically when the jjroclud of the extreme
segments is equal to the proih(ct < if the ivhole line and the middle segment.
Homograpliic Systems of Points.
1058 Definition. — If x, a, h, c be the distances of one
variable point and three fixed points on a straight line from a
point on the same ; and if x, a\ b', c be the distances of
similar points on another Hne through ; then the variable
points on the two hues will form two homographic si/stems
when they are connected by the anharmonic relation
mKQ (cT'-r/) (h-c) _ (x-a) {b'-c)
J-WO» (.i^-c) {(i-h) {.v'-c) {a-b'y
Expanding, and writing A, B, C, D for the constant coeffi-
cients, the equation becomes
1060 A.t\v'-^B.v-\-av-\-D = 0.
From which
^r>«^ C.r-\-D -, ' Bd-\-D
1062 Theorem. — Any four arbitrary points .i\, x.,, x^,a\on
one of the lines will have four corresponding points x[, x.>, x'^, x^
on the other determined by the last equation, and the tiro sets
of points ivlll have equal anliarmonic ratios.
Pkoof. — This may be shown by actual substitution of the value of each x
in terms of a'', by (1 00 1), in the harmonic ratio | ^vVV^t } •
1063 If the distances of four points on a right line from a
point upon it, in order, are a, a, /3, /3', where a, li\ a\ (5' are
the respective roots of the two quadratic equations
(Lv''-\-2lur+b = 0, aa--\-2h\v+b' = ;
the condition that the two ])airs of points may be liarmonicalh/
roii'/iujtitc is ,
1064 (ib-\-a'b = 2hh.
IXVOLFTIOy. 227
PiJOoK. — The liannonic relation, by (\0o7), is
(a -a') (rJ-l'/) = (a-ly) (<t'-/5).
Multiply out, and substitute for the sums and produrta of the roots of tlio
quadratics above iu terms of their coefficients by (.Jl, ."i'i).
1065 If »,. II, bo the quadratic expressions in (10G3) for two pairs of
jioiiits, and if h represent a third pair harmonically c<iMJiiirate with », and ii.,,
then the pair of points it will also be hai-nionically conjugate with every pair
jjfiven l)y the equation u^ + \ll^ = 0, where A is any eonsiant. For the con-
dition (10G4) applied to the last equation will be identically satisfied.
Inrolufion.
1066 Defixitioxs. — Pairs of inverxi' points 77'', (.>(/, &c., on
the simie right line, form a system in involution, and the rela-
tion between them, by (1000), is
OP . OF = OQ . OQ = &e. = k\
A F Q n Q' P-
I I ^1 [ ^ ,
Tlie radius of the circle of inversion is k, and the centre
i) is called the centre of the sijstein. Inverse points are also
termed conjiujtite points.
AVhen two inverse points coincide, the point is called a
forns.
1067 The equation OP- = h^ shows that there are two foci
J , />' at the distance h from the centre, and on opposite sides
of it, real or imaginary according as any two inverse points
lie on the same side or on opposite sides of the centre.
1068 If the two homogi^aphic systems of points in (1058)
be on the same line, they will constitute a system in involu-
tiun when B = C.
PiiOOF. — Equation (lOCO) maj- now be written
^.r.c' + 7/ (a- + «')+/? = 0,
a constant. Therefore — - is the distance of the origin from the ccntro
A
of inversion. Measuring from this centre, the equation becomes l^' = A',
representing a system iu involution.
1069 Any four points whaferrr of a system in involution on
a right line have their anharmouic ratio equal to that of their
four conjugates.
228 ELEMENTARY GEOMETRY.
Proof. — Let j:>, j/; q, i/; r, r; s, s be the distances of the pairs of inverse
points from the centre.
In the anharmonic ratio of any four of the points, for instance {]'q'i's},
substitute 2^-=h^-^]j\ q = Ic'-^q, &c., and the result is the anharmonic
ratio {p'qr's'].
1070 Any two inverse points P, F are in harmonic relation
with the foci A, B.
A F B F
Pkoof.— Let p, p be the distances of P, P' from the centre 0; then
Jc ^, o p' + h h+p
— , thereiore -^ — - = - — ~ ;
p p —Ic k—p
pp = k\ therefore ^ = — , therefore -- — - - -
^■f ' h p) p —Ic k
that is, |f = If' ^^^'^
1071 If a system of points in involution he given, as in
(1068), by the equation
Axic^E{x^-x)-\-B = (1);
and a pair of conjugate points by the equation
a:i?-\-2hc-\-h = (2);
the necessary relation between a^ h, and h is
1072 Ah^Ba = 2Hh.
Proof. — The roots of equation (2) must be simultaneous values of x, x in
(1) ; therefore substitute in (1)
x + x = and xx' = — . (51)
a Li
1073 Cor.— A system in involution may be determined from two given
pairs of corresponding points.
Let the equations for these points be
ax^ + 2]ix + h = and aV" + 2hx + h' = 0.
Then there are two conditions (1072),
Ab + Ba = 2Hh and xW + Ba = 2Uh',
from which A, II, B can be found,
A geometrical solution is given in (985). C, D ; C, B' ai'C, in that con-
struction, pairs of inverse points, and I is the centre of a system in involution
defined by a series of coaxal circles (1U22). Each circle intersects the
central axis in a pair of inverse points with respect to the circle whose centre
is and radius 2.
1074 The relations which have been established for a system of coUincar
points may be transferred to a system of concurrent lines by the method of
(105G), in which the distance between two points corresponds to the sine of
the angle between two lines passing through those points.
'METHODS OF PnOJECTWX.
oor)
The Method of Projection.
1075 DkI'INITIOXS. — Tlic pntjictioit of any j)(>iiil /' in Sj)aco
(Fi_i>-. of 1()7'.>) is tlio point p in wliicli ;i ri^Hit line ()l\ drawn
from a fixed point called the rcrfcr, intersects a fixed plane
called the phnie of projection.
If all the points of any fio-ure, ])lane or solid, he thus ])ro-
jected, the figure obtained is called the ^injccfiou of the
original fia'ure.
1076 Projective Propertie.'i. — The projection of a right line is
a right hne. The projections of parallel lines are parallel. The
projections of a curve, and of the tangent at any point of it , aro
another curve and the tangent at the corresponding point.
1077 The anharmonic ratio of the segments of a right lino
is not altered by projection ; for the line and its projection are
but two transversals of the same anharmonic pencil. (105G)
1078 Also, any relation between the segments of a lino
similar to that in (1015), in which each letter occurs in every
term, is a projective propertij. [Proof as in (1056).
1079 Tlieorem. — Any quadrilateral PQRS may be projected
into a parallelogram.
CONSTWrCTION.
Produce PQ, SR to ■"'
meet in A, and PS,
Qlh to meet in B.
Then, with any
point for vertex,
project the quadri-
lateral upon any
plane j'xib parallel to
GAB. The projected
figure jhps will be a
parallelogram.
Proof. — The
planes OPQ, OPS in-
tersect in OA, and
they intersect the
plane of projection
which is parallel to
0-1 in the lines pq,
r$. Therefore pq and
rs are parallel to OA,
and therefore to each
other. Similorlj, j)"?,
qr are parallel to OB.
230 ELEMENTARY GEOMETRY.
1080 Cor. 1. — The opposite sides of the parallelogram ji^/rs meet in two
points at infinity, which are the projections of the points A, R ; and AB
itself, which is the third diagonal of the complete quadrilateral FQRS, is
projected into a line at infinity.
1081 Hence, to project any figure so that a certain line in it may pass to
infinity — Take the jilane of lirojcdion 'parallel to the plane which contains the
given line and the vertex.
1082 Cor. 2. — To make the projection of the quadrilateral a rectangle,
it is only necessary to make AOB a right angle.
On Perspective Draifing.
1083 Taking the parallelogram pqrs, in (1079), for the original figure,
the quadrilateral PQRS is its projection on the plane ABab. Suppose this
plane to be the plane of the paper. Let the planes OAB, pah, while
remaining parallel to each other, be turned respectively about the fixed
parallel lines AB, ah. In evei'y position of the planes, the lines Oji, Oq, Or,
Os will intersect the dotted lines in the same points P, Q, R, S. When the
planes coincide with that of the paper, pqrs becomes a ground pilan of the
parallelogram, and FQRS is the representation of it in perspective.
AB is then called the horizontal line, ah the picture line, and the plane of
both the picture plane.
1084 To find the projection of any point p in the ground
plan.
Rule. — Eraw pb to any point b in the picture line, and draw OB parallel
to pb, to meet the horizontal line in B Join Op, Bb, and they tvill intersect in
P, tJce point required.
In practice, ph is drawn perpendicular to ah, and OB therefore perpendi-
cular to AB. The point B is then called the jjoi;;/ of sight, or centre of vision,
and the station point.
1085 To find the projection of a point in the grojind plan,
not in the original plane, but at a perpendicular distance c
above it.
Rule. — Take a new picture line parallel to the former, and at a distance
above it = c coseca, ivhere a is the angle hctween the original plane a)id the
plane of p)rojccti(»i. For a plane through the given ptnnt, parallel to the
original plane, will intersect the plane of projection in the \w\\ picture line
so constructed.
Thus, every point of a figure in the ground ^dan is transferred to the
drawing.
1086 The whole theory of perspective drawing is virtually included in
the fuicgoing propositions. The original plane is conmionly horizontal, aud
the plane of pnjiction vertical. In this case, cosee n = 1, and the height of
iho. pidure line for any point is equal to the height of the jiuint itself above
the original plane.
The distance BO, when B is the point of sight, may be measured along
A B, and bj) along ah, in the opposite direction} for the lino Bb will continue
to intersect Oj^ in the point I'.
Orth oii;on a I I* rojrct io n .
1087 Di:riNiTioN. — In oi-tli()^n)iial ])rojectioii tlic linos of
pi-ojection are parallel to eaeh other, and |)erj)en(licular to the
plane of iirojectiou. The vertex in this case may be consi-
dered to he at infinity.
1088 The projections of pai-allel lines are jjaiallel, and the
piDJected segments are in a constant ratio to the oi-iginal
seu'nients.
1089 Areas are in a constant ratio to their projections.
For, lines parallel to the intersection of the original pinne and the plane
of projection arc unaltered in length, and lines at right angles to the former
are altered in a constant ratio. This ratio is the ratio of the areas, and is
the cosine of the angle between the two planes.
Projections of the Sphere.
1090 lu Strreograjyhic projection, the vertex is on the sur-
face of the sphere, and the diameter through the vertex is
])erpendicular to the plane of projection which passes through
the other extremity of the diameter. The projection is there-
fore the inversion of the surface of the sphere (lUl2),and the
diameter is the constant /r.
1091 Ii^ (Uohular projection, the vertex is taken at a dis-
tance from the sphere equal to the radius -i- \/2, and the
diameter thi'ough the vertex is perpendicular to the plane of
I)rojection.
1092 In Gnomon ic projection, which is used in the construc-
tion of sun-dials, the vertex is at the centre of the sphere.
1093 Mercator's projection , \\h\ch is employed in navigation,
and sometimes in maps of the world, is not a projection at
all as defined in (1075). ]\Ieridian circles of the sphere are
represented on a plane by parallel right lines at intervals
eqnal to the intervals on the equatoi-. Th(> pai'allels of lati-
tude are represented by right lines jierpendicular to the
meridians, and at increasing intervals, so as to preserve the
actual ratio between the increments of longitude and latitude
at every point.
With r for the radius of the sphere, the distance, on the chart, from the
equator of a point whose latitude is A, is = r log tau (40*^ + JX).
232 ELEMENTARY GEOMETRY.
Additional Theorems.
1094 The sum of the squares of the distances of any point
r from n equidistant points on a circle whose centre is and
radius 7' = n (r'^ + OP'-) .
Proof.— Sum the values of FB-, PG\ &c., given in (819), and apply (803).
This theorem is the generalization of (923).
1095 In the same figure, if P be on the circle, the sum of
the squares of the perpendiculars from P on the radii OP, 0(7,
&c., is equal to ^wr^.
PuooF. — Describe a circle upon the vaclius through P as diameter, and
apply the foregoing theorem to this circle.
1096 Cor ]. — The sum of the squares of the intercepts on the radii be-
tween the perpeudiculai's and the centre is also equal to -g-ur. (I. 47)
1097 Cor. 2. — The sum of the squares of the perpendiculars from the
equidistant points on the circle to any right line passiug through the centre
is also equal to \nr'^.
Because the perpendiculars from two points on a circle to the diameters
drawn through the points are equal.
1098 Cor. 3. — The sum of the squares of the intercepts on the same
right line between the centre of the circle and the perpendiculars is also
equal to ^nr\ (I- 47)
If the radii of the inscribed and circumscribed circles of a
regular polygon of n sides be r, R, and the centre 0; then,
1099 I. The sum of the perpendiculars from any point P upon the sides
is e(]ual to 'iir.
1100 n. Ifj) be the perpendicular from upon any right line, the sum
of the j)erpendical;irs from the vertices upon the same line is equal to nj^.
1101 III. The sum of the squares of the perpendiculars from P on the
yides is = n{r-^-\OP-).
1102 IV. The sum of the squares of the perpendiculars from the vertices
upon the right line is = n (p' + ^R').
Proof. — In theorem I., the values of the perpendiculars are given by
r—Or cos (0+ -^), with successive integers for m. Add together the u
values, and apply (803).
Similarly, to prove II. ; take for the perpendiculars the values
7> /n I -'""■ \
2>-Rcos[0+-—).
To prove III. and IV., take the same expressions for the perpendiculars;
square each value; add the results, and apply (803, 804).
For additional ])ro])ositions in the subjects of this section,
see the section entitled rianc Coonlinaic Ucomctri/.
GEOMETRICAL CONICS.
THE SECTIONS OF THE CONE.
1150 Definitions. — A Conic Section or Conic is the curve
AP in which any plane intersects the surface of a right cone.
A right cone is the soHd generated by the revolution of
one straight line about another which it intersects in a fixed
point at a constant angle.
Let the axis of the cone, in Fif^. (1) or Fif^. (2), be in the plane of the
paper, and let the cutting plane PMXN be perpendicular to the paper. {L'ead
either the acce)ited or unaccented letters tJtrouijhout.) Let a sphere be inscribed
in the cone, touching it in the circle EQF and touching the cutting plane in
the point S, and let the cutting plane and the plane of the circle EQF inter-
sect in XM. The following theorem may be regarded as the dfifmbuj properfij
of the curve of section.
1151 Theorem. — The distance of any point P on the conic
from the point S, called the focus, is in a constant ratio to
FM, its distance from the line XM, called the directrix, or
FS : FM = FS' : PJ/' = AS : AX = e, the eccentricity.
\_See next page for the Proif.']
1152 CoK. — The conic may be generated in a plane from
either focus *S', N', and either directrix XM, X'M' , by the law
just proved.
1153 The conic is an EJIips^e, a Parabola, or an JFi/jjrrhoItt,
according as e is less than, equal to, or greater th.-m unity.
That is, according as the cutting plane emerges on both sides
of the lower cone, or is parallel to a side of the cone, or in-
tersects both the upper and lower cones.
1154 AH sections made by parallel planes are similar; for
tlie inclination of the cutting })lane determines the ratio
AI-] : AX.
1155 The limiting forms of the curve are respectively — a
circle when e vanishes, and two coincident right lines when e
becomes infinite.
2h
234
GEOMETRICAL CONIC S.
Proif ok TiiEOKKM 1151. — Join P, 8 and P, 0, cutting the ciicular s-ction
in Q, and draw FM parallel to NX.. Because all tanjjcnts from the same
point or A, to either sphere are equal, therefore HE = I'Q = I'S and
AE= AS. Now, by (VI. 2), HE : NX = AE : AX and NX = PM;
therefore PS : P^f = AS : AX, a constant ratio denoted by e and called
the eccentric ill/ of the conic.
Referring the letters either to the ellipse or the hyperbola in the
subjoined figuie, let C be the middle point of A A' and N any other point on
it. Let Djy, h'li be the two circular sections of the cone whose planes pass
through C and A^; PCP' and PN the intersections with the plane of the
conic* In the elii|)si', J!<' is the common onlimitc of the ellipse and circle ;
but, in the hyperbola, PC is to be taken equal to ihe langeut from C to the
circle DD'.
1156 The fundamental equation of the ellipse or hyperbola
is FN'' : AN. NA' = BC~ : AC.
Proof.— PiV^ = NP . NR' and PC = CD . CD' (III. 35, 36). Also, by
similar triangles (VI. :!, G), NR : CD = AN I AC and NE' : CD' = A'N : A'C.
Multiply the last e(juatiuns together.
I
Tlir: ICLLirSE and UYl'F.nV.OLA.
235
1157 Cor. 1.— P.Ylms
e(|ii;il values at two points
e(|ui- distant from AA' .
Hence tlie curve is sym-
metrical witli respect to
.-LI and inr.
These two lines are
called the Dinjor and lui-
twr a.rcs, otherwise the
transverse and conjuijate
axes of the conic.
AVhen the axes are
equal, or BC = AG, the
ellipse becomes a circle,
and the hyperbola be-
comes reef angular or ./
equilateral.
1158 Any elHpse or hyperbola is the orthogonal projection
of a circle or rectangular hyperbola respectively.
PiioOF. — Aloiifr the ordinate NP, mcnsnrc NP' = AN . NA' ; tliercfore by
tlie tliuorem PN : P'N = PC : AC. Therefore a circle or rectangular hyper-
bola, having AA' for one axis, and having its plane inclined to that of the
conic at an angle whose cosine = PG-^AC, projects orthogonally into the
ellipse or hyperbola in question, by (108'J). See Note to (1-JOl).
1159 TTence any projective j^roperfi/ (107G-78), which is
known to belong to the circle or rectangular hyperbola, will
also be universally true for the ellipse and hyperbola respec-
tively.
THE ELLIPSE AND HYPERBOLA.
J(Hnf propcrtirs of the Ellipse and Ili/pcrhola.
Dfi'INITIons. — The tanrjeuf to a curve at a point P
110(3) is the right line PQ, drawn through an adjacent
(Fi
1160 Dfi'initions.
110(3) IS the rigllU nut; / ^»', Ulcl\>ll im^-Mij^n cm cn4|<n.» iiu
point (?, in its ultimate position when Q is made to coincide
with P.
The normal is the pori)endicular to the tangent through
the point of contact.
236
GEOMETRICAL CONICS.
In (Fig. 1171), referred to rectangular axes tlirougli the
centre G (see Coordinate Geometry) ; the length CN is called
the abscissa; PN the ordinate; PT the tangent; PG the nor-
mal; NT the suhtangcnt ; and NG the sulnormal. S, 8' are
the foci; XM, X'M'the directrices; PS, PS' the focal dis-
tances, and a double ordinate through S the Latus Rectum.
The auxiliary circle (Fig. 1173) is described upon AA' as
diameter.
A diameter parallel to the tangent at the extremity of
another diameter is termed a conjugate diameter with respect
to the other.
The conjugate hyperhola has BG for its major, and AG for
its minor axis (1157).
1161 The following theorems (1162) to (1181) are deduced from the
property PS : PM = e obtained in (1151).
The propositions and demonstrations are nearly identical for the ellipse
and the hyperbola, any difference in the application being specified.
1162
CS : CA : CX, and the common ratio is e.
TJf
P^
M'
//>^
Z
A
ii
\
—
S' A'
X'
Proof.— By (1151), e
= ^ = r^ = g jA'S^AS) ^ CS ^^ CA
AX A'X I {AX^AX) CA
CX
1163 In the elHpse the sum, and in the hyperbola the dif-
ference, of the focal distances of P is equal to the major axis,
or PS'±PS= AA'.
Proof. — With the same figures we have, in the ellipse, by (1151),
PS+PS'
PS + PS'
XX' '
and also c =
AS + A'S A A'
PM + PM" XX' ' AX-tA'X
Pur the hyperbola take difference inatuad oi sum.
XX'
therefore &c.
THE ELLIPSE AND TIYPERBOLA.
1164 CS- = AC'-nr^ in tlio ellipse.
[For nS = AC, by (11G3).
CS- = AC--^BC~ ill the liyperbola.
[Oy assuming JlC. Sco (117(1).
1165 BC'=:SL.AC.
Proof.— (Figs, of 11G2) SL : SX = OS : CA, (1151, 1102)
.-. SL.AC=CS.SX=CS(CX^CS) = CA'^CS' (11G2) = 7?C^ (1104).
1166 If a right line tlirougli P, Q, two points on the conic,
meets the directrix in Z, then SZ bisects the angle QSE.
.17/
PnooF.— By similar triangles, ZP : Z(2 = MP : NQ = SP : SQ (Uol),
therefore by (VI. A.)
1167 If PZ be a tangent at P, then FSZ and FS'Z' are
right angles.
Pkoof. — Make Q coincide M'iih. P in tlic last theorem.
1168 The tangent makes equal angles with the focal dis-
tances.
Proof.— In (1100), PS : PS' = PM : P-V (1151) = PZ : PZ'; therefore,
when PQ becomes the tangent at P, Z SPZ = S'PZ\ by (1107) and (VI. 7).
1169 The tangents at the extremities of a focal chord inter-
sect in the directrix.
Proof.— (Figs, of 1100). Join ZR ; then, if ZP is a tangent, ZR is also,
for nir>7) proves h'SZ to bo a right angle.
1170
CN.CT=AC'.
PKOOF.-(Figs. 1171.) -^^' = -pl (VI. 3, A.) = ^^^^- (1151) = ^,
therefore
therefore
T S'+TS _ NX'+NX 2rT _ 2GX
TS'-TS~ NX'-NX' ^ 2CS '2CN'
CN.CT- CS.CX = AC^.
(1102)
238
GEOMETRICAL CONIC S.
1171 If ^'^-*' be the normal,
GS: PS=^ GS' : PS'=e.
Proof.— By (11 G8) and (VI. 3, A.),
GS _GS' _ GS'+GS
PS
F8'
_2CS
PS' + PS 2GA
(11G2)
But, for the hyperbola, change j:)/ its to viinus.
1172 The subnormal and the abscissa are as the squares of
the axes, or A^G : NC = BC : AC\
Proof.— (Figs. 1171.) Exactly as in (1170), taking the normal instead
p.i . . ^,■ GG CN , GN_GX_CA' ....ry^
ot the tangent, we obtain -— - = -~r, . . ^yT 7^~ T^S'^ v-'--'-"-'/'
CN^CG _ CA^^aS^ or^=i^ (11G4).
ON GA' ' ^^'^ ^<^' ^
NO AG^
1173 The tangents at P and Q, the corresponding points on
the elUpse and auxihary circle, meet the axis in the same point
T. But in the hyperbola, the ordinate TQ of the circle being
drawn, the tangent at Q cuts the axis in N.
OF.— For the ellipse : Join TQ. Then CN. CT = CQ"" (11 70) ; thcre-
>,T is a right angle (VI. 8) ; tlicrclbre QT is a tangent.
Proof.
fore C(J,T is a rigut angi
For the liyperbola : Interchange N and T.
THE ELLTPSE AXD IIYrEKJiOLA.
u
239
1174 PN : QN = liC: AC.
PuooF.— (Figs. 1 1 73). NG . NT = I'N', and CN . NT = QN\ (VI. 8)
TluTefuro NLf : NC = I'N^ : QN-; tl.crcforo, by (1172).
This proposition is equivalent to (115H), and sliows tliat an ellipse is tlio
nrt]io<,'onal projection of a circle equal to tiie auxiliary circle.
1175 ^'oiJ- — The area of the ellipse is to that of tlie auxiliaiy
circle as iy^:. 10 (1089).
1176 PN' : AN.NA' = BC : AC\
Proof.— By ( 1 174), since QN^ = AN. NA' (III. 35, 3G). An indopend-
cnt proof of this theorem is given in (11-">G). The construction for JJC iu
tlie hyperbola in (llGl) is thus verified.
1177 Cii • Ct = nc\
ri:uoF. — (Figs. 1173.)
rt _ PiV. . Cn.rt _ FN' _PN' .... o. _ FN"- .„. _ ...
CT - Wf ' ■ CNTcT - CNTnT ~W^ ^~ AN . NA' ^"^^ ^"' '^^^•
Therefore, by (1170) and (117G), ai.Ct: ACP = BC : AG^.
1178 If SY, S'Y' are the perpendiculars on tlie tangent,
then Y, Y' are points on the auxiliary circle, and
SY,SY' = BC\
Proof.— Let P.S meet SY in W. Then FS = TW (11G8). Tliereforo
S']V=AA' (11G3). ALso, .Sr= YW, and ,S(' = r.S". Therefore CY= iS W
= AC. Similarly CY" = AC. Tlierefoie 1', 1" are on the circle.
Hence ZY' is a diameter (III. 31), and tiu'refore SZ = S'Y', by similar
triiintrles ; therefore 6Y . SZ = SA . SA' (111. 3."., '.W^ = CS' ^ C.V (11. 5)
= IW- (llGi).
1179 Cuii.— If C'^ be drawn i)arallel t^) the tangent at J\
then PE= CY = AC.
1180 Tkoblem. — To draw tangents from any point to an
elhpse or hyperbola.
Construction. — (Figs, 1 181.) Describe two circles, one with centre O and
radius OS, and another with centre S' and radius = ..LI', intersecting in M,
240
GEOMETRICAL CONICS.
jr. Join MS', M'S'. These lines will intersect the curve in P, F', the
points of contact. For another method see (1204).
Pkoof.— By (11G3), PS'±PS = AA' = S'M by construction. There-
fore PS = PM, therefore Z OPS = 0PM (1. 8), therefore OP is a tangent
by (1168).
Similarly P'S = P'M\ and OP' is a tangent.
1181 The tangents OP, OP' subtend equal angles at either
focus.
Proof.— The angles OSP, OSP' are respectively equal to OMP, OM'F,
by (I. 8), as above ; and these last angles arc equal, by the triangles OS'M,
OS'M', and (I. 8). Similarly at the other focus.
Asymptotic Projyerties of the Ilyperhola.
1182 Def- — The asymptotes of the hyperbola are the
diagonals of the rectangle formed by tangents at the vertices
A, A, B, B'.
1183 If the ordinates RN, EM from any point R on an
asymptote cut the hyperbola and its conjugate in P, P', P, P*,
li
t;//; iiYPERnoLA. 24.1
then either of ihe following pairs of equations will define both
the branches of each curve —
RS--p\' = r>r'= /*\--/?.v- (1),
RM--DM'= Ar'= inP-H)P {'!).
Proof. — Firstly, to prove (1): By proportion from the similar triaugles
ENC, OAC, wo have -^.^- = -^ = cN'-XG' '
by (llTiV), since AN. NA' = CN'-AG\ By (II. 6)
Tberefuro ^~^A~f-- = 4^. ^J the theorem (G9) ;
thereforo EN'-FX' = BL'-.
Also, by (117G), applied to the conjugate hyperbola, the axe.s being now
reversed, jyN^-BC' " liC' " Tn'' ^^ ''°''^'''' *"^"Sles ;
therefore P'N'-BC = EN' or P'N'-EN^ = BC\
Secondly, to prove (2) : By proportion from tlie triangles EMC, OBC, we
EiP _ Af^ _ I)}iP
^""^^ CM' ~ BC ~ CM'-BC'
by (11 rC), applied to the conjugate hyperbola, for in this case we should
have BM . MB'' = C^P - BC\
Therefore ^^^,J^ = -^S 5 tl^ereforo EM'-BM' = AC\
BC x»C
Also, by (117C), since CM, D'J/ are equal to the coordinates of D',
/-(ira 7)7'' C^P
-'r = — -r = - -,-r7j ^y similar triangles :
DM'- AC' AC EM-' ^ °
therefore I)'M'-AC- = EM' or B'M'-EM' = AC.
1184 CoE. 1. — If the same ordinates RN, EM meet the
other asymptote in r and /•', then
FR.Rr = nC- and I)R.Dr = AC\ (II. o)
1185 ^''">IJ- 2. — As R recedes from C, /'/i' and />>/? con-
tinually diminish. Ilencc the curves continually ap})roach
the asymptote.
1186 If ^E be the directrix, CE = AC,
Proof.— CE : CO = CX : CA = CA : CS and CS - CO. (11G4)
1187 I^D is parallel to the asymptote.
7,'.V* BC' US'- PS' .,,e^. r.V ,
Therefore EN : PA" = PIT : DM; therefore, by (VI. "2).
242
GEOMETBTCAL CONICS.
1188 The segments of any riglit line between the curve and
the asymptote are equal, or Qli = qr.
QTi : QU = qB, : gZ/'") Compound tlie ratios,
Proof, —
and Qr : Qu = qr : qu ) ' and employ (1184).
1189 Con. l.~PL = PI and QV = qV.
1190 Cor. 2.—CH = HL. Because PD is parallel to 10.
(1187)
1191 QP . Qr = PL' = RV'-QV'= Q V--R V.
Vkoof.— Qli : QU=PL: FE ) Compound theratios. Therefore, by (11 8i),
and Qr : Qu = El : Pe 3 QR . Qr = PL.P1 = PL" (1180).
1192 4PH.PK= CS\
P]iGOF.— Pn : PE = GO : On ") .-. PH. PK : PE.Pe= CC : Oo^
and PK : Pe = Go : Go) = GS^ : AEG''; tberefore, by (1184).
Joint Properties of the Ellipse and Hi/perhola resumed.
If PGP' be a diameter, and QV an ordinate parallel to the
conjugate diameter CD (Figs. 1105 and 1188).
1193 QV: PV. VP = CD' : CP\
Tliis is tlic fuiidruncntal equation of tbc conic, equation (1170) being the
most important form of it.
THE ELLirSE AND HYPERBOLA.
243
Otlierwisc :
In the ellipse, QV: (P-iV = CD-.VPK
Tutliolivpcrbola, QV' : CV'-CP-= CIX . CP^;
and Q F- : C F-+ 67'- = CD' : CP\
Proof. — {EUlps^e. Fig. 1195.) — By orthogonal projection from a circle.
If G, r, P', D, Q, V are the projections of c, p, p, d, q, v on the circle ;
qv' = pv . vp
id cd-
cp
The proportion is therefore trae in the case of
the circle. Therefore &c., by (1088).
(Ilijperhvla. Fig. 1188)—
CTP_ _ nv^ _ PL' _ nv'^pu ^ qf' qt^
CF' ~ CV- ~ CP'~ CV'±CP' GV'-CP^ CV^+CF''
(1101)
1194 The parallelogram formed by tangents at the extremi-
ties of conjngate diameters is of constant area, and therefore,
i'i' being perpendicular to CD (Figs. 1195),
rF.CD = AC.BC.
Proof. — (Ellipse.) — By orthogonal projection from the circle (1089).
{ILiperhola. Fig. 1 1 88.) — GL.Cl = APU . PK = CO . Co (1192) ; there-
fore, by (VI. 15), ALGl = OGo = AG.BG.
If PF intersects the axes in and G\
1195 FF.PCm = EC and FF.FG = AC\
t
1197 Con.— FG . PG = CD = FT. Ft.
By (1194)
244 GEOMETRICAL CONICS.
1198 The diameter bisects all chords parallel to the tangent
at its extremity.
Proof. — {Ellipse. Fig. 1195.) — By projection from the circle (1088)
QV=VQ'. {Hyperhola.) By (1189.)
1199 CoE. 1. — The tangents at the extremities of any chord
meet on the diameter which bisects it.
Proof. — The secants drawn through the extremities of two parallel chords
meet on the diameter which bisects them (VI. 4), and the tangents are the
limiting positions of the secants when the parallel chords coincide.
1200 Cor. 2. — If the tangents from a point are equal, the
diameter through the point must be a principal axis. (I. 8)
1201 CoR. 3. — The chords joining any point Q on the curve
with the extremities of a diameter PP' , are parallel to con-
jugate diameters, and are called supplemental chords.
For the diameter bisecting PQ is parallel to FQ (VI. 2). Similarly the
diameter bisecting P'Q is parallel to PQ.
1202 Diameters are mutually conjugate ; If CD be parallel
to the tangent at P, CP will be parallel to the tangent at D.
Proof. — (EUijJse. Fig. 1205.) — By projection from the circle (1088).
]SI"OTE. — Observe that, if the ellipse in the figure with its ordinates and
tangents be turned about the axis Tt through the angle cos"^ (PC -^ AC), it
becomes the projection of the auxiliary circle with its corresponding ordinates
and tangents.
(Hi/perbola. Fig. 1188.)— By (1187, 1189) the tangents at P, D meet the
asymptotes in the same point L. Therefore they ai-e parallel to CD, CP (VI. 2.)
If QT he the tangent at Q, and QV the ordinate parallel
to the tangent at any other point P,
1203 CV.CT=CF'.
Pkoof.— CP bisects PQ (1199). Therefore PT7 is parallel to QP.
Therefore, by (VI. 2), CV : CP = CW : CE = CP :CT.
Tni'J ELLIPSE AND nYFERBOLA.
245
1204: Cor. — Hence, to draw two tangents from a point T, wo may find
(T tVoni the above equation, and draw QVl^ parallel to tho tangent at 1' to
dL'tennine tho points of contact Q, (/.
Let FN, I)N be the ordinates at the extremities of con-
jugate diameters, and PT the tangent at P. Let the ordinates
at N and I\ in the clhpse, but at T and C in the hyperbohi,
meet the auxiUary circle in p and <l ; then
1205
CN= (in, en =:pN.
Proof. — (Ellipse.) Gp, Cd are parallel to the tangents at d and p (Note to
1'20"2). Therefore pCd is a right angle. Therefore pNC, CRd are equal
right-angled triangles with CN=^dli and CR = pN.
(Eijperbola.) ON . CT = AG' (1170),
and DR.CT= 2ACDT = 2CDP = AG . BG (119-i) ;
CN AG pN ...^,. . GN DR GR . . ., , . , .
• • irR=JfG = FN (^^^^^' • • ,^:V = -FN= IN ^^^"^^"^ '"^"°^"^^-
But ^=^^(VL8); .-. GR = pN. Also Cp = Gd; therefore tho tri-
2)N 1 N
angles GpN, dCR are equal and similar; therefore CN = dR and dR is
parallel to pN.
1206 Cuij.— DR : (in = BC : AC.
Proof. — (Ellipse.) By (117-1). (Hijperbola.) By the similar right-angled
triangles, we have dR : fN = GR : TN = DR : PN;
therefore dR:DR= pN : PN= AG : BG (1174).
In the same figures,
1207 {Ellipse.) CX'-j- Cn' = A (" ; /> A'-'+ PS' = BCK
1209 {Uiipcrhola.) CN'-CR' = AC; DW-PX' = BC\
Proof. — Firstly, from the right-angled triangle CNp in which ^'^V = CR
(li2<.:.).
Secondly, In the ellipse, by (1174), PIi'- + PA'' : dR' + pN' = BC- : AC^,
and dR- +pN'— A C, by ( 1 205) . For the hyperbola, take difference of squares.
246 GEOMETRICAL COKICS.
1211 {Ellip.^e.) CP'-^CD' = AC'+BC\
1212 {Ihjperhula.) CP'-CD' = AC'-BC\
Proof.— (Figs. 1205.) By (1205—1210) and (1.47), applied to tlie
triangles CNP, CRD.
The product of the focal distances is equal to the square
of the semi-conjugate diameter, or
1213 PS . PS = CD\
Proof. — (Ellipse. Fig. 1171.) 2P8 . PS' = (PS + PSy - PS'- PS"
=z4^AC'-2CS'-2CP' (922,1) = 2(AC' + BC'-CP') (1104) = 2CI>-(1211).
(Ryperhola.)— Similarly with 2PS . PS' = Pg' + PS''-(PS'-PSy = &c.
1214 The products of the segments of intersecting chords
Q0(/, (}'0q are in the ratio of the squares of the diameters
parallel to them, or
OQ.Oq : OQ'.Oq = CD" : CD\
Proof. — (Ellipse.) By projection from the circle (1088) ; for the propor-
tion is true for the circle, by (111. 35, 3C),
(Byperlola. Fig. 1188.) Let be any point on Q^. Draw lOi parallel
to Ee, meeting the asymptotes in I and i ; then
OR.Or-OQ.Oci = QR.Qr (11.5) = PV (1191) (1).
„ on PL .Or PI , OR . Or _ PL' _ CD' ,,.on
^°^ OI = p¥'^^'^ 07=Pi' '■-dTOi-pETPe-^C'^^^''^^-
^, . OR.Or-PU CD' _, n^ OQ.Og _ CD'
Therefore ^iToTII^O^ = ^ ' ''^' ^^ ^^^' 0/. OZ-i^'C"^ " W
Similarly for any other chord Q'O'/ drawn through 0.
Therefore OQ.Oq: OQ' . Oq = CD' I CD''.
1215 Cor.— The tangents from any point to the curve are
in the ratio of the diameters parallel to them.
For, when is without the curve and the chords become tangents, each
product of segments becomes the square of a tangent.
1216 If from any point Q on a tangent FT drawn to any
conic (Fig. 1220), two perpendiculars (}R, QL be drawn to the
focal distance PS and the directrix XM respectively ; then
SIl : QL = c.
Proof.— Since QU is parallel to ZS (1107), therefore, by (VI. 2),
SE : PS =QZ:PZ= QL : PM;
therefore SR : QL = PS : PM = e.
Cor. — By applying the theorem to each of the tangents from Q, a proof
of (1181) is obtained.
777/; i:LLirsi: axd mi'innioi.A.
17
1217 '^'/"' iJii'crfur Cirrlr. — The locus of the point of iiitcr-
Btcrioii, 7', of two tangents always at right angles is a circle
called the Dlnrfor Circle.
Proof. — Perpendiculars from S, S' to tlic tancfents meet them in points
y, Z, y, Z', which lie on the auxiliary circle. Therefore, by (II. 5, ('>) and
(III. 35, 3G), TC ~ AC = TZ.TZ' = SY . S'Y' = BC^. (1178)
Therefore W = A(T ± BCT; a constant value.
Note. — Theorems (1170), (1177), and (1203) may also be deduced
at once for the ellipse by orthogonal projection from the circle; and, in all
such cases, tlie anahigous projieity of the hyperbola may be obtained by a
similar projection from the rectangular hyperbola if the property has already
been demonstrated for the latter curve.
1218 n the points A, S (Fig. 1102) be fixed, while C is
moved to an infinite distance, the conic becomes a parabola.
Hence, any relation which has been established for parts of
the curve which remain finite, when AC thus becomes infinite,
qrlJJ he a property of the j^arahoht.
1219 Theorems relating to the elli])se may generally bo
ada])ted to the parabola by eliminating the (juantities which
become infinite, cmi)loying the ])rinciple thntjlulfe ililj'rreiirea
may be neglected in consideriiKj the ratios of injhiite quantities.
Example. — In (1193), when P' is at infinity, IT* becomes
in (1213) FS' becomes = 2C'P. Thus the equations become
2CP: and
Therefore QV
-pF=7t ""' '•' = 2CP-
; \rS .rV in the parabola.
248
GEOMETRTCAL CONICS.
THE PARABOLA.
If 8 be the focus, XM j^j
tlie directrix, and P any point
on tlie curve, the defining -pro-
pert ij is
1220 PS = PM
and e = \. (1153)
1221 Hence
AX = AS.
1222 The Latus Rectum = 4^AS.
Pkoof.— SL = SX (1220) = 2 AS.
1223 If PZ be a tangent at P, meeting the directrix in Z,
then PSZ is a right angle.
Proof.— As in (1167) ; tbeorem (1166) applying equally to the parabola.
1224: The tangent at P bisects the angles 8PM, 8ZM.
Proof.— PZ is common to the triangles PSZ, PMZ ; PS = PM and
/.PSZ = PMZ (122;J).
1225 Cor.— ST = SP = SG. (1. 20, 6)
1226 The tangents at the extremities of a focal chord PQ
intersect at right angles in the directrix.
Proof. — (i.) They intersect in the directrix, as in (1160).
(ii.) They bisect the angles SZM, SZM' (1224), and therefore include a
right angle.
1227 The curve bisects the sub-tangent. AN = AT.
Proof.— ST = SP (1225) = PM = XN, and ^IX = AS.
1228 The sub-normal is half the latus rectum. KG = 2 AS.
Pkoof.— ST = SP = SG and TX = ,S'.V (1227). Subtract.
Tin: PAUAnoLA.
10
1229 rN'= iAS.AN.
Puoor.—FN' = TX. Nd (VI. 8) = AN.2N(} (1227) = •l-.l.'^ . AN (1228).
Otherwise, by (117(;) imd (11G5) ; making .-IC infinite. See (1210).
1230 The taii.cronts at .1 and P each bisect S}f, the latter
l)isi'eting it at right angles.
Proof.— (i.) The tangent at A, by (\fl. 2), since AX = AS.
(ii.) FT bisects SM at right angles, by (I. 4), since PS = PM and
/. SPY = MPY.
1231 C.m.-
SA : SY : SP.
[By similar triangles.
1232 To draw tangents from a point to the parabi^la.
CoNSTKUcnoM. — Describe a circle, centre
and radius OS, cutting the diiectrix in
M, M'. Draw MQ, M'Q' parallel to the axis,
meeting the parabola in (2, Q'. Then OQ,
0(/ will be tangents.
Proof.— OS, SQ = Oif, MQ (1220);
therefore, by (I. 8), Z OQS = OQ^f■, there-
fore OQ is a tangent (1221). Similarly
OQ' is a tangent.
Otherwise, by (1181). When S' moves
to infinity, the circle MM' becomes the
directrix.
1233 C.R. 1.— The triangles SQO, SOQ' are similar, and
SQ : SO : SQ'.
Proof.— z SQO = MQO = SMM' = SOQ'. (III. 20)
Similarly SQ'O = SOQ.
1234 Cor. 2. — The tangents at two points subtend equal
angles at the focus ; and they contain an angle equal to half
the exterior angle between the focal distances of the points.
Proof.— z OSQ = OSQ\ by (Cor. 1).
Also z QOQ' = SOQ + SQO = n-OSQ = '^QSQ'.
1235 Def.— Any line parallel to the axis of a parabola is
called a iViamctcr.
1236 The chord of contact QQ! of tangents from any point
O is bisected by the diameter through O.
Proof. — This proposition and the corollaries are included in (1108-1200),
by the principle iu (1218). Au independent proof is aa follows.
2 K
250
GEOMETRICAL CONIC S.
The construction being as in (1232), _, (^
wo have ZM = ZM' ; therefore QV^VQ'
(VI. 2).
1237 Coi^ 1. — The tangent
liW at P is parallel to Q(2' ; and
OP = PV.
Proof. — Draw the diameter RW.
QW= WP ■ therefore QR = RO (VI. 2).
Similarly Q'R' = R'O.
1238 Cor. 2.— Hence, tlie dia-
meter through P bisects all chords parallel to the tangent at P.
If QFbe a semi-chord parallel to the tangent at P,
1239 QV' = 4^PS.PV.
This is the fundamental equation of
the parabola, equation (1229) being the
most important form of it.
Proof. — Let QO meet the axis in T.
By similar triangles (1231),
Z. SRP = SQR = STQ = FOR ;
and ZSPJ^= OPB (1224). Therefore
PB''= PS .PO = PS .PreindQV=2PR.
Otherwise: See (1219), where th(
equation is deduced from (1193) of the
ellipse.
1240 CoE. 1. — If V he any other point, either within or
without the cm-ve, on the chord QQ' , and iw the corresponding
diameter
vQ .vQ' = 4/>8 .jyv-
(11. 5)
1241 CoK. 2. — The focal chord parallel to the diameter
through P, and called the parameter of that diameter, is equal
to 4>ST. For PV \n this case is equal to PS.
1242 The products of the segments of intersecting chords,
(lO<i, Q'Oq', are in the ratio of the parameters of the diameters
which bisect the chords ; or
OQ.Oq : OQ'.Oq = PS : PS.
Proof.— By (1240), the ratio is equal to 4PS .pO : 4P'/S . j^O.
CIP PS PS'
Otherwise: In the ellipse (1214), the ratio is = - „ = jy.,' ,,.',
L JJ 1 o . 1 o
■when >S" is at iutinity and the carve becomes a parabola (1219).
(1213)
PS
PS'
ON CONSTRUCTINO Till: CONIC.
251
1243 (^oiL — The squares of tlie taiip^eiits to a parabola from
any point are as the focal distances of the points of contact.
Proof.— As ift (121o). Otherwise, by (1233) and (VI. 19).
124:4 The area of the parabola cut off by any chord Q(/ is
two-tliirds of the circumscribed parallelogram, or of the tri-
angle formed by the chord and the tangents at (j, (/.
PROor. — Througla Q, q, q', &c., adjacent points
on the curve, draw right lines parallel to tlio
diameter and tangent at P. Let the secant Qq
cut the diameter in 0. Then, when q coincides
with Q, so that Qq becomes a tangent, we have
OP =z PV (1237). Therefore the parallelogram
17y = 'lUq, by (I. 43), applied to the parallelogram
of which 0(^ is the diagonal. Similarly vq = 2iiq',
&c. Therefore the sum of all the evanescent par-
allelograms on one side of PQ is equal to twice
the corresponding sum on the other side ; and
these sums are respectively equal to the areas
PQV, PQU.—(NE\\Wii, Sect. I., Lem. II.)
Fracfical methods' ofcotisfntcrmg the Conic.
1245 To draw the Ellipse.
Fix two pins at S, S' (Fig. 11G2). Place over them a loop of thread
having a iierimeter SPS' = ,S',s" + .Lr. A pencil point moved so as to keep
the thread stretched will describe the ellipse, by (llGo).
1246 Othprir;,f>.—(Vig. 1173.) Draw PHK parallel to QC, cutting the
axes in II, K. PK = AC and PR = EC (1174). Hence, if a ruler PHK
moves so that the points II, K slide along the axes, P will describe the ellipse.
1247
To draw the Ilijperbola.
Make the pin S' (Fig. 11G2) serve as a pivot for one end of a bar of any
convenient length. To the free end of the bar attach one end of a thread
whose length is less than that of the bar by A A' ; and laslen the other end of
the thread to the pin S. A pencil ])oinL moved so as to keep the thread
stretched, and touching the bar, will describe the hyperbola, by (11G3).
1248 Otherwise: — Lay off any scale of equal parts along both asymptotes
(Fig. 1188), starting and numbering the divisions from C, in both positive
and negative directions.
Join every pair of points L, I, the jimdiicf of whose distances from C is
the same, and a series of tangents will be formed (irj2) which will detiue
the hyperbola. See also (12b9).
252
GEOMETRICAL CONICS.
1249 To drcnv the Parahola.
Pi'oceed as in (1247), with this difference: let the end of tlie bar, before
attached to S', terminate in a " T-square," and be made to slide along the
directrix (Fig. 1220), taking the string and bar of the same length.
1250 Otherivise: — Make the same construction as in (1248), and join
every pair of points, the algebraic sum of whose distances from the zero
point of division is the same.
Peoof. — If the two equal tangents from any point T on the axis (Fig.
1239) be cut by a third tangent in the points Ji', r ; then EQ may be proved
equal to rT, by (1233), proving the triangles SBQ, SrT equal in all respects.
1251 Cor. — The triangle SRr is always similar to the isosceles triangle
SQT.
1252 To find tlte axes and centre of a given central conic.
(i.) Draw a right line through the centres of two parallel chords. This
line is a diameter, by (1198) ; and two diameters so found will intersect in
the centre of the conic.
(ii.) Describe a circle having for its diameter any diameter PP' of the
conic, and let the circle cut the curve in Q. Then PQ, P'Q are parallel to
the axes, by (1201) and (III. 31).
1253 Given two conjugate diameters, CP, CD, in
and magnitude : to construct the conic.
On CP take PZ = Glf'-^CP ; measuring
from C in the ellipse, and towards C in the
hyperbola (Fig. 1188). A circle described
through the points C, Z, and having its
centre on the tangent at P, will cut the
tangent in the points where it is intersected
by the axes.
Proof. — Analysis: Let AC, BC cut the
tangent at P in T, t. The circle whose
diameter is Tt will pass through C (III. 31),
and will make
CP. P.^ = PT. Pi (III. 35, 3G) = CD' (11 97).
Hence the construction.
Circle and Radius of Curvature.
1254 Definitions. — Tlie circle wliicli lias the same tangent
■with a curve at P (Fig. 1259), and which passes through
another point Q on the curve, becomes the circle of curvature
when Q ultimately coincides with F; and its radius becomes
the radius of curvature.
CIRCLE OF cunvATirnE.
253
1255 Otherwise. — The curie of rurratitre is the circle which
passes through three coincident points on the curve at /'.
1256 Any chord PIT of the circle of curvature is called a
elionl oj' niri'dfure at P.
1257 Tlirough Q draw PQ' parallel to Plf, meetinpr the
tangent at P in P, and the circle in Q' , and draw C^F parallel
to pp. J?Q i& called a .subtense of the arc PQ.
1258 Theorem. — Any chord of curvature PII is equal to
the nl ti unite' iml lie uf the square of the arc PQ divided by the
subtense HQ parallel to the chord : and this is also equal to
Proof.— 7?Q' = RP-^RQ (III. 36). And when Q move.s up to P, RQ'
becomes PH; and RP, PQ, and QF become equal because coincidoit Uuci^.
1259 In the ellipse or hyperbola, the semi-chords of cur-
vature at P, measured along the diameter PC, the normal
FF, and the focal distance PS, are respectively equal to
err err err,
CP' PF ' AC '
the second bcimr the radius of curvature at P.
vr . cr>'
CP'
(ll'J3)
UP
Pkoof.— (i.) By (1258), PU = ^^
limit when T'P' becomes PI' = 2l'l'.
(ii.) By the similar triangles I'lfU, PFC (III. 31), wc Lave
PU.PF = CP . PU = 2CD\ by (i.)
(iii.) By the similar triangles PIU, PFE (I1G8), wo have
PI.PE = PU.PF = 2C'D-, by (ii.) ; and PE = AC (1179).
tho
254 GEOMETRICAL CONTCS.
1260 In tlie parabola, tlio chord of curvature at P (Fig.
1250) drawTi parallel to the axis, and the one drawn through
the focus, are each equal to 4SF, the parameter of the dia-
meter at P (1241).
Proof.— By (1258). The cliord parallel to the axis =QV^-^PV= 4PS
(1239) ; and the two ehoi'ds arc equal because they make equal angles with
the diameter of the circle of curvature.
1261 Cor. — The radius of curvature of the parabola at P
(Fig. 1220) is equal to 2SP'- ^ SY.
Proof.— (Fig. 1259.) ^PU = ^PI sec IPU = 2SP BecPST (Fig. 1221).
1262 The products of the segments of intersecting chords
are as the squares of the tangents parallel to them (1214-15),
(1242-43).
1263 The common chords of a circle and conic (Fig. 12G4)
are equally inclined to the axis ; and conversely, if two chords
of a conic are equally inclined to the axis, their extremities
are concyclic.
Proof. — The products of the segments of the chords being equal (HI.
35, 36), the tangents parallel to them are equal (1262). Therefore, by (1200).
1264 The common chord of any conic and of the circle of
curvature at a point P, has the
same inclination to the axis as
the tangent at P.
Proof. — Draw any chord Qq parallel
to the tangent at P. The circle circum-
Bcribing PQq always passes through the
same pointy (1263), and does so, there-
fore, when Qq moves up to P, and the
circle becomes the circle of curvature.
1265 PrvOBLEM. — To find the centre of curvature at any given
^oint of a conic.
First Method. — (Fig. 1261.) Draw a chord from the point making the
same angle with the axis as the tangent. The perpendicular bisector of the
chord will meet the normal in the centre of curvature, by (1264) and (III. 3).
1266 Second Method.— Draw the normal PG and a perpendicular to it
from il, meeting either of the focal distances in Q. Then a perpendicular to
the focal distance drawn from Q will meet the normal in 0, the centre of
curvature.
MISCELLANEOUS THEOREMS. 255
Vroov.— (Ellipse or Hyperbola.) By (1259),
the radius of curvature at V
= l^F = ^$ ^^ ^^^''^^ = TF^ ^^ ^^^•'■^•>
= PG sec' S'PG = PO.
For AC = PE (by 1170).
(Parabola.) By (rJOl). The radius of curvafn
= ^SP'^SY = 2SP sec -S'PG = PO.
For 2SP = PQ, because SP = SG (11-25).
Miscellaneous Theorems.
1267 lu the Parabola (Fig. 1239) let QD be drawn perpendicular to PV,
tlien QL^ = 4A.S.PV. (1231,1239)
1268 Let liPIt be any third tangent meeting the tangents OQ, OQ' in
Ji', ii''; the triangles SQO, SPR', SOQ,' are similar and similarly divided by
SR, SP, SR' (1233-4).
1269 Cor.— OR.On; = RQ.R'Q'.
1270 Also, the triangle PQQ; = 20RR'. (12-il)
With the same construction and for any conic,
1271 OQ: OQ'=^RQ.RP' : r:Q'.PR. (1215,1243)
1272 Also the angle RSR' = IQSQ'. (1181)
1273 Hence, in the Parabola, the points 0, R, S, E are concyclic, by (1234),
1274 In any conic (Figs. 1171), SP \ ST = AN : AT.
Proof— f| = f ^^ = "^ (69, 11G3) = |^ (1170) = j^ (09).
1275 Cor.— If the iangent PT meets the tangent at A in R, then SR
bisects the angle PST (VI. 3).
1276 In Figs. (1178), SY', S'Y both bisect the normal PO.
1277 The peipendicular from S to PG meets it in CY.
1278 If Ci) bo the radius conjugate to CP, the ptrpcndicular from D
upMii CY is equal to PC.
1279 SY and CP intersect in the directrix.
1280 If every ordinate PN of the conic (Figs. 1205) be turned round N,
in the plane of the figure, through the same angle PNP\ the locus of P' is
also a conic;, by (ll'J3). The auxiliary circle then becomes an ellipse, of
■which AC and PC produced arc the equi-conjugatc diameters.
If the entire figures be tlius deformed, the points on the axis AA' remain
fixed while PN, IJR describe the same angle. Hence CP remains parallel to
PT. CP, CD are therefore still conjugate to each other.
256 GEOMETRICAL CONICS.
Hence, tlic relations in (1205-G) still subsist when CA, CB are any con-
jugate radii. Thus universally,
1281 FN: GB = DB: CN or FN. CN = DB . CB.
1282 If the tangent at P meets any pair of conjugate diameters in T, T',
then FT. FT' is constant and equal to CD\
Proof. — Let CA, CB (Figs. 1205) be the conjugate radii, the figures
being deformed through any angle. By similar triangles,
FT : CN = CD • ^'B ] ' ^^^refore PT. Pr' : FN. CN = CD^ : DB . CB.
Therefore FT. FT' = Clf, by (1281).
1283 If the tangent at P meets any pair of parallel tangents in T, T,
thou FT. FT = CI)\ where CB is conjugate to CF.
Peoof. — Let the parallel tangents touch in the points Q, Q'. Join PQ,
PQ', CT, Cr. Then CT, CT' are conjugate diameters (1199, 1201). There-
fore FT.FT' = CB' (1282).
1284 Cor.— ar. Q'r = CB\ where CFf is the radius parallel to QT.
1285 To draw two conjugate diameters of a conic to include a given
angle. Proceed as in (1252 ii.), making FF' in this case the chord of the
segment of a circle containing the given angle (III. 3B).
1286 The focal distance of a point P on any conic is equal to the length
QN intercepted ou the ordinate through P between the axis and the tangent
at the extremity of the latus rectum.
Proof.— (Fig. 1220). QN : NX = LS : SX = e and SF : NX = e.
1287 In the hyperbola (Fig. 1183). CO : CA = e. (11G2, 1164).
If a right Hue FKK' be drawn parallel to the asymptote CB, cutting the
one directrix XE in K and the other in K' ; then
1288 SF = PK = e. CN-AC; S'P = FK' = e.CN+AG.
Proof.— From CB = e . CN (1287) and GE = AC (1186).
1289 Cor. — Hence the hyperbola may be drawn mechanically by the
method of (1249) by merely tixing the cross-piece of the T-square at an
angle with the bar equal to JJCO.
1290 Definition. — Confocal conies are conies which have the same foci.
1291 The tangents drawn to any conic from a point T on a confocal conic
make equal angles with the tangent at T.
Proof. — (Fig. 1217.) Let T be the point on the confocal conic.
SY : SZ = S'Z' : S'Y' (1178).
Therefore ST and S'T make equal angles with the tangents TF, TQ ; and
they also make equal angles with the tangent to the confocal at T (1168),
therefore &c.
1292 In (ho construction of (1253), FZ is equal to half the chord of
cui-vature at I' di-awn thi'ough the centre C (]25;»).
DIFFERENTIAL CALCULUS.
IXTRODUCTIOX.
1400 Ff()irfiou.<i. — A quantity wliicli depends for its value
upon another quantity x is called afiinctiou of </'. Thus, sin ,r,
log.v, n-^, a^-{-ax + x' are all functions of x. The notation
y =/{,!') expresses generally that y is a function of ./■. y = sin«
is a particular function.
1401 /CO is called a confinuoiis function between assigned
limits, when an indefinitely small change in the value of x
always produces an indefinitely small change in the value of
/OO-
A transcendental function is one which is not purd^-
algebraical, such as the exponential, logarithmic, and circular
functions a"", log.r, sin.'B, cos,/', &c.
If /(.c) =/( — a?), the function is called an even function.
If /(,,) — — /( — ir), it is called an odd function.
Thus, ./■- and cos j- are even functions, while a;' and sin x are odd functions
of x; the latter, but not the former, being altered in value l)y changing the
sign of X.
1402 Diffrrcniinl Coefficient or Der'wnth'e.—Lct y be any
function of x denoted by /(-r), such that any change in the
value of X causes a definite change in the value of // ; then x
is called the Independent variable, and // the depend mt rarialde.
Let an indefinitely small change in x, denoted by dx, produce
a corresponding small change '/// in//; then the ratio --• , in
the limit when both dy and dx are vanishing, is called the
dlffrre)itlal coefficient,' or derivatin', of y with respect
to X.
2 L
258 mFFERENTTAL CALCULUS.
1403 TiiEOEEM. — The ratio c/// : dr is definite for eacli valuo
of <?', and generally different for different values.
Pkoof. — Let an abscissa ON
(Defs. 1160) be measured from
equal to .r, and a pcrpendicalar or-
dinate NP equal to y. Then, wliat-
ever may he the form of the function
y =f(x), as X varies, the locus of
P will be some line PQTj. Let
OM = x', MQ = //' be values of x
and 7/ near to the former values.
Let the straight line QP meet the
axis in T; and when Q coincides with P, let the final direction of QP cut the
. axis in T'.
Then II or --^f^ = |^. And, ultimately, when QS and SP vanish,
they vanish in the ratio of PN : NT'. Therefore '-^ = =— , = tan PT'N, a
dx NT
definite ratio at each point of the curve, but different at different points.
1404 Let NM, the increment of x, be denoted by li ; then,
when h vanishes, ^ = /C^+ZQ -/(.Q ^ ^, ..
iLv h »/ \ /'
a new function of. a', called also the first derived function.
The process of finding its value is called differentiation.
1405 Successive differentiation. — If ^^ or/'(<6') be differ-
dcC
entiated with respect to x, the result is the second differential
coefficient oif(a'), or the second derived function ; and so on
to any number of differentiations. These successive functions
may be represented in any of the three following systems of
notation : —
(Ijl (Pjl cJ^ (l^ iV^
d.v da'' fAr^' dd'' ^/> '
/G^O, /"CO, /'"Ci"), /^(.^O, /"GO;
y^^ Vtx, i/s^; //la-, //«...*
The operations of differentiating a function of x once,
twice, or n times, are also indicated by prefixing the symbols
jL jH j!;;_ d (dv (dy
da?' d.i^' •" d.e^' "'^ d,v' W7' - \dr) '
or, more concisely, d,, d.^, ... d„^.
* See note to (1487).
TXTnoDrr'Tinx. 2't\}
1406 It', nj'trr (liffoivutiatin^ ;i function for .r, x ])e niado
zero in the result, tlio valiio may bo indicated in any of tho
following Avoys : -j'~, f'W^ //.ro> -7— ? ^^.o-
If any other constant a be substituted for x in ?/^, the
result may bo indicated by. v/.,.^ .
1407 Infniteslmals and Differentials. — Tlio eviinescent
([uantities dx^ thj are called iujinito^linah ; and, with respect
to X and //, they are called diffcrentiah. <lr, (I'-y are the second
differentials of x and ij ; dx^, d^y tho third, and so on.
1408 The successive differentials of >j are expressed in terms
of d I' by the equations
,ty =/' (.r) fhv ; d'\f/ =f"{.v) dv' ; &c., and d'\,/ =/"(.r) d,v\
Since /"(<r) is the coefhcient of dx in tlie value of di/, it lias
therefore been named the differential coefficient of y or /(.'0-*
For similar reasons /' {x) is called the second, and /" {x) tho
n^^ differential coefficient of f{x), &c.
1409 Two infinitesimals aro of the same order when their
ratio is neither zero nor infinity.
If dx, dij are infinitesimals of the same order, dx', dy^^, and
dxdy will bo infinitesimals of the second order with respect to
dx, dy; clx^, dx'dy, &c. ^vill be of the third order, and so on.
dx, dx^, &c. are sometimes denoted by x, x, &c.
1410 Lemma. — In estimating the ratio of two quantities, any
increment of cither which is infinitely small in eonji)arison
with the quantities may be neglected.
Hence the ratio of two infinitesimals of the same order is
not affected by adding to or subtracting from either of them
an infinitesimal of a higher order.
Example. — ■^~ = ^ —dx = -^, for dx is zero in comparison with
, dx ax dx
the ratio ^-[K Thus, in Fig. (1403), putting P*S = dx, QS = d>j ; we have
dx
ultimately, by (12.'')8), QR = Jcdx^, where A- is a constant. Therefore
FN _ Wl _(hi-hd^_ chi_ .^^ ^j^^ j.^^^.^^ ^^^ ^j^^ principle just enunciated;
NT' I'S dx dx
that is, QR vanishes in eonqmrison with PS or QS even loh-n those lives them-
selves are iufinitehj small.
* The name is slightly misleading, as it seems to imply that /'(•'') is in
some sense a coefficient of /'(•'")•
260 DIFFERENTIAL CALCULUS.
DIFFERENTIATION.
DIFFERENTIATION OF A SUM, PRODUCT, AND QUOTIENT.
Let u, V be functions of x, then
tAtn d(uv) du , dv
1412 -T— = V -^ +w ^— .
d.v ax ax
1413
(/ / u \ _ ( du _ _^\ _:_ 2
</ct' \ V / \ dx dx J
Proof. — (i.) d(n + v) = (ii + dH + v + do) — (u + v) = du + dy.
(ii.) d(Hv) = (u + du) (v + dv)—uv = vda+udv — diidv,
and, by (1410), dudv disappears in the ultimate ratio to dx.
,... . 1 / u\ u + du u vdu — udv
(ill-) d [ = ; = — r— ,
\ i; / v-{-dv V {y + dv)v
therefore &c., by (1410) ; vdv vanishiBg in comparison with v^.
Hence, if u be a constant = c,
1414 !iM=o4l and ^l£) = -^^.
dx dx dx \v / L- dx
DIFFERENTIATION OF A FUNCTION OF A FUNCTION.
If 7/ be a function of z, and z a function of x,
1415 dy__dti_ dz_
dx dz ' dx
Proof. — Since, in all cases, the change dx causes the change dz, and the
change dz causes the change dj ; thcrefoi'e the change dx causes the change
dy in the limit.
Differentiating the above as a product, by (1412), the successive differ-
ential coefBeients of y can be formed. The first four are here subjoined for
the Rake of reference. Observe that (^j)j: = l/a^x'
1416 7A = y.z^.
1417 //..,■ = //..^:.H-//.^2...
1418 //:,. = //:,,-:. + ;{//..-.c^...+//.-U-
1419 uu- = Uu^l + %..4;^.c + //.. (»5^L + A^..-...) + U^-^^'
DIFFEBENTTATION. 2 CI
DIFFERENTIATION OF A COMFOSITE FUNCTION.
If n and v be explicit functions of .i', so tluit a = (^ (./•) and
^^"^^ dTv TT^T dx' "^ (//' iLv '
Here f77^ in tlic first term on tlio riglit is the cliangc in
F{u,v) produced by da, the change in u; and dF in the
second term is the change produced by dv, so that the total
change dF{i(j v) may be written as in (1408)
dF, + dF,= ^dii + ^du,
die do
DIFFERENTIATION OF TEE SIMPLE FUNCTIONS.
Since -^ ^ /('^ + ^^)~/W ^hen h vanishes, we have tho
dx lb
following rule for finding its value :
1421 Rule. — Expand f(x+h) hy some hiown theorem in
ascending poivers of h; subtract f (x) ; dliido hi/ h; and m
tlic result imt h equal to zero.
The differential coefficients which follow are obtained by
the rule and the theorems indicated.
1422 y = ^i'"- ^ = "^^""'•
PKOOF.-IIero • -^ = 1^ 1
(12o) = nx"-^-]-C(n, 2) x"-'h+... = )ix"-\ wliou h vauishcs.
1423 Cou.-
i^=«(»-l)...(M-r+l),,--'. ^=k-
1424 z/ = l<'g..'-; 7^ = ,7k^-
PKOOP.-By (145). '"''-^■■- + ;-^-'"'^-' = ^ \ log. (l + {) I -^ 7-
J ^ ■" It X lug, at \ X I } X
Expand the logarithm by (15-5).
,.„^ ,, il-,/ (-1)"-'1«-1 rat« = -lin(l.i23)
1425 U...- -^= a.'\oga"~ - andr = ,.-l.
262:
DIFFERENTIAL CALCULUS.
1426
y =
'■■■ %'
rt''loge«.
Proof.—
li ~ h
^. Expand a" by (149).
1427
Cor.—
|^ = «'(log.
a)\
Fimdion.
Derivative.
Method of Proof hy Bute (1421)
and Limits (753).
1428
sin a?.
COS tl\
Expand by (627, G29), and
1429
cos a?.
— sin X.
put 1 — cos /i = 2 sin'' -^.
1430
1431
1432
1433
tan cV.
cot 0?.
sec .V.
cosec<r.
sec^ .r.
— cosec^ <r.
tan ct- sec x.
— cot.i coseccr.
Expand by (G31), observing
(1410).
By cota; = -^, and(1415).
By secx = -^, and(1415).
cos/;
Similarly.
1434
1436
sin~^ tt'
COS~^ cV ) '
tau-^iO
cot-i ^^. ] •
± 1 .
If sin'^a; = y, x = sin y,
therefore
dx /-, ?
--- = cos 7/ = V i — x' :
J//,
therefore '^}'^ = -r; -•
dx Vl—x^
1438
sec~^ X
COSeC"^ W:
■ "^.vO^-lY
Similarly for the rest.
1440
Examples.
(1422)
1441
(;^),= (■'-")- —-"-•--.-•
(1422)
1442 {(a + x"-y(h + xy^} = 3 (a-^u-y- 2x (h+xy- + 2 (b + x^) 3.r Or+,r)»
= 6z(a+zyih + ,v') ( h + ax + 2.1-^) . ( 14 1 2, ' 1 5, '22)
1443
/ ^^-^- ^\ ^ ('''' + e-'')(e' + e-')-(e'-e-')(e'-e-') ^ 4
(1413, 142G)
1444
'/^ (log tun ,*■) = 2 lo'' tau x sec' x = ^^ . (1415, 24, 30).
tun X sin zx
sucCESSivr: inFVKRv.xriATjns. 2r.:]
Sonio ilifTeivniiations nro rciulorcd cnsioi' by liikinj,' tlio lo^'arithm of tlio
function. For example,
1445 y = ^/ J~l. ; tbcrcforo logy = \ log i\-x-)-l log(l + x') ;
. P 1 '''/ _ 1 -2.« 3 1x .
tlierctoro ,-- — — y. .rr ^ 77-; — jT >
X, r f?!/ -2.>-(2-3'-) _ -2.rr2-x'')
thcrofore -/-=:?/ 7 = :, r-
(Zo! 1-x* (H-a;^)»(l-x-)*
1446 2/ = (si" ^y ; theroforo log ?/ = » log sin x ;
thereforo — V* = log sin a:+ -^ cos a; ; (1 il 5, '24, '28)
1/ • siu «
therefore 1/, = (sin a-)' (log siu aj+a; cot x).
Otherwise, by (1420), y, = x (sin a-)'"' cos a; + (sin a;)' log sin x (1420)
= (siu .r)"" (.7; cot a; + log sin a;) .
SUCCESSIVE DIFFERENTIATION.
1460 Leibnitz s Theorem. — If n be any integer,
... +C(«, »•)//(„_,),;:;,,+ ... 4-.V^«x.
PnooF.— By Induction (233). Differentiate the two consecutive terras
C («, '•) i/(,.-r)x2«+ C (?l, r + 1) 2/(,.-r-l)x^(r.I)x,
and four terms are obtained, the second and third of which are
C{n, r) ?/(,._„, «(r.nx+C(?i, r + l) .'/(,.-.) x ^/r.i)*
= [C{ih r)-^G{n, r+l)}y(,.-r)x2(r.i)-= C{n + l, r+l) ?/ ,m-7n x 2 ..1,,^
This is the general term of the series with n increased by unity. Similarly,
by differentiating all the terms the whole series is reproduced with n in-
creased by unity.
DIFFERENTIAL COEFFICIENTS OF TUE n"' 01^ DEE.
1461 (s»W «'0«x = «" siu {(LV-\-lmr). By Indnotion
1462 (cos (U')„,. = a" cos {fLv-^UTr). ^"^'^ ^^•^-'^•
1463 {('"■''),.. = (i"t''-'' (1^'^G)
1464 {ff\f/)„.. = c-'-{a + (r,.y',/,
where, in the expansion l)y the Binomial Theorem, d'^i/ is to
be rephiced l)y //,..c. (1100, '03)
264 DIFFERENTIAL CALCULUS.
1465 {e"" cos 6cr)„^ = rV'^ cos (6.r + w<^),
wliere a = r cos <^ and 6 = r siu (j).
Proof. — By luduction. Differentiating once more, we obtain
r"e'"' {a cos (bx-\-n(p) — h sin (bx + n(p)}
__ yn + igO'lcos^ cos (hx + n^) — Bm<{> sin {hx + n<}>)}
— /''len'cos (tx + n + l^).
Thus 71 is increased by one.
1466 i^'"'-' log ^)n. = liLzJ. -^ ^^'. (1-^60), (283)
1468 (tan-i ^)nx={- 1)""' 1^-1 siu** ^ sin n0,
where ^ == cot"^ a.\
Proof. — By Induction. Differentiating again, we obtain (omitting the
coefficient)
(>i sin""' cos 9 sin nd + n cos nd sin" 6) 6^
= w sin""^0 (sin«0 cos0 + cos?i0 sin0) ( — sin-0).
Since, by (1437), 0, = -(l + x')-' = -sin^O.
Therefore (tan-^ «),,..!) « = (—1)" l«_sin""' d sin (n + l) d,
n being increased by one.
1469 ( ^ ,, ) = (-1)'^ I^sin'^^i ^ sill (n + 1) 6- (1-^36,
1470 (j^) =(-l)"|;^sin'^^^^cos(>i + l)^.
\l-rct' //jj;
PuoOF.-By (14G0), (i|^)„^= ^ (xi:.)^^" (r^O.-./
Then by (1469).
1468)
1471 Jacohi's Formula.
c/(.-a). (1-^r)"'^ = (-1)'*"' 1 • 3 ... 12/1-1) sin {n cos-\r) -h «.
Pkoof. — Let )j = l—x^; therefore
(^"- i)„. = -(2h+1) (.r7/"-i)(„.„x. Also (//"-i),. = (.'///"-•O-.x.
Expand eacli of these vahics by (1460) and eliminate (y'"^),„-2}x, the^dcriva-
tivc of lowest order. Call the result equation (1). Now assume (M7U true
for the value n. Diilerentiate and substitute the result, and also (1471) on
the right side of equation (I) to obtain a proof by Induction.
TIinOPiY OF OriJRATlOXS. 2(15
1472 Tlii'orcm. — If y, :: are finictions of .r, and ;/ a positive
iiiri\u-('t',
~//«.r = (//^),.r-" ('/^..•)(«-l).r+^' («» i^) (//^2.r)(«- 2) .r •••+(- 1 )" //--..r-
PnooK. — By Tiulnction. DilTcrontiato for a;, substituting for z^y,,^ on tlio
rip;ht its viiluo by tlio formula itsolf.
PARTIAL DIFFERENTIATION.
1480 If '' =/X'^'» ?/) ^^"^ ^ function of two imloppiulinit vari-
ables, any difforcntiation of n with respect to x rrr/irlres tJuif ij
should be considered constant in that operation, and rice versa.
Thus, -'— or Wjx signifies that n is to be differentiated succes-
sively twice with respect to x^ y being considered constant.
1481 Tlie notation ., ^ or 7/2^3^* signifies that u is to bo
differentiated successively twice for x, y being considered
constant, and the result three times successively for y, .c being
considered constant.
1482 'I'lio order of the differentiations does not -affect tho
final result, or 7/^,^ = Vy,..
Froof.— Let u=f(xy); then y^ =fJ^:±]i:^!lzdL(j^J!l in limit. (1181)
,h>^ f (x + Ik 7/ 4- /.-) - /' (> •. 7/4. /,•)-/ f.r + /^ y)-^f(x,y) . , . .
"- = -.ij =• ' ' — —hir^ '" •'"''*•
Now, if Vy bad boon first formed, and then iiy^, tbe same result would liavo
been obtained. Tlio proof is easily extended. Let u^-=.v;
then v^y = v^,j = i\^ = iivis ; and so on.
THEORY OF OPERATIONS.
1483 Let the symbols <^, ^, prefixed to a quantity, «lenoto
operations upon it of the same class, such as nudtiplication or
differentiation. Then the law of the operation is said to be
distrUjutice, when
<i>(.r+//) = -i>GO+^K//);
* Sec note to (1487).
2 .M
266 DIFFEnEXriAL CALCULUS.
that is, tlie operation raay be performed upon an undivided
quantity, or it may be distributed by being performed upon
parts of the quantity separately with the same result,
1484 The law is said to be commutative when
that is, the order of operation may be changed, ^ operating
upon ^x producing the same result as ^ operating upon <^x.
1485 ^""^ denotes the repetition of the operation 4> m times,
and is equivalent to <t>* ... a? to m operations. This definition
involves the index law,
which merelv asserts, that to perform the operation n times in
succession upon x, and afterwards m times in succession upon
the result, is equivalent to performing it m-\-n times in suc-
cession upon X.
1486 The three laws of Distribution, Commutation, and the
law of Indices apply to the operation of multiplication, and
also to that of differentiation (1411, '12). Therefore any
algebraic transformation which proceeds at every step by one
or more of these laws only, has a valid result when for the
operation 'of muUijjlication that of differentiation is sub-
stituted.
1487 111 making use of this principle, the symbol of dif-
ferentiation employed is —r-, or simply r/,, prefixed to the
ux
quantity upon which the operation of differentiating with
respect to x is to be pei-formed. The repetition of the opera-
d^ d^ d^
tion is indicated by -7^, —r^, -, '-, ^ , &c., prefixed to the
•^ dx^ rt.r dx^dif
function. An abbreviated notation is d^., d.y^^, d.^, (/o.,.,,^, &g.
Since d^Xd^ = dl in the symbolic operation of multiplication,
it will be requisite, in transferring the operation to differentia-
tion, to change all such i)idiccs to suffixes when the abbreviated
notation is being used.
Note. — The notation 17,, y^, w^^a^, cZa^rSy, &c. is an innovation. It lias, how-
ever, the recommendations of defiuiteness, simplicity, and economy of time in
•writing, and of space in printing. The expression -■ .^ requires at least
fourteen distinct types, while its equivalent if;,^.;,^ requires but seven. For
THEOIiY OF OPEHATloXS. 2Ci7
such reusous I have introduced tho shorLur notation experimonUiUy in tliuse
pages. , , „
All such abbreviated forms ofdiffbrential coefiicienta as y'y"y"'... or y // //...,
thou<?h convenient in practice, are iucoaiplete expressions, because the inde-
pendent variable is not specified.
The operation (f^^iv^ n^d tho derived function »i^s^, would bo more
accurately represented by (./;)J and ("l)], the index a.s usual indicatinj^ tho
repetition of the operation. JJnt the former notation is simpler, and it has
the advantage of separating more clearly the inilex of dillerentiation from tho
index of involution.
In the symbols ?/' and y^ji t'lo figure 2 is an index in each case: in tho
first, it shows the degree of involution ; in tho second, the onfrr of differentia-
tion. The index is omitted when the degree or the order is unity, since wo
■write )/ and //,.
The sTilUx takes precedence of the superfix. yl means the square of y,.
djc(,y*) would be written (//■), in this notation.
As a concise nomenclature for all fundamental opoi-ations is of great
assistance in practice, the following is recommended : J^ or y, may be read
"V /or *," as an abbreviation of the phrase, "the ditfereutial coefficient of (/
71 )2 7S
for, or with respect to, .t." Similarly, -^, -j—, TTTl' °^ ^^^® shortened
forms j/ix, "xy, «ix3v, niay be read '' y for two x," ''ii for xy," ^^u for two x, three
y," and so forth.
The distinction in meaning between the two forms y,,^ and y„ is obvious.
The first (in which n is numerical and ahoays an inleijer) indicates n succes-
sive diflerentiations for x; the second indicates two successive ditierentiations
for the variables x and z.
The symbols -^^ or y^, and - '{- or y^^, may be read, for shortness,
"y for X zero," "y for two x zero"; d.^^^^ (^ {xy) can be read "fZ for two x
three y of </> (xy)."
Although the notation r, is already employed in a tot.illy dilferent sense
in the Calculus of Finite Ditferences, my own experience is that tho double
signification of the symbol does not lead to any confusion: and this for the
very reason that the two meanings are so entirely distinct. Whenever tho
operation of differentiation is introduced along with the subject of Finite
Differences, the notation ^ must of course alone be employed.
Thus, in differentiation, wo liavo
1488 Tin: i.isTKir.iTivi: LAW (I_^(n-\-r) = (/_,i( + (i.r. (Mil)
1489 The COMMUTATIVE L.\w i1^{d^u) — (l^{(l^u)
or d,,u = (ly.u. (11^2)
1490 The iNi.KX LAW </>/:« =.(lT'u,
that is, ifm^d.^u = (f (,„+„). r^f- (l-^^^')
268 DIFFERENTIAL CALCULUS.
1491 Example. —
{ch-d,y = (ch-d,) (d^-d,_,) = d^d^-dAly-d,d^ + d,d, = d,-2d,, + d,,.
Here d^d^—dyd^ or d^y = d^^, by the commutative law. (U89)
(?A = f^2x by the index laio. (l4i»U)
Also {d.,-'ld^y-\-d.y) u = di^u — 'ld^yU + d^yU, by the distributive law.
Therefore, finally, {d^—d^Yu = d.:,u — 2d^yU + d2y^t.
Similarly for more complex transformations.
1492 Thus d^ may be treated aa quantitative, and operated upon as such
by the hiws of Algebra ; d" being written d,,^, and factors snch as d^dy, in
which the independent variables are different, being written d^^, &c.
EXPANSION OF EXPLICIT FUNCTIONS.
TAYLOR'S THEOREM.— EXPANSION OF f{x + h),
1500
where 9 is some quantity between zero and unity, and n is
any integer.
Proof.— (i.) Assume f(x + ]i) = A + Bh+CP + &G.
Differentiate both sides of this equation, — first for x, and again for /;,— and
equate coefficients in the two results.
1501 (ii.) Cox s Proof. — Lemma. — Iff(x) vanishes when a; t:= a, and also
when x—h, and if f(x) and f'(x) are continuous functions between the
same limits ; then f'(x) vanishes for some value of a; between a and h.
For /'('■) must change sign somewhere between the assigned limits (see
proof of 1403), and, being continuous, it must vanish in passing from plus to
minus.
1502 Now, the expression
f(a + x)-^fia)-xf'(a)- - j^/" 00
-^'=[/(-+/0-/(«)-V''(«)— -|^/"(")}
vanishes when x = and when x=h. Therefore the differential coefficient
with respect to x vanishes for some value of x between and h by the lemma.
Let dh be this value. Differentiate, and apply the lemma to the resulting
expression, which vanishes when x = and when .e = 06h. Perform the
pame process n+l times successively, writing Oh for Odii, etc., since merely
stands for some quantity less than unity. The ivsult shews that
/""('^+-^0-4^[/(" + /O-/(")-/'/OO---|-^/"('O|
vanishes wheu x = Oh. Substituting Oh and equating to zero, the theorem
is proved.
EXl'ANSION OF L' XT LICIT Fi'XrTlUXS. 260
1503 The last term in (1500) is called the reinaiii(h>r after
'/I terms. It may be obtained in either of the sii]>ji)ined forms,
tlie first being due to Lagrangt*,
l^f-ir+0h) or |_^(l-e/' •/»(,,■ + «/,).
1504 Since the coelHcient ,— ^ diminishes at last without
limit as n increases (230, ii.), it follows that Taylor's scries
■is conrergcnt if f " (x) remains finite for all values of n.
1505 If in any expansion of f{x-]-h) in powers of // some
index of h be neijatice, then / (,c) ,/'(<«),/" (.v) , &g. all become
infinite.
1506 If tbe least fractional index of h lies between n and
// + 1; then /"^^ (.'•) and all the following differential coeffi-
cients become infinite.
Pkoof. — To obtain the value of /"(.'), (.lifTercntiato the expansion 7i times
successively for /(, and put /< = in the result.
MACLAURINS THEOREM.
Put 07 = in (loOO), and write d' for // ; then, with the
notation of ( 1-1-0 G),
1507 fV) =/(0)+.rr(Oj+^/'(0)+... + ^/"(^^r),
where 0, as before, lies between and 1.
Putting y =f(,r), this may also be wiitten
1508 // = //o+' 777- + T^ 777^ + 1 ,, .. T^ + ^^^•
1509 XuTE. — If any function f(x) becomes infinite with a fuiite value of
X, then fix), /"(x), &c. all become infinite. Thus, if /(.r) = sec"' (1 +.'),
f'(x) is infinite when x = (14o8). Therefore /"(O), /'"(O), &c. are all
infinite, and /(■«) cannot be expanded by this theorem.
ncrnonlli's Scries.— Vnt h = -.v in (1500); thus,
1510 /(O) =nr)-.rf{.r) + ^f'\,v)-j^f'\A') + &c.
270 DIFFERENTIAL CALCULUS.
1511 If 1> (// + /'•) = and <p (//) = x; tlien
Proof.— Let y = tp-' (x) = f (x) , and let 7/ + /.- =/(.c + /0 ;
therefore «' + h = (j>(y + k) = 0.
Therefore y + h =/(0) =/(.'c)-ac/"(a;) + p^/'Ca^) -&c., by (1510);
which proves the theorem.
EXPANSION OP f{x + h, y + h).
Let f{xfj) = u. Then, with the notation of (1405),
1512
/(c*'+/j, 7/+A-) = u+(hu,+ku,) + ^{hhi,,-\-2hku,,-^k'u,J
1513 The general term is given by , — {h(l^-\-kdy)"u,
where, in the expansion by the Binomial Theorem, each index
of d^ and dy is changed into a suffix; and the coefficients
d^, d.^, &c. are joined to 21 as symbols of operation (1487) ;
thus ul is to be changed into u^.
Proof. — First expand f(x + h, y + ^c) as a function of (x + h') by (1600);
thus, f(x + k, y + Jc) =f(x, y + k) + hf, (ic, y + k) + ^^^ h%, (x, y + k) + &c.
Next, expand each term of this series as a function of (y + k). Thus,
writing u for /(,<'//),
f(x,y + k)= u + ku, + r^k'n,, +|y^X +n-^^X +
J>fr(x,y + k)= hu, + hku,, + r^hJc'ii,,, +Xhk'u^,^+
,-^ f2. (.^, y + /'•) = rj ^^'^ir + I ,^ 1^'ku.^,^ + — — h-k-a.^^^ +
|^/3.(.^-,y + /0=^/^V + j-^/^'A-».3.. +
r^/u(A2/ + ^)=^ A'«..+
i± Li
The law by which the terms of the same dimension in h and 1: are formed,
is Feen on inspect ion. They lie in successive diagonals ; and when cleared
of fractious the numerical coefficients are those of the Binomial Theorem.
EXrAXSTON OF EXPLICIT FUNCTIONS}. 271
The theorem may be extended inductively to a function of
three or more variabk'S. Thus, if u =/(.'', //, ;.'), we have
1514 fU-\-h. //+A-, z + l) = u + (hn,,+h'n^-\-lu,)
the general term being obtaincnl as before from the expression
UL
1515 C'oR. — If 21 = f{xnz) be a function of several inde-
pendent variables, the term {hn^-\-lcii,j-\-JuS) proves, in con-
junction with (1410), that the total change in the value of ?/,
caused bv simultaneous small changes in a', ?/, z^ is equal to
the sum of the increments of u due to the increments of x, ij, z
taken separately and supt'rposrd in any order.
This is known as the principle of the siq)crpositi'on of small
quantities.
1516 To expand f{x, ?/) or f{x, y, z, ...) in powers of x, //,
t^c., put x, ?/, z each equal to zero after differentiating in
(1512) or (1514), and write x, y, ... instead of h, k, &c.
1517 Observe that any term in these series may be made
tlie last by writing x-{-dh for x, y + dh for?/, &c., as in (1500).
sy:^ibolic form of taylor's theorem.
The expansion in (1500) is equivalent to the following
1520 /Gr+/0 = c-'ri^v).
PuooF. — By the Exponential Theorem (150), writing the indices of J, as
suffixes (1487),
e'"^'/(') = (1 + /,J, + iW,,+ .,.)/(^) =f(x)+],f,(x) + lh%(.r)+..., by (U88).
Cor.- a /(.r) =fi^v+h)-f{.v) = (e"'''-l)/(.r),
therefore A'^ (.r) = (f'"'' - 1 Yf(.v) ,
and generally A"/ CO = (c'"''-l)"f(.r),
the index signifying that the operation is performed n times
upon f{x).
1521 Similarly /(./+ A, //+/.•) = e'"''"'VC^^ //)•
Froof. —
= j\.e + h,y + l:), by (150) and (1512).
272 DIFFERENTIAL CALCULUS.
1522 And, generally, with any number of variables,
Cor.— As in (1520),
1523 If u = f (■/', //) = </) (r, 0) , where x = r cos 0, // = r sin Q ;
and if x = r cos (0 + ^), y' = r sin (0 + w) ; then f(x'y') is
expanded in powers of w by the formula
/(.»■',//) = «"""'-■"'''/(..•,?/).
Proof. — By (1520), r being constant,
^ (r, + 0,) = e'*"'^ ^ (r, 0) = e'^^VC^', 2/).
Now a; and y are functions of the single variable ; therefore
u, = w^«„ + My//9 = u^ ( — r sin 0) +Uy (r cos 0) = xa^—yu^.
The operation d, will be transformed by the same law (1492) ; therefore
d^ = xdy — yd^; therefore
f(x\y') = e"'(-^''v-2'^'x) j(,., y) = l + w{xu,-yu^) + h''X^''ihy-^^y''h,j + f'h.) + &o.
1524 Examples. — The Binomial, Exponential, and Loga-
rithmic series for {l-\-xT, a'', and log(l+,^), (125, 149, 155),
are obtained immediately by Maclaurin's Theorem (1507) ; as
also the series for sin « and cos a^ (764), and tan~^a3 (791).
The mode of proceeding, which is the same in all cases, is
shewn in the following example ; the test of convergency
(1504) being applied when practicable.
1525 tail .V = .V + 1 .v' + ^ .v' + ^ a^ + J$. .iH &c.
«» lO »)1.) J,r»t)«)
Obtained by Maclaurin's theorem, as follows : — Let
f (a;) = tan x = y 1 Therefore 2/, = « and z^ = 2yz ;
f ^^^ _ gp^jS^ = z ) y and ;; being used for shortness.
f (.b) = 2 sec^ X tan x = 2yz,
f" {x) = 2 (.^, + ?/..) = 2 (.^ + 2/;;),
fix) = 2 (%;>;'' + 4//2'' + V2) = Q{1yz- + yh),
f {x) = 8(2z^ + 8yV + Zyh'' + 2y*z) = 8 (2z' + Uy"-z"- + 2y'z),
r'(x) = 8 (I2yz' + 22yz'+ ...) = 272yz'+ ...,
r\x) = 272z*+...&c.,
the terms omitted involving positive powers of y, which vanish when x is
zero, and which therefore need not be computed if jig term of the expansion
higher than that containing .t;^ is required.
iJXPAXSioy OF Expjjcir Frxrrins<i. 273
Hence, by making aj = 0, and tlioreforo y = and 2=1, wo obtain
/(0)=0; /C<>) = 1; /'(0)=0; /"(())=2; r(0)=0; /''(0) = 10;
r'(0)=0; r\0) = 27-Z.
Thus tlio terms up to x^ may bo written by substituting these valued in
(1507).
In a similar inainicr, may bo obtained
1526 •^^'^••'■-i + 4--'' + in''' + 7!^r'''"' + ^^'
Methods iff i\i pdii.shni Inf lnd('f( nuindir Cor/licicnt.s.
1527 Rl'I'E 1.— Assume f (x) = A-f Bx + Cx"4-&c. Diffrr-
rntiate both sides of the eqiiatlou. Then e.rpniid f'(x) hi/ smne
known theorem^ and equate coefficients in the two results to
determine A, B, C, ijv.
1528 Ex. ^.-K. = .+ ^^ + ]^^ + ]^< + &o.
Obtained by Rule I. Assume
siu-'.r = A+Bx+Cx' + Dx'' + Ex' + F.c'+
Therefore, by (1-101), (l-.«-)'^ = i? + 2Cu; + 3DxH4LV + 5F.«' +
But, by Bin. Th. (1-28), (1-..^/^ = 1 + V+ 1^ .^*+ .j^ x«+
1 13
Equate coefficients ; therefore 7? = 1 ; C = 0; D = — ; E = 0; F= ^^ ^ ;
&c. By putting x = 0, wo seo that .1 =/(0) always. In this case
^l = sin-'0 = U.
In a similar manner, by Rnle I.,
lo^y c - I -t- ^' -1- pj 1^ ^ (^ -r
1530 Rl'I-T' it.— Assume the series, as before, with iinhiown
coefficients. Diffierentiate successireh/ until the function re-
ap2)ears. Then equate coefficients in the two rquindcnt scries.
1531 K^- — To expand sin.^• in powers of x.
Assume sin a; = A + Bx+ Cx^-\-Dx^ + Ex*+ Fx^+
Differentiate twice, co3.-e= B+ 2Cx+ 31)..-} -\- 4Ej'-i oFx*+
-sinx = 2C+3.2Dx + 4.3/;.cH5.4Z'V+
Put x = in the first two equations; tborcforo A — 0, B=l.
Equate coefficients in the first and third series.
1 N
&c.
274 DIFFERENTIAL CALCULUS.
Thus -2G=A, .■.G=^0; -S.2D = B, --D=-^;
-4..dE = C, .•.E = 0; -6AF=D, .-. F= \,
Tlierefoi-0 sin.-c = «- .—^ + , ., o . r — 'S:c-> as in (7G4).
1.2.3 1.2.0.4.5
1532 Rule III. — Differentiate the equation y = f (x) ta-icG
witJi respect tu x, and combine the results so as to form an
equation in y,j^, and j^x- ^^^^-^ assume y = A + Bx + Cx"^ + &c.
'Differentiate tiuice, and substitute the three values of j, y^, y.^
so obtained in the former equation. Lastly, equate coefficients
in the result to determine in succession A, B, C, Si'c.
1533 Ex. — To expand siu mO and cos mO in ascending powers
of sin or cos 9.
These series ai'e given in (775-779). Tliey may be obtained by Rule III.
as follows : —
Put a; = sinQ and y = smmd = sin (m siu"^a;).
Therefore y^ = cos (m sin'' x) (1434) (i.)
. , • \ \ in"^ , / --IN "tnx
?/2x = — sm (»i sm ' X) :, + cos {m sm x) :—^.
1 — x- (1 — .r)2
Therefore, eliminating cos (^rtsin~^);), (l — x") yo,—xy^-\-in^y = (ii.)
Let y = A + A^x + A^x''+ + A„x"+ ("i-)
Differentiate twice, and pnt the values of y, y„ and ?/2^ in equation (ii.) ;
thus 0=:vr(A + A^x + A,x^ + Ay+...+A,X + )
-x{A, + 2A.,x + SA^x-+...+nA„x"-^+ )
+ (\-.-){2A, + 2.SA,x+
+ {u-l)nA,,x"-"- + n{n-\-l)A„,^x"-'+(n + l){)i + 2)A„^2-v"+...}.
Equating the collected coefficients of x" to zei'o, wc get the relation
A.o= "^'""' A (iv.)
Now, when .v = 0, y = 0; therefore ^1 = 0, by (iii.). And when x = 0,
y^ = vi, by (i.) ; and therefore A^ = vi, by diftcTeiitiating (iii.). The relation
in succession.
Cos md is obtained in a similar way.
1534 Rule IV. — Form the equation in y, y^, and y,,^, as in
Jiulr J II. Take the n"' derivative of this equation bij applying
Leibnitz's formula (14G0) to the terms, and an equation in
ExrAxsiox OF ExniniT functions.
3x0)
y(n+2)x» y(n+i)x» ^'^^^ ynx ''^ ohtn'nirtl. Piif x = ill this; and
i'mploij the rcsnltlinj formnln In riilnihttc in succession y
y.uo» 4"''- i^i Machiurins cximnsiou (loOZ).
1535 1^-x. ('""" ' = 1 + fi.r + — ,/- + ^ .7.. ■^'
Obtained by Rule IV. Writing y for the function, the relation found is
(1 — .S-) yix—xy^—a-y = 0.
Differentiating n times, by (14G0), we get
(1— a;') //(,.. 2) r- C2>t+1) .«//(„. i;r — ('t- + n=) !/„x =.0.
Therefore ?/,„*2)xo = (<i* + w") y,wroi a formula which produces the cocfli-
cients in iMaclaurin's expansion in succession when y^ and y-,r,) have been
calculated.
1536 ARBOGAST'S METHOD OF EXPAIsDlNG 4){z),
T^-hcrc ;:: = ^/ + ^^.»' + j^,.*'' + Y^j-*''+ ^^'C (i.)
Let // = «/) (::). When x = 0, y = i> {") ; tlicreforc, by
Maclaiirin's tlieorem (1508),
1/ = 4>{fl)+^^\'/.rn-{- jh^Z/-2.-o4- j-V[]//.vo + ^*^'C (ii.)
Hence, in the values of y^, y.,,, &c., at (141()), ■/; has to bo
put = 0.
Now, when x=0, z=a; therefore y., y.y., &c. become «|)'(^;,
<f>"{a), &c. ; and 2;.ro» ^2xo» '^u-or &c. become a^, (h, rtg, &c. Hence
1537 KxAMi'Li:.— To expand \os(a + hx + c..r + .l..:' + &c.).
1 1 2
Here «, = ^, (/,= 2r, </, = GJ, ij,' (a) = —, <^"{i,) = --^, f (a) = ---.
It It U)
Therefore 2/^ = — , y..^, = r, H , ^j^ = ^ — .. H •
a a- a a <t «
Therefore, substituting in (ii.), we obtain as far as four terms,
276 DIFFEREXTIAL CALCULUS.
1538 Ex. 2.~To expand {a + aiX-\-a2x'^+ ...+a^x"y in powers of x.
Arbogast's metliod may be employed ; otherwise, we may proceed as
follows. Assume (a^+a^x ■^-o.^x^i- ... a„x"Y = Af, + A^x + A^x^+
Differentiate for .r ; divide tbe equation by the result; clear of fractious,
and equate coefficients of like powers of u'.
BERNOULLI'S NUMBERS.
1539 ^ = l-|^+C.|-iJ.|+i?„|-&c.,
wliere B2, B^, &c. are known as Bernoulli's numbers. Tlieir
values, as far as B^^, are
7. _ ^'^^ -P -^ V _^G17 J. _ 4.3867
■^^'^~273U' ''~ 6' ''~ 510' ''~ 798 *
They are found in succession from tlie formula
1540 nB,,_,-\- C (n, 2) n„_,+ C {n,V>) B^,_,+ ...
... + C{n,2)B,-in + l = 0,
tlie odd numbers 7^3, B^, &c. being all zero.
Proof.— Lot 2/= 7^- Then, by (1508),
,r" r* X*
y = y^-^y^^x-^-y-iro ,— ■ + 2/3x0 ,-7 + 2/4-0 ,-7 + &e.
\A [1 \±
Here y,,^^{-\y'^B,,. Now 7/0 = 1 and y^ = -l, by (1587). Also
ye' = y+x. 'J'lierofure, by (14G0), differentiating n times,
e^ {y.. + '"y,.-r>. + G(n, 2) y^n-2).+ ... +ny, + y] = y,.,.
Therefore ny^„.■,^M■\- G (n,2) y^„.o^M+ ...+ny^ + yo = 0.
Substitute I)„.i, Bn.2, &c., and we get the formula required.
1>3, J/5, J)j.. &c. will all be found to vanish. It may be proved, a j^riori,
that this will be the case : for
1541 e---T + l> -^7i:i-
Therefore the series (1530) wanting its second term is the expansion of the
expression on the right. But that expression is an even function of .i' (1401) ;
changing the sign of x does not alter its value. Therefore the series in ques-
tion contains no add powers of x after the first.
1542 The connexion between Bernoulli's numbers and the sums of the
powers of the natural numbers in (27G) is seen by expanding (I— <?')"' in
povvei-8 off', and each term afterwards by the Expoucntial Theorem (150).
EXrAXSIOX OF EXrUCTT FUNCTIONS. 277
1543
1544 "
— 1 *> T
Pkoof. — ^ — = — '—- T——: fi^d — — = 1 ■' - , and by (1 539).
111 0271 — 1 —.2)1
1545 1 + :^::: + :^ + :^ + = , ,, , ^4.
o-i j-< 4-« ]^ 2 2;/
Proof. — In the expansion of -^ -^ — p (1540) substitute 2i0 for a*, and it
becomes the expansion of cot 6 (770). Obtain a second expansion by difFer-
entiatinj? the higarithm of equation (815, sin in factors). Expand each
term of tlie result by the Binomial Theorem, and equate cocCQcicuts of like
powers of in the t\vo expansions.
STIRLING'S THEOREM.
1546 4>{.r^h)-4>{.v) = h<t>'{,v)^A,h {f (.r+//)-f Or)}
wliere .4 ,„= (-])"/?,„ -h [2n_ and A.„^, = 0.
Proof. — ^„ .^Ij, A^, &c. are determined by expanding each function of
x + h by (1500), and then equating coeliicients of like powers of x. Thus
To obtain the gencial relation between the coefficients: put (.r) = r',
since Jp J.^, &c. are independent of the form of (p. Equation (LVIC) then
prod u ces —j^ — = 1 — AJi — A.Ji^ — A J J' —&.C.;
and, by (1539), we see that, for valuL;s of?! greater than zero,
A ,,. . , = and A,,. = ( - 1 )"7>\,. -^ [^.
BOOLE'S TlIEORiaL
1547 t^(.r+/0 -</»(.') = AJi {4>' (.r4-//) + f (.r)!
Proof. — A^, A.,. A^, Ac. arc found by the same method as that employed
in Stirling's Theorem.
278 DIFFEL'EXTIAL CALCULUS.
For the general relation between the coefficients, as before, make ^ (.<.') = e',
and equation (1547) then produces
-f~ = A,h+A,Jr + A,h' + &c. ;
and, by comparing this with (1544), we see that
A. = and ^,._i = (-l)»-'i?,„ ?^.
EXPANSION OF IMPLICIT FUNCTIONS.
1550 Definition. — An equation f{x, y) = constitutes y
an itiq^licit function of x. If y be obtained in terms of c« by
solving the equation, y becomes an explicit function of x.
1551 Lemma. — If y be a function of two independent
variables x and ,^,
Proof. — By performing the differentiations, we obtain
F'(y)yryz + F(y)y,, and F'(y) y.jj, + F (y) y^,..,
which are evidently equal, by (1482).
LAGRANGE'S THEOREM.
1552 Given i/ = z-\-cV(f){f/), the expansion of u=f{i/) in
powers of x is
/(.'/) =/(«)+.'''^W/W+-+j^,-£^[!'^W}"/W]+
Pjioof. — Expand u as a function of ■(■, by (1507) ; thus, with the notation
of (140G), n = nQ + xu,o+ ,-^«2xo+ ••• + ,— ?f«xo+&c.
Here 7/^ is evidently /(;.')•
Differentiating the equation y = z-\-a'(p (y) for x and z in turn, wc havo
y^ = </' (y) + •'■f (//) 'Jr and y, = l+.r^j^' (//) y,.
Therefore y,. = <!> (//) //, ; and, since v^ =/'(//) y^ and », =j"(ij) y,,
therefore also ?/> = (j) (y) x^ (i-)
The following equation may now be proved by induction, equation (i.)
being its form wlicn it, = 1.
Assume that m,,^. = f?(„_i), [{/^ (i/)}" "J ("•)
Therefore «(,..i„ = d^.._,^,ih [{V (y)}"".] (1482)
= ^.«-.).^.[{t(2/) }•'".] (15^1) ='/..[{V'0/)}"*'^'.], by (i.)
EXPANSION OF IML'LK'IT FUNCTIONS. 270
Thus, n becomes ?i-f-l. But equation (ii.) is true when n = 1 ; for then
it is equation (i.) ; therefore it is universally true.
Now, since in equations (i.) and (ii.) the differentiations on tlie rif^ht are
all etlected with respect to z, .r may be made zero 6»;/"';/v diirerLiitiatiii^' instead
of after. But, when ./■ = 0, u,=j'{u) and ^ (//) = V (-'). therefore e<iua-
tious (i.) and (ii.) give
».« = 9(-0/(--); «....= '^-.)=C{9 00}"/ CO].
1553 l'^>^- 1- — Given >/—aij + h = : to expand logy in powers of — .
Here >J '= - + •^-; therefore, in Lagrange's formula,
a a
_,3 1
x = -; z = —; f(ij) = \og>j; cj> (y) = >/; and y = z+,,j\
Therefore u^ = log :: ;
»„^= J,._„, (.■^''-^) = (3,^-1) (3u-2) ... (2u + l) z^\
Therefore, substituting the values of x and z, (1552) becomes
log, = ,ogA + Jll+...+ CS'-'Ho»-2)...(^" + l) ^l+
° •' *= rt a- a 1.2 ...n tr" ct"
1554 Ex. 2. — Given the same equation : to expand y" in powers of — .
f(y) is now y", and, proceeding as in the last example, we find
^" f 1 _._ ^' 1 ^ n(n + b) i>* i_ , »('n + 7)0> + 8) h" J_
'•^ =^ r'^";?^"^ 1.2 a* a*"^ 1.2.3
nr» + 0)0> + 10)(»4-n) _^ J_ . ^.,. I
' -^ CI \ tt^ a a* a- or uT ii" a*
CAYLEY'S SERIES FOR — -.
1555
^=...-^r.^«,..+...+^lfi^[.[f;^)]"-a.+....
wliere ^l == -, ,-.
cp (U)
Proof.— Diflercntiato Lagrange's expansion (1552) for z, noting that
^ = L . Replace x by ^. Put /' (y) = •' ^ ; and therefore
dz l-.T<?.(i/) ^ 9(^/) V'(^)
/'(r) = — -, since/ is an arbitrary function. Then make y = 0.
280 DIFFERENTIAL CALCULUS.
LAPLACE'S THEOREM.
1556 To expand /(//) in powers of x wlien
I^ULE. — Proceed as in Lagranrje's Theorem, merel)/ suh-
st'dntlii'j F (zj for z in the formula.
1557 Ex. 3. — To expand e" in powers of x when ij = log {:: + x sin?/).
Here /(^) = e"; F{z) = logz; ^ (//) = siu 7/ ;
In the value of m„^o (1552), f (z) becomes f {F(z)} = siulogz;
/(z) becomes f{F(z)} = e''"^'' = z; therefore f'(z) = 1.
Thus the expansion becomes
e" = g + a; sinlog2+ ... + f^tZ(„_i), (sin log r)".
1558 Ex. 4.— Given sin y = x sin (y + a) : to expand y in powers of x.
Here y = sin"^ (x sin y + a), with 2 = 0.
f(y) = y; Fiz) = sm-'z; <j> (y) = sin (y + a).
(j>{z) in (1552) becomes <l>{F(z)} = sin (sin"' z + a).
f(z) becomes f [F (z)} = F (z) = sin-'z; therefore F' (z) = {l-z')-^.
Thus y = ^sin(sin-'2 + a) (l — z-)~^
+ lcK{sin'(sni-'z + a)(l-^r)-''\+ix%J,s{n'(ism-'z + a){l-z')-^\ +
with z put =0 after differentiating. The result is, as in (796),
^ = a; sin ti + 1-*^' sin 2a + \x^ siu oa + &c.
BURMANN'S THEOREM.
1559 To expand one function /(//) in powers of another
function ^ (ij) .
Rule. — Put x = xp (y) in Lagrange's exjiansion, and there-
fore <^ (y) = (y-z) -^^^ (y) ; therefore
1560 /(/y) =/W+V'(//) {f^/'(^) },..+•••
Here 7/ = z signifies tliat after differentiating ;: is to be sub-
stituted for //.
EXrANSTOX OF TMriTCTT FUNCTTOX^^. 2^1
1561 CoH. h— Since .'■ = ^(.y), ij = xp-'{.r) ; therefore (15G0)
becomes, by \vritiiig ./• for ^(//),
/{,-u.o!=/«+... + f|^{(-^])>0/)}^^+...
But since tlie variable // is changed into z after differentiating,
it is immaterial what letter is written for ;/ in the second
factor of the oreneral term.
1562 <^'^i?. 2. — If /'('/) he simply //, the equation becomes
'^-(..•)=~-+.-(:y),..+-+,T\|^{(-^y},.+
1563 Cor. 3. — If .-: = 0, so that )j = X(},{//), we obtain the
expansion of an inverse function,
1564: Ex. 5. — The scries (1528) for sin"' aj may be obtained by this
formula ; thus,
Let sin"' ^ = 'J, therefore x = sin y = \p (>j), in (loGS) ; therefore
sin" X = X
.sin7//,!/=o 1 .2 Vsin^ ;//yo 1 .2 .3 Vsiu' /// ayo
1565 Ex. (3.-If y = ,~—-^, r, = ^ ^^^ ''\ tlien, by Lagrange's
theorem (1552), since y = -^ + ^J'. we find
Put a; = 2 v//, thus
1566 ll^^}-::m'=r+,.r'+... + 'll^2^i'-+
\ 2 / I r \ 7i + r
Change the sign of «, thu.s
1567 ('±^^M)-=i_,„+...(-,).»i£^r±
This last series, continued to — + 1 or — -— terms, according as u is even
or odd, is equal to the sum of the two scries, as appears by the Binomial
theorem.
2 o
282 BIFFEIiENTlAL CALCULUS.
Also, by Lagrange's Theorem,
1568 K'i/ = iogf + (;^)'+^^.4g(f)%
or, by putting x = 2 \/t,
1569 .ogi-^4i^ = <+...+lgr+
1570 Ex. 7. — Given xy = logy ; to expand y in powers of x.
The equation can be adapted as follows :
y = e'", therefore xi/ = xe"".
Put xy = y\ therefore y ■=■ x&\ from which, by putting z = in (15.52),
y may be expanded, and therefore y.
Ex. 8. — To expand e"" in powers of yi"-' .
Here x = yi'\ f (jj) = y ~^ = e-^\ if we take z = 0. Therefore
1571 e'"'=l + «ye^^ + a(a-26)'^' + a(a-86)^^^+
ABEL'S THEOREM.
1572 If ^ (') be a function (developable in powers of e^\ then
+ «("-'-^)'-' ^-(.,+,.i) +
J. . .w ... )
Proof.— Let (^ (//) = A,,^ A^(^' \ AJ-^ -^ A^e^'-' -\- (i.)
Put 3/ = 0, 1, 2, 3, &c. in (1571), and multiply the results respectively by
ylfl, A^^., A^e^, &c. Then the theorem is proved by equation (i.).
1573 Cor. — If ({> (,*') = x", Abel's formula gives
-\-C{n,r)a{a-rby-'{,r+rhy-'--^&c.
INDETERMINATE FORMS.
1580 Forms A, — • Rule.— 7/' ^44 ^^ « fraction irhirh
if X i/- (x)
takes cither of these forms irhen x=:a: tJtoi ^-i^ — ^,)l or
^„ . , the first determinate J raction obtained hij differentiating
IXLETEinflXATK FOn}fS. 283
the numerator and rJeuominafor simuIfdnpoiisJi/ and suhxt'ttnthig
Q. for X in the result.
1581 But at any stage of the process tlie fraction may be
reduced to its simplest form before the next differentiation.
See example (1589).
Proof. — (i.) By Taylor's theorem (1-500), since (a) = = v// (a),
(p (a + h) _ <!>(a)-\-h<p (<i±Bh) ^ tj/Ja + Oh) _ (f>' (a)
\P{u + l,) 4.{;) + l4''(<i + 0li) xp\a + Uh) ^//"'(u)'
wlien // vanishes.
^ ^ ^^ ^ ^^ ^ ^(a) xPia) ,p{a)'
which is of the first form, and therefore
_ f (a) _^ <t>'(a) ..,,,. _ {0 (a)Y ;^>) m, . <p (a) _ ^J^t)
- {4.(a)Y ■ .;<^ (.) ^ ^^^'^^ - (H-n' 9(ny ^^'"'"' ^OO " 4^\'-0'
1582 Vanishing fractions in Algebra are of the indeter-
minate form just considered, and may be evaluated by the
rule, or by rejecting the vanishing factor common to the
numerator and denominator.
^ ^, a-»-a« (.r-a)(.r"- + a.r + <r) Scr 3
Ex. — When x = a; , = - = ^^ ; -' = ^ - = _ a.
.c--a' (x-a){.c + a) '2<i li
1583 Form Ox x. Rule. — If (^ (x) X ;// (x) takes this form
U'hen x = a, j"^^ 1* {^) ^ ^ (=^) = i" (=') "^ Tl — ' ""^"'^^^ ^"^ ^f ^^^^
form — .
1584 Forms 0^ x", 1". Rule.— 7^ {<^(x)]*^'> takes any of
these forms irJicii x = a, find the limit of the to(/arithm of the
expression. For the loga r it tun r=: xp(ji") log (pi^n), irhlrh, in eaeh
case, is of the form X oo .
1585 Form X — X. Rule.— If <{>{x)~xf.{x) takes this form,
when x=a, tee have e'''^"^-*^"^ = ^^^ = 77' ^^"^ ^f ''*'' '-"^"^ ^^
^^I's expression he found to he c, hy (1580), tlie required value
ivill he log c.
1586 Otherwise : ./,(a)-i^(a) = ./>(a) U - '^f^]-, u-/i/c/i /s of
L (p{-d))
the form oo X (1583).
284 DIFFERENTIAL CALCULUS.
1587 Ex. L— WitL X =0, y = -^ = 1 = ^ (^^^^^ = 1-
Also, with a; = 0,
e'-l-xe' e'-e'-xe' -x 1
1
^'^ (e'-l)' 2(e'-l)e^ SCe'-l) U 2e'
2"
1588 Ex. 2.— With x = l;
.. ^^ loff-T cos'' (t,7;) ^1 t.oA^ 1
cot (tt/O lOirX = ^- =: — = > i (J 580) = — .
1589 Ex. 3.— With a; = ;
X'" (log.!-)" = ^ t,,/ = — = „. = 7 ^-^r^7n = ^>
X 00 — m.i; '" { — m)x
by (1581), differentiating n times and reducing the fraction to its simplest
form after each difPerentiation.
1590 Ex. 4.— With x = 0; y = (i + axy = r.
By(im), logy-^-^Kil+J^-^- <^ (1580) -a;
X l-\-ax
••• y = e".
1591 Ex. 5.— With x = ir; y = (tt-.-c)""' = 0°.
By (1 584) , log 7/ = sin x\og{iT-x)= ^QgC'^"-^) ^
z sin X cos 3^
coseca; oo {tc — x) cos*
(1580) = --= . ^.u....u... ^Q. _._ ^^-^_
U — cos .^• — (tt — .-c) sm x
1592 If /('^O ^i^tl X become infinite togetlier, then
Pkoop,- ^-i'i = ^- =£M (1S80) ^ /(■'■ + ])-/fa) (1404),
a; 00 1 1
since, when a* = oo , h may be taken = 1.
Indeterminate forms invoicing two variables.
1592 Rule. — First : If tlie values x=r, y=b make the frac-
tion 2*^1-?-^=: — ; the true value is = y^, 'f 1>y <^"^^ ^y ^'^^^^
vanitih.
1593 Second 1 1/ : If f,. : i//.,. = ^^ : t//^ = k, the true value of
the fraction is k.
Proof.— (i.) By (1703) %^^ = ^J±hlji, and y being an arVdrary
function of x, — that is, independent of x, — the value of the fraction is inde-
terminate unless (})„ and xjy,, both vanish.
(ii.) If we suUstituto f^ = k\p^ and <p,^ = l\\p^, the fraction becomes = Jc.
JACOJiFANS.
28.'
JACOBIANS.
1600 I^^'t //, r, ir be n functions of n vanables a', ?/, ." {n = 3).
The fi)ll(nviii^ determinant notation is adopted : —
II, Uy u,
'•.r '^ '*.-
u\^ u\ w.
(I (i(rtr)
•*V '^'p "^f-
//» //<• //»
</(.r//.t)
(/ [unr)
The first determinant is called the Jacohlan of ?r, r, ir with
respect to .r, //, ;:, and is also denoted by J{iirir), or simply
bv J.
1601 Theokem.—
(/{.n/x) d {nrir)
Proof.— If the product of the two tletenniuants be formed by the rule in
(57U), first changing the cohiuins into rows in the second determinant (669),
the first column of the resulting determinant will be
110
= 010
1 1
u^Xu + u^y„+v,z„ = u,,^
and the whole
«„ V„ IV,
U^X^+U^I/v+Uz^i = "v (
, determinant
u„ v^ w,
u^x„+Uyy„ + u.z^ = «,r )
will be
7f„ i;„ IV
= 1.
1602 If fi, ^S w are n functions of n variables a, ft, y (/< = 3),
and a, /3, y, functions of ,r, //, ?; ;
djuvw) f/(ai8y) _ d (tine)
d{a^y) d{.vi/z) d[.n/z)'
Proof— Form the product of the two determinants, changing columns
into rows in the second as iu (IGUl). The first column of the resulting
determinant will be
«.a,+ «<^/3x + «,yx = Ur') and the whole de-
n,a^ + iiflliy + 2i,y, = u^ > , terminant will
«.a.4-«^/3. +«^y. = u. ) be
since rows and columns may be transposed (560).
Ux
1'x
^Vx
U,j
t\
W,
u.
Vz
w.
_ d (iivw)
d (xi/z) '
1603 Cor. — If a, ft, y are only given as implicit functions
of u; y, z, by the equations .^ = 0, ;^ = 0, xP = ^), involving the
six variables ; then
d{afty) "^ >l{.r,/::) ^ ^ ^.n/z) '
ruuoF.— By (1737), V'„a,4-<?.,,/^, + f,y, = - «/'x, whore 9, is the partial
derivative of ^. Thus »^, v„ u\, in the determinant, arc now replaced by
— ?>x. — X't — 4'x ; so with y and z ; and by changing the sign of each clement,
the factor ( — 1)* is introduced (6G2).
286
DIFFEBENTIAL CALCULUS.
1604 If u, V, w, n functions of n variables x, ?/, z (n = ^), be
transformed into functions of K, v, I by the linear substitutions
X = (hi-^(hr)-^a,l^ ^^^^^^^ (/(unr) _ ^^ (J (urir)
z = (\$ + c/T) + (',C ^ or J = MJ,
where M is the determinant (ai^h.,r.^) called the modal us of
transformation (573).
Pkoof.
/ =
Uz ^ly ":
M =
a^ &i Cj
J' =
V, V,j V.
Oj &2 ^2
WX Wy W^
rts &3 C3
■?*„ U/
Form the product Jf/ by the rule in (570).
v^ ^ ^^
w^ w^ w^
The first element of the
resulting determinant is u^a^ + iiyb^ + uxi = u^^r^ + ti'yij^ + u.z^
Similarly
for each element. Then transpose rows and columns, and the determinant
J' is obtained.
1605 AVlien the modulus is unity, the transformation is said
to be vnimodular.
1606 If, in (1600), (^ {umv) = 0, where <^ is some function;
then J (uvio) = ; and conversely.
Proof, — Differentiate ^ for x, y, and z separately, thus
similarly y and z ; and the eliminant of the three equations is J(>(viv) = 0,
1607 If ^^ = 0, ?; = 0, w = be a number of homogeneous
equations of dimensions m,n,p in the same number of vari-
ables x^y^z; then J (uviv) vanishes, and if the dimensions
are equal J^, Jy, Jg also vanish.
Proof. — By(lG24), xu^+yuy+zu^ = mu'^
xv^ +yVy +ZV. = nv > ; .*. Jx:
xu\ + yWy + zio. = pw )
By (582), A^, I?i, Oj being the minors of the first column of/,
if u, V, to vanish, J also vanishes.
Again, differentiating the last equation, J+xJj. = A^mU:, + Binv^+ Cipiv^.
Therefore, if m. = n= p, J+xJ^ = m (A^zi^ + BiV^+ C^iv,) = nij.
Thet;efore J^, vanishes when J does.
A^^nu + Biiiv + Ci].nv.
Therefore,
1608 If « = 0, v=0, ir = are three homogeneous equations
ol" the second degree in x, y, z, their eliminant will be the
determinant of the sixth order formed by taking the eliminant
of the six equations u, ?', w, /,., /„, Jg.
QUANVfr^^
287
Pkook. — / is of tlie third degree, and therefore J,, .7^, J, are of the second
degree, and they vanish because u, v, w vanish, by (1G07). Hence n, i\ w,
Jt, «7y, /, form six equations of the form (;r, //, i-)' = 0.
tl
1609 T^ " variables a?, y, « (7?- = 3) are connected with n
otlier variiibU'S I, »;, l^ by as many equations ?( = 0, t' = 0, 7(' = ;
d.y (hi fh (Jiiirn') . fl(Krii')
J^ tq Tie
dx dij dz
li'd^T^
iI^iyjQ ' (({^rf/z
)
«x«f n.Ur, ^=2<r
«f % \
^'x^-f V.Ur, ^--^i
=
"i ^ ^^
V\X^ li\jll^ W._Z^
«V ""r, ^'^i
QUANTICS.
1620 Dellm'iidn. — A Quanfic is a homogeneous function of
anv number of variables : if of firo, three variables, &c., it is
called a Unarij, ternary quantic, &c. The follo^ying will
illustrate the notation in use. The binary quantic
iLt^ -\-Ui-y -^Zcxif -^df
is denoted by (</, ?>, c, d\x, [/Y when the numerical coefficients
are tliose in the expansion of (,t' + //)\ When the numerical
coefficients are all unity, the same quantic is written
{i',h,c,(l\,v,yy. W[\e\\ the coefficients are not mentioned,
the notation (.r, yY is employed.
EULER'S THEOREM OF QUANTICS.
If u =f{.r, y) be a binary quantic of the ?i^'' degi'ee, then
1621 ^vu,-^ijUy = nu.
1622 .r-'/..+2.r//»/,,4-/r'^.. =.n{n-\) it.
1623 {.nl,+ f/f/;yu = ii (/<-l) ... (/*-/•+!)«. (1102)
Proof. — In (lol'J) pat h = ax, k = ay ; then, becauKc the function is
homogeneous, the equation becomes
(l + uy H = u + a(xu, + (/Wy) + i't" (x^u,_, + 2x>ju^ + i/u.^) +
Expand (l + a)", and equate coefficients of powers of a.
288
DIFFERENTIAL CALCULUS.
The theorem may be extended to any quautic, the quan-
tities on the right remaining unaltered. Thus, in a ternary
quantic u of the n^^' degree,
1624 ^Tit^-\-yify-\-^if= = »itf', and generally
1625 {.r(h+i/d,-]-zd,Yu = n {n-l) ... {n-r-^l) u.
Definitions.
1626 The Elhninant of n quantics in n variables is the
function of the coefficients obtained by putting all the quantics
equal to zero and eliminating the variables (583, 586).
1627 The Discriminant of a quantic is the eliminant of its
first derivatives with respect to each of the variables (Ex.
1631).
1628 An Invariant is a function of the coefficients of an
equation whose value is not altered by linear transformation
of the equation, excepting that the function is multiplied by
the modulus of transformation (Ex. 1632).
1629 A Covariant is a quantic derived from another quautic,
and such that, when both are subjected to the same linear
transformation, the resulting quantics are connected by the
same process of derivation (Ex. 1634).
1630 A Hessian is the Jacobian of the first derivatives of a
function.
Thus, the Hessian of a ternary quantic u, whose first
derivatives are li^, n.
d{u^u„uj)
d {wyz)
U,r. u.
(I.,. U,„ II.,,
1631 Ex.— Take the binary cubic u = ax^ + Sbx^y + dcxy^ + dy\ Its first
derivatives are
u^ = Sax^ + 6bxy + Scy' ,
u,j = Sbx^ + 6cxy + ody'.
Therefore (1627) the discriminant of m
is tlie annexed determinant, by (587).
1632 The determinant is also an invariant of u, by (1G38) ; that is, if u
bo transfDrmed into v by putting x = nl^fii) and // = n'£-|-/3'»/; and, if a
corresponding (h;terniinant be formed with tlie cocilicients of r, the new
determinant will be equal to the original one multiplied by (u/3' — a'/3)\
3a 66 3c
3a 66
3c
36 Q>c M
36 Gc
3(Z
QU ANTICS. 289
1633 Again, «2, = GtUJ + G&//, Wjy = Ccc + O'/y, u^,^ = 6bx + 6ci/.
Therefore, by (1630), the Hessian of w is
v.^n,^-nl^ = (ax + hii)(cx + dij) - {h.c + r//)»
= {ac-lr) x-+(ad-bc) x >/ + (h<J - r) if.
1634 Anil this is also a corariant ; for, if u bo transformed into v, as
bftcut.', then the result of transforming the Ilessiiin bj the same equations
will be found to be equal to V2zCiy — i%. See (lL'io2).
1635 If a quantic, 7( =/(.?•,//,:: ...), involving n variables
can be expressed as a function oF the second degree in
Xi, X., ... A",,.!, where the latter ai-e linear functions of the
variables, the discriminant vanishes.
PuooK. — Let u = (f>X\ + \//X,A'j + x-^'A^s + «^c.,
where A', = a,a; + h^H + c^z + &c.
The derivatives u„ u^, &c. must contain one of the fjiotors XjX, ... X,,., in
every term, and therefore must have, for common roots, the roots of the
simultaneous equations X, = U, X.^ — 0, ... X,,., = ; u-l equations being
required to determine the ratios of the n variables. Therefore the dis-
criminant of a, whicli (lGl2") is the eliminaut of the equations Hr = 0, n^ = 0,
&c., vanishes, by (oS8).
1636 <^^<"^- 1- — If '^ binary quantic contains a square factor,
the discriminant vanishes ; and conversely.
Thus, in E.xample (1G31), if u has a factor of the form {Ax + J)i,')'-, the
deterniinaut there written vanishes.
1637 Cor. 2. — If any quadric is resolvable into two factors,
the discriminant vanishes.
An independent proof is as follows: —
Let M = Xy be the quadric, where
X= {ax + by + cz+...), Y= (ax + h'y + c'z+...).
The derivatives u^, u^, u. are each of the form i>X-\-qY, and therefore have
for common roots the roots of the simultiineous equations X = 0, F=0.
Therefore the eliminant of u^ = U, u^ = 0, &c. vanishes (IG27).
1638 The discriminant of a binary quantic is an invariant.
Pkoof. — A square factor remains a square factor after linciir (ransforma-
tion. Hence, by (1G3G), if the discriminant vanishes, the discriminant of
the transformed equation vanishes, and must thenforo contain the former
discriminant as a/«r/or (see 1G28). Thus the determinant in (1G31) is an
invariant of the quantic ii.
The discriminant of the ternary quadric
1639 u = aa^'-\-bf/'-{-rz'-]-'2fyz+2gz.r-^ 2hxy
2 p
290
DIFFEIiENTIAL CALCULUS.
is the eliminant of the equations
1640 iu. = a.i-\-hi/-\-i>'Z = ^
tliat is, the
' determinant
a
l> g
iu^ = h.v + bi/±fx =i)i
h
!> f
-h'. = ^^^v-j-ff/-\-rz=i))
<r
f f
1641 = abc^'Ifii-h-af-
-b^'-clr = A.
1642 The following notation Avill frequently be employed.
The determinant will be denoted by A, and its minors by
A, B, G, F, G, IT. Their values are readily found by differ-
entiating A with respect to a, h, r,f, y, h ; thus,
A = hc -f- = A,, B = ca-QT = A,, C = ab-Jr = A„
F = irh-cif=: ^A,, G = hf-bg' = \A^, H=fg-ch = \A,.
1643 The reciprocal determinant is equal to A^ or
A H G
a h ^
H B F
=
// b f
G F C
^ f c
By (575).
1644 The discriminant of the quaternary quadric
-^2pd n- -\- 2qi/ic -\-2rzi(7
is the eliminant of tlie equations
1645 hf:c = ((d'+h?/-\-gz+]m' =
%+/-+7
tr =^{)l
i^'v = ' ... -
hfw= pd'-{-qf/-\-rz-\-(/{r = 0^
that is,
the
deter-
minant
a
h
P
h
b
/
n
^
f
c
r
P
fl
r
d
The determinant will be denoted by A', and, by decom-
posing it by (568), Ave have
1646 A' = (/.A-Ajr-B(f-Cr-2F(jr-2Grp-2fIpq.
1647 A' = ^ (;>A;,+ryA; + rA;)+rf.A.
1648 Tiif:i>i;i;m. — If ^ (/n/) be a ciiiantic of nn even degree,
is fin iiivai-iaiit of llie quantic.
QUAXTICS. 291
Proof. — Let the linear substitutiims (1(528) be
x = ai-\-l'>i, y = a'i + h'tj (i.)
Solve for I and »?. Find l^, £^, Vx, '/.,. ""f^ substitiitn in tlie two equations
d, = d.l^^d,,,,^; ' ,l, = d.E, + d„„r.
The result is
(/, = {aJ^ + h(-.J.)]^M; -d,= {a'dr, + h'(-dt)}^M (ii),
where M = ah' — a'h, tlie modulus of i ransformation. Etjuations (i.) and (ii.)
are parallel, and show tiiat the operations d^ and —d^ can be transformed in
the same way as the (juanlilies x and // ; that is, if 9 (a;, ?/) becomes v/* (I, n),
then f(d^, — (/,) becomes v/- (</,, , — Jf ) -f- J/", where 71 is the degree of the
quantic 0. But ^ (d^, — c7,) (.c, //) is a function of the coefficients only of
the quantic 0, since the order of differentiation of each term is the same as the
degree of the term ; therefore the function is an invariant, by definition (1628).
1649 Example. — Let {x, y) — ax*+bx^i/ + ex!i^+fy*. The quantic must
first be completed; thus, f (xy) = ax* + bx^y + cx-y^-\-exy'^+fy\ {c = 0) ; then
<p (d^, -J,) f (x, y) = (uti,, — 5t?3j,z + cc/2y2x— «^!,ax+/(?4x) 9 (a;, y)
= a.2if-h.Ge + cAc-e.{Jb+f.2~ki = 4:(l-2af-dhe + c").
Therefore 12af-Sbe-\-c- is an invariant of ^, and = {\2AF-^BE + C^)-i-M*,
■where A, B, C, E, F are the coefficients of any equation obtained from ^ by a
linear transformation.
But if the degree of the quantic be odd, these results vanish identically.
1650 Similarly, if (p(,r,y), ^P {x, ij) are two qualities of tho
same degree, the functions
<t>{fI,,-(L.)rl^{.v,i/) and rp{d,,-(l.,)<t>i.r,,,)
are both invariants.
1651 Eyi.—Uc(> = ax' + 2bxy + cy' and 4. = ax--^2b'xy + cy' ; then
{ad..^ — 2bdr^ + cd.i^) (ax'' + 2b'xy + c'y^) = ac+ca'-2bb', an invariant.
1652 A Hessian is a covariant of the original quMitic.
Proof. — Let a ternary quantic u be transformed by the linear substitu-
tions in (1604) ; so that u = (p (x, y, z) ■=■ ■^ (£, »j, ;). The Hessians ol tiio
two functions are ^11"^'>3) and 'l^W'il (1^30). Kow
d (xyz) d (i»/;')
d(n^u^u^) ^ ^^d{ n^ u^ u. ) ^ ^^ djj^.n^ ^ ^^, djn, 11^ »,)
d{inO ' d[-rj,) ' c/(^<;0 ' '/'•'■y^')
The second transformation is .seen at once from the form of the determi-
nant by merely transposing rows and ci)lumns; the first and third are by
theorem (l()04j. Theret'ore, by definition (itJ2'J), the llessiau of tt is a
covariant.
1653 Coij^rcdicnts. — Variables are cogredient when they are
subjected to the same linear transformation ; thus, .r, ;/ are
292 DIFFERENTIAL CALCULUS.
cogredient witli x', y' wlicn
y — ('i + <h ^ y — ^'^ + ^h
,r = a^^-hrj \ ^^^ a' = ai'-\-hr)' | _
1654 Emancnts. — If iu any quantic 2i = (p {x, ij), we change
X into x-\-px' , and y into y-\-py', where x , y' are cogredient
Tvith X, y; then, by (1512),
<l>(.r-\-p.v',y+py')
= n-\-p (x'd,-[-y'dy) u-\-iy (.rV/,+//V/,)- m + &c.,
and the coefficients of p, p^, p^, ... are called the first, second,
third, ... emanents of u.
1655 The emanents, of the typical form {x d^-[-y' dyY u, are
all covariants of the quantic u.
Proof. — If, in (.r, y), we first make the substitutions wliich lead to the
emanent, and afterwards make the cogredient substitutions, we change
X into x-\-px', and this into a$+ J»; + |0 (a£'+ 6»?').
And if the order of these operations be reversed, we change
cc into al-^-hri, and this into ail-\-pt)+l> {r)-\-prjf).
The two results are identical, and it follows that, if ^ {x, y) be transformed
by the same operations in reversed order, the coefficients of the powers of p
in the two expansions will be equal, since p is indeterminate. Therefore, by
the definition (ltJ21>), each emanent is a covariant.
loOO For definitions of contrngredieiits and cotitravariants, see (1813-4).
Fur other theorems on invariants, see (179-i), and the Article on Invari-
ants in Section XII.
IMPLICIT FUNCTIONS.
IMPLICIT FUNCTIONS OF ONE INDEPENDENT VARIABLE.
If y and z be functions of <t, the successive application of
formula (1420) gives, for the first, second, and third deriva-
tives of the function ^ (//, ::) with the notation of (1405),
1700 4,A!iz) = ^:,!i, + j>..z...
1701 <t>: (!>^^ = 'k,,.vl+--i4'...'ir~.+<t>u^:+<t',ir,. + 't'.~u-
IMPLWIT FUNCTIoXS. ^i)'.)
1702 K d/^) = *.,//^+;?^..//;~.+'?</»...//x-;+<^:..4
By iii:ikiii<j^ ;. = ,r in tlic last tlifcc formnlji}, and (-(.nsc-
finoiitly •.',.= I, ::.,^=0, or else by tlilTcrciitiating independently,
we obtain
1703 <f>J.n,) = <l>„,/,,. + cf>_,..
1704 <P:.r (■>■,/) = (^,,/y:.+2(^,„,//,.+(^,,.+(^,//,...
1705 K (,'//) = (^,,//^+;?(^,,...//:.+i?<^,.,.//.+<^3x.
1706 In these formnlff! tlie notation <f>^ is used where the difforentiation
is partial, while ^^(.r, y) is used to denote the complete derivative of (}>(x,y)
with respect to x. Each successive partial derivative of the function ^((/, z)
(1700) is itself treated as a function ol' ij and z, and differentiated as such by
formula (14-20).
Thus, the differentiation of the product cp^y^ in (170(1) produces
The function <p^ involves y and z by implication. If it should not in fact
contain z, lor instance, then the partial derivative 0„, vanishes. On the other
hand, y^, y,^, &c. are independent of z; and z^, Zjj., &c. are independent of?/,
DERIVED EQUATIONS.
1707 If ^ (•''7/) = 0, its successive derivatives are also zero,
and the expansions (1708-5) are then called i\\Q firt^t, second^
and third derived equations of tlie primitive equation ^(.r//) = 0.
In this case, those equations give, by eliminating //j.,
1708 !hi = — i^', ^JJL= 2 j>..„ j>.,. (f)., — <!>:,■ <f>; — <^,y (t>i _
1710 Similarly, by eliminating ij^ and >/.,,., equation (1705)
■SN'ould give y^^ in terms of the partial derivatives of ^ (.^7/).
See the note following (1732).
1711 Jf 4>{'n/) = i) niHl i^=0; p{=:-hr.
di dd' </)^
1712 and ^ ^ H,,4.-<I>:A.^
Proof.— Bj (1708), ^, = 0. Therefore (1704) and (J70o) give these
values of t/o, and y^,.
201 BIFFEUENTIAL CALCULUS.
1713 If </'a- f^Tid (\>y botli vanisli, 7/.^ in (1708) is indeterminate.
In tins cape it has two values given by the second derived
equation (170 I), which becomes a quadratic in y^.
1714 If <\>-i.ri fl>.r,n and (poy also vanish, proceed to the third
derived equation (1705), which now becomes a cubic in y^,
giving three values, and so on.
1715 Generally, when all the partial derivatives of (j> {,r, y) of
orders less than n vanish for certain values x = a, y = h, we
have, by (1512), <}>{a, h) being zero,
i> {a + h, h + Jc) = — (/<4 + K)"'^ (A'i/).a,i+ terms of
higher orders which may be neglected in the limit, {x, y are
here put = a,h after differentiation.) Now, with the notation
of 1406), H..a-\-Hv.b = 0;
therefore JL = -t^ = ']!.;
the values of which are therefore given by the equation
1716 ihcl + kd,Y<l>{.r,y),^,,= 0.
1717 If Vjc becomes indeterminate through x and y vanish-
ing, observe that -^ — ^L in this case, and that the value of
° dx X
the latter fraction may often be more readily determined by
algebraic methods.
If X and y in the function (j) [x, y) are connected by the
equation \p (x, y) = 0, y is thereby made an impHcit function
of X, and we have
1718 M-,!/) = '^^^!^^-
1719 K {■>', .'/) = { i<i>,„t-'i'^M * + iK'i'-<i--^<i>.) r,
Phoof.— (i.) Differentiate both f and \p for x, by (1703), and eliminate y^.
(ii.) Differentiate also, by (1704), and eliminate ?/, and y^x-
If ?/,?/, z are functions of x, then, as in (1700),
1720 <t>r (^y^) = <^. »/.,.+(^,.v.,.+(^.r...
iMi-rjrrr functions. 205
1721 <t>.r ('///-) = <!>■:,. »/;+(^,,//;+<^,.4
+ Cf),, U.,, + <l>„ //,., + </). ^i r .
1722 To obtain <p^{i'ir:) and f j.^ (.''.V"-) , iiiako //=,/• in tlio
abovo cMjnations,
Let U= (p (.?', //, ;:, I) be a function of four variables con-
nected by three equations u = 0, v = 0, iv = 0, so tliat one of
tlie variables, ^, may be considered independent.
1723 ^^^e have, by differentiating for ^,
X . 1 ./ ,/f I - . I s I where J — .
1 ^VOA '^''' — — 'f ^'"'"'^ 1 ^/ _ _ (Ufn'i(') _1_ .
^^'^^ ^ - 715;^ ./ ' ^// ~ r/ (.f..) ./ '
(li (/{•n/i) J '
Observe that V^ stands for tlie complete and </)f for the
'partial derivative of the function U .
Proof.— (i.) U^ is found by taking the eliminant of the four equntions,
separating the determinant into two terms by means of the element f^—U^,
and employing the notation in (IGOO).
(ii.) Xf, i/t, and z> are found by solving the last three of the same equa-
tions, by (582).
IMPLICIT FUNCTIONS OF TWO INDEPENDENT VARIABLES.
1725 If the equation <p {.v, y, z) = alone be given, y may
be considered an imj)licit function of .r. and ;:. Since .r and z
are independent, we may make z constant and differentiate
for ■/'; thus, for a variation in x only, tlie ecjuations (1703-5)
are produced again with ^ (Xyy,z) in the ])lace of <p (.'',//).
1726 It" ■'■ he made constant, z must replace x in those ecpia-
tions as the independent variable.
Again, by differentiating the equation <p{ryz) = first for
:r, making z constant, and the result for z, making x constant,
296 DIFFERENTIAL CALCULUS.
vre obtain
1727 (/>.,.,+ (^,,.//.-+(^,.-//.,.4-f>.//.//.+<^.//.,-. = 0.
From tills and the values of y^ and //,, by (1708),
1728 f/ = ^"'' ^" ^'' "^ ^"' ^" ^'' ~ ^ ''" "^ ^ ~ ^ '' ^' ^'" ■
^_^^ "^
1729 If 3S, Ih - i^^ tlie function <^(,f, //, 2) be connected by the
relation i/- (,r, //, ::) = 0, // may be taken as a function of two
independent variables x and ;?. A¥e may therefore make z
constant, and the values of f^ {.r, y, z) and «^2.r (^\ Ih ^') are
identical with those in (1718, '19) if x, y, z be substituted for
a?, y in each function.
1730 By changing x into ;: the same formula give the
values of ^, (,2", y, z) and ^o. (-f, 7/, ,;•).
1731 On the same hypothesis, if the value of <^^ {x, y, ,?), in
forming which z has been made constant, be now differen-
tiated for z while x is made constant, each partial derivative
^j., xpy, &G. in (1718) must be differentiated as containing x, y,
and z, of which three variables x is now constant and y is a
function of z.
The result is
1732 f.(.r,,v,=:) = {(<l>..,<!,-^...,4'M-{hA,-^:,A,HAl
In a particular instance it is generally easier to apply such rules for
differentiating directly to the example proposed, than to deduce the result in
a functional form for the purpose of substituting in it the values of the partial
derivatives.
1734 Example. — Let <p (x, ?/, z) = Ix + ;»// +• uz and \p (.»•, y, z) = x- + 7/ + z^
= 1, X and z being the independent variables ; <p^ (;<•, y, z) and <p_^ (.c, y, z) are
required. Differentiating 0, considering z constant,
4>x (x, y, 2) = l + m -•- = l — m — ; since -~ = — , = ;
dx y dx V'y y
, , y — xijr tr + x"^
02X (.r, y, -) = -VI -^—^ = -m ^,— ;
a result which is otherwise obtained from formula (1719) by substituting the
values <p^ = Z, </>y = '"> ^s = « ; <p2x = ?»2j/ = ^2: = ;
^P. = 2x, ^, = 22/, ■>P. = 2z; ,/.^ = ^,, = ;/.,, = 2.
Again, to find ^^. (x, y, z), differcutiato for z, considering x constant iu
the function
-JL—l-m — ; thus — -^ = +?» --j-= — w-r-, since —= - = .
ax y dxdz y y^ dz \p^ y
nn-i.K'iT FrxrrioxFf. 297
1735 T^ct U = <p (;v, ?/, ^, ^, ij) be a function of five variables
coiniectetl by three equations vt = 0, v = Oy w = 0; so that two
of the variables ^, »? may be considered independent, ]^^aking
?} constant, the e([uations in (172:3), and llie values obtained
for [/f, X(, //f, r^ff, hold p^ood in the ])resent case for the varia-
tions due to a variation in S, observing that <f>, a, i\ ir now
stand for functions of rj as well as of ^.
1736 The corresponding values of Z7,, .r^, y^, z^ arc obtained
by changing ^ into rj.
IMPLICIT FUNCTIONS OF n INDEPENDENT VARIABLES.
1737 The same metliod is applicable to the general case of
a function of n variables connected by r e(piations « = 0, r = 0,
w = 0...&c.
The equations constitute cuiij n — r of the variables wo
please, indeprndcnf : let these be ^,r),l.... The remaining r
variables will be dependent: let these be u:,ij, ::...; and let
the function be U=(l>{riy,z ... ^, tj, 2^ ...).
For a variation in ^ onhj^ there will be the derivative of the
function U, and r derived equations as under.
1738 (^.,,r.+(^,;/.+f,-, + ... + ^. = L^^,
U^.V^-\-Uj,?/^ + U^Z^. -f ... + ;/,== 0,
«'x.*'f + ^^.'A+i'.— f + ... + 7V = 0,
&c.,
involving the r implicit functions x^, v/^, z^, &c. The solution
of the r equations, as in (1724), gives
-\MOQ (ft' (/(unr...) 1 (/// (/(unc...) 1 n
1740 where J = -ri f • Also -rp — -r-^ — g- -r-
The last value being found (wactly as in (172:3).
1741 With ^ re])laced ])y »j we have in like manner the
values of x^, i/^, ,v^, U^ ; and similarly with each of the inde-
pendent variables in turn.
1742 If there be n variables and but one equation
^ ('j .'/, ■- . . .) = 0, there will be ?? — 1 independent and one depen-
208 DIFFERENTIAL CALCULUS.
dent variable. Let // be flependent. Then for a variation in
X only of tlie remaining variables, the equations (1 703-5)
apply' to the present case, </> standing for (^ {x,y,« ...). If x
be replaced by each of the remaining independents in turn,
there will be, in all, n—1 sets of derived equations.
CHAXGE OF THE INDEPENDENT VARIABLE.
If II be any function of x, and if tlie independent variable
ic be changed to i, and if t be afterwards put equal to y, the
following formulas of substitution are obtained, in whichi p = t/j,:
1760 !k = 2lL = l.
(Lv a\ .Vy
differentiating these fractions, we get
■J'
oV ^^ ^ -^ ■" ^ ~ ^ ^ '*
1766 = ^"^'''Z^''''' =Pir+inhr
'^;
j.^^: Ex. — If X = r cos 6 and y = r sin B ; then
1 7AQ ^ _ ^'^^ ^<^^ ^+^' <'o^ ^ f/'V _ r-H-2>-^ — rr..^
^ '^° f/r ~ >', cos e-r sin ^ ' iLv' ~ {r, cos ^- >• .sin df '
'■^'I'roof.— Wiitiiig for t iu (17C0) and (17G2), we have to tiud ,)•„ y,,
aJi« 2/29 ; thus,
Xg = r^ cos fl — r sin ; X2e = »";■« cos — 2i\ sin — r cos ;
7jg = 7-^ sin + r cos ; yie= ''•« sin + 2/-^ cos 6 — r siu 0.
Substituting tliese Viilues, the above results are obtained.
'", l^o cliange tlie variable from x to / in ('/ + />.'■) '\v„.r> where
"(r/'-^'/A/-) = ('\ employ the formtda
1770 {a-^hrr,/„, = h^{d,-7T^\) K->7=2) ... (^/ -1)//..
i^'' Which, multiplication by ^7^ by the index law signifies the
repetition of the operation d, (1102).
CnANGE OF Tin: JSniWENDEyfT VAIUAlifJ:.
299
Pk(K)K.— <M('i + '/J')''y„,} = {n (" + t.r)""'/;//,„ + (a + ij;)".V(...i;r} -P.-
Nuw hj', = r' =■ a + l>.r,. Substitute this, and iluiioto (n + hj-)"i/„, by U,,
thcreforo </, ( T.) = « ?/,. + J f/',. . i, or T,, . , = /. ('/. -n) f^,..
Thorefoi-e L\ = i (</,-«-!) U
,d ?;„., = Z, (,/,_„_ -J) r„ 2, &c.,
and finally T, = ^ (,/,-!) ir,.
13ut r, = ((Z. + //J-) .V, = hx^ii', = hij,.
Thercfdi-o Un = h" (J, - IT^) (J, -TT^) . . . (<?, - 1 ) lu-
1771 ^'<^Ti. — ('/+.'')".'/"..• "^^^^ '^'"y^ix '»i'L' transformed l)y tlio
Bamo fonuula by putting /^ = 1.
1772 Ijf^fc F £.-/•'('■> //)> wliere x, u aro connected witli ^, ij
by tlic c(jnations ?/ = 0, r = 0. It is reciuired to cliaiige the
independent variables <r, y to ^, rj in the functions F^. and F,,.
1773 Kn>K. — To find the value of F^ — Difforrntiate V, u, v,
cflc/i- M"//7i respect io x, considerinfj ^, tj functions of the inde-
pendent variables x, y ; and form the (diminant of the residtiiKj
equations; thus,
Siruilarly, to find Vy.
I'i
r.
-]
*/c
".
u
?*,e
t'n
V
= 0.
1775 Ex.— Let cr = r cos 6 and // = r siu ; then
,- ,. a I' sin ^ 1- ,' . /] , 1' cos ^
I ^ = ] ^ cos ^ — I ,, — — , 1^ = I ;. sin ^ + I <, .
Proof. — u = r cofiO—x, r = rsinW — y,
and tlie deteruiinant in (1774!) takes tlie form
annexed by wrijiuj' r and d instead of i aud rj. I . ,, ,, f.
A similar determinant gives Y ^. '
To tind Tir, substitute FrCos(i— I'/'" in the p'.. ce of V in the value of
r
v.; and in differentiating for r \\\u\[), conaider l'^ and 1', as functions of
both rand ^. Similarly, to lind V-,^ and \\^. Thus,
)sy — rsinO —1 = 0.
1777 r.=:]'.c..s=(; + (ir.-r.)'^
1778 i.= F.sin'«-(h;-i;.)^
9 cos W ,. siu' i) , ,, sin'
-r y r r" ' » — i"
r r r
2 Pin cos . ,r cos' S , Tr cos'
"T > r r y It' i"
r r*
By addition these equations give
1779 )^4-F,,:- r,,-f--i-F. + 4i
300
DIFFERENTIAL CALCULUS.
1780 Given V = f{x,y,z) and ^,v,l known functions of
X, y, z; \% F„ V^ are expressed in terms of F,, Fj„ T", by
the formult^
dV^ ddVO ^j dV^ di^VQ .J dV _ d{^V) ^j
di d{,v,/z) ' ' dr) d[A'f/z) '*' dC d [.vt/z)
Proof. — Differentiate V as a function of ( T^s +V^y]^+ V^i^^ = V
f, t], ^ with respect to independent variables
a', y, z. The annexed equations are the result.
Solve these by (582) with the notation of (1600).
1781 Griven V = f{x, y, z), wliere x, y, z are involved witli
^, rj, I in three equations ?i = 0, -y^O, w = 0, it is required
to change the variables to ^, rj, ^ in F^j F'^,, and F,.
Applying Rule (1773) to the case of three variables, we have
n
y.
^h
u^
v^
^\
ICp
IL\
F- -F
= 0.
F,^.,+ F,7;,+ F,L-F. = 0-
The determinant gives V^ in terms of T""^, T\,, F^ and the deri-
vatives of n, V, ID. Vy and F^ are found in an analogous manner.
1782 Similarly with n equations between 2n variables.
TIV
COS
0.
1783 Ex.— Given
a: =z r siu ^ cos (^ ; i/ = r sin 6 sin <^ ; z = r a
The equations ^6, v, ?« become *
rsiuQ cos^ — .« = 0; r sinO sin i^— ?/ = ; rcosO-a
"Writing r, d, f instead of ^, »?, f , the determinant becomes
v; i; i; -f
sin cos r/) r cos cos — r sin (^ sin </> — 1
sin sin ^ r cos sin ^ r sin cos ^
COS0 — rsin6
From which V^ is obtained. Similarly, V,^ and V, ; and, by an exactly similar
process, the converse forms for T'„ l',, and V^. The results are
* In writing out a determinant like the above, it will be found exjieditious
in practice to have the columns written on separate slips of paper in order to
be able to transpose tliera readily. Thus, to lind the coefficient of F„ bring
the second column to the left side, and, since this changes the sign of the
determinant, transpose aitij two other columns, so that the coofliciont of V^
may be read off iu the standard form as the minor of the first element of the
determinant.
CUANGE OF THE INDEPENDENT VAUIMlLi:. '\0\
1784 1 X = 1 r Sin COS f + F, ^ - F^ — r-^.
■«MrtK T 1' • n • , , I' COS sill </> , -,r COSA
1786 F.= F.cosO-IV"';;'.
1787 T; = F.siu cos </)+];. sin sin Y'+F cos 0.
1788 V, = V^ )• cos cos ^ + F,, r cos sin if> — F, r sin 0.
1789 1^# = - ^"x r sin sin + F^ r sin cos 0.
1790 To find V^ directly; solve the equations ii, v, ?v, in (1783), for r, 0,
and f; the sulution in this case being practicable ; thus,
Find r„ 0^, ^^ from these, and substitute in F^ = F,)-^+ T',0^+ V^'P^. Simi-
larly, Fj, and F.. Also F,. = F^.i;,+ Fyy,+ F^v Similarly, F, and \\.
1791 To obtain F^^, substitute the value of F^ in the place of F, in tho
Viiluu of I'j,, in (1785), and, in difi'erentiating Vr, F^, F^, consider each of
these quantities a function of r, 0, and (p.
To change the variables to r, 0, and (j>, in V.^-\-Voy-{- ['^,
the equations (178;3) still subsisting. Result —
1792 V2.-^V,,+ V,,
= V,,.-^l T;+-V(r>'ot6>+i;,+ T;,cosec^6').
r >•"
Proof.— Put r sin = p, so that x = p cos ^> and jj = p sin 0,
therefore, by (1770), 1'^+ F,, = Fv + ^ i;+ -\ F,, (i.)-
Also, since « = r cos and p = >• sin 0, wo have, by the same formula,
]:,.+ ]\= V,r+~V,+ \ F., (ii.)-
J. ^.-
Add together (i.) and (ii.), and eliminate F„ by (177(;), which gives
T • fl , T COS0
r
If r l)r a function of ii vnrial.U'S ./', //, :: ... connected by
the single relation, d--\-tf-\-z'-\- ...= >*'" (!•)•
1793 »;,•+ 1'.,+ 1'..+^^- = ''-+^ »V.
302 DIFFERENTIAL CALCULUS.
— , by differeutiating (i.),
r
therefore V-,, = V„ - + \\ '-—^ = V,r ^- + F, . „ , .
Similarly F,, = F,. l^ + F. (i- - -^ ) , &c.
Thus, by addition,
T2,+ F,, + &c. = T,. ■-, +^(^7 ^8 — j - F,,+ -^T,,
LINEAR TRANSFORMATIOK
1794 If I^ = /('^» i/j '^)5 and if the equations u, v, w in
(1781) take the forms
y = a,i-\-b,7] + c,C [ , then j Srj = B,.v+B,,/-}- B,z,
by (582), A being the determinant {(h^'2<^z)i and J^ the minor
of ^1, &c.
1795 The operations d^, dy, d^ will now be transformed by
the first set of equations below ; and d^, d^, d^ by the second
set.
d, = {AM,+B,d,+C.Ak) - A [ , r/, = h,d^^h,d,-^h,d^ .
d, = {A,d,+B,d^+t\d,) - A^ d, = c,d^^c,d,+c,dj
Proof. — By c^^ = d^^^ + d^ij^ + J^C^ and d^ = d^x^ + d^y^ + d,z^; and the
values of s„ Xt, &c., from the preceding equations.
1797 From^ (1705), V^ = a,V^ + aJ\ + a,V,. Operating
again upon V^, we have
and by substituting the value of V^, and similarh; with j'a,,,
Vi^, we obtain the formula^,
1798
V,, = b; V,^-\-l/; V,^-\-lK^ V,,-\-2bA]\,+2hA y..r^'2bA F,., [ .
cnAxni'j OF THE iNDErENDr:xT vAj^fAnm.
803
ORTHOGONAL TllAN.SFORMATION".
1799 If the traiisfoi-mntion is oi'tliOLr<)n:il (-^S-l), wo liuvo
and sinco, by (582, 584), A=l, J^^a^, etc.; (Mjuatioiis
(1701) now become
1800 ^r=a,^-^b,r}+r,C-
And equations (1705) become
1802 (i,=:a,d,^-h,(l,^^c,(l;
The double relations between x.
equations of (1800-1), and the similar relations in (1802-8)
between dj.dy(L and d^d^J^, are indicated by a single diagram
in eacli case; thus,
1804 i V I fh (fn (k
\(f;\ dc = (i\d ^-\-<i.d„-\-(i(1.^
'aI; V, f/„ == h,d,-\-h.M„-^h,d, >.
., d^ ) d- = (\ </, -h (', </, + r, d, J
?/, z and ^, 7?, t, in the six
tt'
ill
h. c.
ilr
(I I
fh
(\
y
a.
h (\
^/.
a.y
b.
(-'z
^
(h
h'i ('z
ih
a.
h
('■i
1808 llonce, when tlie transformation is orthogonal, the
quantities x, ij, z are co<jredient with d^, dy, d,, ])y the defini-
tion (1(35.3).
1807 Extending the definition in (1020), it follows that any
function ?< = (/)(./•, //, ::), wlien orthogonally transformed, has,
for a covariant, the function <|) {<lj,, d^, d,) n. That is, if by
the transformation,
u = (/)(.r,.V, x) = \/»(^, ri, 0,
then also <l> {d^, dy, d,) u = xj/ {d^, d^, d-) u.
1808 But if n 1)0 a quantic, then, as shown in (lOlM),
</) ((/,., dy, t/,) u is always afnnrtion of the coefficients onhj of »,
and the covariant is, in this case, an invariant.
1809 Ex.— Let u or ^ (.r, t/, z) = a.'c^ + %' + c>>H'2///^ + 2j;;x + 2/,.ry,
.-. ip {i.l„ il^, J.) u = au.^-\-bu.i^ + cu.i^ + 2j\, + 2gu„ + 2hu^^
= 2 {a' + Z/- + c' + '2/' + Ly + -/('}, and this is an invariant of «.
304 DIFFEUEXTIAL CALCULUS.
1810 When Y=f{d\ y, z) is orthogonally transformed,
n.+ T^2.+ F,,= V,,-\-V,,^V,,.
Proof. — By adding together equations (1708), and by the rehations
Oi + hi + c'l = 1, (fee, and aia„ + hih„ + c^Ci = 0, <fec.,
established in (584).
1811 If two functions u, v be subjected to the same ortho-
gonal transformation, so that
u = (t>{.v,f/,^) = ^{^,y],0 and v = r}^{.r,i/,-) = ^{tv^Q'^
then (^ {d^, d^, d^) v = ^ {d^, d^, d^) v.
1812 Ex.— Let ^^ = ax'' + hf + cz^ + 2fyz + 2gzx + 21ixij = ^
= aV -^h'n'-^ cC + 2/ v^ + Ig'Kl + lliiri = *,
and let v = .(;' + / + 2= = ;H»r + ^'= 4/ and ^.
Then (f7„ cZ^,, cZJ v = ar,^ +?^i',j, +ci-,, +2/^^, +2r/y,, +2/(V^y ,
and $ ((7^, (Z^, cZ^) v = a'v^^ + Z/'ug^ + c'v^^ + 2/'t?^^+ 25r'v^j + 2/i'i)^^.
But t'oj. = 2, and v,,, = 0, &c. Hence the theorem gives a-\-'b-)rC
= a' + i' + c'; in other words, a-\-h + c is an invariant.
1813 Contrnfiredicnt. — AVhen the transformation is not
orthogonal, (1795) shows that d^. is not transformed by the
same, but by a reciprocal substitution, in which %, &i, c^ are
replaced by the corresponding minors A-^, B^, Oj. In this case
d^, dy, dg are said to be contragredicnt to a', ^, z.
1814 Contravariant. — If, in (1G29), the quantics are sub-
jected to a reciprocal transformation instead of the same, we
obtain the definition of a contravariant.
1815 AVhen ;<; is a function of two independent variables x
and 2/, the following notation is often used :
dz _ dz _ dp _(Px
Tv^^'' Ty-'^' d.v- da}
dp ^ (hi ^ dz ^ ^^ f!l^fll = t,
dif d.v d.vdij ' djf dif'
Lot (/) (-/', ?/, z) = 0. It is required to change the inde-
pendent variables from .r, y to z, y. The formulas of trans-
= r
}rAXIMA AM) MIMMA. 3
05
formati
an are
1816
f/ :■ ~ p ' (hi p ' dr.' ;> ' *
1819
(/',r _ 'lsp(i — tjr—r<i\ d'r _ qr—ps
(iif- p^ ' (/t/(/.z jt'
Proof. — Forinuloa (17G1, 1703) fjive x. and jj^., because, since y remaina
constant, (}» may be considered a function of only two variables, x and z.
Formula) (1708-9) give .r^ and x.^, in terms of partial derivatives of 0,
since z is now constant, and ^ may be taken as a function of the two variables
X and y.
But «/. (.r, //, z) = is equivalent to \p (x, y)—z = 0; and the partial
derivatives of (p with respect to x and y are the same as those of \p ; and
therefore the same as those uf z when x and y are the independent variables.
Hence z may be written for f in the formula).
Lastly, f-'] =(-^'JA =(-'l) 'I± = 'U^^-J^. 1. = ^ILTH,
The independent variable is here changed from z to x, without reference
to the equation 9 = 0; and this is allowable, because y is constant for the
time being in either case.
MAXIMA AND MINIMA.
Min'una and Mininui rnlucs (if (i function of
in drpcn den t va rid hie.
1830 DEnxiTiox. — A function <!> (./•) has a
when some value x = a makes </> ('i') (ji'catei
by any vahie of .r, indefinitely near to a.
minimum value, reading lei<s for (jreatcr.
1831 Illustration. — If the
ordinate y in the figure be al-
ways drawn =f{x), it has
maximum value
than it is made
Similarly for a
maximum values at A, G, E,
and minimum at B and D
(1403).
Note. — For the algebraic
determination of maxima and
minima values by a quadratic
equation, see {J>'S).
1832 Rule I. — A function </> (x) is a maximum or minimum
when (p' (x) vanishes^ and chanfjcs its siijn as x increases from
plus to minus or from minus to plus rrspcctivcly.
2 u
306 DIFFEREXTIAL CALCULUS.
1833 Rule II. — Otherwise <|> (x) is a maximum or minimum
ivlien an odd number of consecutive derivatives of </> (x) vanish,
and- the next is minus or plus respectively.
Pj^OOF. — (i.) The tangent to the curve in the last figure becomes parallel
to the X axis at the points A, B, C, B, E as x increases ; therefore, by (U03),
tan 6, which is equal to /(.!•), vanishes at those points, while its sign changes
in the manner described.
(ii.) Let /"(,«) be the first derivative of /(.?') which does not vanish when
a; = a, n being even; therefore, by (1500), /(a ±70 = f{a)+ ^-^f'{a^Hh).
The last term retains the sign of /"(a), when 7; is small enontrh, whether h
be positive or negative, since n is even. Therefore f {x) diiiiini.shes for any
small variation of a; from the value a if /"(a) be negative, but increases if
/"(a) be positive. Hence the rule.
1834 Note. — Before applying the rule for discovering a
maximum or minimum, Ave may evidently —
(i.) reject any constant factor of the function ;
(ii.) raise it to any constant poioer, paying attention to sign;
(iii.) tahe its reciprocal ; maximum becoming minimum, and
vice versa ;
(iv.) tahe the logarithm of a positive function.
1835 Ex. 1.— Let <p {x) = x'-7x*-d5x + l,
therefore f Oe) = T.^^- 28a;» - 35 = 7 (x'-5) (,r^ + 1) .
Also (j)"(x) = 7(6x^ — l2x^). Therefore a; = V5 makes ^'(.v) vanish, and
f"(x) positive; and therefore makes (p(x) a minimum.
1836 Ex. 2.— Let <i>(x) = (x-2y'(x-2y\ Here
f (ao = u(,x-^y'(x-2y'+u(x-sy\x-2y'={x-sy\x-2y\2r>x-6i),
and we know, by (444) or by (14G0), that, when x = S, the first thirteen
derivatives of ^ (c) vanish ; and 13 is an odd number. Therefore (x) is
either a maximum or minimum when « = 3.
To determine which, examine the change of sign in (^'(x). Now (,r — 3)"
changes from negative to positive as a: increases from a value a little less
than 3 to a value a little greater, while the other factors of <p' (x) remain
positive. Therefore, by the rule, cp (a;) is a minimum when a- = 3.
Again, as x passes through the vakie 2, 0' (x) does not change sign, 10
being even. Therefore x — 2 gives no maximum or mininuim value ot [x).
Lastly, as x passes through the value -*.- the signs of the three factors
in <p'(x) change from (-) ( + )( — ) to ( — )( + )( + ); that is, (/.'(•«') changes
from + to — ; and, consequently, f (a;) is a maximum.
1837 Ex. 3.— Let ^ (,r, y) = x* + 2x-y-i/ = 0. To find limiting values
ofy.
^r.\xl^^.\ Axn mimma. 307
Here y i.s <,nvon only as nn implicit fuiicLion of x. niUi'ieutiiiLiiig, in
order to employ ronnulto (17uH, 1711),
f , = •l-.c» + Iry, if-^ = 1 '2x- + •!■//, 0, = 2.r - -uf ;
7/, = makes 0^=0. Solviiif,' tins eiiuatioii with f (.c, i/) = 0, wo got
a' = =fcl, // = — 1 wlieu i/r vauislu's.
0., \l-\-
And then ij..^ = — - = ^^ — - = 8, positive ; therefore, when a; = ±l,
y has —1 for a minimnni value.
Similarly, hy making >/ the independent vaiiaMe, it may be shewn that,
when 2/ = , .c has both the maxinmm and miniinum values ± v^G.
1838 A limitino- value of <^ (.*', //),
subject to tlie condition ^ (,r, //) = i) (i.),
is obtained from tlic equation ^,i//,, = ^,^x//, (ii.)
Simultaneous values of x and //, found by solving equa-
tions (i.) and (ii.), correspond to a ni ixiuium or minimum
value of (p.
Proof. — By (1718), (j> being virtually a function of x only ; and, by (1832),
<t>r(-v'j) =
1839 Ex.-Let f{x,y)=X!, and 4^ (x, y) = 2x' -,•>, + y' = (i.)
Kipiation (ii.) becomes // (:>//" — .'■) =x (G-r" — ?/).
Solving thi.s with (i.), we Hnd //* = 2x^ and x' (I./-— y^).
Therefore .i;=^\'"2, y=\l''\- are values con-esponding to a, viaxinnim
value of (}). That it is a uiaxiinum, anil nut a ii-iiiimum, is seen by inspecting
e(piation (i.)
1840 Most geometrical problems can bo treated in this way, and the
alturuative of maximum or minimum decided by the nature of tho case.
Otherwise the sign of <p,^{xii) maybe examined by formula (1710) for tho
criterion, according to the rule.
Ma.i'imn and Minima raJue.s of a function of two
inilcpcndrnt variohlcs.
1841 Rl'LE I. — .1 faacfioii (j> (x, y) Is a maximum or minimnm
irlirii (p^ find <p^. hofh vanish and changr fhcir si(fns fnun jilns to
minus or from minus to plus rrsjirrflrrh/, as x aud y nicrraxr.
1842 Rri'E II. — Other wise, 0, ami (j>y must ranish ; </>., ^o,. — </)';y
must ho positive, ami ^^x or ^^y ^'^''t*'^ '^'' negative for a ma.vimum
and positive for a minimum value of <p.
Pkoof. — By (1512), writing A, B, C for i^^,, (p^^, fi,j, we have, for small
changes h, Ic in the values of x and y,
f i-c+h, y + k)-(l> (x, y) = % + % + ! (A}r + 2DUc + CIc') + terms
which may be neglected, by (lilO).
308 mFFEUEXTJAL CALCULUS.
Hence, as in the proof of (1833), in order that changinc^ the sign of/; or
Tc shall not have the elYect of changing the sign of tlie right side of the equa-
tion, tlie first powers must disappear, tlierefore f^ and ^y must vanish. The
next terra may be written, by completing the square, in the form
-— \ iA—-+B\ +AG—B- i ; and, to ensure this quantity retaining its
sign for all values of the ratio h : /.-, AC—B^ must be positive. ^ will then
be a maximum or minimum according as A in the denominator is negative
or positive.
It is clear that A and B might have been transposed in the proof. Hence
B must have the same sign as A.
1843 A limiting value of <^ (.v, y, £),
subject to the condition ^ (<r, y,z) = (i.),
is ol)tainccl from tlie two equations
1844 <#>A = M, (ii-). M. = <^A' 0"-);
1846 or, as they may be written, 2j: — xi' = i-£ (iv.)
V'.r V*-/ V'.
Simultaneous values of ,t, y, ,?, found by solving equations
(i., ii., iii.), correspond to a maximum or minimum value of (p.
Proof.- — By (184-1), being considered a function of two independent
variables x and z, and, by (1729, 1730),
^^C*', ?/, 2) = gives (ii.), and (p.^sc, ij, z) = gives (iii.)
The criterion of maximum or minimum in (1842) may also be applied
■without eliminating y by employing the values of (j>2x and <p-,. in (1719, '30).
1847 Ex.— Let cp {.r, //, ;:■) = /- + //- + :-
and i^i.r,!/,::) = a:r + hr-^cz' + 2fi,:: + 2r,xx-\-2hvi/-l = 0..(i.)
Equations (ii.) and (iii.) licre become
r = , 'i r = '. = i, sav, (iv.)
Therefore, by proportion (70) and by (i.), .r + .'/' + ■-" = ^''"^ = */*•
From equations (iv.) we have
a — li ]t cj I
// l-Ti f = 0,
q t c-n
1848 o,vi-/ni + r,:: = Rx^
hx-\-hij+/z = By [ ;
gx-^fy+cz = li::)
1849 or {R - a) {R - h) (R - r) + 2l]ih
-{R-a)f-{R-h)</-{R-c) Ir = 0, or (see KUl)
MAXIMA AXn MIXIMA. 309
Tliis cul)ic m R is tlio climinaTit of the three equations in
(T, //,::. It is called the (h'srriiniinitiinj nihlr of the (juadric (i.),
and its roots are the I'eeiprocals of the niaxinia and niiniiiia
values of x- -\- if- -^ ::'- .
1850 To show that the roots of the disciiminating cubic
are all real.
Let 7i'i, li, be the roots of tlic quadratic equation
1851 M'T^ ,/ \ = {j^-b){K-c)-f = o (v.)
Ji'-r
J^^ > h and r, and h and c > /?.,.
Intake R = it', in tlio cubic, and tlic result is rerrative, bcinp^ minus a
pqnare tiuantily, l)y (v.). ^lako 7i* = h'.,, and the result is j)ositive. There-
fore the cubic has real roots between each pair of the consecutive values 4-00 ,
Ji'p 7?.,. —00 ; that is, three real roots, liut since the roots are in order of
magnitude, the first must be a maximum value of JJ, the third a minimum,
and the intermediate root neither a maximum nor a minimum.
M(tA''nu(i (lud Minima values of a fuuctiitn of three
or more variables.
1852 I^t^t <{> {■''!/■-) he a function of three variables. Let
1>±r, «/>2,/' </>2--. </>,/--> </>.-.r, fr;, bc dcuotcd bj a, h, r,f, g, h ; and let
A, B, C, l\ (r, 11 be the corresponding minors of the deter-
minant A, as in (1G42).
1853 Rule I. — ^(x,y,z) is a maxiimnn or minimvm wlirn
<^x) ^y> ^7. ^^^ vanish and change their signs from plus to minus
or from minus to plus respccticehj, as x, y, and z inrreasc.
Other'^'ise —
1854 Ri'Ti: II. — TJir first dcriraflrrs of (j> nnisf rani.di ; A
and its cocfiirimt iu tin' reciprocal determinant of A nmst Iw
jwsitire ; and «/> u-ilt tie a maximum or minimum according ax
a is negatire or jiositice. Or, in the place of .1 and a, read Ji
and b or C and c.
Proof. — Punsning the method of (181-), let t, »/• C be small changes in
the values of .r, //, z. By (lolt),
fix + l, y + n, 2 + 0-9 ('. 2/' 2)
= c>, + 19, 4 ^9. + i ("i' + W + cr- + 1H + 2;/;^ + 11<in) + Ac.
For constancy of sign on the right, 9^, <p,j, 9. must vanish. The quadric
may then be re-arranged as under by first completing the square of the term.s
in t. and then collecting the terms in C, V. ft»d completing the square. It
thus becomes ] (ai, + Jir} + g^) + ^—^^ ~^ ^ — -7,— "
'2.1.1 K. Li L
310
DIFFE Z? ENTTA L CALrUL US.
Hence, for constancy of sign for all values of E, »?, i^, it is necessary that G
and liC—F'^ should be positiVe. Tliis makes B also positive. By symmetry,
it is evident that A, B, G, BG—F", CA-G', AB-R- will all be positive.
The sign of a in tlic first factor then determines, as in (1842), whether ^ is a
maximum or a minimum.
1855 The condition may be put otherwise. Since
BG—F' = a/^ by (577), the condition that BG'—F"- must be
positive is equivalent to the condition that a and A must have
the same sign. Hence we have also the following rule : —
1856 R-ULE III. — (j>^, (^y, (f>^ must vanish ; the second of the four
determinants below must he positive, and the first and third
must have the same sign : that sign being negative for a max-
imum and posltlnefor a mliilinnm value of (^ (x, y, z).
1857 4>..:
f-
>■!.■ <l>..;, <#>,,--
5
<^-2.r ^.-7/ <^.r.- <^.r,c. •
K. 4',, f ,--
<^,ya- <^:;.v <^//-- ^ino
!-,-, i.,, 4'..
<t>zx 4>zu ^tz <t>.'W
<l>w.c i>wy 4*tcz 4>>iv
1858 The theorem can be extended in a similar manner to
<l> («, y, ;<!, n', . . . ) , a function of any number of variables. Form
the successive Hessians of (1630) for one, two, three, &c.
variables in order as shoAvn above ; then —
1859 RubB. — In order that <^ (x, y, z, w, ._..) may he a max-
imwiii or minimum, <^^, «^y,«^„<^w5 &c. must vanish; theHesslaiis of
an even order must he positive ; and those of an odd order must
have the same sign, that sign being negative for a maximum and
positive fur a minimum value of the function (p.
For a demonstration in full, see Williamson's Diff. Gale, 4th Edit., p. 433.
1860 Ex. — Required a limiting value of the function
The condition in the rule produces equations (1), (2), (3). Equ:i
results from Euler's theorem (1624), thus; introducing a iuiuih v;i
as in (1()45), we have a;z(, + ?/?/,; + 2"r + ?'""»■ = 2«,
which reduces to (4) by means of (1), (2), (3), and the value of n
ir = 1 .
1861 };><,.= ax + hy + fjz^p = (1) ^ a /' .'/
Ih; = Lc + hy+p+q = (2) I h b J
\,a, = <j.c->rfy +cz +r = (3)
p.« + r^v/f »•.-+./ = V (4)
'1
fj f c '•
j) q ?• d—U
tion (I)
riable w,
, putting
= 0.
^f^XT^fA axd .vf.wv.t. nil
Tlie ilctcrniinant is tho elimiuiuit of «lio four r(iii!iiiotis, l)y ^•''.^3), nn<l ig
c<|uiv:.U-iit, l)V tl.o inothod i)f (1721, Proof, i.), to A'-A/<=0, or u = A'-^ A
(Notation of'ir.lG).
To dL'tennine whether this value of ii is a nmximtim or minimnrn, either
of the conditions in (185 1, T.) may be npplicd ; and since, in this example,
«2,= )la, Uiy — '2b, Ac, tho letters a, h, c, f, <j, h may be considered identical
with those in the rule.
1862 To determine a limiting: value of </> (,r, //, ", ...), a ftnic-
tion of 111 vai'iaMcs connected by n eijuations '^i = 0, //o = 0, ...
R^'j^E, — Axaiimn n inidetcrmlncd mnUiplu'rs Xi, X;;, ... ^n
iciih the JoUoiriiKj m liquations: —
</)x-fXi(Ui)x + X.,(u,\+ ••• +X„(u„), = 0,
fv + Xi(ni), + X,(n,\-h ... +A„(ii„), = 0,
mal'ing in all m-\-n eqxdfions in m + n quant if i(\'i, x,y, /, ...
and X,*, X., ... X„. The valves of x, y, z, ..., found from these
equations, correspond to a maximum or minimum value of (p.
Pkoof. — Differentiate f and ?/,. ii„_, ... «„ on the hypothesis that x, ?/, z,...
are arbitrary functions of an independent variable /. Multiply the resulting
equations, excepting the first, by A„ Aj, ... X„ in order, and add them to the value
off,. The coefficients of x,, y„ z„ ... may now be equated to zero, since the
functions of t are arbitrary, producing the equations in the rule.
1863 Ex. 1.— To find the limiting values of r^ = x* + y^ + z^ with the
conditions A.c^ + By- + Cz^ = I and lv + my-\- nz = 0.
Here m = 3, ?!- = 2 ; and, choo.sing X and /u for the multipliers, the equa-
tions in the rule become
2x + 2AXx + nl =0 (1)-^ Multiply (1), (2)i (^) respectively by
2y + 2]i\y + fim = (2) > . x, y, z, and add; thus ^ disappears,
2z + 2C\z+ttn =0 (3)) and we obtain
a;2 + y» + 2= + (.'1.1-' + By' + Cz') X = 0, therefore \ = -r\
Substitute this in (1), (2), (3) ; solve for a-, y, z, and substitute their values
in lx-\-vuj + nz — 0.
1864 The result is —f— + -"'--- + -^^ = 0, a quadratic in r'.
Ar' — i JJr' — 1 Lr — i
The roots are the maximum and minimum values of the square of the
radius vector of a central section of the quadric A.v' + Hy'^A- Cz' = 1 made by
the plane Ix-\-my + nz = 0.
1865 Ex. 2.— To find the maximum value of u = (x + 1) (y + l) (^ + 1),
subject to the condition N = a'b^c*.
312 BIFFFnEXTTAL CALCULUS.
This is equivalent to fiuding a maximum valua of
log (x + 1) +\og (y+l)+\og (z + 1),
subject to the condition logN = x\oga + y log h + z\og c.
The equations in the rule become
JL+Xloga = 0; -L.+\\ogh = 0; -l--+k\ogc = 0.
By eliminating X, these are seen to be equivalent to equations (\SW)).
!Multip1yingnp and adding the equations, we find X, and thence x + \^ y + 1,
z+l; the values of which, substituted in u, give, for its maximum value^
u = {log (Nahc) }«-=- 3 log a^ log b'' log c\
Compare (374), where a, b, c and x, y, z are integers.
Continuous Maainia and Minima.
1866 If fr ai^^^ •/>-/' "1 (18-i-)j liave a common factor, so tliat
where P and Q may also be functions of x and 7/ ; tlien tlie
equation 4< {x, y) = determines a continuous series of values
of X and y. For all these values <p is constant, but, at the same
time, it may be a maximum or a minimum ivith respect to any
other contiguous values of cp, obtained by taking x and y so
that xp (xy) shall not vanish.
1867 In this case, i>2^<p.2y — <t>% vanishes with ;/., so that the
criterion in Rule II. is not appUcable.
Proof. — Differentiating equation (i.), we have
^,„ = Q,^+(H, ) ' fv = QA + Q-hi
If from these values we form ^2^0^^ — ^.y X ^„^, -^y will appear as a factor of
the expression.
1868 Ex. — Take z as ^ (.nj) in the equation
z"- = cr-lP + 2by(.c' + y')-;,r-y' (i.),
^' = -(77^-1) -^^ - = 2/(
The common factor equated to zero gives xr -\- y" = b', and therefore 2 = ±a...(ii.)
Here a is a continuous niaxinmm value of 3, and —a a continuous minimum.
Equation (i.) represents, in Coordinate Geometry, the surface of an anchor
ring, the generating circle of radius a having its centre at a distance b from
the axis of revolution Z. Equations (ii.) give the loci of the highest and
lowest points of the surface.
For the application of the Differential Calculus to the
Theory of Curves, see the Sections on Coordinate Geometry.
INTECxRAL CALCULUS.
INTRODUCTION.
1900 The operations of different laf ion mid integration are
the converse of each other. Let /'(/*•) be the dm-ivative of
</)(.7;); then «/>(./■) is called the integral oif(x) with respect to x.
These converse relations are expressed in the notations of the
Differential and Integral Calculus, by
!Mii =/(,,.) and by \f{,r) ,/.,■ = ^ (.,).
1901 Theorem. — Let the curve y =f(,r) be drawn as in
(1403), and any ordinates LI, Mm, and let OL = a, OM=b;
then the area LMuil = <p(l>) — <p{a).
Proof. — Let OX be any value of x, and PN
the corresponding value of y, and let the area
ONPQ = A ; then ^1 is sunie/iniction of x. Also,
if NN'=(lx, the elemental area NN'FP = dA
= ydx in the limit ; therefore — - = y. Thus A
ax
is that function of x whose derivative for each
value of X is y or f{x) ; therefore A = (f> (x) + C,
where C is any constant. Consequently the area
LM)nl = <}) (i) — v> («). whatever C may be.
The demonstration assumes that there is onl}' one function f (r) corres-
ponding to a given derivative /(x). TliLs may bo formally proved.
If possible, let \l^(x) have the same derivative as <{>(-r); then, with tlio
same coordinate axes, two curves may bo drawn so that the areas detined jw
febove, Hke LMinl, shall be (j>{.r) and \p(.i-) respectively, each area vanishing
with x. If these curves do not coiiieide, tiien, lor a given value of x, they
have different ordinates, that is, f (x) and 4^' i-^) are different, contrary to the
hypothesis. The curves must therefore coincide, that is, 9 (x) and ^ (x) aru
idtutit'al.
J\,W M
314 INTEOJiAL CALCULUS.
1902 Since <p(h) — (\>{a) is the sum of all the elemental areas
like NN'P'P included between LI and Mm, that is, the sum of
the elements ydx or /(.-r) d.v taken for all values of x between
a and h, this result is ^Titten
f/(.r)rf.. = .^(6)-.^(«).
1903 The expression on the left is termed a definite integral
because the hmits a, h of the integration are assigned.*
Wlien the limits are not assigned, the integral is called inde-
finite.
1904 By taking the constant (7=:0 in (1901), we have the
area ONPQ = 4> (.r) = (/(.r) dv.
«/
Note. — In practice, the constant should always he added to
the result of an integration u'hen no limits are assigned.
MULTIPLE INTEGRALS.
1905 Let f{x, y, z) be a function of three variables ; then
r«-2 f*v2 r*^2
the notation j \ \ /(.r, ;/, z) d.v dy dz
J.ri Jyi *. zx
is used to denote the following operations.
Integrate the function for z between the limits 2 = ^i, z^z^,
considering the remaining variables x and y constant. Then,
whether the limits z^, z.^ are constants or functions of x and t/,
the result will be a function of x and // only. Next, consider-
ing X constant, integrate this function for y between the limits
7/i and 7/2» which may either be constants or functions of x.
The result will now be a function of x only. Lastly, integrate
this function for x between the limits x^ and x^_.
Similarly for a function of any number of variables.
1906 The clearest view of the nature of a multiple integral
is afforded by the geometrical interpretation of a triple in-
tegral.
* Tho integral niivy be read " Sxiin a to b, j (x) fix''; | signifying "sum."
jXTnnriii'Tioy.
lUTj
Taking rectangular coordinate axes, let tlie surface
z = <i> (,/', //) (A'ro7)i in the figure) be drawn, intersected by the
cylindrical surface y —. xp {,r} (UMNiim), and by the ])lane
x = a\ (LSmI). The' volume of the solid OLMNohan bounded
by these surfaces and the coordinate planes will be
», «. .. c (I «.
Proof. — Since tlie volume cnt off" by any plane parallel to OYZ^ and at a
distance x iVom it, varies continuously with r, it must be smue J'nnct'ioti. of x.
Let Fbe this volun:e, and let (/ T be the sniidl change in its value due to a
change dx in x. Then, in the limit, JV= I'Qqp X d.c, au element of the solid
shown by dotted lines in the figure. Therefore
^=r(2'ii'=J^ ?>0^.'/)^///, by (1002),
X being constant throughout the integration for y. The result will be a
function of a; only. Making x then vary from to x, we have, for the whole
volume, \Jl t(..-..v).'y|.'' = J, !j,, [J„ .'-].'."(./..
since f (x, //) = z. With the notation e.vplained in (l'.iO:)j, the brackets aro
not required, and the integrals aro written as above.
1907 Tf the solid is bounded by two surfaces z, = (/), (.»', y), ?s = 0j («• !/)•
two cylindrical surfaces //, = \//,(.r), i/^ = \p.i(x), and two planes x = x^, «=;•'■».
the volume will then be arrived at by taking the diffV-rence of two similar
integrals at each integration, and will be expressed by the integral in (I'JUS).
if any limit is a constant, the corresponding boundary of the solid
becomes a plane.
316 INTEGRAL CALCULUS.
METHODS OF INTEGRATTOX.
INTEGRATION BY SUBSTITUTION.
1908 Tlie formula is ( (^ (.r) (Lv = ( cf> {.v) ^' dz,
«, »y UZ
wlicre z is equal to/(.T), some function cliosen so as to facili-
tate the integration.
Rule I. — Put x in terms of z in the given function, and
multiply the function also by x^; then integrate for z.
If the limits of the proposed integral are given by x = a,
X = b, these mast be converted into limits ofz by the equation
^ = t-'W-
The following rule presents another view of the method of
substitution, and is useful in practice.
1909 Rule II. — //"^(x) can be expressed in the form F(z)Zj;
then ^ <t> (.r) (Lv = I V (z) zA^' = f ^ (^) d^-
Ex. 1.— To integrate ,_," — —• Substitute z =^ x + ^/(.r + a") ;
tbei-efore --- = 1 -| ■ — ; = ^— ^^ ; — = -- — „ -,
\vw^> t " = i? = '°«^ = '"^ f" + ^^■''+'-'»-
Ex. 2. —^^'^ dx = l^i--±:^ dx = - log (x-' + x-').
J x + x' J x'' + x-' ov -r y
Here z = x-'-\- x-\ F (z) = - --, z,= - (rxv-' + 2x--).
•go r 1— ^ -^^ d^ r 3'"'— c dx
] l+rx' y{i + ax'' + c'x*) ~ ]x-' + cx ^(c-x^ + x-'' + a)
dJx-'^ + r.r)dx
1
1 , XK/(a-2c) + ^/(l + ax'■ + rx*)
V{a-2c) ^^ l+cx^
_ 1
Vi2c-n)
Bj (1027) or (1026). Here k = x-'+cx
1 _,«N/(2r-rr)
or — / o ^ cos ' — ; 7, — .
Vi2c-a) i + cx'
METHODS OF IXTECHATIoX. 31
In Examples (2) ami (3) the process is analytical, and leads to the dis-
covery of the particular fiuietioii ;:, wifli respect to which tlie intejrrution is
effected. U z be known, Ride 1. supplies the direct, thoujjh not always tho
simplest, method of integrating the function.
INTEGRATION \)Y PARTS.
1910 T\v differentiating 7(V with respect to ./', we obtain tlio
general formula \ i(,r(Lv = iir — \ ur.d.v.
The value of the fii-st integral is thus determined if that
of the second is known.
Rule. — Separate the quantity to he integrated into two
factor.^i. Integrate one factor, and differentiate the other with
\'esprct to X. If the integral of tlie resulting quantity is
known, or more readily ascertained than that of the original
one, the method by " Parts" is applicable.
1911 Note. — In subsequent examples, where integration by Parts ia
directed, the factor which is to be integrated will be indicated. Thus, in
example (10-">1), "By Parts |6'"'(/.c" signifies that e" is to be integrated and
sin h.c differentiated afterwards in applying the foregoing rule. The factor 1 is
more frequently integrated than any other, and this step will be denoted by \dx.
INTEGRATION BY DIVISION.
1912 A formula is
^{a+hry d.v = a ({a + Lv"y-' + h f.r" (« + />.<•")"-' d.r :
The expression to be integrated is thus divided into two terras,
the index p in each being diminished by unity, a step which
often facilitates integration.
Similarly, j (a + bx" + ex'")'' dx
= a (a + hx" + cx'"y-^ + hx" (a + i.c" + cj;'")''-' + ex'" (a + hx" + cx''-y-\
1913 Ex.— To integrate \^(x' + n')dx.
By Parts | J.r, | y (.r' + cr) dx = x ^ (.." + a') - | -^f^^^y
By Divi.sion, I v/(.r' + a') dx = j ^^^:^^^^ + j v/(xHa')-
Therefore, by addition,
= ;iv'(i'+(i') + J.<Mog{.c+ v'(r + a')}, by (1900, Ei. 1).
318 INTEGUAL CALCULUS.
INTEGRATION BY RATIONALIZATION.
1914 In tlie following example, 6 is tlie least common de-
nominator of the fractional indices. Hence, by substituting
z = x^, and therefore x^ = Oz^, we have
z^-l dx '•-'-
dz
1 \
dx.
Each term of the result is directly integrable by (1922) and
(1923). For other examples see (2110).
INTEGRATION BY PARTIAL FRACTIONS.
1915 Rational fractions can always be integrated by first
resolving them into partial fractions. The theory of such
resolutions will now be given.
1916 If </>('^0 and F{x) are rational algebraic functions of x,
<}>{x) being of lowest dimensions, and if F {x) contains the
factor {x — a) once, so that
F(a>)-(.r-«)t/,Gi') (1);
1917 «.n|« = _l_ + |gana^ = iM (,,
Proof. — Multiply equation (2) by (I), thus
(j> (x) = A4^ {x) + (x-a)x (x).
Therefore, putting x = a, (p (a) = A4^ (a). Also, by differentiating (1), and
putting x = a afterwards, F' (a) = \p (a). Therefore A = <p (a) -^ F'((i).
1918 Again, if F{x) contains the factor (x — a), n times, so
that F{d^ = (.{—«)"</» Or).
Assume ^V^ = 7 hr, + 7 T:r-i + ---H rT7~T-
F{.c} {,v~a)" id—a)" d-a xli[.r)
To determine A^, A^ ... A,,. Mulliplij hi/ (x — a)"; put
X^a and, dijD'crentiate, alternately.
1919 If ^'(^0 = ^ ^^^ ^ single pair of imaginary roots
.97'. 1 XD. I /«■ D TSTIU! U.\ LS. 3 1
a±li5; tlu'H, applying (1017), lot
and the partial fractions corres[)on(ling to these roots will bo
A-ili A-^ili _'2\(,-a) + 2li§^
For practical methods of resolving a fraction into partial fractions in the
different cases which occur, see (235-238).
INTEGRATION BY INFINITE SERIES.
"When other methods are not applicable, an integral may
sometimes be evahiated by expanding the function in a con-
verging series and integrating the separate terms.
Ex. j - dx = logx + ax + j-^^ + ^-^, + 3-2-3-^, + &c.
(150)
STANDARD INTEGRALS.
1921 Some elementary integrals are obtained at once from
tlie known derivatives of sim})le functions. Thus the deri-
vatives (1422-;)8) furnish corresponding integrals. The fol-
lowing are in constant use : —
1922 j.vrf..=^. }- = ''"'-'■■
1926 f_^ = lc„s-^or-lsi,.-.^. [.SU..1
320 TNTEGRAL CALCULUS.
1928 r ,/^:'\_ ... - log {^v^V{^i'±a-)}. (1009, Ex. 1)
1929 ( -^-4^^ — - = siu-^ — or - cos-' — . (1434)
1930 (-yf^ = ±^^{a'±.v-).
1931 ^y Parts, Division, and adding results (1913), we obtain
[ \/(cr^ ± a-) dx = U v/(ci''- ± a') ± i«' log U + V{.v^ ± «") } .
1932 By Parts, Division, and difference of results,
J 7^^ = i.i-N/Gf'±«^) T i«^ log {.r+ v/(a'^ ±a'-)} •
1933 f v/(«'-ci-) d.v = ^a' siu-^ ^ + ict' V'(«--cr^).
-^ " [As in (1031)
1934 f /5' .,, = i«'- «m- ^ - i.r v/(«-.ro\
J V{(r—cV-) a [As in (1032)
1935 r _^, = 1 tan-^ ± or - 1 cot"^ i- (^36)
J .r--h« « a a a
1936 r _^, = J- log ^^-;:^. [By Partial fractions
1937 f^!^, = ^log'i±i-. [ Do.
1938 1 sin -I' t?'*^ = — cos X, \ cos .v dx = sin a;.
1940 \ tana dr = — log cos .i\ j cot a: dd' = log sin <r.
1942 \ seCcvrfcr = log tan (-4- -J. 1 coscCcvrAr^logtan '-.
Meiiiou. — (1940, '2), substitute cos.c. (1941, '3), substitute sin .c.
1944 I «in"' <^' d.v = .V sin-^ .v + v^(l —d") .
1945 \ cos"^ .r d.v = .r cos"^ .v — v/(l — .r") .
(7/7? (• I 'L A Ji Fl 'XC TJONS. 321
1946 \iin\-' a- ill = .r tnir' .v - ] lo- (1 +.»").
1947 \voi-' Jill = .r1j»ir'.r+ I l()-(l+.r-).
1948 (*soc-^ .r (/.r = .r soc"' .r - loir [r + v^^C' '- M } •
1949 \ co.scc"^^^/.r = jeosec'\r + log [.v-^ ^'\d--—l)\ .
^Ieihod.— (104-i) to (1949), integrate by Parts, J .?.c.
1950 1 log .r d.V = .r log X^.V. [By Parts, J Jx
according as a is > or < h.
[Siil).s. tan o.i;, and integrate by (1035 or '37)
VARIOUS INDEFINITE IXTF.GRALS.
GENERALIZED CIRCULAR FUNCTIONS.
1954 f sill" (Lv. \ COS" dr. \ coscc" (Ia
«. « » .
Method. — When n is integral, integrJite the expansions in (772-4).
Otherwise by successive rednction, see (2uG0). For i cosec" i^.e, see (2008).
^nrw i\ n / t^nl"-^r tan"-'.r , tan''-^r n
1957 W«-"-^'- = ^7^r--^;^Tr+^;^r-^^^-
Phoof. — By Division; tan" .r = tan"'-x sec* x — tan""". c, tlie first term of
•which is integrable ; and so on.
1958 i-^r^ — n; = , . V-.-i U"-'' '-'"'->'" ''-■
^Method. — By substituting tan ^^ = ^^" TT \/ ( "XT ) ' Similarly with
pin .r in the place of cos-r, substitnte -if — t.
1959 ] d'l- \ d I . \ dd\ \ djr.
*^ cos w.r Jsin;/r J cos n.f^ J siii /m
322 INTEGRAL CALOJJLUS.
Method.— By (809 & 812), when p and n arc integers, the first two
functions can be resolved into partial fractions as under, p being < n in the
first and < n — \ in the second. The third and fourth integrals reduce to one
or other of the former by substituting ^tt — x.
1 q63 ^^^^ = - 2'-" (-1)-' sin(2r-l)Qcosn2r-1)0 ^.^^^ ^ ^ .
•*-^"*^ cos«.^ n '"^ cos.f-cos(2r-l)0 2u
1964 2^- = -^- S—'C-l)'-'^^"'^'^""^''"^ with = -^.
■*-*^ * sin «,(; visin.u ''"' cos .f — cos rO 7i
The fractions in (10G3) are integrated by (1952) ; those in (19G4) by (1990).
Fonmdce of Bednction.
1965 {'^^^^civ= 2f '^''^("-J)-^ rf..-r '-"^^"7-^''' rf.i-
J COS",l' .' COS"-'.l- J COS''.!-
1966 r'-:?^^ rf.r = -2 f'^'".^"-!^''- rf.r+ ('£^(£llll£ rf...
1967 l'^i^'./.= 2f ^^l^^^-^^'" ^aH-f '^"^"7^^'" ^/^i-
J cos'\f J cos'^-\v J cos'\r
Proof. — In (1965). 2 cos.i- cos (n— 1) .t? = cos 72.^4-cos (?i— 2) .r, &c.
Similarly in (1966-8).
1969 \ siu^' X sin n.v dx
^ __ sin^.reos7i.r _j^ ^sm-^^ cos (»-l) .r r/.r.
1970 I cos'' cr sin »,r f/.r
= — '- — I eos^ \r sin (n — i) x a.r.
1971 i sin^'.r cos 7i<r (/.r
^ si"".. sin«.r _ _?>_ r ;,,,., ^, ^i^, („_i) ,. ,,,,
p-\-n i> + « ^'
1972 1 cos'' .r cos ?*.r (Lv
»-'
^I-5llii!i^ + --ii- iVos"- .r cos (;2-l) .r (Lv.
p-\-7i /> + ;/ J
cos"
P
CIRCULAR FUNCTIONS. 323
PiiOOF. — (lOtil*). By rnrts, ^ sin )/.»• (/./■. In tlio new intcf^'ial change
cos >/a; COS ./• into cos (;< — 1) .*• — sin >/./• sin .r. By sncci'ssivo reduction in
tbis way tlio integral may be found. .Similarly in (llTU-'i).
Otlierwise, expand t-in" x or cos''.e in mnlliplo angles by (772-4), and
integrate the terms by the rollowing formulie.
1973-1975
r . . . 1 /sin (/J — ;0,r sin (/> + /i) .r\
\ sin p.r am ua' (Ij = -- -^ '-—^ .
J ' 2 \ p — n ;> + /* /
and so with similar forms, by ((jijG-i)).
1976 \ ^111^^ I ^'^'^ lOLiud trom
J cos n.v J cos ud'
r i GOspx—&m]Kv 7 , _ o f -"-'^"^""^ '^'^
J cos )ui J 1 + .■-•■■"
wlicn j) and u are integers, by equating real and imaginary
parts after integTating the right side by (2023).
Proof. — Put cos.r + i sin x = z ; therefore izdx = dz. Multiplying nume-
rator and denominator of the fraction below by cosH.i; + i sin /laj, we get
cos/ix + i' siny)j; _ j, cos (p-\- » ) x + i sin ( p + ?i.) a; _ n ^•" ' " .
cos nx " 1 + cos 2?iU! + i sin ^.nx 1 -t z'"
therefore f "'" ^'^ + ^ ^'" ^^'^ dx = - 2/ f '^^-^.
J cos nx J i + ^'
1978 f<:i^^P:!:^' and r!!iy^:I^' are found in the same
J sill wr J sin ;hr
way from ^—r-' '— dx = 'J — j —-.
Pi;ooF. — As in (107<)), by multiplying numerator and denominator of
cos W.C + 1 sin I'.r , • •
^ . '— by co3«A' — I sin?ix'.
1980 |-^:21^ ,,,,1 r^iiiii^
Putting y ^ cos .7^ + I sin .c ^^.^ g^^j tan «x = i (1-^/"), J"i'l therefore
"^(cos «.»■)
ydx = - — -•-. Hence, mulli:»lyiiig by /, we have
2-7/"
r /cos.r-si ,Kr^^r J>^^
J C-'Ccos^a-) J 2-;/'
The real part and the coefficient of;' in the expansiou of the integral on
the right by (2021, '2), are the values required.
824 IXTEGBAL CALCULUS.
1982 [ o "^^ ■ , = -^/-IT^a^'' (tan,r /^). [Subs, tan a.
j a con' X + b am- X \/{i-ih) \ \J a I
19o3 7— = —c~~r-> {h \o^( a cos X -Y li &\nx)-\-ax\. [Subs, tana;
} a-^h tau X a--\-b' ■"
1934 f —^ = -TT^r-Tl tan- ^f^^+^^l [Substitute cot.
J a -\- b am- X ^/{a -\-ab) cot .c v/a
1985 ".''," dx = — - tan"' (a cos a;) ^— . [Substitute a cos a'
198d :; Ty 5- = 3 ,,T tan - — — ~. [Subs, sin a;
J l-a^cus^« a\'^(l — a-) ^(1— a') a'
1987 cos a; s^(\ — a-&m^x)dx = J sin »•/(! — a" sin" x') + —- sin'' (a sin a;).
[Substitute a sin.o
1988 sin.-B v/(l— o^sin-.r) (Z.y = — |cos.? v/(l — a-sin^f)
^- — log j a cos x + V ( 1 — a" sin'-' x)\. [Subs." a cos «
1989 sin .T (1 — a'sin^a;)- cZ.r = — ^cosa;(l — frsiu^a;)^
+ 1(1 — a') sina; v/(l — a" sin-.r) tZaj. [Subs, acosa?
1^^^ J bill .<;C't + /; COS*)
By
log { (<t cosec x-\-h cot a;)^ tan" |a;}
1 (7 — b cos a; ?>^ sin a;
sin a; (a+ 6 cos a;) (tr — b^) sin a; (a''' — b"^) {a + i cus .v) '
1991 [ t"" •'' ^•'' = ^
J v/('t+i tau^'t) V'(fc-a
1 cos ,7' x/f?) — 0~)
- cos"' ; -.
) Vb
[Subs. COS a' ^/(h - «)
-^992 [ Via + l^m'^x:) j^ ^ ^^ ^^^_, ^hccx
J sin a; \/{a + b)
— ^/a log { \/a cot a; + ^/(a cosec'a^ + 6)}.
Method. — By Division (1912), making tbe numerator rational, and in-
tegrating the two fractions by substituting cot a; and cos a; respectively.
2993 I" 'li = c f r d i' _ f '^-^ I ^
J a + -//cos.<; + cco.s2.i; r« (. J 2ccos.t; + 6^/» J 2c cos * + ^ i- v/i j
wliere m = ^\\j^—1c {a—c)\. Then integrate by (1053).
1994 f /.'• . ^ = f , ..^/f ,^ . (i'«3)
Method. — Sul)stituto = x — u, where tan a = .
EXrOXEXTIAL AND LOaAlUTlIMlC FUNCTIONS. 325
1995 ^ F(.\n.v..n.,),l. ^
J a cos ,r-\-b sin .v-\-c.
F bring an integral algclmiic function of sin.r and cos r,
!MKTii(in. — Subslituto = x — n as in (1991), and the resulting integral
,, ,, . f /'(sinfl, t>nsfl)(/0 [ <h (cos fi) iJd , f sin 0»i/ Tens ^0
takes the form --^ — ,. — = ^,- rr—r, + — : : tt,
J A cos + Ji ] AcosO + B J Acosd + n
since /contains only inteii:«"il j)o\vcrs of the sine and cosine, and may Iherc-
f .re be resoivod into the two terms as indicated.
To lind the lirst intet,n:il on tlie ri^'ht, divide by the denominator and
integnite eaeli term se|)anitely. To tind the second integral, substitute tho
denominator.
1996 r F(co^,r)(rr _^
J (^^ + '>i t-os.r) ((t2-\-t)., COS.*) ... {a„-\-b,^ cosct')'
Avlicre F is an int(^gral function of cos x.
Mrthod. — Resolve into partial fractious. Each integral will be of the
1997 C Ac„..v + lls\n.r + V ^^^,
J a COS d-\-o sill d-\-c
Method. — Let </»(.}•)=« cos a; + /^ sin j' + c ; .'. <;»'(.>•) = —a sin. r + Z> cos ar.
Assume .1 Cvis x-t- JJ .sin .J -f- C = \<p(.r)+fx<ft'(j:) + t'. Substitute the values of
<f> (.r) and <}>'{x), and equate the coefiicients to zero to determine A, /u, r. The
integral becomes
I C ^' v- (•'■)?> (-'J 3 ^ °^^ ^^J 9{x)
and the last integral is found by (1991-).
EXPONENTIAL AND LOGAIITTILMIC FUNCTIONS.
1998 * ("^F(.r) (Lv can be found at once wlien F (,r) can l)o
expressed as the sum of tiro f (ructions, one of ivhirh in tlic
dcrivatii't' of the othrr, for
j"'-!<^(.'-)+f(.'))'/..='-<;t(.').
1999 * f"-" cos" La (Lv and | <'"■'' sin" Lr(l,v are respectively =
— ^ — — c^'^cos" bd.\- -^- — :. - I €'■" cos" -Oddu-.
a -\-no- a-j-M 6- J
326 INTEGRAL CALCULUS.
and
g sill &.r-»& cos fta> ^,. ^j^„_, ^^, ^K^^-l)^^ f..^. ,i^.-2 j^^^.^.v^
Proof. — In either case, integrate twice by Parts, j e"^dx.
Otherwise, these integrals may be found in terms of multiple angles by
expanding sin" x and cos" x by (772-4), and integrating each term by (iyol-2).
2000 I e-^siu'^ci'cos'^crdr is found by expressing sin^d' and
cos"aj in terms of multiple angles.
Ex. : J e^ sin^a: cos" x dx. Put e'^ = z in (7G8),
(2i sin a;)"^ (2 cos ;7;)' = (z-z-'y (z + z-y
2''e^ sin* a; cos'* — e"" (sin 7x — '6 sin o.f + sin 3.1' + 5 sin a-).
Then integrate by (1999).
2001 Theorem. — Let P, Q be functions of x ; and let
^Pdx = P^, iP^Jx = F,, ^P,Q.dx = P,,&c. Then
(pQ"(lv = P,Q'-nP,Q-'-i-n {n-l) P,Q-'- ... ±^/\+i.
Proof. — Integrate successively by Parts, J P dx, &c.
2002 Theokem. — Let P, Q, as before, be functions of x ; and
fP , P I\
J ^^ '"'" - (>*-lj (/.//-^ (^^-1) {n-'l) Qjr-^
i\ 1 r/Vi^/^
(/<-i)(«-2)(vi-;5)g/»>"-^ | /i-i J g
Proof.— Integrate successively by Parts, — ^^^,„ — ^■'"^^■'^•
Examples.
2003 f.r--^(log.r)'W/.r
:^ ^(/''-^/"-V^^i^^ /''-.. .+(-1)'''^)-
Method.— By (2001). P, = •'""', i', = -'", P, = """. '•tc
ALGKVUAh' FrXCTfOXS. 327
2004
and cacli term of tliis result can be integrated l)y (200:^).
2005 fiP^.
'*' \n-l^ {u — l)[n-'2)'^ {u-i){u--2){n—l.i) ''''
1 1 liwr .. •
+ — ,
Method.— By (2002). P.r = j-", P, = wx-'"', P, = vrx"-\ etc.
The last method is not applicable when n = 1. In this
case, Avriting / for log.v,
METnOD: = — . Expaiul the numerator by (1''0), and inte-
k)g X X log X
grate.
See also (21G1-G) for similar developments of the expo-
nential forms of the same functions.
PARTICULAR ALGEBRAIC FUNCTIONS.
n(v + ^)... 2(11-1) ,^_ X
\-x
« being even. (1018)
2008 f^^_,,^^,^^^ = -^
[Subs. ^('-^'^
log
2009 J ^T" ^r^V(TT7) " V'l '"*= y(i-*'')
oniA f <?•'• 1 ,._ gy2 4-v/(»'-l)
328 INTEGRAL CALCULUS.
2011 = ^sm-\/(-H:^^r^'^^), aobc
V{b<r-acc) ° V{c + ex^)
2012 [ -r^ = -^ log ^(-^^'^n-^^a ^ ^S,b3. 1
2013 i(a^5lfT^, = ;^,--^^^£^- [--r^.
2014 f a^:^Ai:..') =;^-'°-f^- ^""^-'r^
2015 l^q^- = ;^[.o.^^^H6*^--iai~
[Substitute ^ n/(1 + .(■")= a; s/2 in (2015 -f.)
c]^ _ 1 1 .„ (2^^-11*-^.- _ 1
2017 f , /% ,, = l^o,(^ff^
tan"
(2,r-lji
[Substitute a' = ,-(2.f2-l)*
2018 f n
J (l-\-x'){^/{l + x')-x'^^
tau'
[Substitute — -
L'-'""-^'^'-"
v'{>
/(i+.
c*;-.t
n
2019
73 '^"'
.^ 2^1 -,.«--
X v/3
^. [S
ubs. ^
a;
r fZa;
= log V .(•+ yi — a;^-
!•')
J t^(L-
■^«)
2020
{;
dx
(i+x)l/{l + 'Sx + SJ)
reduces
to (2010) by
substit
uting
X
.r'-l
INTEGRATION OF
If / and n are positive integers, and / — 1 < n;* then, ti
being even,
* If l = n, the value of the integral is simply — log (.r" — 1),
,.'-1
2021
INTEGUATION OF -i-— . 329
i-.r^ ^ i log(.,_l) + (zi)'log(.r+l)
,' .r — 1 // n
where 3 = '^- , and Z denotes that the sum of all the terms
n
obtained by rnakiii*,^ r = 2, ^i, G ... )i — 2 successively, is to
be taken.
If ?i be odd J
2022 Ji^' = llogGr-l)
ti >- n\ ' n 8111 rp
with r = 2, 4, 6 . . . /i — 1 successively.
If n be even.
2023 \ ^^r^' = -LtvosrI/3\og(.v—2.rco.rfi+l)
J .{'" + 1 n
. - < • //0 4- -1 «*'— eos>'/3
-4 S sm rlB tan ^ — ; ^,
n sill >'p
with r = 1, 3, 5 ... » — 1 successively.
If n be o(irf,
2024 f£^=. til^logGr+l)
- i ^ cos W)81ogGi--2.i^cos ry8+ 1) +? S sill W/8taii-^ l^^:i^l^^
/2 ^ n sill rytJ
with r = 1 , 3, 5 ... » — 2 successively.
Proof. — (2021-4). Resolve -'^ - into partial fractious by the method
of (1917). We have -f-^^ = ^ = — , since a" = ^ 1. Tho diirorent
1' in) ?/<«""' 7i
values of a are the roots of x" ±1 = 0, and these are given by x = cos r/3 ±
t sin r/5, with odd or even integral values of r. (See 4-80, 481 ; 2r and '_*>•+ 1
of those articles being in each case here represented by r.) The first two
terms on the right in (il021) arise from the factors .r ± 1 ; the remaining
terms from quadratic factors of the type
(y — cos rfi — i sin r/3) (a- — cos r/3 -t- ( sin r/3) = (.r — cos rp)' + piir r/3.
These last terms are integrated bv (10l^3) and (1935). Similarly for the
cases (202-2-4).
2u
830 INTEGRAL CALCULUS.
2025 If, in formulae (2021-4), + — S Q-tt - 7-/3) sin W^ be
added to the last term for the constant of integration, the
integral vanishes with x, and the last term becomes
=F — ^ sin W/3 tan ^ ^,
reading - in (2021-2), and + in (2023-4).
2026 f4^=i(«)^f^,,/.
where az" = Ihif. Then integrate by (2023-4),
202^ J ,,T+1 ' ■ = T ^ "" ""■^ *"" -TmTT'
where )3 = tt-^v?, and r = 1, 3, 5, ... successively up to n — 1
or 71 — 2, according as n is even or odd.
2028
1 £ .1 ^^^, ^ -^ Scos?«ry8.1og(.r--2.rcosr^+l),
with the same values of r; but when n is odd, supply the
additional term (--1)'" 2 log {x^l)^n.
Proof.— Follow the method of (2024, Proof).
Similar forms are obtainable when the denominator is x" — 1.
2029
— ^ = S 008 (»—/)</). tan ^ — ; — -X
— .^2 sin (h — /) <^.log (ci- — 2.r cos <^+l),
where <p = Q-\- "^ and r = 0, 2, 4, ... 2 (»-l) successively.
But if the iiiteo'i-al is to vanish with x, write tan"^—^'^ — - ,
• /oAorA 1— .rC0S(/)
as m (2025).
Proof. — By the method of ("2024). — The faf'tors of the denominator are
given in (1^071, putting y equal to unit}-.
TNTEGRA TION OF J x'" (a + 6^" ) « . 831
2030 r^-^^^T^' ^^- = =^ - ^' I ^"-^ (*''^+>-)
Xlog(a'-2.rcosr)8+l)--'si»(W)8+i//*7r)tair':^^^=^^ ,
with the values of /3 and /-in (2021-4).
Proof. — Differentiate the equations (2021-4) lu times w ith respect to /,
by (1427) and (1461-2). If m be negative, integrate m times witb respect
to I, and tlie same formula is obtained by (21.!>5-0).
In a similar manner, from (2027-8) and (2029), the general
terms may be found for the integrals
2032
.•;.,,,.-.±(_lV,,.^.-.;(l.,^.,).^^^ ^^^^ r .r'-(lopr..-)'V^
J .r'dtl J .r-" — 2.t"cos/j^+l
INTEGRATION OF ^ .v'" {cLV-\-b.v"y^ (Lr.
2035 Hulk I. — When "^"' /s a positive integer, integrate
n
1
% sh6s/ i7 ;/ / /^?.v z = (a + bx") "^ . Th us
Expand the binomial, and integrate the separate terms by (1922).
2036 But if the positive integer be 1, the integral is known
at sight, since ra then becomes = v — \.
2037 r.uLE II.— When I^+i + J^- is a negative integer,
^ n q
substitute z = (ax-' + b)". Thus
Expand and integrate as before.
2038 But, if the negative integer be -1, tlie integral is
found immediately by writing it in the form
/• "P p r V
= _— i— -(..«- + ^0^^'-
nn{pJrq)
332 INTEGRAL CALCULUS.
Examples.
2039 To find J.c*(l + «*)*cb. Here m = \, « = i, p = 2, g = 3, '-^ = 3,
a positive integer. Therefore, substituting y =. {\ -\- x^ )^ , x = {tf — \f,
Xy = 6?/" (i/'— 1), and the integral becomes
the value of which can be found immediately Vjy expanding and integrating
the separate terms.
2040 [ '^-^ (« + fe-0* ^^^ = ^1 (a + hx*)^.
For '-^^^ = 1 (2036) ; that is, m + l= n, and the factor x^ is the derivative
n
of lx\
2041 I '^^ "^ Z'"'^' cZ.r, or [ X-'- (l + x^f^ dx. Here m = -l,n = ^,p = 2,
q = 3, '''^^tJ: 4- J^ = _ 2, a negative integer. Therefore, substitute
n q
y = (a;-* + l)% X = (i/'-l)"', a-y = -6y^ (z/'-l)'"- Writing the integral in
the form below, and then substituting the values, we have
Sx-'(x-i + l)^x,dy = -6^y'(y'-l)chj,
which can be integrated at once.
2042 f , t; M =[-■'(" + ^-") -' dx. Here 'ii±l + ^ = - 1 ;
therefore, by (2038), the integral
= I .«-"-' (ax~" + b)-' dx = -~ log(a.c-"-f?0-
REDUCTION OF J .r'" (a+6cr")^ dr.
When neither of the conditions in (2035, 2037) are ful-
filled, the integral may be reduced by any of the six following
rules, so as to alter the indices m and p, those indices having
any algebraic values.
2043 I. To change m and p into m-fn and p — 1.
Integrate by Parts^ Jx'^dx.
2044 n. To change m and p into m — n and p + 1.
Integrate hy Parts, Jx""' (a + bx")'nlx.
2045 ni. To change m into m + n.
Addl to ^. Then integrate hy V<irfs, jx™dx; and al^o
by Dii'isiov ^ ajid crjunfp Ifir yrsii/ls.
REDUCTION OF j u:'" {a + Lc")" dx.
333
2046 IV. '/'<' rhini>j<^ 111 ////.' 111-n.
Add 1 to p, and siihfnict n from ra. Thrri integrate by
Parts, fx"'ilx; and also hi/ ])lrif<ion, and equate the results.
2047 y- '/'-' rhaiKje p Into \)-\-\.
Add 1 to p. 7'Ar/i, iufojrnf,- Inj Division, and the new
integral bij rarfs, (x''-^ (a+"bx")".
2048 VI. To change ^ into p-1.
Integrate hg Division, and the new Integral hij Parts,
J x"-^(a + bx")".
2049
Mnemonic Table for the same Rules.
I.
II.
III.
IV.
V.
VI.
111 + 11,
p-1
Bij Parts (m).
m — n,
p+1
]}>/ Farts (p).
ra-l-n
(p + 1), Parts (m) and Division.
in — u
(p+1, m — n), Farts (m) and Divisio7i.
P + 1
(p + 1), Division, and the new integral by Parts (p).
p-1
Division, and the nev) integral by Parts (p).
By applying the rules, Formulce of reduction are obtained.
Thus, any of the six values below may be substituted for the
integral ^v'" {a-]-kv''y (Lv.
2050-2055
I.
[a-^bA'")''-' d.v.
■ r"'-"^'(V/ + ^.>'")-" _ »n-n±l f ,.-.^,,_^^,,..^,-i ,/.,.
jjj . v"-Hn + b,v"Y^^ _ bin^ + n^nn+\) T ,., n^^^^f^^y ,,,,
a(ni-\-\) a{tn-\-l) J
^^. .r"'-'"'u/ + A^")^'^' _ a(n,-n^\) T ,...-«(,, + /,.,,'.);- ,/.,,
b {nt-\-njt+\) /m>" + "/'+1)»
^. _ . ^-"(^^4-^^"r" . n, + n+np+] ( y {a + hv'r'^ (Lv.
«/<(;>+ 1) (i>i(p-\-i) J
Yl. • r'"'Ua^br'y' nnn y,. [a +by')'-^ d,v.
in-\-)iir'{\ ni~\-njt't I »
334 INTEGRAL CALCULUS.
Examples.
2056 To find f ^'^p'^^ dx. Apply Rule I. or Formula I. ; thus
^x-'(a'-x-)hlx = -lx-\a'-x'y- + l ^x'' (ar-or)-^ dx (1927).
2057 To find I — -dx. Apply Rule II. or Formula II. ; thus
^x' (a'-x'yUx = a^a'-x'r'-S ^ x' (a'-x'r^ dx (1934).
2058 fcosec"'^fW. Substituting sin = a;, the integral becomes
j" siu-'»0 ^ dx = { .«-'" (1 -x-)-^- dx.
Apply Rule HI. ; thus, increasing j^ by 1 and integrating, first by Parts
\x'"' dx, and again by Division ;
{ x-'-(l-^)idx = '''"" }^ -'^^- + -^^ {x'-"(l-x-)-^dx,
J i — m 1 — 9/i J
[ .c-"' il-x')hlx = {x-"'(l-x')-Ux- {x'-'"(l-x"-)-idx.
Equating the results, we obtain
2059 j*-(i-«r'A. = fr^.(i--')'+ fES |-=^-a-''')-'<i-
By repeating the process, the integral is made to depend finally upon
{x-'(l-x'y^dx or [(1-xT^dx,
according as m is an odd or even integer (1927, "29).
2060 '\sm'"ddd is found in a similar manner by Rule IV. The integral
to be evaluated is ^x"' {l—x^)~' dx ; and the integral operated upon is
Ja-m-z (i_a;2)^ dx. Otherwise apply Formula IV. See also (1954).
2061 To find ., '^ ,. . Apply Rule V. p = — r, and increasing p by
J (y;--|- a')
1, we have, first, by Division,
[ (,r + ((-)'-'• dx = { ,r (..- + a')-" dx + a- j (.r + «') -'' dx.
Integrating the new form by Parts, j x (x'^ + a")-'' dx, we next obtain
Substituting this value in the previous equation, we have, finally,
2062
C d.r ^ .V , 2r-a i d,r
REDUCTION OF (sin"* cos" OdB. 335
This e(|uation is i/wvu at oiiro ])y Foriimla \'. Thus /• is
clmncred into r-1, and by ropeatiii^r the process of ivduction,
the original integral is ultimately made to depend upon (1935)
for its value if r be an integer.
Another formida Un- this integral is
20^3 \ ^Tq:^. - 1.2... 0-1) W^'\, iS , /3 '•
Prook.— Write ft for a" in (10;!")), and differentiate the eqniilion /• - 1
times for /> by the principle in ('JJ.")5).
2064 To find j' (a" + .r )''%/./■. Apply Rule V[. By Division, we have
The last integral, by Parts, becomes
f ,r (a-+/-)^'^-\lr = 1 .. («r + /-)''" - -^ [ (a^^'^;*"-
J 'i " J
Substituting this value in the previous equation, we obtain
2065 |(«=+..-^)'« </..• = ■ '•'';;+f"' + ,-;:fi j"(«H..=)'"- rf.'-.
a result given at once by Formula VI.
If n be an odd integer, we arrive, finally, by successive
reduction in this manner, at \ {(i'^-\-,r)^ d^v (1031).
2066 The integral j'sin'"0 cos^Oc/0 is reducible by the fore-
going Rules I. to VI., if, in applying them, n he ahrays put
equal to 2; if p be changed info p±2 iuiitead of p±l; nnd
if Division he always effected by separating the factor
cos-e = l-sin^e.
Proof. Jsin" cosPOf/fl = J x"'(l-a:')* "-' (/.r, where .>• = sin fl. Thus
n = 2 always, and the index i (jJ — 1) is increased by 1 by adding 2 to p.
Thus, Kule I. gives the formida of reduction
2067
.' )u-\- 1 )n-\-\ »'
336 INTEGRAL CALCULUS.
But the integral can be found by substitution in the fol-
lowing cases : —
If /• be a positive integer,
2068 1 cos-''^^ct^siu''crf/cr = yl—zY^^dz, where z = sin X.
2069 I siir''^\r cos^'ct'^Ai' = — \ {\—z-Y^^fi^, where z = cosx.
If m -{-}:) = -2r,
2070 \ sin'"cr cos'\i'^/cr = \ {l-\-z'y'''^"'(f^, where z^tarix.
FUNCTIONS OF a + b.v±Cd'\
The seven following integrals are found either by writing
2071 a + bx + ex' = { {2cx + by + Uc -W\^ 4^c,
and substituting 2cx + 6 ; or by writing
2072 a + bx-cx" = {A.ac + b'-{2cx-bY] -r 4c,
and substituting 2cx — b.
om'i ( ^-^' - ^ ^ 2c.r+6-v/(6^-4ac)
2 - _i 2rcr+6
O^ -771 Tn '^" —771 7n>
according as // > or < 4ac (2071, 1935-6).
2074 r ^^-^^ - 1 log v/(6--+4..) + (2..r-6)
(2072, 1937)
2075
2076 ^^r-J^ :r, = 4- ''^i"-' :7^rv
(2071-2, 1928-9)
FFXCTIONS OF a + bx ± ex}. 83 7
2077 f X '{n-^hr-VcA^) (iv = .lr-5 I' ,/(//H 1^/^—^^) 'In-
2078 j\/(<f+/>,r-r.r'-') dv = ^c"^ \ ^/{lur-^/r-jr) fh/,
wliere y = 2cx + b. The integral^ are given at (1931-3).
2079
^ _ O^.P-irn-^ I' 'IlL
J {(i-{-Li+i\r)" J {ir-\-luc — h')"
[By (-2071), tlio integral being reduced by (20G2-3).
2080 f /^+'"j%
I i {'2('A'+b)(Lr ( bl\ C <h'
~ 2c J {a+Lv+c.ry "^ V''' 2c/ J {a-\-Ou'-\-c.r)"'
The value of the second integral is {a-\-hx-\-rx"y-'' -r- (1— p),
unless p = l, when the value is log [a -^ bx -\- cx^) . For tho
third, see (2070).
Method. — Decompose into two fractions, making tlic numerator of the
first 2rx + b ; that is, the derivative of a + hx + cai\
2081 I — ^^'^ '^ — r/,r may be integrated as follows : —
J (i-\-h.r--\-Cd'^
I. If /r > 4>ur, put « and jS for z±±jyi^z:^:!l^ ^ ^^^^^
by Partial Fractions, the integral is resolved into
c{a-fi) U .t--a J .r-fi S
y (1036)
II. If b' < 4rtr, put ""- = n' and ^^(''') ^' = nr, and tho
integral may be decomposed into
2082 ^ \ i* fa-;>>0^r+</m ^^ ^, _ r (^ -j^^7- ^,, ^ ,
the value of Avhich is found by (2080).
III. If h' = 4ar,
2083 f ~ = — V-.+ \ .>n-|>JA). (2062)
338 INTEGBAL CALCULUS.
2084 (—£JL}-—--—^
^^^^ C y-(Lr 2a ( 1 , _,/ /h\ .v )
2085 J ,+,,.-.+e..^ -T-l7(2^)^"^ \-V2;i)-2;r+^-j
REDUCTION OF J a?'" (a + ?Ai''' + cx^'^' dx.
Note. — In the following Fnrmulce nf Iiedudiov, for the sake of clearness,
iv"'(a + hx" + cx'-"y is denoted by (m,p), and the integral merely by j (w, p).
2086 (m^-l)Jo>^.7>) = 0^*+l,7>)
— 6mj9 I (7n-{-n,2) — l) — 2cnp I (m+2»,;> — 1) (1).
2087 h7i (;> + !) f (m,i>) = (m-» + l,7>+l)
2088 2c>j (;> + l) f (m,7>) = (m-2>i + l,;) + l)
-(m-2« + l) j (m-2w,;)+l)-6/i(i)+l)J (m-»i,7?)...(3).
2089 (m+//7; + l) J (m, />) = (m+1, 7>)
+ a«7) i {m, j) — l) — cnp ^ (»j + 2>j,;> — 1) (4).
2090 (m + 2;i/> + l) j 0», p) = {m-^l,p)
-{-2anp \ {m,p — l)-\-bnp I {m-\-n, p — 1) (5).
2091 ^m + ;,y>+l) f(m, />) = (»*-;* + !, />+!)
^ ^ ('>)■
2092 /»'(/> + !) |'(m,/>)=-(m-;^ + l,/>+l)
^ (0-
-REDUCTION OF f .f'* (a + ^f^-h wr')" ^^'*''- ^'^^
2093 rN{p-\-\) \ {m,p) = im-'2n^\,l>-\-\)
-\-an(i>-]-\) \ [)n-'2n,i>)-{in-^nj>-n + \) \i in-'2n, p-\-l)
2094 an (/>+l) \ (m, p) = -(m + 1, ;>+l)
2095 -V;/(/>+l) ((/>», ;>)=-(//< + !, /> + l)
^ (!<')»
2096 «(»' + !) f (>",/>) = (^>» + l,7>+l)
--b{n,-^)^p+)l-]-^)\(m-\-n,p)-c(m+'2np-^'2il^l)\{)n + '2n,p)
^ (11).
2097 c{in + '2up-{-l) ^{m,p) = (m-2n-^l, p-\-l)
— b()u-\-)ip-)f^}) \ iin — n,p)—(i{m — 2ii-\-l) \ {)n — 2n,p)
^_ ' (1^).
Proof. — By diftereutiatiun, we have
2098
Formulw (1), (2), and (3) are obtained from this equation by altering the
indices m and p, so that each integral on the right, in turn, becomes j {))i,p).
Again, by division,
2099 J('«.i') = a\l(in,p-l) + hj^(m + v,p-l)+cyni + 2n,p-l)...iA).
And, by changing m into hi — )i, and j) intnp + l,
2100 JO"-", i' + l) = "3 (»i-«,iO + '\l"(»'.i') + ^. ('"+". iO (.^)-
Formula! (1) to (12) may now bo found as follows: —
(4), by eliminating j(j» + n, p — 1) between (1) and (A);
(5), by eliminating ((//I +2m, p — 1) between (1) and (A);
(6), by eliminating 3 (m-H, j)-hl) between (2) and (B) ;
(7), by eliminating J (ju + », j>) between (2) and 0');
340 INTEGRAL CALCULUS.
(8), from (4), by clianging in into rn — 2n, and p intop + 1 ;
(9), from (4), by changing p into p + 1 ;
(10), from (5), by clianging p into p + l;
(11), from (6), by changing on into yn + n;
(12), from (G), by changing ?» into vi — n.
If o and /3 are real roots of the quadratic equation
a-\-bx" + c\r" = 0, then, by Partial Fractions,
2101 f -"::;-■ .„. ^ i^K4^_f£^{,
J «-f /At'"+ect-" e{a—/3) iJ cV" — a J a'—p )
and the integrals are obtained by (2021-2).
But, if the roots are imaginary,
where cos?i0 = — -— -^^ — r and z = l—Y''x.
2 Viae) \a/
2103 f- f"'^^^'^ , is reduced to (2079-80) by (2097).
^^^^ J (.,^+/Ov(«V^'cr+e.t^) =~J y(-44-%+Cy)'^"^'^^
where y rrr (.i^ + /;)-i, ^r=c, B = h-2ch, C = a-hh+ch\
2105 f ^^£-=^ = --l-cos-'i±^=-i=cosh-I±$.
2106 f ^^iL__= l_sin-'l±^. [By (2181).
Method.— Substitute (.i; + /0"\ as in (2101). Observe the cases in which
A=l.
'^^"' J (.r+/ov(«+^>''+^'0 ~ ^' V u+7i//+(y)'
with tlie same values for A, B, C, and y as in (2104). The
integral is reduced by (2097).
91 OR r ( ^'■+»') '^■''
INTEGRATION BY RATIONALIZATION. 341
:Mktiiod.— Substitute 1) liy puttint,' .>■ = p tan (0 + y), and (lotcrrninc tlio
constant y by equating to x-cru the coenicicnt of sin 2« in the denominator.
The rcsu
.. . , ,. „,, f. f // cos « -f iV sin ,
Iting mtcgi-al is of the (orm . ,, — tT"''"-
Separate this into two terms, and integrate by substituting sin in tlie first
anil COS y in the second.
2109 C__±u)jrr
wlicrc «/)(,'') mid F{,r) arc rational alo:('l)raic functions of .t,
the former being of tlie lowest dimensions.
Method. — Resolve ^, '^ into partial fractions. The resulting integrals
are either of the form (2107), or oIfs they arise from a pair of imaginary roots
of F(x) = 0, and are of the type f .-V^'-'l // '^"l! 7 _l, n ' Substitute
a- — o in this, and the integral (21U8) is obtained.
INTEGRATION BY RATIONALIZATION.
In the following articles, F denotes a rational algebraic
function. In each case, an integral involving an irrational
function of x is, by substitution, made to take the form
]' F (z) d::. This latter integral can always be found by the
method of Partial Fractions (1915).
2110 f;.'|„..,(_^:f,(5+|i;)Uc.( </..-.
Substitute v = '', where / is the least common de-
nominator of the fractional indices ; thus,
h-fjz" dz~ (b-fjz'y ' \f+gxl
the powers of z being now all integral.
2111 j>'^1..->(5^)M5$^)-.^^-('^'-
Reduce to the form of (2110) ])y substituting .r\
342 INTEGRAL CALCULUS.
2112 ^ F{^(a + \/nLV-i-n)} ilv. Subs, ^{a-^^/mx-\-n).
2113 (V{.r, y/{hx±c.v~)\(lv. Substitute x=^-^.
J z -\-c
AncULereforc y(^x ± c.^ = ^, g = _ _^^.
2114 f ^ Iv, ^{a-\-hv+c.v')} dx.
Writing Q for a-\-hx-\-cx^, F may always be reduced to
tlie form ^ + ^^^ , in wHcli ^, 5, 0, D are constants or
C-\-V\/ Q
rational functions of x. Eationalizing tliis fraction, it
takes tlie form L-\-M \/Q. Thus the integral becomes
\ Ldx-\-\ M\/Qdx, tlie first of wliicli two integrals is
J J . . r MQ
rational, wliile the second is equivalent to —~z dx, wMch
is of the form in (2075). ^ ^
2115 Otherwise. — (i.) When c is positive, tlie integral may be made
rational by substituting
a — cz'
dx _ 2c (hz-cz^-a) y(^ + j., + ,.,.) ^ y, {B^ + ''
2cz-b' dz (2cz-hy ' " ' ' V2cz-h
(ii.) When c is negative, let a, /3 be the roots of the equation
a+l>x — M^=0, which are necessarily real (a, b, and c being now all
positive), so that a + hx — cx' = c (x — a) (fi—x). The integral is now made
rational by substituting
In each case the result is of the form j F(z) dz.
2116 f .r"'F lv\ Va-\-bA--{-c.r"} dv,
when IS
substituting x^
when ^' is an integer, is reduced to the form (2114) by
2117
f r [.r, y^{a^h,), , /(/+ -.r)} r/.r. Substitute r} =
a + hx
TNTEOUALS EEDUCIBLE TO ELLIPTIC INTECHiALS. 343
and, therefore, .r = ., ' . , V{"-\-'>-^) — — 77"'-'" /\»
The form J 7^ [", ^(f/.r-//)] (/.v is obtained, Avlilch is
compreliended iu (2114).
when ^^^ is an integer, is rednced to tlic form J /•' (v) (h:
by substitnting ;: = /u-" + \/{a + h'\c-"), and therefore
2119 \ d"' (fi + b.r") '' F (,r") iLr is rationahzed by snbstituting
either (<i-\-Li:"Y or (ax-"-\-hY according as or "- + ^
is integral, whether positive or negative.
2120 1 d^''-^ F [d''\ .r", {(i-^1hv")'^\ (Lr, when — is eitlicr a
positive or negative integer, is rationalized by substituting
(rt + ia,'")".
INTEGEALS REDUCIBLE TO ELLIPTIC INTEGRALS.
2121 f F{.r, ^{a-{-b.v-\-Cd''+(Lv'-\-r.r^)\ fLr.
Writing A fur the quartic, the rational function /■' may
always be brought to the form \.^ y ,
and this again, by rationalizing the denominator, to the form
344 INTEGRAL CALCULUS.
M-\-N\/X, where P, Q, P\ Q\ M, N are all rational functions
of X. ^ Mdx has already been considered (1915).
(n^X(Lv = f-^ civ or f^^\ where B is rational.
By substituting^ a^ = ?l±Vi/^ and determining j; and q' so
•^ ° l + ;y
that the odd powers of // in the denominator may vanish, the
last integral is brought to the form
B being a rational function of ?/, may be expressed as the
sum of an odd and an even function; thus the integral is
equivalent to the two
91 oq (• ?/F.(.v^)rfy r_F^{!rUhi__
^^'^'^ J y(«+6r+<-/) "^ J V(''+*/+<-/)'
The first integral can be found by substituting \/y.
The second, by substituting f^ foi
depend upon three integrals of the forms
2124 f ,, ^"; ,, ., . fv^lES^^
The second, by substituting /., for 7/^ can be made to
J {\-^nar)Vl—M'.\—lraj
By substituting (/> = sin ^x, the above become
2125 r ,, '^^. ,^, fx/l-A:^sm^^#,
«^ Vl — ^- sm"9 •^
(• H
J (1 -{-n siir^) v^l— /r sin-<^
These are the transcendental functions known as Elliptic
Integrals. They are denoted respectively by
2126 y{.K<i>), E{k,<t>), li («,/.-,</>).
TNTEGT?ATS nEDUCIBLE TO ELLIPTIC INTEGIiALS. 345
APPHOXIMATIONS TO F{1-, ^>) AND A' (/•, ^,) IN SEKIES.
AVIuMi /• is loss lltiiii unity, llio Valiums of /''(/.-,«/>) mid
E{Ji, tp), from the oriL;-in <s> = <>, in converging series, aro
2127 /'(A%<^) = </>-| t.+ ^^.-^445^''+
2128 /<: (A-, « = </>+ J ^1^-2^. ^'+ iTi:^^ -^'■'-•••
+ (^^ 2.1.0.../* 2^^ '*'" '^^•'
^ ^^^ . .^ [vi being an even integer.
. sill \6 i sill 2(f) , ., ,
. sill 0(i (I sill If/) , 15sin2<i i^jl
. _ siiw/(/> ;/ siii(;/ — 2) j > C (/?, 2) sin (» — i) </>
n n — 1 n — h
_^(;^;{)sin(»-G)<^ _^(_^).„,^.(.^^^-,^,)^^
Proof. — In each case expand by the Binomial Theorem; substitute from
(773) for the powers of siu^, and integrate the separate terms.
The values of F(l',cp) and E{Jc,(}>), between the limits
^ = 0, ^ = ^TT, are therefore
2129
2130
But series Avhich converge more rapidly aro
2131
2 Y
)
346 INTEGRAL CALCULUS.
2132
l-^/(l-A-)
0133 r ^ (-'^) ^^'^^ _, ^Yhen i^(.') can be ex-
pressed in tlie form (x ;)fi'^+-T)' ^^ integrated by
substituting x-^—.
If h is negative, and i^(r(,) of tlie form U-\- ^ )/ (^'— ^j 5
substitute x
x
2134 ^(•'■)"'-
y(a+i»u'+cu-4-^At''+fu^'+6cr'+«a''')'"
Substitute <r+ — =
Hence -^^ = I =t- — , ) ^Z^,
x/cf' "" V2 V ^ 2 2 y ;H^ ^
and the integral takes the form
J ^ {«(:s^-;k)+Z' (;s'-2)+c;^+^/} " 2v/^--4
where P, Q are rational functions of z. Writing Z for the
cubic in z, we see that the integral depends upon
f_il±_ and f^|±^:,
the radicals in which contain no higher power of z than the
fourth. Tlie integrals therefore fall under (2121).
2135 ( ^^■•■^"
Expressing F {x) as the sum of an odd and an even func-
tion, as in (2i23), the integral is divided into two; and, by
substituting .r, the first of these is reduced to the form in
(2121), and the second to the form in (2134) with a = 0.
iNTEcnALS ni'DrrrniE to eluptjc iXTEaRMs. 317
2136 iV
r i.v) fh'
Put .v = i/-\-a, oheino-ii root of tliooquation a + //,»- -f nr + (/,(''= 0;
niul, in tiie rcsulliiiu- iiitooral, siibstituto ;:i/ for the duuomiiuitor.
The form lliiallv obtained will bo
f(// + r»>\/a + y8^V/-
wliicli falls under (i!!:^), /'and Q bein.^r ration,.! functions
of ^.
2137 r IMjLH .
Expressinir F(.r) as tlie sum of an odd and an oven func-
tion, as in ('2123), two integrals are obtained. By puttin.cr
the denominator eciual to z in the first, and equal to .cz in the
second, each is reducible to an integral of the form
which falls under (2121).
l+.f' 2 \\ :
PhOok.— Substitute cos'x in (21.38), iuul 'Jtau-'.r in (l!l:'.0).
2138 j ^ J_ ,,^ =-^2^' (72' V- ^'''''""' ''^' ^"^''
2139 f_^, = i/.-(-L,.^
2140 ,, ,, ,
i' ill- i ,./ rill ,\ l-i. ri ' '' k\
according as // is > or < 2*/.
Pkoof. — Substitute aceurdiu^jly, x =■ 'la siii^ ^ or .v. = b siu'^.
Pkoof.— Substitute .r = a-(a-6) 8in->, * being < a and >h.
343 INTEOBAL CALCULUS.
SUCCESSIVE INTEGRATIOX
2148 In conformity with the notation of (1487), let tlie
operation of integrating a function v, once, twice, ... n times
for X, be denoted either by
\v, \ V, ...\ V, or by rf.^-, d_.2x, ••• (^-nx^
«, X J 2.C «. nx
the notation d_^ indicating an operation which is the inverse
of 4. Similarly, since y^., y..^, y^.r, &c. denote successive
derivatives of y, so y_^, 7/_2«., y^x^ &c. may be taken to repre-
sent the successive integrals of y wdth respect to x.
2149 Since a constant is added to the result of each in-
tegration, every integral of the n*^' order of a function of a
single variable 'x must be supplemented by the quantity
^ + ?kl!_=+...+«,._.r+«,.= f 0,
\7i — l \ n — 2 Jnx
where a^, flg? (h ••• '^n are arbitrary constants.
Examples.
The six following integrals are obtained from (1922) and
(1923).
When p is any positive quantity,
When p is any positive quantity not an integer, or any
positive integer gi^eater than oi,
2151 f J- (-J)" I f
When 7) is a positive integer not greater than n , the fol-
lowing cases occur —
2152 f — = ^"-"^T' log.r+f 0.
2153 f i = tiili:(.,iog.r-uO + f 0,
J(y'+l).i-.*'' L/^~i J(p+l)x
SUCCESSIVE INTEGHATION. 310
2154 (• -i=^^'{f (..■i..,..)-^}+r •••
lor the integral witliiu the brackets, see (21 GO).
The following foriuula is analogous to (llGl-2)
2155 tt]"..=J.tl('"-^-)+j>
SUCCESSIVE INTEGRATION OF A PRODUCT.
Leibnitz's Theorem (14(;0) and its analogue in the Integral
Calculus are briefly expressed by the two eipiations
2157 />«x(«0 = (^/r+8,-)"''is i>-«.r("'') = (^/., +8,)-" ffr ;
where D operates upon the product ur, d only upon u, and S
only upon v. Expanding the binomials, we get
2159 J)Jifr)=:i(„,i' +«W(«.i),r, + " ^J~ ^/ („-■■:) x^:. +&C.
Proof. — The first equation is obtained in (14G0). The second follows
from the first by the operative law (1488) ; or it may be proved by Induction,
independently, as follows —
AVritiiig it in the equivalent form
f (uv) = f uv-n f ».,+ 'iI;^ f .a,.-&c (i.),
(,/(■)= pry— nv^+\ nv.^ — &c (ii.),
a result which may be obtained directly by integrating; the left member suc-
cessively by Parts. Now integi-ate equation (i.) once more for .r, integrating
each term on the right as a product by formula (ii.), and eiiuatiou (i.) will
be reproduced with ('i + I) in the place of?;.
2161 f e"''.v"'= c'''-(a-\-(i.)-\v'"-{-\ 0. Or, by expansion,
2162
]nx a" L a 1.2 a* ) ] nx
■ If m be au integer, tiie series tcrminuleb with ( — l)*" ?i "" -i- a".
ikc u = 1 ; til
350 INTEOBAL CALCULUS.
Siniilarlj^, by cliangiug the sign of m,
2163
f e^ _ e^ ( _1_ , nm n(n + l) in(m + l) , ^^ 7 , f q
]„x a;'» ~ a" la;'»'^aa)""i'^ 1.2 aV'-^ ' ') J«x '
Proof. — Putting u = e""", v = x"' in (2158), tLe formula becomes
J Ha:
Here e"^ is written before ^^ within the brackets, because ^ does not
operate upon e"^. Observe, also, that the index —n affects only the opera-
tive symbols d^ and ^j., but it therefore affects the results of those operations.
Thus, since d^e"^ produces ae"-^', the operation d^. is equivalent to aX, and is
retained within the brackets, while the subject e""", being only now connected
as a factor with each term in the expansion of (a + ^a,)"", may be placed on
the left.
2164 P-."V/. = ^ f,.»-^,...->+!il^) ...»-.c.(
Proof.— Make w = l in (21G2) and (21G3).
2166 f cr''(log.r)'" = f e^^-*-"^•^^•'^ [Sabs. log...
J nx «. n.v
Hence the integral of tlie logaritlimic function may be
obtained from tliat of the equivalent exponential function
(2161).
For another method, see (2003-5).
HYPERBOLIC FUNCTIONS.
2180 DE^INlTI0^•s. — The hyperbolic cosine, sine, and tan-
gent are written and defined as follows : —
2181 cosh .V = }j {e'--\-c-') = cos (/.r). (7G8)
2183 siub .V = I {v' — c-') = -/ «iu (/'.<).
2185 ianh.r^: '—^ =.-iii\n{h). (770)
6'' 4" <-'
TTYrElUJOLIC FUNCTIONS.
351
"By tlicsc equations the followinu^ relations are readily
obtained.
2187 0()>li 0=1; siiili = 0; cosli x = siiih j: = r. .
2191 eosh-.r — sin]r.r= 1.
2192 S'ii»l» ('^H-//) = siuh.i' eosh//4-cosli.r siiih//.
2193 cosh (t -f//) = ooshii' cosh ;/+siiihcf siiih//.
2194 tanh(,r4-//)-
fanh.rH-tnnh//
l-ftaiihct' taiihy*
2195 ^iii^^ -•^' = - siiili '^' cosh .i\
2196 cosh 2.1' = cosh'.r+sinh'cr.
2197 = 2cosh'-.r-l = l + 2sinh-.r
2199 siiih ar = o siuh .1+4 siiih'' .r.
2200 coshiU = 1 cosh\t -3 cosh .r.
2201 tanh2.r =
2tanh.r
1+taulr.f
2202 tanli.J,r= — , , ., ' . , •
1 4". J taiili.r
2203
2208
. 1 .?' /cosh.r — 1 1 .r /coshr+l
siiih — = y ry ; cosh— = \j
2205 taidi— = W j —r = — r-j = -, — j-T.
2 vcosh.r+l smli.r cosh.r+l
2 tanh Iv
cosh .r =
■+
l+lanlr^r
l-lanlr.'„r
^iuli
I— taiilr .\.t''
INVERSE RELATIOXS.
2210 T.et u = cosh.r, /. .r = cosl|-' t( = lo^r (,/_|_ ^/„-i_l).
2211 r = sinh ,r, /. .r= sinlr^ r = h)g (r+ \/rM- I)-
l+»c^
2212
7r = ianli,r, .'. .r = tanlr^c= .lloirf _ )•
352 INTEGRAL CALCULUS.
GEOMETRICAL INTERPRETATION OF tanh /S^.
2213 2V<(^ tangent of the angle which a radius from the centre
of a rectangular hyperbola snakes with the principal axis, is
equal to the hypcrholic tangent of the included area.
Proof. — Let 6 be the angle, r the radius, and S tbe area, in the hyperbola
i>
isec20cW = ilogtan(i7r + 0). (1942)
Therefore e^« = I±li^ ; therefore tan = ^^^-^ = tanh 8. (2185)
1 — tan y e -re
VALUE OF THE LOGARITHM OF AN IMAGINARY QUANTITY.
2214 log {a+ib) = ilog {a-+b')-^iimi-^^^.
Proof.- log ."t^",.,, = log J ^ ^ [ = i tan"^ A. By (771).
/ 1+^-
, a -\-%b , / a
DEFINITE INTEGRALS.
SUMJklATION OP SERIES BY DEFINITE INTEGRALS.
2230 (/Gr)f/.t>= [/(.OV(«+^^^0+-.-+/(«+^*^/aO]^^^^^
where n increases and dx diminislics indefinitely, so tliat
qidx = h — a in the limit.
2231 Ex, 1. — To find the sum, when n is infinite, of the series
1 + _.l. + JL + _1_. Put >i = -^; thus,
n n + 1 n + 2 n + n dx
i£ + _^£_+ ^--^ + + -'- = r— = log2.
a a + dx a + 2dx 2a J„ x
2232 Ex. 2.— To find the sum, when n is infinite, of the series
^ + ^ + '' + + —!L- Put « = ^, then
wH 1' »i' + 2* «"^ + 3- »-' + 'i' f^-«
^?a> . da: , . ^^'^^ = f ^''L = 5.. (1935)
l + (Jx)''^l + (2(?,ry^^ ^l + (>a/.0=' 1^+'^' 4
TnEOn TIMS R ESVECTING LIMITS OF INTEQRA TION. .'5 5 ;i
THEOREMS RKSPECTIXa THE LIMITS OF
INTEGRATION.
2233
I "^ (.,) (hv = fV (^'-'^O '''»'• [Substitute a-z.
2234 f>(,r)r/.r = 2fV(.r)r/.
or ::t'ro, according as '/>('') = i '/'("—'*') ^^^' '^^^ values of aj
between and a.
Ex.— [ smxdx = 2 ['
*'o Jo
dx = 0.
If ,/,(,r) = </)(—. r), that is, if <|.(.r) be au even function
(1401) for all values of x between and a.
2236 ("" <t> (■'■) ^/'^' = fVO*') '^' = T i ' ^ ('"^ ^'^'■-
J -a Jo ^J-a
Ex.— „ cos .i'f/x = '^ cos .i;rfa! = ., T „cos a;(7j.
J~2 "" " ~2
If <^(,,.) = _^(_a>), that is, if </>(.r) be an odd function
for all values of ,/' between and a.
2238 r */>(•*■) 'f'' = -( V W <^'^' a^^^"^ i *'*(''') ^^''' = ^^-
J -a * •- -a
Ex.— j „ sin X d.r = — r «iii •«' '?•*' f^n^ "^ „ ^'" •'' '■^■'' ~ ^^
J ~ i *' "^ ~ ■>
Given ^/<c<^, and that x = c makes */>('•) infinite, the
value of [<p{x)dx may bo investigated by putting /t = 0,
after integrating, in the formula
2240 \ V ('*■) <^''' = r "^^^ ^-''^ ^^"''+ i *^ *^''^ ^'''''
If the function rp{r) elianges sio-n on b.'ccnung iniinito, tliis
expression, when ^ is an indelinitely small (puiutity, is called
the j>ri'»c/j>a/ value of the integral.
2 z
354 INTEGRAL CALCULUS.
which is the principal value. If, however, n be made to vanish, the expres-
sion takes the indeterminate form oo — co .
2241 Given a < c < 6, the integral I ' —3^' will always be
finite in value while n is less than unity.
Ppoof. — Let ^ in (2240) be taken so near to c in value that \// {x) shall
remain finite and of the same sign for all values of x coinpi-ised between
c ± iu. Then the part of the integral in which the fraction becomes infinite,
and which is omitted in (2240), will be equal to ' '^ . „ , multiplied by
a constant whose value lies between the greatest and least values of ;// (x)
which occur between 4' {c — f^) and 4' (c + /")• By integration it appears that
the last integral is finite in value when « is < 1.
2242 \"f{^) d.v = {b-a)f{aJrO{h-a)},
•. a
where 6 lies between and 1 in value.
The equation expresses the fact that the area in (Fig. 1901), bounded by
the curve y =f(x), the ordinates f (a), f{h), and the base h — a\s equal to
the rectangle under I — a and some ordinate lying in value between the
greatest and least which occur in passing from /(a) to f{h).
If ^ (,T.) does not change sign while x varies from x = a to
2243 f/G^') ^ W d^^^ =f{n + e(h-a)} fV Gr) (Iv.
2244 If ^(^j ~) is ^ sj^mmetrical function of x and — ,
|->(,..,i)l^ = 2fV(,,.,i)l^.
PiJOOF. — Separiitc the integral into two parts by the formula = +
,,,.,, 1 ..,,.-. , Jo Jo Ji
and substitute — in the last integral.
METHODS OF EVALUATING DEFINITE INTEGRALS.
2245 Rule I. — fiiih^titute a new rariahle, and acljust the
li/nii ts a ceo rding ly .
For examples, see nvmhers 2201, 2808, 2;342, 231-5, 241G, 2425, 2457,
250G, 2605, &c.
METHODS OF EVALUATISC, DEFINITE INTEGRALS. 355
22-16 Rl-hilU.—liilr./ralr //// rarl,i (lUlO), so as li> intro-
duce a kiLoivu dfjiniic inlfijiuil.
For ej:ai,i2>les, see numbers 2J^:3, 21'J0, 21:30, 2-I-5:i, 2-4Go, 2484-5,
2608-13, 2(32o, 2G25, &c.
224:7 Ivii.i: 111. — D ijjc rent III tc or integrate ivith respect to
i<onie tiiiantltif other than the carlable concerned; if a known
integral is thus obtained, evaluate it, and then reverse the
operation of differentiation or integration before performed
with respect to the secondari/ variable.
For examples, see nnmhers 2346-7, 2364, 2391, 2417, 2421-4, 242G, 2428,
2407-8, 2502-4, 2571, 2575-6, 2591, 2604, 2614, 2617-8, 2632, &c.
2248 IvULE IV. — Sntjstitute iniaginary values for constants,
and thus transform the expression into one capable of inte-
gration.
For examples, see numbers 2430, 2404, 2577, 2504, 25J8, 2603, 2606,
2615, 2641-2.
2249 'Rule V. — Expand the function, if possible, in a finite
or converging series, and integrate the separate terms.
For examples, see numb'^rs 2395-7, 2402-3, 2418-9, 2479, 2506, 2571,
2593, 2598, 2614, 2620, 2625, 2629, 2630-2, 2639.
2250 Rule YI. — Decompose the integral into a numbrr of
partial integrals, and change all these by some substitution
into integrals having the sa)ne limits. Bij summing the
resulting series^ a new integral is obtained which mag be a
known one.
For examples, see numbers 2341, 2356-61, 2572, 2638.
2251 Rule VII. — Separate the function to be iitf<-grard
■into two factors, and replace one of them by its value in the
form of a definite integral taken between constant limits with
respect to some new variable. The doutde integral so obtained
may frequently be evaluated by changing the order of integra-
tion as explained in (22G1).
For examples, see numbers 2507, 2510, 2573, 2619.
2252 Rule VIII. — Mnltiply a known definite integral u-hich
is discontinuous between certain values of a constant which it
356 INTEGRAL CALCULUS.
contains, hy some function of that constant, sucih that the
integral of the j^t'oduct with respect to the constant is hnoiun.
A new definite integral may thus he obtained.
For exaviples, see ntnnhers 2518, 2522.
Particular artifices uot included in the foregoing rules are employed in
2293, 23U5, 2310, 2314-5, 2317, 2367-9, 2404-15, 2422, 2429, 2456, 2495,
2514, 2518, 2585, 2000, 2626, 2635, 2637.
Additional formulae for integration will be found at 2700, et seq.
DIFFERENTIATION UNDER THE SIGN OF
INTEGRATION.
Let 11 = f(x) dx, where a, b, and f{x) are iudepeudent
J a
of each other ; then
2253 ^=/W and ^ = -/{")■
Proof. — Let u = (j>(b)—f{a).
Therefore % = 0' (&)=/(&) and n„ = —<j>' (a) = —f(a).
rb
Let u = f{x, c) dx. Then, when a and h are inde-
J a
pendent of c,
2255 u, = f {/(.r, c) } (Iv and n„, = f {/(.r , c) } „. d.v.
Phoof.— ^ = j \f{x,c + h) dx- ["/ (x, c) dx I ~ h
= --^ '^— dx (since h is constant relatively to x) = ( ^•''' dx.
But if a and h also are functions of c,
225V ^=j:irq£-)j<..+.m,.)f-/(«..)S.
Proof. — The complete derivative of u with respect to c will now be
u,+ a,b^ + u^a^. But III, =f{b, c) and u„ = —f (a, c), by (2253-4).
ArvnoxuiATE ixTEauAriox. 357
INTKCl RATION BY DTFFKRKNTIATTNd UNDKll THE SIGN OF
INTFCi RATION.
2258 i:x. 1.- \y'>^'-^ir = ^(c'-),,,,.h = j.,.^.-dr (2250)
by (Mr> I), a and .r boiiif,' transposed.
2259 i:^- 2.— {x"c"-^\uh.rd.t = il,Jc"ii{nh.rJx.
The last intet,M-al is given in (1999), putting n= 1.
2260 Ex. 3.-
INTEGRATION UNDER THE SIGN OF INTEGRATION.
When the liiuits are constant,
2261 ("" ^"\nv, y) dvdj, = T" r7(.r, ij) fh/flr.
That is, the order of integration may be changed.
But an exception to this rule occurs wlien, at any stage
of the integration, an infinite value is produced. The double
intesrrals above will not then have the same value.
ArrrxOxiMATE integration.
BERNOULLI'S SERIES.
2262 J/('0^/.^^= ^{/X^O-j^^/X^O + y-^^
Proof. — Integrate successively by Parts, J -/.c, l.c/r, Ac. Or cliange
/'(aj) into/(j;) in (1510).
2263
358 IXTEGRAL CALCULUS.
Proof. — Put/(a) for V)'(tt) iu the expansion of tlic right side of equation
(19U2), by Taylor's theorem (1500) ; viz.,
\^f{x)dx = ^(h)-<^{a) = (6-a)^'(a)+fc|I%"(a)+&c.
The following is a nearer approximation : —
Let {h — a) = nh, where « is an integer ; then
2264 |7W civ = h {lfib)-\-if{a)^f{a+h)+,..Jrf(h-h)}
-7j!rw-r(«)}+&c.
Proof. — Expand (<?" " -^ — 1 ) -4- (e"-'' — 1) by ordinary division, and also
by (15oi)), and opciate upon J\x) with each result; thus, after multiplying
by h, we obtain, by (1520),
h{f(x)+f(x + h)+f(z + 2h) + ...+f(ix + l^^h)}
■which expression, by changing .-« into a and x + n]i, into b, is equivalent to
the above, since d_-^{f(x + nh)—f{.c)} = / (,7.) dx.
2265
Proof. — Assume a; = ce''^. Then a; is equal to the coefficient of —
in the expansion of —log f 1 — — ]. Tliug
^ 2//C-- , 3-//-r' , 4V,V ,
" = ^+ 1:2+ 1:273 + 1:^:3:1. "^-
- /Id .
Substitute d^ for .t, and therefore dj.e •'■ for c in this equation, and operate
with it upon \ f (x + h) dx, employing (1520). Finally, write f(x) for f (x).
and a An- x.
A more general result, obtained iu the same wa^-, is
i*a+nh 7,9.
2266 \ /Or) (Lv = nhf{n-h)-^rn {n^-2)^f {a-2h)
Tin: L\Ti:i!i:.\Ls h{l,i,i) asij V(,i). 350
THE INTEGRALS B (/, m) AND r(w).
EULER'S FIltST INTEGRAL h {!,,„).
Till' tlirco ])fiiici|):il fornix; arc —
2280 I. B{l,nt)= \ \r'-\\-.ry'-'(h'=Ti{m,l). [ny(2233)
2281 II. lUl,m)=( .'!''' / f'^'- [By substituting-^- in I.
Jo (l+.r) " -L •'■
2282 111. lUl,m)=\ -4^1-1— fAl-. [By substituting!—^ in I.
Jo (1+cl') •*■
"Wlieu I aud m arc positive, and / is au iutcgcr,
2283 B (/, w) = i|^.
If rti be the integer, iutercliangc / and ui. If Ijotli / and
m are integers, the forms are convertible.
Proof. — Integrate (2280) by parts, thus,
f .r'-'(l-.^0"'"''^-^'= — f .^•'-'(l-a;)"•.
Jo m Jo
Repeat this step successively.
EULER'S SECOND INTEGRAL T{n).
n being a real and positive quant it v,
2284 y{n) = \\-\i--\Lv = \ '(logiy \Lv.
The second form being obtained by substituting c ' in the first.
2286 r(l) = l, r(2) = l.
2288 I'(7' + l) = >'!'(") =^n{u-\) ... (n-r) V{n-r).
2290 r (>* + l) = ! « , ^vhen n is an integer.
Proof.— By Parts, f Cx^'-'dx = ^1 + — f r^'x^lx.
■^ ' Jo "e'Jo n Jo
The fraction becomes zero at each limit, a~s appears by (1580), dilTerenfiiiiiug
the numerator and denominator, each r times, and taking r>n.
360 INTEOBAL CALCULUS.
2291 Ce-'V-'d., =^ = f ^'- (logi)""'rf.r.
Jo A, Jo \ tl /
Proof. — Substitute l-x in tlie first integral, and so reduce it to the form
(2284). In the second integral, substitute — log.t;, reducing it to the former.
When n is an integer, C2291) may be obtained by differentiating n — l
times for k the equation I e~'''^dx = — .
AYhcn in is an indefinitely great integer,
2293 r{u)= . \;f /, ^ /^^
1 1
Proof. — log— = lim. /u (1— x*^) (1583). Give it this value in (2285),
X 1
and then substitute y = .f^ ; thup, in the limit,
r(n) =/.«-' r(l-.rM)"-'cZx = ^" ['>/-' (l-yy-'dy. Then, by (2283),
changing fi finally into /i + 1 in the fraction.
logr(l+?2) IN A CONVERGING SERIES.
2294 Let n he <1, ^ an indefinitely great integer, and
/Sf, = l+^ + J7+...-^, then
log r (! + »).
2295 ={\ogii-S,)n+lS,n'-lS.y-\-iSy-iS,7i'-\-&c.
2296 = ilog^^^^ + (log/.-.SO n-iS,n^-iS-X-&c.
sill HIT
+ |(l-S.) + |(l-S.) + '!tc.
2298 =i\og-^^-i log I +i' + 1227813«
^aiunir 1 — n
P.OOF.-By(2203), ^ d + n) = ^^^^ ^^;^;^^ ^^^ ^^^ ,
1, when /M = CO . Whence
ii-tfi + i
iogr(i+H)=«/^.-Ki+n)-/(i+ ;^)-z(i+ ;;)-... -/(i+^j.
THE INTEGRALS B(/, m) AND l{ii). 301
Dovelopinf? the lopfarlthms by (155), the series (2295) is obtained. The next
series is deduced from thi.s by substituting
<S>* + -i.V* + iV + i'V + &c. = hifr«7r-lo{,'sin«7r,
a result obtained from (815) by putting = mc and expanding the logarithms
by(15(;).
The series (2207) is deduced from the preceding by adding the expression
= -J- log J-"ti' ^n^—-^^- +«!^c., from (157).
\—n 3 5
2305 B('.'") = M^.
Puoor. — Perform tlic integrations in the double integral
first for .r, by formula (2201), and then for//, by (22^1), and the result is
B (/, m) r (i-\-m). Again perform the integi-ation, first for //, by (2201), and
the result is r(/) T{m), by (2284).
Note. — The double integral may be written by the following rule : —
Write xy for x in r(0, 'i«t^ imdtiplij hij the factors of r(»i-|-l). We
thus obtain [ \ e-^^(.njy-'x-c-\rdxd>/,
jo J I)
which is equivalent to the integral in question.
2306 B(/, m) B(/+m, n) = B(m, «) B(/** + «, /)
= B(>j,/)B(;i + /,»0
2307 =%SW- f^^<-->-
2308 ('\v^-\a-.v)""'(U' = a'^"'-'B{l, w). [Substitute ^.
Jo "'
If j; and q are positive integers, i'<'/, and if m = "^^ — .
2309 rr^. ^^* = ^r-^ —
Jo l+.t' ' -7 siii>/<7r
2310 I V^ (f> = 1-7^ —
Jo l—ar'' 'Iqiiuxmn
Proof.— (i.) In (2023) put / = 2p + ], » = 2-/, and take the value of tho
integral between the limits ±oo. The lirst term becomes log 1 = U ; llio
sec"ond gives tlie series
q L 2q 2q Iq ) qsmniir
by (800). The integral required is one-half of this result, by (2237).
(ii.) (2310) is deduced iu a similar manner from (2021).
3 A
362 iNTEnnAL calculus.
2311 ii:,/,,. = ^j^, fi_,/,.=^L-,
Jo l+.r siiu/<7r Jo 1—0,' taii/>i7r
"wliere m lias ainj value between and 1 .
^p + 1
Proof.— By substituting ;r'^ in (2309-10). Also, since w = =^ — , by
takino" the integers p and q large enough, the fraction may, in the limit, be
made equal to any quantity whatever lying between and 1 in value.
2313 r(m) r(l-»0 = . '^ , m being <1.
siii/><7r
Proof.— Put l+m = 1 in the two values of B (/, w) (228-2) and (2305) ;
thus, r {in) r (l-»0 = f '^ <lr = —"—, by (2311).
Jo l + x s\n iiiTT
2314 CoE.- r(i) = v/'r.
The following is an independent proof:
r(l) = [ e-'\c-^cU = 2 [ e-'-'dij = 2 | er'-dz.
Now form the product of the last two integrals, and change the variables to
r, d by the equations
y = rs{nd\ ^^,^^^ ^^.^^ by (1 009), cluch = 'l['^'-)-drdd = rdrJe. Hence
{r(i)}- = 4l \ e-'" *'''dydz = 4[ \' e-'\drdd = tt;
Jo .'o Jo Jo
the limits for r and being obtained from
r^ = i/ + ::\ tan0 = ^.
(27r)"-^
2315 r(l)r(l)r(|)...r(^l) = V^
Proof. — Multiplv the left side by the same factors in reversed order, and
apply (2313) thus
-^^—, by (814).
. TT . 27r . (l!— n TT
bui sm - ...sin
n n 11
2316 -^, r(,,) r(,,-+i) ... vL+'^ = v^i^^^'.
THE INTEGRALS B (/, ?//) AND T{n). 3G3
Pljoor. — Call tlif cxincssiuii on the Icfl '//(.r). (!liaiij^o x to ./• + r, wlicro
r is niiy iiifc^'tT, mid cliaiifrt' each (iaimiia fuiicti.'ii by ilic forinula
V(.c-\-r) = J-"" V{x) (2'JHS). The result aftor rcilucrtion is (/.(.r). II<-iico
*f>{x) = ^ (.(•-!-/•), liowi'ver great r may be. Tlii-relore </>(.r) is iinlrjimdvul of
a*. But, wlu'U x =: — , 'l'{x) takes the value in ciut-stiuu by ("iolo). There-
fore tp (x) ulwa^-s lias that valui\
The formula may also be obtained by means of (220 I).
NUMKiaCAL CALCULATION OF r(.r).
2317 All values of r(,i') may he found in terms of values
lying hetween r(0) and r(^).
"When X is > J, foi-inula (2280) reduces r(.e) to the vahie
in which x is < 1 ; and wlien x lies between 1 and ^^ formula
(2818) reduces the function to the value in which x lies
between and ^.
Values of T(,r), when x lies between and 1, can also be
made to depend upon values in which x lies between J and J-,
by the formuh\3, . v
, !■(£)
2318 rw = 2'-=VTT.!5^. r(,.') = — ^^^^ f^-
Proof.— To obtain (2:U8), make n = 2 in (2:11 T.). To obtain (2310),
put i»=:|(l+,r) in (-:31o), and change x into ^-c in (2318), and then
clinunatc I" (— 7— J-
Mcfhods of employing the forniuhe —
2320 (i.) When x lies between | and 1, reduce r{x) to
V{l-,v), by (2818).
2321 (ii.) When x lies between -J and I, reduce by (2:}K»),
the limits on the right of which will then be I, and -J.
2322 (iii.) When x hes between (» and ' , rr.hicc by (2818) ;
r(^+.'') will then involve the limits .} and J, and will bo
reducible by case (ii.)
If 2x is <i, reduce r(2./) by (2818), writing 2,r for ./•. If
this gives 4,/'<^V, i-cduce again by the same formula, writing
4x for Xf and so on.
364
INTEGRAL CALCULUS.
2323 The figure exhibits tlie
curve wliose equation in rect-
angular coordinates is ?/^r(,T),
Let the unit abscissa be OA =
AB=1. Then the ordinates
AD, BG are also each — 1 by
(2286-7).
The minimum value of T(x)
is approximatelyO'8556032, cor-
responding to a? = 1 '461 6321.
The values of logr(a;) in
the table at page 30 correspond
to ordinates taken between
AI) = T{\) and BG = V{2).
INTEGRATION OF ALGEBRAIC FORMS.
2341
f/Sw^^^'^' = ^(^'^'^)-
Pr.oor.— Add together (2281) and (2282). Separate the resulting in-
tegi'al into
+ , and
substitute — in the last part.
.r'-^(l-.r)'"-^
Bilm)
2342 y ^^'-^'-^y: d. = 4i\^ [Substitute ^-±i^
2343 rV-Vl-.r")"'-'^Ai' = —B(L^ m\ [Substi
Jo n \n J
tute x".
2344 f'^igi^^^,/,. = -Lz,(/,,0.tS"b.^;^3f^
Jo \a.v-\-b{l—d')\ uo ^ ^
The integral is also equivalent to
Jo (a sur
(M.siii-^+6cos-^)"
(19, and siniilarlv in other casi
2345
(/.V
'o {H-\-OA-y"^
nab"
[Substitute — ~.
INTEORATIOX OF ATJlKBl^Air FDiniS. 3G5
2346 , 1/1.. = -^^ -^/
PuooF.— Differentiato (234d) hi— 1 times for a. (2255)
Pkoof. — Substitute .); = aij in (2311), aiul tlicn difreirntiato n times for a.
2348 \ -j- — - = - c'osec — , ) z= - cot — -,
»',. l+.r' ;i « Jo 1—.* /* n
where m and n are a?i^ positive quantities, and in is < ?«.
Proof. — Change m into "^ in (2311-2), and then substitute x".
"When n is positive and greater than unity,
2350 1 -T-, — = — cosec-. \ r = — cot — .
Jo l+.r" n n Jo 1—.*'^ m n
Pkoof. — Substitute x'" in (2311-2) and change m into — .
2352 \ r=— COWOC— . 1 -^^ =— col— .
Jovl— U" ^< " J" V l + .« ' >' «
Proof. — Substitute — - — ^ in ^2350), and ' in (2351).
When m Hes between and 1 ,
^ -I n^'t^ *^ — ' «.o-*-ct «^ ^
Proof. — Make n = 2 and write 7)i + l for m in (231-8-9).
2356 \''!^::;±^.i.,=^^. f'l!^
Jo 1 +cr smnin Jo 1— d' tan>/<7r
where ')ii hes between and 1.
Proof. — Se2)arate (2311-2) each into two integrals by the formula
I = + I > ^^d substitute .f' in the last integral.
Otherwise, iu (2G01) substitute t'"^, aud change a Into Tra — jtt.
3G6 INTEGRAL CALCULUS.
Pjjoof.— From (2354-5) by tlie method of (235G).
2360
I — ^ (Lv = — cosec — .
Jo l+.r" n n
2361 j _^^_^ ,/.,. = -cot _.
Proof. — In tlie same way, from (2348-9).
2363 J / !_;> '^-|,^""l^-
Proof.— In (2601) substitute e 2« and put a = ^. In (2595) substi-
tute e « and put a = — .
71
/»00 T -J (»-l)
2364 j„ f-^ = 5t^ 2;^. « l^eing an iBteger.
Proof. — By successive reduction by (2002), or by differentiating
f „ ' = — -, w— 1 times with respect to a^. (2255)
x^ + a^ 2a '
2365 f ^^^'-^1^^ _ 7reosoe»;7r ^gubs. V^^ (2311)
2367 ^^{fL-^,. = lo,n.
Proof. — The value when a = 1 is \ogn. The difference, when the value
is a, is dx — n ' — '- — dx,
Jo i--« Jo i--'-;'
which, by substituting .<;" in the second integral, is seen to be zero.
F(x) being any integral polynomial,
2368 1 ^' At^^^ = ^'^^ ^vliere J is eciual to tlie constant
J-:v/(l-.r-) / 1\-^
term in the product of F^i) and the expansion of M r, ) .
Loa.\nmniir axd expoxenttal fo7?vs'. 1^07
Pkoof. — By successive redaction by (2053), we know that
i^^ = *(')^<'-">-'[7(f^, <•>•
■wliere <p (r) is some inti'Sjral polynomial and A is a constant. Therefore the
inti'cral in question = Avr. To detcrmino -1 write the last equation thus,
f^(^-;;^)"'--'*«('-7)-'-'U.{'-'=) -
Expand each binomial; perform the integrations and equide tlio coenirients
of the two logarithmic terms in the result.
F{.r) being an integral polynomial of a degree less tliaii h,
2369 f;^,.,. = |^"i-{.(,.)lo.^:}.
J,i (.'•—<■/ I »- I J a <.'^' \X — Cl )
But El£} = fix, c) + ^--^, where f is of a dimension lower than n — l
i--c x — c
(421), and therefore d,,,.^^ , / (.r, c) = 0. Hence the integral on the right
INTEGRATION OF LOaARITHMIC AND
EXPONENTIAL FORMS.
Pkoof. — These are cases of (2202). Otherwise; to obtain the first
integral diflerentiate, and to obtain the second integrate, the equation
2391 f;,.''iog,../,,.=^,. j;;^>iog(/<+i).
Pkoof. — These are cases of (2202). Other
integral diflerentiate, and to obtain the second
j x" (Ix = — Y with respect to ;< (2255 and 22<)1).
Jo /''-'•
2393 j"'.,"(log,r)",/., = (-i)»Il|+ll.
Proof. — Sec (2202). Otherwise, when » is either a positive or negative
integer, the value may bo obtained, as in (2:i01), by performing the difler-
entiation or integration tliere described, n times successively, and employing
formula) (21GG), and (21G8) in the case of integration.
Jo log.r *!+*•
368 INTEGEAL CALCULUS.
2395 <'"•-''' rf.,. = -^«„y = - i— ^rf.r.
Jo 1— cr >< ..0 c' — 1
Jo 1+.^' 2/t Jo e' + l
Proof. — Expand by dividing by l=Fa;, and integrate by (2393) ; thus
The first series is summed in (1545). The difFerence of the two series mul-
tiplied by 2'^"" 'is equal to the first ; this gives the value of the second series.
2399
ri2££rf,. = r^jiKii^ = -1-1 1 1 &c. = -^.
Jo l—t Jo ^v 2- 3- 4- b
Proof.— As in (2395-7), making n = l.
The series (2399) may also be summed by equating the coefficients of 6
in (764) and (815).
2401 fM£rf. = -l-l-i, -l-&c.=-^.
Jo 1— .1^^ tii)^ «
Proof.— The integral is half the sum of those in (2399, 2400).
Proof. — Expand the logarithms by (155) and (157) and integrate the
terms. The series in (2400-1) are reproduced.
2404 Let r ^""^^"'^'^ (Lv = <t>{(i), a being < 1.
Jo <(■
Substitute \ — x = i/; therefore, writing Z for log,
u( \ = C ^''^ = r ^^ "^y - C'"^1L^
"^^'^ ].-al-2/ l^-'J I 1-'/
The second integral by (2399), and the third by Parts, make the right side
= _ .^l + Ja I (1 - a) - f"" ^-^—^ '?i/- Therefore
LOQARlTUmO AND EXPONENTIAL FORMS. 8G9
2405 cf>(^a)-{-<i>[^i-a) = \oira lo^- ^ I _a)- 'tt.
Again, V (■'•) = r '-^^^^ '/•'•, ••• i^y -^-^3) 9 (r) = iii— ^ ... (i.)
Put , ■ - for .<: ; then
2406 ^(-•'•)+^HrT:r') " Hiog(i+.r)}"^.
Also, 9(.r) = - [ (l+ 1+ ^ + |- +&C.)
Henco ^ (.0 + ?. (-x) = - -^ (•«'+ fr + |r +'^c.) = 7^-?. (.c'j.
Eliraiuate (-.»•) by (2 tufi) ; thus
2407 <^(;^j)+i'^(.'-0-^(.') = 4{log(H-,r)}'-
Let — ^ = a;", and therefore x = tt + ~t v^^ = /' say,
x+1 2 2
.-. by (2407), !<? 05^)-'/' (/3) = ^ {^ (l+/3)r,
or |-9(l-/3)-9 03) = l(/y3r-; [v /5-^ = l-/> and l + /3=~
and by (2405) (1 -ft) + -^ (/'O = 2 (//3)^ - -j^ 7r=,
<p (,5) = (//3)'^- Itt' and 9(1-/5) = (//5)=- j^. tt', that is,
2408 jT'i^S^^ 'l-^ = (<«?^=-')' - W-
2409
2410 Let a be > 1, then 9 («) contains imaginary elements, but its
value is determinate. We have
^ („) = I' ilk:^) ,i.+ 1° Ul=s) ,,. = - ^ + [ " + '^-" .
3 R
370 INTEGRAL CALCULUS.
the integration by 2300, and l{-l) = ^i by (2214). The last integral
Substitute — =y in the last integral, and it becomes
X
]± y io y h y 6 \ a /
Hence, when a is > 1,
2411 <^ («)+</» (^) = -^+^nog«+i(iog«)^
If a = 2, tills result becomes, by employing (2405),
2412 f ^ lo^(l-v) rf,, ^_^ +,; ,og2.
Jo .V 4
2413 Let ^(a.)=£^logi±|(Z*.
l+.-B 1 , IX
Therefore f C^) = ^ Z (i^), cZ,^ ^:^^ , - -^^.„
therefore 4/ (a;) + 4/ l^^^] = T (l ^-^ — + -~] dx, therefore
\l + x/ Jf, \ 1 — x 2x 1— »"/
2414 ^ W +^ ([=f ) = ilog.r- log l±f .
The constant vanishes, by (2403) and (2401), putting x = 1.
Let X = — '-, and therefore x = v^2 — 1 ; then, by (2414),
1 -\- X
2415 T'l^gy^ 4?- = -* {logiy-'-i)}'-
2416 f'M+iiOrf,_-,„g2.
Jo l-|-cl' o
Proof. — Substitute ^ = tan"'.r ; then, by (2233),
2417 l^^y (liffereiitiating or intooTating tlie equations (234.1)
to (2;]G3) with respect to the index w, the integrals of func-
LOGAlUTmilC AND I^Xl'oyhWTlAL L'(WMS. IWl
tioiis involving logc are produced ; thus, from {'2-\o(')), by
integrating for ))t between the limits ! and in, we have
2418 f' /i'I'Vr"'" ''•'■ = '<•& *"" ^- ('■'«>
Otlierwisc, this vesult may be nrrived at by farming the expansion of the
fraction in powcM-s of.c, and integrating the terms by (2o'J2) ; the reduction
is then efFected by 815-G.
In a similar manner, we obtain the more general formula
2419 f ^•fri";',". '^ ''' ' = '"S- ^ ^ ^ -
Jo (l+ci' j log.f ii in-\-p n-\-'2j) m+*^p
2420 ] ,, ■ „,, (f'V = log tan —-.
Jo (l+.r") logci' "^ 2n
Proof. — Integrate (23G0) for m from 2« to m.
2421 f' ""-''"Y^'"7^'"^'' ''^''
Jo (log.r)"
= (7.H-l)log(y>+l)-(i-+l)log(r+l) + (r-7>)|l + log(v+lj}
Proof. — Integrate (2394) for j) between tlie limits r and j).
2422 I" (v-.-)..-+(>--/>) ..-+(;>-,) .r- „.
Jo (log.rj-
= log{(y,+ l)0^+')(7-.)(,^_^l)('/+i)('-i^)(,.4.1)(.+ n(/.-.)}.
Pkoof. — Write (2421) symmetrically for r, ;; ; /», y and 7, r. ^Multiply
the three equations, respectively, by ry, r, j>, and add, redu<.-ing the result by
2423 r '"^^^.^+"''% /.f=:^log(l+«i).
Proof. — Differentiate for o, and resolve into two fractions. T^ffect the
integration for x, and integrate finally with respect to n.
2424
Proof. — In (2423) put a = 1, and substitute - = ij ; multijdy up by b,
and intei^rate for I between limits U and -, and intho result substitute 6y.
372 JNTEGBAL CALCULUS.
2425 fV*-\/A= 1^^
Jo 1 ^ k
[Substitute lix^.
2426 f %-'-\i'^" chv = ^•^••:if;^~^^ A^" '^ y^.
«.. Z
Proof. — Substitute Avr. Otherwise, differentiate the preceding equation
n times for I:
2427 j ^ ^^—d.i=loga-\ogb.
2428 f"'(^L^^^' _ (£=^)^')rf.. = .-6+iog*;^.
rate this for a
Proof.— Making ?i = 1 in (2291), [ e-"' ch = --. Integ
between limits b and a to obtain (2427) ; and integrate that equation for h
between hmits b and c to obtain (2428).
2429 r('-i:^:=ir_(^£^*)^),,,, = „_i+HogA.
Jo \ .r" .r / ® a
Proof.— Make c = a in (2428),
Otherwise. — Integiating the first term by Parts, the whole reduces to
[£:!:n£:l']- + ,,p-'-^-",,.
The indeterminate fraction is evaluated by (1580) and the integral by (2427).
2430 |--j£!:z^_(«-^V-_(«-y>-"V^,^.
Jo ( ,v^ .r- 2.V )
= \ {(r-^'^lr-4.ah-\-2lr{\oga-\ogh)].
Proof. — By two successive int(^grations by Parts, '\x''^dx, &c..
Also {'L^d. = '^'"'-[^-dr.
Substitute these values, and make e = 0. The vanishing fractions are found
by (1580), and the one resulting integral is that in (2427).
In a similar maimer tlie value of the subjoined integral may be found.
TNTEOnATION OF CIRCULAR FORMS. 373
2431
Jo i~~? .r' 1.2.6- 1.2.;t,r' S •
INTEGRATION OF CIRCULAR FORMS.
Notation. — Let a^"^ signify the continued product of n
factors in arithmetical progression, the first of which is a, and
the common difference of which is h, so that
2451 a^;^ = a{a + h){a + 2b) ... {., + 0^-1) /.} .
Similarly, let
2452 a':l = a{a-h){a-2h)... [a-in-l) b].
These may be read, respectively, " a to n fartors, differ-
ence b"; "a to n factors, difference minus b."
2453 ■ sin".r^Ar = \ .sin"-.r^Ar.
Proof.— By (2048) ; applying Rule VI., \vc have, by ili\ ision,
I sin".i;tfx = I sin""^rtf«— I sin" "^i; cos"
1 1 Ti i f ■ n-2 7 sin""'.)' cos .V . 1 L,;„n
and by Parts, sin" ^x coa-xdx = ■ H , sni
•^ J 71 — 1 11 — I }
i'l' If*".
Tliereforc h'ui" ' ■ .r con' x dx = sm xdx.
Jo n — l Jo
The substitution of this value in the first equation produces the formulu.
t/.r,
vdx.
If n he an iiite<^vr, with the notation of (2I")1),
ri' ' 2"'> i-i- 1*"' 7r
2454 siir"^\rr/r= -^ and sin^ " .rf/.r = -^ ^
PnoOF. — By repeated application of formula (•21:'>3).
Wall is' s Formula.— Ji m be any positive integer, we liavc
2456
374 INTEGRAL CALCULUS.
And siucc the ratio of these limits to each otlier constantly
approaclies unity as m increases, the value of either of them
when m is infinite is -Jtt.
Ex. — With m = 4, ^v lies iu magnitude boUveen
2^4^^^8 , 2-A-.C,K7
and
Proof. — Put 2m = n, then
fin nrr „_1 fl^r
sin"'^ xdx, I sm"xdx, and sin""';
Jo Jo »^ Jo
■dx
are in descending order of magnitude ; the first and second because sinx i.s
< 1 ; tlie second and third by (•24-33) ; then substitute the factorial values by
(24.54-5).
2457
Jo 2sinm7r
2158
r\iu\rf/.i' = 2 r\in\rf/a\ [By (2234).
Jo Jo
2459 I sm'\v(ios^a:(Lv = 2\ siu^.v cos^ j? dcC or zero,
Jo Jo
according as p is an even or odd integer. [By (2234).
/•iTT Mir
2461 I siu'Si' cos" .vdv = \ siu^cr cos'^.r cLv. [By (2233).
Jo »'o
2462 \ "siu\r COS" ,i(Lv = i^(^^, '^^)- t^^^'-"^^
.siu-.r(228u)
Let either of the integers 7i and j9, in (2461), be odd, aiitl
the other either odd or even ; thus, let n be odd and
= 2m + 1 , then
2463 \ «m'"-'^-oos^rrf.,-= , -' „.^„ . (2151)
Proof. — Transposing the indices by (24G1), we have, by Parts (2007),
e COS"'" " ' X dx.
2m r*" „;..«.:;.„...:.«, -1
['"sin^s; cos'"'*^^•(?.^; = -^ { "sin''^
Jo i'+ljo
By repeating the reduction, the integral finally arrived at is
2464 I ^^^i"'' ■'" a." ^'us .r dx = ^
ji + -m -h i
TXTEanATTOX OF CIECULAB FORiTS. 37'-
If both the indices ni-c cvon, then
2465 ^i"'-''^'' t'^*-"^-''' •'•'/•'•= ■Tn;rr7^ (2i:.i)
Proof. — Ileduce hy Parts as Ix'forc. Tlic liinl intiLrnil is siii-""'''a;'/j;,
the value of which is given at ("2 -too).
2466 Sliould eitlier of the indicos bo a nco^ative integer, the
vaUie of the integral is infinite, as the foregoing reduction
shows, for the factor zero vn\\ then occur somewhere in the
denominator.
2467 * sin^j.r sin/).rr/<r = ] cos iLV cos jhtdv = 0,
••0 *
when n and p are unequal integers.
2469 \ ^\nnr cos jhv (I, V = -^ — -, or zero ^
»'o n—p-
according as the difference of the integers n and j) is odd or
cnm. [By (1973-5).
2470 1 sin-/Mv/r = I cos-na(Lv = Itt,
Jo Jo
when u is an integer.
Proof. — E.xpress in term.s of cos2iix', and then integrate.
2472 \ shi\vd,i = \ cos'\v(f.v= \ {l—,v') ~ (h\
The following four integrals (2473-9) all vanish for in-
tegral vahies of }i and }) excepting in the cases here specified.
2474 fJsin^r sin n.vd.v = (-l)"-'\;(^/>,'-^)-;^,
wlien }) and n are both odd, and n is not greater than p.
2475 But if 7^ be even, and u odd, the value is
^ 2"-'hr-ir 7,-(p--2r'^ fr-{p—ly
.:.i-^y'^^l
376 IKTEORAL CAWUIjUS.
2476 Csi»" .r COS n,vd.v = ( - 1 ) " C (,,, 2^) ^,,
when j; and n are hoth ecen, and '?i is not greater tlianj;.
2477 But if j) be otZri and n ecoi, the value is
, i^"-^' 1 ( ;> C (/),!)(?>-:;) . C(,., 2) (/,-<.)
*■ ' 2"-H/)'-H- (/)-2)--«- "^ (;)— 1)--«-
2478 i Jcos'' .1' cos «.i J.i^ = C (;>, ^') ^,
when 2^ and 7i are either both odd or hoth even, and ^i is not
greater than j)-
2479 rcos^is smnxdx, when j; ~ ?i is odd, takes the vakie
n\ 1 C(y>,l) C(/),2) ^., ?
the last tonn Avithin the brackets being
^v~2~/ w^^®^ P is ^^^^^ ^^^^ ^' V '"2") ^^^^^^ i' is ^^"^^^
5 — i 71 even, or s and n odd.
/r— 1 ^'"
Proof.— (For 2474 to 2470.)— Expand by (772-4), aud apjily (2467-2470)
to the separate terms.
CoROLLAEiES. — 11 being any integer,
2480 i "t*os" '^^ cos nddv = ^, \ cos" .f cos udd.v = -^^.
.'o -- •• ^
2482 \ sin-"ciCOs2/?crf/.r= ( — 1)";^,
JO
Jo""
m-"^\rsin(2w + l).rr/.r=(-l)".-^.
2484 I cos^.f cos >ia7/ci' = ^—^ — J- \ cos"--.f cos nd(Li\
Jo jr-tr Jo
JXTKCIiWrioS OF I'lUCFLAR FOHMS. 377
2485
i*i' i> r /> — 1) i' . • , fi
\ cos'\rsiiUKr(/.r = '— ? r- ) cos'^-.r siunrd.r—-, — -,.
Jo p—tr Jo p —n
Proof.— (For either fonuuhi) I5y Piirts, j e<)S.j-(/.r ; an.l the new iiiti-gral
of highest dimensions in cos x, by l\irts, j cos'' "'a; sin a- (/j;.
2486) eos''~-cr(M)s»,rf/,r = 0. j cos" ^r sin /J.r^/.r = -— j.
Proof. — ^lake p = u in (2184-5).
When k is a positive integer,
Mir
2488 ) cos'-'-^r co^n.vdv = 0,
2489 |%«os--^^r COS »u(Lv == ^^^±^ ^::^.
Proof.— The first, by putting ;) = n — 2, n — 4-, ... ?; — 2/.: successively in
(2484) and employing (248t)). The second, by putting ^i = u + 2, 7i + 4,
... n + 2k successively and employing (2481).
When k is not an integer,
2490 ( "co.s'-'^r co.s )U(Lr = 2' -""^^ sin /.-tt /i (/i -2/, + 1 , A).
P
efficient
f;noF.— In (270G) t;.kc /('0!= """'^ a"*^ transfurm by (7t'.t^). ^The co-
mt of I vanishes by (22'jO), and the limits are changed by (22o7).
2491 \ cos".rsin>M</,r=,^ 2+^+^ + ... + - .
Proof.— By successive reduction by (1070), making vi = n, and tho
integral definite.
2492 Wlien ;) and n iire integers, one odd and the other even,
C^^ , (_lV("+"*'M<''' (+, with n odd,
J„ I"— y').y <^ ""' with^) e\en.
Proof.- Reduce successively by (2 IS 1). The final integral, according as
J) is odd or even, will be
rW , cos'.J^Tr (-l)*" |'4* , Rin?.)nr (_l)i(-')
1 cos.rcosji.rcAc = - = - or I cosv.r fix = = — .
Jo 1-n' i-" Jo '* »
3 C
378 TNTEilRAL CALCULUS.
2493 ) cos^\icos)hvcLv = ^-g-^\ sm^\r cosP-".r(/cr,
•.0 I2 •- "
where n and p are any integers whatever such that p—n
is > 0.
Proof. — "When p — n is odd, each integral vanishes, bj (2478) and (2459).
When jjj-n is even, let it =2k; then, by (2488),
(n + 2k)ii 'T _ (n + 2/,-)_, 1, I2 IT
I cos «(
•dx
V" 2" '■''■'' IJ"
= (ji±^^ [^\h,^>'xcos''xdx, (by 2405).
Bat n + 2l-=]), and by (2234) the Umit may be doubled. Hence the result.
2494 \ ^cos'^'^a^ mn)Li(Lv = -—, =^.
Jo 2"{n — l)
Proof. — Tn (2707), put /.•= 1 and /(.'■) = a;""-. Give e'"^ its value from
(76G). The iinaginai-j term in the result vanishes, and the limits are changed,
by (2237). Finally, write x instead of 0.
2495
\ /"(eoscr) »'m~'\rdd = l.ii ... (2>? — 1) \ f {cos .v) cos rid (Lv.
Jo Jo
Proof. — Let ;: = cos.-c. By (1471), we have
f7,^_,^,(l_,Y-^=(-l)""'l-3...-(2n-l)^'- (i.)
n
Also, by integrating n titnes by Parts,
f f"{z)(\-r)"-Kh = (-1)"!' n,)d„,(l-zy'-idz
= -l.S... \-2. - i ) [' ^/(.) .7, (^) dz, by (i.)
Then substitute z = cos.i".
Othcrw!s''.—LL't f(z) = J„ + A^z + AS- + &c. = ^A^z",
••• f"i^) = ^p(l'-l) ... (r-n + l)A,z^-",
I /(cos.r) cos7ixdx = ^Ap cos''.); cos nxdx
Jo Jo .
= — }- [7"(cos.^) sin'"J,r, by (2403).
1.0... (-/I — 1) Jo
2i96 C -• — .. '\'', = ^ (i'->''-)
.(( <r COS' .v-\-b- >^\n- ,v lab
INTEGRATION OF CIUCULAR FORMS. -^70
2497 p *;""'/;^'. . . = -r^. : I)ilT,n.n.iM(..rJ.l.:H;)fora.
.'n {(I- i'Os-,r-\-fr Sin . I)- la h
2498 r ' ^'"''^7^('^'. . -, = ~,. [ DifftTontiato (2VM\) f..r h.
[A.id iuLTctlicr d'-i-lir-S)
^^^^ Jo («^ t'Os^r+/r .siir .r/ ~ iJl^i^t «+^/ />^ + ah' "^ A'/'
(2o00) and (2501) are obtaincil l)y rcpeidinp^ upon ('il'.*!*) the operations
by which that integral was it>clf obtained from (2490).
Jo .r v' (1— •♦■") -
Proof. — Denote the intej^ral by u.
[by (2008)
</" Jo(l+.«V)v/l-..^ 2 v/(l+<r)'
2503 fii;il^,/,, = ^lo,«o+").
Jo .r(l+.r-) 2
Proof.— DinVrentiate for a. Integrate for x by partial fractions, and
then integrate for a.
2504 J>-f-'T^ = ^'"''[('+^)\' + 7:)'l-
Proof. — From (2.j";{) we ol)t;iiii
i' tiin-'.r / _ 'T loL^(l +'0
Jo a' ("' + •'•'/■'" '-^
Integrate for a between liniit.'^ j and x , and in the le.sult sub.stituto hx.
380 INTEQBAL CALCULUS.
2505 1 tnn- «--tni.-'/M- ^,^. ^ ^ a
Pkoof. — Applying (2700), <p (0) liere vanishes. Also, by Parts, we have
J h_ X 2 J A «
since Ix is infinite and therefore tan"' (hx) = — in evenj element of the
integral. Hence the required value is
V ['' dx TT 1 a
Proof. — (i.) Substitute tt — a*, and the integral is reproduced, and is
thus shown to be
2506 rT4^^''.'=T
Jo l + COS-.l' h
TT f^ sin?/ , 1" /. _, 1-1 A\ ''^^
z= — i— dii ■= (tan ' cos tt— tan cosO) = — -,
2 ]^l + cus,-y -^ 2 ^ ^4
xpand by dividing by the denominator, and i
It by Parts. Employing (247H) we obtain the
(ii.) Otherwise, expand by dividing by the denomniator, and integrate
each term of the result by Parts. Employing (247H) we obtain the series
6 ij
ncnrt f cos.r , / tt ("^ filler ,
Ju ^d- V 2 Jo v^^'
Proof.— By the method of (22.j1), putting
1
, e-""" dii,
^x v/'tt Jo
the integral becomes
— [ [ e-''^'cosa;(7.r(7^= — f \ c-''^ co^xdridx (2201)
The second integral is obtained in a similar manner.
2509 \ COS ?/- (lu = ^^-^ = \ siiw/Vv.
Jo Z^ Z Jo
PitOOF. — Substitute >/, and (2507-8) arc produced.
INTEGRATION OF CIRCULAR FORMS. 881
"Wlien n and j> are intcg^ers,
2510 r^rf,. =-i^^-i r\\'-\--sm'.,.h,h:
The integration for x in tlie double integral is given in
(2608-9), and the original integral is thu3 reduced to tho
integfral of a rational fraction.
Proof. — Bj the method of (2251), putting
[By (2201).
[By (2510).
2511 ) — -T- ^^^ = - -rXi = ^
nc-in r^«^in\r , ^ f ' f/::: tt [Bv (25
2510)
081).
2513 ^— !—(Li=hv^J-.
PijooF. — By (2700). Transforming the numerator by (G73), and putting
I (r + 'l) = '^ ^ (I' — 'l) = ^'' ^'"s becomes
2514 f!iliI^!-2illA.%/,..= liog^J.
2515 p oo.,.r-eos;M ,,,.^:.(^,_^)
p,;ooF.— Integrate (2572) for r between the liniit.s p and q.
2516 r ^"'"'''-"^^'''' rf.r = ^ orO,
If u and /y are po.>itive quantities,
l" ."^iiw/.r eosA.r ,/ , _ ■""
Jo .<■
according as a is > or < f>.
Pkoof.— Change by (GOO), and employ (2572).
2518 r shwusinA,,- ^,,^^^ „^ ^<_
Jo '*' •^ **
according as a or b is the least of the two numbers.
382 INTEGRAL CALCULUS.
Troof. — From (2515), exactly as in (2513).
Otherwise, as an illustration of the method in (2252), as follows. De-
noting the integral in (2516) by w, we have, (i.) when 6 is > a,
that is, ^ = r r ^^""^'""^^-^ dhdx = f ^^R^pRJ^dx. (22G1)
2 JJo X Jo x'
(ii.) When t is < o, {"ndh =C^db='^.
Jo Jo - '^
If a is a positive quantity,
2520 r!i^IJ2^\iv = ^{2-a) or 0,
according as a is or is not less tlian 2.
Proof. — sin^a; cos ax = \ sin a; {sin (1 + a) a; + sin (1— a) a;} ;
and the result then follows from (2518), the value of the integral being in
the two cases -^—^ = and J^ — ^ (a — 1) = — (2 — a).
4 4 4 4 4
2522 J^*2ilL™rrf,, = |
according as a is > or < 2.
Proof. — Denote the integral in (2520) by u ; then, when a is > 2, the
present integral is equal to
\%,da= r J (2- a) da + To
2, rnda = {"^(2-a)da='^-
Jo Jo 4 2
And, when a is < 2, | nda = \ ^ (2 — o) da = ^ — ^
INTEGRATION OF CIRCULAR LOGARITHMIC
AND EXPONENTIAL FORMS.
2571 r£::!lf!nzi',A, = tan-^.
Jo .f tl
Pkook. — Difroroiitiato for r, and integrate by (2584).
Oihcni'iae. — Expaul since by (704), and integrate the terms by (2201).
Gregory's series (7'JIJ is the result.
CTBCULAn LOCAUlTllMir ASD i:.\ l'(i\ h'STlA I. mirMS. 383
2572 ('^•,/.,=^.
Proof. — (i.) By making a = in ("2571).
(ii.) Olhencise. By the method of (22oU). First, obsorvinp that the in-
tegral is imlepeudent of r, which may be proved by substituting rx, let r = 1.
Then Jx = \ ilv+\ dx+l iU + &c.
Jo a; J„ X ]„ X i'n X
Now, n being an integer, the general term is either
r-'" ^J^d,. = r -■-^"'.'/'^.V , by substituting x = (2,i-l) 7r + y,
^^ r---!il--J.= r sin.,/, ^ by substituting. = (2.-1).-,;
r sin .r , f- . C 1 1,1 1 . t f.,. "i j„
■' Jo » Jo Ctt-j/ 7r+// Stt-i/ oTT + y ott—ij )
= I {^myUn^dy (2910) = \\m-^dy = ~.
Jo - Jo -^ "^
Proof.— (i.) By (22;j), putting -— . = 2 [ e-^^-^ s^'ydy (2291),
i + •'■ J I)
the integral takes the form
2 I [ cos /.f e " " ' -^ ' "' y dx dy = -[ j e " ""'y e ' "' -^ cos 7-.r J(/ dx
= ^^^\^ c'"'' *y' dy (2G14) ='";^''(2G04).
„ , , „ . r* sinaj*cns/Ar ,
(ii.) Olherxvise. By tlio mutliod of (22o2), putting ?t = dx,
Jo *
it follows from (201 G) that
^ uc-^da = C Oc-" da + r I c-^da = ^ t-*.
Therefore «"'' = c"dadx=\ j, dx, by (2583).
•^ Jo Jo « Jo i + **
2575 r'T£^'''^=T^"
2576 r,7ff^"'=T(l-'-')-
Proof. — For (257.'>) dillcrentiate, and for (257G) integrate equation
(2573) with respect to r.
384 INTEGRAL CALCULUS.
Proof.— By (2-201), re-'-^x^-'dx = ^^.
Put /c = a + ih, and a = r cos 9, i = r sin ^ ; thus
rg-(a.ii)J-_^n-l^^_p = (cos «^-isin»)^)^^
by (757). Substitute on the left side tor e"''-^ from (767), and equate real
and imaginary parts. Otherwise, as in (2259).
2579 r~v-' "'" (A^) '/..• = ^^ '"'( ^).
*^*^ ' ^ Jo COS ^ ^ b' cos\ 2 /
Proof.— Make fl,= in (2577).
Sill /, N
2581 A^^V- ./.r=— ^ ""
^. N ,^ sill/ »/ir
Tim) 2 -^
^ ^ cos\ 2
Proof. — Put n —. 1 — »? in (2579), and employ
Vnr{l-n)
SlU?i»r SUU^TT
2583 1 e-''^ sill «».r(/cr = -TTT-TJ. \ c""" cos fearer = -:r^,
Jo a-\-b' Jo «-+6-
Proof.— Make n = 1 in (2577-8).
Othcnoise. — Directly from (1999), putting n = 1, and —a for a.
■\v1ktc n is a positive integer > 1.
Proof. — In (2577), nut tan"'— = i), thus, uriting p for »i,
a
[ >r"\c^-'^mb.c<hv = ^-0'^^ cos" 6^ sin ?.f,'.
Jo ""
Multiply this equation by h"'^<lh = a" tau"-'(i fccedd.
and integrate from h = to oo , by (2579). Then the corresponding limits
in the integration for will be and }^w.
CTUCULAR LonMurinnc Axn i^xpoxenttal forms. 385
2587 r'-.-«:>^)''''=;:r,:(f),7^-
Proof. — Put n=p — l in (2585).
2589 f .-'..-■ f^ I {.V Um 6) ,lv = r(») cos" e "^^ „e.
Proof.— In (2700), let ^ (.»•) = cos (.r tan d) ;
ian-^ . _ tan*
&c. ; Ai, Jj, &c.
.-. by (7G5), A^=l, -^* = ~Y^' ^*~ 172^3.4'
vanishing. Therefore
j e-'a;"-' cos (x fan 0) dx
1 _ <L(^}±y) taa> + " ;; tan* - "4' tan" + ... = ^^ p •
1.2 1 i j e-'x'^-'^dx
The series on the left = i (1 + t tan 0)-" + i (1 -'"^f^" S)"". which by the
values (770) and (708) reduces to cos a^ cos"^ 0. Then change a into n.
Similarly, with sine in the place of cosine.
2591 r ^"^^~^"'% m6T(Ar = tan"^^ -tau"^-^.
Jo -Jc
Proof. — Integrate (2583) for a between a=:-a and a — (i.
2592 \ e"^"^ cosa.r sm"crr/cr,
Jo
where n is any pcsitivc integer.
See (2717-20) for the values of this integral.
2593 \ L-ZJ__sin»M'f/.j .= 4
yrx — irx
cO f' — c
c"'-{-'lv()S(i-\-e '"
a being <7r.
Proof. — The function expanded by divi.sion becomes
(^" + e-")sinm.r (e-" + e-'" + c"" + &c.)
Multiply in an<l integrate by (2583). The result is
\{2n-\)ir-aY-\-m^ |(2»-1) 7r + a}' + m»
But this series is also produced by dilTerciitiating the logarithm of pqu.ition
(2i:t53). Hence tiie result.
;i 1)
386 INTEGRAL CALCULUS.
2594 \ COS ))hV(Lv = — -r-; \ :
Proof. — Change m into iO in (2503), thus
p (,,-»x^,-<xx)(,gx_^-ex) ^^^^ ^ sing _
j^^ e7rx_e-7ra! COS a + COS y
Now change a into hm and write a instead of 6.
^^^^ Jo e--6>-- 2^' Jo ^--,,— -4e^'«+e-^'
Proof.— Make ?u = in (-iSOi), and a = in (2593).
2597 \ sm 7n.v (Iv = — J-.
Proof.— Make ri=7r in (2593).
2598 ^ — '1± = ±-^B.,.
*.'o e —c
Proof, — Expand sin'm« on the left side of (2596) by (7G4). The right
side is = — |i tan {ihn) by (770). Expand this by (2917), and equate the
coefficients of the same powers of m.
2(c'" — C""') sillrt
e-'"+2cos2«+6'-"'
2599 \ -T ^r- sin m.:p(Za?
Jo ei-'-+c-^^"-'^
2S00 f " 4^^±^ COS „M.d. = ,2^";+'-;) cos«
Proof. — To obtain (2599), put a + ^ir and a—W successively for a in
equation (2593), and take the ditference of the rcsufts. (2G00) is obtained
in the same way from (2594).
2601 1 4^1±^.fAi-=secw/.
Proof.— Make m = in (2600).
2602 f ''sin (ri)-' f/.v = f ^M).M {crY (h = :/^y
CIRCULAIt LOOARITnMIG AND EXl'ONENTIAL FOliMS. a»7
PnooK.— Bv (2125) | c'"'' d.c = ^^
•By (2125) \\-"'
Put a= -c. Substitute on tho luft from (700), and equate real and
imaginary parts.
2604
Jo i!
Proof. — Denote the integral by u. Diircrcntiate (lie equation for a, and
Bubstitute — in the resulting integral to prove that -- = — 2«, and there-
fore u = Ce'-"^. AVhen a = 0, we get
re-\lv=C, .-. C = }j^/n (2-125).
2605 re-("^-)%/.r = ;A^c>--,
Jo -v/Zt
Pkoof.— Substitute x ^n:, and integrate by (2G0-i).
2606 J>-^''^'-^-":-K.'=+|)-^]'^r
2 siii\ 2
Pi;ooF.— In (2005) put /.• = cos + / sine ; substitute from (700), and
equate real and imaginai-y parts.
r" 1 2 i\ f2;j + l)
2608 e-'^^ur"^\v(Lv = '"T ."'^ . >— rr^ -
Jo (^r + l)(fr + 'V-)...('' +2« + r)
].2.l\...'2n
Jo n (^r + 2-)(</- + l-) ... {(r-{-2n )
2610
1'" a (i(2n-\-\)2n
'., //--U- / 'ii -4- I r I
+ a(2» + l)27i(2»-l)(2>t-2) _^ _^
<r + (2/i + l)-"^(/r + 2// + l'j(^r + 2;*-l')
.T., ON (7 2)( 4- 1
(""
o;^Y^a' + 2n-V){a--{-2n-'S') (,r+-'» + l-) . (u'+l)
38S INTEGRAL CALCULUS.
r* „ , a (i2n{2n — \)
2611 \ ^"""^ cos-'* xdx = ., , . . V' + r .. . ^ -X / •> , 7> o-A
aOn(2n-\)j2n~2)(2n- ^ , , _"J ^'- , •
(a= + 27i') (tr + 2m - :^) (a- + 2n-V} ' ' ' {a' + 2h) . . . (a- + 2'-)
Proof of (2G08-11). —Reduce successively by (1999). The integral
part after each reduction disappears between tbe limits in the cases (2608-9),
but not in tbe cases (2t)10-l). See also (2721).
2612
J^in {a--\-l){a-+&)...((r-\-2n-[-l )
2613
J_^. a i(r-\-2^){crJt4^)...{cr^2n)
Proof. — By successive reduction by (1999).
2614 i ^"''"' cos 2hi(Lv = ^ e~''\
Jo 2a
Proof. — Denote the integral by n, then
'-h = - [^-«^x'^ 2.. sin 2hx dx = - f^ ^ e-^^' cos 2h.c dx = -^ ,
db J„ J, a' a'
the second integration being effected by parts, j c" ""•'■" 2a! dx. Therefore
log u = log C- ~ ; and b = gives G = ~^~ (2-125).
a -<■(>
Othenvise. — Expand the cosine by (765), and integrate the terms of tho
product by (2426). Thus the general term is
= (-1)" -'^^-'^ ( — )'" i^p-^^ . which gives the required result by (150).
2615 Tf-" •'' cosli 2hj = ^ <'('') . (2181)
,'o 2(1
Pkoof.— Change b into ih in (_2G14;.
CIIiCULAR LOOARITmrW AND EXPONENTIAL FORMS. 8S9
2617 f ^•-'^r sin 'ILrdr = '^ v ''\
2618 l^'-'.r"^^ sin (2/Ar+ Inn) ,Lr = ^ ^ ('>'•''')•
Proof.— To obtain (2617), put a=l in (2014), and (lUreiviitiiiie for /v.
To obtain (2018), dilVurentiate, in all, n + 1 times for b.
2619 r£2££^£r,/,, = iog„.
Jo J'
PitOOF. — By (2261) putting .- ~ c'^ Jy (22'Jl), and changing the
order of integration, the integral becomes
r r (cos x-e-"'') e-'" dy dx = [ r (e'^^cosx-e-^"'"'') dy dx
Jo \1 + !/ '^ + ^''
2620 H"?^ (!--" <*"^ •'+"') ^^'^^ = ^^'
Jo
when a is equal to, or less than, unity ; but is equal to
27r log a, when a is greater than unity.
Proof. — (i.) a=l. By (2035), since
log 2 (1 — cos x) = log 4 + 2 log sin -J .r.
(ii.) a < 1. By integrating (2922) from to if.
(iii.) a > 1. As in (2920), integrating from U to tt.
2622 y\og{l-nco^a)iI.v.
When ?i is less than unity, the values of tliis integral depenil
on those of (2620). See (2933).
according as a is less or greater than unity.
PnuOF.— Integrate f log (1— 2a cos ./r + a"') J^ by Parts, \dx, and apply
(202U). Jo
2625 \ cosr.i log(l — 2acoy.i+^/') (/.'
according as a is less or greater than unity.
r ' r
390 INTEGRAL CALCULUS.
Proof. — Substitute the value of the logarithm obtained in (2922). T
integral of every term of the resulting expansion, excepting the one in which
u = r, vanishes by (24G7).
^r«/%M T" siu ^? sill r.r r/,t' mi''''^ irfr'-'"^^^
2627 J^ i-2„cos,,^ = -2-' °' —T-'
according as a is less or greater tlian unity.
Proof. — Integrate (2625) by Parts, J cos rx dx.
2629
f cos VcVcIj? TTCf 1 . . -,
\ 1 — n T— 2 = 1 -.5 « l^emg < 1.
Jo 1 — 2acosa7+a 1— «-
Proof. — The fraction = cos ra? (1 + 2a. cos a; + 2a^ cos 2^3 + 2a' cos 3a; + ...)
-f-(l-ft2), by (2919), and the result follows as in (2G25).
Jo 1+.:^' ■ l-2a cosCcT+tt' "" 2(l-a-) l-ae"^'
Proof. — Expand the second factor by (2919), and integrate the terms
by (2573).
2631 r log' (l-2« cose.^+a^) r/.r ^ ^ ^^^. (i-^.e-Q.
Jo 1 -|~ t^'
Proof. — Expand the numerator by (2922), and integrate the terms by
(2573).
^ Jo (l^x%l-'la cos c.r+«0 ~ 2 (<"-«)'
Proof.— By differentiating (2G31) for c.
Othenvise. — Expand by (2921), and integrate the terms by (2574).
2633
r^'^loiifri -l-rcos,r) , 1 ('"'' / _i N9
\ —^ — ^ (Iv = -k ]— — (cos ^ c)-
Proof. — Put « = 1 in (1951), and take the integral between tlio limits
and ^TT, then integrate for b between limits and c; the result is
fi^ lo^d+ccos.) j^ ^ 2 r -1 tan- Jl^ db,
Jo t^^'«^'' Jo v/i-6- V 1+6
and the integral on the right is found by substituting cua~' b.
CinCVLAU LOGABITHMIO AND EXPONENTIAL FORMS. 391
2634 r log(l + 0COS£) ^,,. ^ ^ ,;„-, ^
Jo COScl?
Proof. — As in (2633), by taking and tt for the limits of x.
2635 J""log sin ^-d.,- = ^ log i = f ;75^ <?.'■•
Proof. — I siniCtZ.tf= cosa;cZ.r (2233). Add these integrals and sub-
stitute 2,1', applying (2234) to the result.
2637 ^^ log siu ci f/.i^ = ^ log i.
Jo -^
Proof. — i?' log sin a; c?^; = (tt— .«)Mogsina; (it;, by (2233). Equate the
Jo Jo
difference of these integrals to zero.
2638 t ^v log siu^ xdcc = — ;iV log 2, 7i being an integer.
Jo
Proof.— Method of (2250),
xl sin^ xdx = xl sin^ x dx + xl sin' x dx + . . . + xl sin' x dx
Jo Jo Jt J{)!-llir
= a'Zsin'a;t?a;+ (tt + t/) Z sin" ?/ fZ?/+ ... + {(n — l)Tr + y\lshrydi/.
Jo Jo '0
Each integral reduces by (2635) and (2037) ; for example,
(ir + y) I sin'^ ydij = 2 \ (ir + y) I sin y dy = ^tt i ls'mydy + 2\ ylsinydy
= - 27r' log 2 - tt' log 2 = - 37r2 log 2.
The result is -{1 + 3 + 5 + ... + (2?i-l)} rMog 2 = -«Vnog2.
Proof. — Develope sin mx by (764) ; integrate the tei-ms by (2396), and
sum the scries by (1539).
2640 r™rf,,= 1
Jo e^ + 1 2)11
392 INTEGRAL CALCULUS.
Proof.— Develope sin mx by (764) ; integi-ate the terms by (2398). The
1 IT
resulting series is = o — ^ 2"-^°^^° "'^'^' ^J (2918), wliicli is equivalent to tbe
above by (769).
2641 C cos(mlog.r)-cos(nlogcy) ^ ^, _ ^ j^^ l + >>r
Jo logcf ' ^ ° l + zi'-^'
o^yio r^siu (m loo'cr) — sill (w loader) , ± -1 i -1
2642 1 ^^ ^— r ^^ — d.v = tan ^ m— taii ^ n.
Jo log- ci'
Proof. — Put p = im and q = in in (2394), and equate corresponding
parts. See (2214).
2643
Jo log.v Jo logci^ ''l + n-
Proof.— Put w = in (2641) and (2042).
MISCELLANEOUS THEOREMS.
FRULLANI'S FORMULA.
2700 r J>(«-^-)-<^(M ,/^, ^ ^(0) log^+ rilM ^.r,
Jo '■V ^' J — '^'
/i being = 00 , find the last term generally = 0.
Proof. — In the integral rAi2 — zA-J-dz substitute z = arc and z = h,
Jo
and equate the results thus,
I ^ Jo ^ Jo ^ Jo '^
[^lOL^0:r:lI?i^:)^,_ p 01^) ,7, = (" 9lO) d, = ^ (0) log^.
J„ X J A a! J»_ a; ' ^ " a
a 6
Then make h inliiiite. For applications see (2513) and (2505).
MISCELLANEOUS THEOREMS. 393
+<^(0)(iog|:-«+*)-(«-6)^+jt«<^.
with h = CO .
Pkoop.- rf„ { r ^ ,;. I = I « £lHl cfa - »fi
(2i57)=r'S^<ix-f(0)Za-l('l>,
Jo
by making 6 = 1 in the proof of (2700). Integrate for a between limits
aand^tbus [' t^l^ d. . fi tSM d.
Jo ■*-" Jo ^
and the left is = j' tiill^ilM^.- J"J*i|Hlfe
POISSON'S FORMULA.
c being < 1 .
Proof. — By Taylor's theorem (1500), and by (2919), the fraction is equal
to the product of the two expansions
2 [/ («) +/'(«) cos X + p^ /" (a) cos 2x + ^-^ /'" (a) cos 3x + . . . |
and { 1 + 2c cos x + 2c^ cos 2a; + 26^ cos 3a; + . . . }
divided by (1 — c^). By (2468) the integral of every term of the product
vanishes, except when it is of the form 2 I cos^ nx, and this is ^ tt, by
(2471). Hence the result. ^^
2703
r/(^^+e")+/(a+e-'-) (i_, eos .r) d.v = rr {/(« + e)4-/(«)} •
Jo 1— 2c cos.i^+c-
2704
Jo 1 — 2ccoScr+c^ ^*
Peoof.— As in (2702), adding unity to each side of (2919), and employing
(2921, 2467, 2470).
3 E
394 INTEGRAL CALCULUS.
ABEL'S FORMULA.
Given that F{;v-[-a) can be expanded in powers of e"",
then
2705 f " P(-+MyVF (.-«,) ^j, ^ ^p( ,.^„),
Jo 1 + ''"'
Pkoof.— Assume ^(« + a) = A+A^e-"'-irA^e-'''' + A.,e-^'' + &c.,
:. F (x + iaf) + F (x — iat) = 2A + 2AiCos at + 2A^cos2at + &c,
Substitute and integrate by (1935) and (2573).
Ex.— Let F (.r) = — , then f ^■, . f, , ,. = ^ /_^ , •
KUMMER'S FORMULA.
2706 {^^ f{^^ Qos0e'') e^'^'dO = sin kn (\l-zY-'f{.vz) dz.
J-i^ Jo
Proof.— If h = .re-'(>, then ,r + h = 2.r cos Oe'^ by (766). Substitute these
values in the expansion of f (x + h) by (15U0) ; multiply by e-'''^ and in-
tegrate ; thus, after reducing by (760),
f'j(2.cos e»», ^'-<W = sin A, |.^ - fif + ^ig^ - fe. I
Again, putting h = — xf in (1500), multiplying by f'^'^ df, and in-
tegrating, we have I (pf''~^f(x — x<f)d<p=^ the foregoing series within the
Jo
brackets. Equating the two values and changing ^ into 1 — 2, the formula
is obtained.
For an application see (2490).
2707 Wlien I' is an integer,
f^ f\2.vQ^o^ee')e''''ede = 'L^^^{\\-zf-'f{a^z)dz.
J-l^ ^l .'o
Proof. — Divide equation (2706) by sin A-tt, and evaluate the indeterminate
fraction by (1580), differentiating with respect to k.
For applications see (2490), (2494).
2708 If ^ ^e a function of x so chosen that
rV(*, Z:) ./.' = G, f Xr(.r, 0) .7.. (i.),
and if the series
Af{-r,0)+AjX.,>, l)4-^1./(.>', 2) + &c. ... = .K-r) ... (ii.),
MISCELLANEOUS TBEOBEMS. 395
wliere ^ is a known function, then
^Co+J,C,+.4,a+&c. ... =-4^- (iii.)
\ Xf{,r,0)(lv
Proof. — Multiply (ii.) by X, aud integrate from a to h, employing (i.)
2709 If tlie sum of tlie series
be known, then
— Jo
- 'a
Proof.— In (2708) let X = e-^.i!^-i and f {x, k) = a;\ Then since, by
Parts, we have j e-^a;"*'^-'tic = a (a + 1) ... (ci + h—1) e-''x"-'^dx,
it follows that C/, = a (a + 1) ... (a + Z; — 1). Hence, conditions (i.) and (ii.)
being fulfilled, result (iii.) is established.
For an application see (2589).
Theokem.— Let /(<« + /?/) = P + iQ (i.)
2710 Then£f^rf,«?^= rrs"^''" ("•)
2711 {"^''^dady^-^'^'^dyda: (iii.)
JaJa (ly J a. J a ll<-l
Proof. — Differentiating (i.) independently for x and y,
f'(x + ii/) = P., + iQ,, ifix^iy) = Py+iQy,
••■ -Pa-f iQ^ = Qy-iPy, .-. P^ = Qy and Q^ = -P^.
Hence by (2261) the equalities (ii.) and (iii.) are obtained.
Ex.— Let f(x + iy) = e~'^-'>^' = e'^V (cos 2xy-isin 2xy).
Here P = e-^V cos 2xy, Q = — e'^V sin 2xy, therefore, by (iii.),
[\-^' (e^'cos 2/3 a; -e''' cos 2a x) dx = (^ e^ (e-^\in 2by — e-""' sm2ay) dy.
Put a = a = 0, 6 = co; therefore
( e-^' (e^' cos 2/3a; - 1) dx = 0, .-. e^' [ e"^' cos 2p.r dx = e"^^ f7.-».
Jn Jo •'o
396 INTEGRAL CALCULUS.
CAUCHY'S FOFtMULA.
2712 Let \ x~''F{.v^) dx = A.^,,, n being an integer, tlien
P 1
Proof.— In the integral z-'' F {z") dz =: 2A.,,„ substitute z = x——^
anditbecomes f (.« - J-V" (,.+ -1) i^ [ (,^ - -1)' | ^^ = 2«,„ (i.)
Jo ' '
Let the integral sought be flenotcd by C^,,, then
This is proved by substituting - in the first integral. Therefore by addition
I^--^0-[(^-^)'lf-^" -f^
Now, in the expansion of cos (2u + l) (776), put 2 cos = ^+ — and
2i;8in^ = ;i;— — , where x = e'^ by (768-9), and multiply the equation by
X
_p j /a3— — ] ' [ '-^, and integrate from x = to a; = oo . Then, by (i.) and
in?
(ii.), the required result is obtained
2713 Ex.— Let F(x) = e-"^ then
r"e-"\L-. = ^-^ ^-^-'^-^V 7r and A ^"^
Therefore
A..= \^^^.^-1. = '^^^^V^ and A=^.
FINITE VARIATION OF A PARAMETER.
2714 Theorem (2255) may be extended to the case of a
finite cluingc in tlic value of a quantity under the sign of
integration.
MISCELLANEOUS THEOREMS. 397
Let a be independent of a and ?>, and let A be the differ-
ence caused by an increase of unity in the value of a, then
CA(J>{.v, a) ilv = A \ VCr, a) flv.
2715 Ex.1. re-'^dx=—, .-. [ ^e—^cJx= A~-, ihatis
I?
(e-a-_l) Jx —
«(a^l)
Also, by repeating the operation,
I A"e-'^dx = A"—, that is
Jo
2716 r V-(e- -1)^^ d.V = i7/^^^^_^ V
Jo a(a+l) ... (a+zj)
2717 Ex. 2.— In (2583-4) put h for a and (2a-m) for h, then
-'''' A sin (2a- m)xdx = A ,., , ""~'"' , (i.),
f e-^'"" A cos (2a — m)xdx = A ^-r, z^ r, (ii-)-
J^ A- + (2a-»0-
In (ii.) let m = 2jj, an even integer, then
A'^cos (2a-2^) X = cos (2a + 22j) a;-2p cos (2a + 2jj-2) .r+ ...
... -f cos (2a — 2^) X
= cos 2ax [cos 2^x-2p COS (2j)-2) .c + (7 (2p, 2) cos (2p-4) a;-...
. . . + cos 2j^.t]
— sin 2ax [sin 2j.).« — 2jj sin (2^j — 2 ) o- + . . .
... —sin 2px^,
The coefficient of cos 2ax, in which equidistant terms are equal, xh.
— (— l)^2-J'sin-^a; (773) ; while the coefficient of sin 2a« vanishes becausQ
the equidistant terms destroy each other. Therefore
A=^ cos (2a -'2p) a; = (-1)^ 2"^^ cos 2a.7; sin=^^.
Hence (ii.) becomes
2718 £e-^- COS 2a.r ^\n'^adx = ^-^ ^'' \.-^ + (2a-27>)^ ^
2719 Again, in (i.) let m = 2p + '[, an odd integer, then
A-^^i sin (2a-2p-l) X = sin (2a + 2p + l) a7-(2jj + l) sin (2a + 2jJ-l) x
+ C(2p + 1, 2)sin(2a + 2i)-3)a?-...- sin (2a-2p-l) a;
= sin 2ax [cos (2p + l) s- (2p + l) cos (2j)-l)a!+ ...-cos (2^ + 1) a;]
+ cos2aa; [sin (2j; + 1) a3-(2i) + 1) sin (2p-l)a;+ ... + sin {2j} + l) x].
The coefficient of sin 2ax vanishes as before, while that of cos 2ax is
= (-l)''2-''^'sin2^*^r (774).
398 INTEGBAL CALCULUS.
Therefore equation (i.) becomes
2720
Ce-'^ cos 2a.r ^'m'^^Krdv = i^^^ A'^^' 2a-2;>-l
Jo 2-^^-*-^ A;-+(2a-2;>-l)^
To compute tlie right member of equation (2718), we have
A^^ ^ = 7c r ^
2p , C(2p,2) 1 -]
Jc' + (2a +2p- 2f /r + {2a + 2^-4)- ■ ■ A= + (2a - 2pf] '
Let a = 0, then the equidistant terms are equal, and we obtain in this case
07P1 A^. fe _ (-l)n.2...2/).2^^
Thus formula (2600) is proved.
Similarly, by making a = in (2720) after expansion, formula (2G08) is
obtained.
Let p be any integer, and let q and a be arbitrary, but
q<2p in (2722), and <2j9 + l in (2723).
2722
r cos2a^-sin^^..- , _ (-1)^ r ^,, z^^^ ^,
Jo x^^' ~ 'I'n^ (r/+l) Jo z'+ i2a-2py
2723
Jo ^^ ''
(-^y r^...: (2a-2;)-l)^^
= 2^^+^r(ry + l) Jo z'-\-{2a-2zy ^'
where A lias tlie signification in (2714).
Proof. — Employing the method of (2510), replace
q being integral or fractional ; therefore
r c_os^a.. sin^^« ^^ ^ 1 r r 2^,^ si^..,.,-.r^., J, j^^
Jo, ^'" , in2 + i)]oJo
by changing the order of integration. Substitute the value in (2718) for
the integral containing x, Avriting the factor z'' under the operator A, since it
is in(k'])endeut of a.
Similarly, with 2j7 + l in the place ofp, we substitute from (2720).
MISCELLANEOUS THEOBmiB: 399
It may be shown tliat, whenever a > p, formula (2722)
reduces to
2724
For a complete iuvestigation, see Caucliy's " Memoire de I'Ecole Poly-
technique," tome xvii.
2725 Ex.-Let a = 2,p = l,q = i,
r cos4aW^^^_^ = 1 A= (_2a-2)K
h ^ srmsin^
6
and A2(2a-2)*= (2a + -2)*-2 (2a)* + (2a -2)^ = r>-2.4* + 2i
FOURIEE'S FORMULA.
2726 r~<^(.)./.. = -|<^(0),
Jo sill cl £i
when (X= CO and li is not greater than -^tt.
Pkoof. — (i.) Let <^{x) be a continuoas, finite, positive quantity, de-
creasing in value as x increases from zero to li.
(i-),
a a
— being tbe greatest multiple of — contained in h. The terms are alter-
a d'
nately positive and negative, as appears from the sign of sin ax. The fol-
lowing investigation shows that the terms decrease in value. Take two
consecutive terms
r " sin ft » ^ / X 7 r " sin aa; / ^ ,
Inn Sm .B ^ ^ ^ J (n + l)7r Sin X
a a
Substituting x— — - in the second integral, it becomes
f a Sin ax / , T \ ,
-1 — r<ph+~]dx,
sin (x-\
\ a
and since f decreases as x increases, an element of this integral is less than
the corresponding element of the first integral.
400 INTEGRAL CALCULUS.
N'ow, by substituting ax-=y, we have
-. — f Or) (ii^ = (p[-]iii/ = <piO)\ --— fZ// . . . (11.),
n a
■when fl is infinite, because then (p { -] = ^(0) and a sin ^ = y.
' \a I a
Hence the sum of n terms of (i.) may be replaced by 0(0) ' ^^^cZv,
Jo y
which, when n is infinite, takes the value </)(0) ^tt by (2572) ; while the sum
of the remaining terms vanishes, because (the signs alternating) that sum is
less than the ?t-|-l"' term, which itself vanishes when n is infinite.
(ii.) If ^(a;), while always decreasing, becomes negative, let (7 be a con-
stant such tihat (7+0 (») remains always positive while x varies from to h.
The theorem is true for G-\-(p (x), and also for a function constant and equal
to (7, and it is therefore true for the decreasing function whatever its sign.
If (j){x) is a function always increasing in value, —(f>{x) is a decreasing
function. The theorem applies to the last function, and therefore also to {x).
2727 CoE. — Hence tlie same integral taken between any
two limits lying between zero and -^tt, vanislies when a is
infinite.
2728 C^^^^[a:)(lv
Jo sm <r
= 7r{i^(0)+^W+<^(27r)+...+<^r«-l)^ + ^(«7r)},
when a is an indefinitely great odd integer, and mr is the
greatest multiple of ir less than h. But when a is an indefi-
nitely great even integer, the second and alternate terms of
the series have the minus sign.
P sin ax / ^ j /""'^sin a.r ^ / \ 7 , ['' sin ax ^ , ■. j ,. .
—. (x) dx — — ^ f (.)■) dx+\ -0 (.r) dx (1.),
I sin a; „ sin x I „ sin .1;
Jo Jo J "f
decompose the second integral into 2n others with the limits to ^tt, ^tt to tt,
TT to |7r, ... (2;) — 1) gTT to UTT ; and in these integrnls put successively x = ?/,
IT— _;/, TT + y, 2v — y, 2w-\-y, ... mr—y. The new limits will be to \ir, \tc to
alternately, with the even terms negative, so that, by changing the signs of
the even terms, the limits for each will be to ^tt. Also, if a is an odd in-
teger, — ; — '- is changed into — ; — - by each substitution, so that (i.) becomes
sm* &n\y
r"^^{0(y) + 0(— !/) + 0(-+:/)+... + 0("— 2/)}./.'/
Jo ^^^y c" ■
, sm ax /■ •. J /•■■ \
+ --. ?'(.^)(Z.c (ill.)
J,,^ sm.c
But, when a is even, the substitution of m =F ;/ for
smy
Peoof.
MISCELLANEOUS THEOREMS. 401
r
Jo
whenever r is odd. The limit of the first part of (iii.) is
— {<!> (0') + 2<p(w)+2cp (27r) + ...+2^ (n-l)7r + ,p (nir)}, by (2726).
In the last part of (iii.) put x = nw + y, and the integral becomes
^JRm^(n^ + y)dy=^cf,(n7.), if /,-«,r is > -^, by (2725).
sin 1/ '^ ^
If h—7iir lies between -Jtr and tt, decompose the integral into two others ;
the one with limits to ^tt will converge towards ^tti^ («t), while the other
with limits ^tt to h — mr becomes, by putting y = Tr — z,
the limit by (2727). Hence the last term of (iii.) is iir(b (nir). Substi-
tuting these values, (2728) is obtained.
2a '
2729 Ex.— By (2614), [ e-^'^' cos 2hx dx =
Put & = 0, 1, 2 ... n successively, and add, after multiplying the first equa-
tion by i, thus
f,
-a2ar> (i + cos 2ar + cos 4^ + . . . + cos 2n«) c7a;
The left side = J [%-.'»' ^'° (^." + ^) " <;., by (801),
sma;
Jo
and, if w = 00 , becomes
JL jl + e-Tr2a2_^e-4^2aS_^e-9;r2a=^_,..}, by (2728) ;
/ _1 _i -i
Put Tra = a and — = /J ; therefore
a
2730 v'a{4+<'~°'+«"*'''+<'~'°'+-}
= y/3{i+e-«'+e-«'+e-'^'+...},
with tlie condition a/3 = tt.
Jo <r -<i
when a is an infinite integer.
Proof. — The integral may be put in the form
f ^ sin ax
J„ sin.T
^^° '^•'^ - (.r) c^,., where ^(x)=^<t> (.r).
3 F
402 INTEGRAL CALCULUS.
therefore, by (272G), when h is > Jtt, and by (2728), if /t is >.^7r, the value
is Itt* (0), since in (272«) <i> (ti), a> Cln), Ac. all vanish. But * (Oj = cp (0).
Hence the theorem is proved.
Wlien o and /3 are both positive,
J a .V J -a cV
PROOF.-(i.) f = f - f" = f^O)- ' 1, (0), bj (2729).
.L Jo Jo ^ ■^
(ii.)|' =1° +j[=|*(0)+f*(0),
by substituting — ;<; in the second integral.
2734 I 1 <j) {.1} cos ux du dcV = -r-(l>{0), when a = oo .
PiiOOF. ^IIL^ — oos«.i;c?«. Substitute (his in (2731).
X
Jo
AYhen a and /3 are positive, the limit when a is infinite of
2735 I I <l>Lv) cos in COS UcvdudiV,
Jo Ja
or of I 1 <^(,r) sill /« sin //,rf/wr/cr,
.0 ».'a
is ^7r0 (/), if / lios between a and /3, :{;?(/) (/) if f = a, and xH'o
for any other vahie of t.
Pr')OF. — When a = cc) we have, by (668), and integrating with respect to «,
f rr,)oo,nxco.Mrclu = :L f/^ sin a (.-/) ^ ^^.-^^^_^ , [^ lillWllti^ ^ (,,) cZ..
JaJfl Jo ^~^ J« '•'"^^
= xr'^i^c.+o,.+ir'^^^^Q:-,),, (i.),
by substituting z = x — t and 2 = .r + / in the two integrals respectively.
"When a is infinite, the limit of eacli integral is known.
When a and /3 are positive and Hies bt'i\vc(Mi tlieiu in value, the
limit of (i.) is V0(O, by (2782 8) (ii.)
When a ami /3 are positive and / does not lie between them, the
value is zero, by (2732) (iii.)
MISCELLANEOUS THEOREMS. 403
If a = Hu (i.), the first integrnl becomes = ^ir<p (i) by (2731), and
the second vanishes as before ; so that the value, in this case, is ^tt cp (t) ... (iv.)
The same demonstration applies in the case of (2736), transforming by
(6G9) instead of (GG8).
Hence, by (ii.), if t be always positive,
/100 /-tec
2737 ] \ <t> ('t^) cos fu cos iLV (In d.v = — <l){t)
/»0D /loo
= 1 \ (l){.v) iiiu til sin u.vdiidcV.
Jo Jo
2739 Ex.— Let <p (x) = e-'^^
e~"^ cos tu cos ^lxdudx ■= — e~"^
Jo Jo ^
Therefore-, by (2584), f^^^?^ du = ^ e""*,
which is equivalent to (2574), with t = 1.
The expressions in (2737-8) being even functions of ii, we have, supposing
t to be always positive,
I ^ (x) cos til cos uxdu dx = 7r(j)(t) = \ <p (x) sin tu sin ux du dx ... (i.)
J -00 Jo J -X J
Replacing <p (x) by <p{~x), and afterwards substituting —x, these
equations become
(p (a-) cos tu cos ux du dx = tt^ ( — ^)
J _x J -x> f* f
= — (() (x) sin tu sin uxdu dx (ii.)
From (i ) and (ii.), by addition and subtraction, we get
2740 i <j> (x) COS tu COS ^lxdudx = irlfp (t)+(l> ( — t)],
2741 ^ (•'^) sin tu sin ux du dx = tt [^ (J)—<f> (~0]'
Whence, by addition,
2742 \ f " 4> (ciO cos u {t-x) dudx = 27r<l> (/),
the original formula of Fourier's.
404 INTEGRAL CALCULUS.
THE FUNCTION t/i(a7).
The function dj,\ogT(x) is denominated ■(p{x).
2743 V,(.r) = ]og^- 111 1
wlien /t is an indefinitely great integer.
Peoof.— By differentiating the logarithm of (2293).
2744 Cor. ^(l) = iog(>.-l-i.-i-...-^^,
wlien /I = CO ,
= -0-o77215,664901,532860,G0 ... (EuJcr).
All other values of ^p{x), when a? is a commensurable
quantity, may be made to depend upon the value of ;//(!).
When X is less than 1,
2745 ^ (1 -.^0 - V' i-^) = ^ cot TTcr.
Proof. — Differentiate the logarithm of the equation
r (x) r (l-.r) = TT -f- sin™ (2313).
2746 .^(..•)+'A(.'-+i)+'^(^-+|)+...+V'(..-+^^)
= nxjji^nd^^nlogn.
Proof. — Differentiate the logarithm of equation (2316).
2747 '-To compute the value of ^(—) when ^ is a ^proper
fraction. ^ ^
Find ^ ( - ) from the two equations
2748 t/»(l— ^)-tA(^) = 7rcotii7r, (2745)
2749 '^ ^ ^
V'(l-i^)+t/,(-^) = 2]t/,(l)-log</4-cos?^log(2vers??)
+ cH>8^1og(2vors4^)+cos^log(2vors55) + ,^c.|.
THE FUNCTION y\i{x).
405
The last term within the brackets, when q is odd, is
(ild)j^lo,c.^2Yers(i::i5>);
cos ^ '-^— log ( ^
9. \
and when q is even, the last term is + log 2 according as _p
is even or odd.
Proof. — Equation (2743) may be written
fi being an indefinitely great integer.
-D 1 • 1 u 1 2 3
Keplace x successively by — , — , — ...
teger; thus 9. <1 1
q/ ^ 2 + 1 ' 2^ +
. + 1
1 ; where q is any in-
^(i
i + '*-3^^+'*-
-±+l,.
1 +7a_ ? +ZA
g + 2
22 + 2
32+:
__1
32 + 2
2 >> " Fl^'"'' 22-1"^'^ 32-1
V 2 / 2-1
;P(1) =-1 +12-
+ 11-
Now, if 6 be any one of the angles ^=^, — , —
... ilinllzr, we shall have
(i-)
1 = cos q(p = cos 220 = cos ^q<p = &c
COS0 = cos (2 + 1)0 = cos (22 + 1)0 = cos (32 + 1)0
(ii-),
&c (iii.)>
cos0 + cos20 + cos30 + ...+cos (2 — 1)0 + 1 = by (803) (iv.)
By means of the relations (ii.) and (iii.), equations (i.) may be written
cos0;p(— ] = - 2 COS0 + cos 0^2 ^cos(2 + l)0+ cos0Zf — ,
cos 20 1//
^\=-^ cos 20 +cos20Z2 2_cos(2 + 2)0 + cos20Zf-,
q I 2 q + ^
^^ X cos 30 + COS30Z2 2_ cos(2+3)0 + cos30Zf-,
3 ^ q + o
Hi)
cos(2-l)0^ (^— ) =--5jCos(2-l)0 + cos(2-l)0Z2
^cos(22-l)0 + cos(2-l)0Z|-,
4/(1) = -! + 12 - h + Zf -.
Upon adding the equations, the coefficient of each logarithm vanishes, by (iv.)
The remaining terms on the right form a continuous series, and we have
cos0;/^(— )+cos 20 ;//(-)+...+ cos (2-1)0^ ('^^-)+^(l)
= — 2 { cos + 1 cos 20 + i cos 30 + in inf. }
= 12 log (2 -2 cos 0) by (2928) (v.)
406 INTEGRAL CALCULUS.
Let — = w. Then, by giving to (p in equation (v.) its different values w,
2w, 3w ... (q — l)<*), -we obtain q — 1 linear equations in the unknown quanti-
ties \p ( — ], yp [ — ) ■•■ '^ ( )• To solve these equations for \// [~j
p being an integer less than q, multiply them respectively by
C0SJ3W, COS 2p(o . . . cos {q — \)pio,
and join to their sum equation (2746), after putting x= — and n = q.
(k \ ^
— j in the result, /.; being any integer less than q, is
cos jJw cos Ji(o + cos 2j:)<i» cos 2/iw -|- . . . + cos (q — l)p(i) cos (g — l)ku)-rl.
By expanding each tei'm by (6C8), we see by (iv.) that this coefficient
vanishes excepting for the values k = q—p and k = p, in each of which
cases it becomes = ^q. Hence, dividing by ^q, we obtain
yp(9-:P\+^ (£.\ = 2xP (I) - 2lq + cos pio I (2 -2 COS co)
+ cos2^wZ(2-2cos2w) + +cos (/? — l)27wZ {2-2 cos(g-l) w}.
The last term = cospwZ(2 — 2 cos w) = the third term; the last but one
= cos 2^wZ(2 — 2cos 2a^ = the second term, and so on, forming pairs of
equal terms. But, if q be even, there is the odd term
cos iqpoj log (2 — 2 cos ^quj) = dh 2 log 2,
according as 2^ is even or odd.
Examples. — By (2748-9) we obtain
2750 ^(f) = ^(l)-31og2+^, 4^ (I) = ^ (I) -Slog 2- j-,
2752 ^(l) = ^(i)-Hog3+^^, ^(i) = v/^(i)-fiog3-^,
2754 ;/.(i)=;/.(l)-21og(2).
DEVELOPMEN'TS OF t/»(rf+.r).
Wlien ,1' is any integer,
2755 -/,(«+,.•) = ^(«) + J- + -l,+-L.,+ ...+ ^
Proof. — By (2289), putting n = a + x — l and r — x—1,
r (a + x) = (a + x-1) (a + x-2) ... (a + 2)(a + l) a r(a).
Differentiate the logarithm of this equation with respect to a.
2756 <l,(a+.v) = ^ia)+^-pf:zll + £ip^kl^
a 2a{a-\-l) .ia (« + !)(« 4- 2)
,r{,v-l)(,r-2)(,r-i\) ,
THE FUNCTION \fj{a^. 407
If 33 be a positive integer, the number of terms in tliis
series is finite, and the value of ^p{a-\-x) can be found from
that of xp{a).
Hence, by this or the preceding formula, in conjunction
with (2747), the value of ^{N), when JV is any commensurable
quantity, may be found in terms of ^(1).
Proof.— Let \|/ (a + x) = A + Prx+Cx (x-l)+Dx (x-l) (x-2) + &c.
Change a; into x + 1; then,
A4^(a + x) = ^lJ{a + x + l)-^l^(a+x) =:f?^{logr (a + x + l)-logV(a + x)}
= ch\og (a + x) (2-288)=-^^,
Ax = 1, Aaj (x-l) — 2x, Ax {x-Y) {x - 2) = ox (.« — 1), &c. Therefore
— ^-B^20x^'iJ)xix-V)^^Ex{x-V){x-2)\,
a-\-x
A — = 2(7+2.3jDj; + 3.4E«(a;-l) + ,
a-\-x
A^^— = 2.3D+2.3.4.i;a- + ,
a-\-x
A*— - = 2.3. 4^ + .
a-\-x
Put a; = in each equation to determine the coefficients A, B, G, D, &c.;
thus A = 4^(a), B = -, 2G=a'^ ^ ^ ^
a' a a + 1 a a(a + l)
1 —1 2
2.3D = A^ — = A -
(a + 1) a{a+l){a + 2y
1 2 ^3
2 3 4^ = A'— = A = ■- — — —, and so on.
a a(rt+l)(a + 2) a (a + l)(a + 2) (fl + 3)'
SUMMATION OF SERIES BY THE FUNCTION t/»(cv).
2757 Fonnulal. ^ + ^^ + ^^ + ... + ^^
Proof. — Let S„ denote the «. terras of the series to be summed. We have
S„.,-8„ = f-^(^^+n+\)=^[^[-^ + n+2)-^[^ + n + l)']{2288)
or >S«,i- — 4/ (-^ + n + 2] = S„- ^ 4/ (^ + n + l).
c \ c I c \ c I
Hence the difference is independent of n, and therefore
408 INTEGRAL OALOALUS.
2758 Ex. 1+ 1 + 1 + - + 2;;^ = ^-^"^ (^^+-^^ '^"+^)-
2759 Formula IL ^
7> + 3o
= .lFT^-|:{*(4^')-*('^>HHf^-)-^(^-)}-
Proof. — The series is equivalent to
+ r-^+,-V+...+ ,-A-- ^7^ + , . "', ., ■■■ +
and the result follows by Formula I.
2760 Formula III. —
1 1 • 1 ininf.
b i+c^i+2c
i !*(¥)-*(*-!?)!
Proof. — Make n= 00 in Formula II. The last two terms become equal.
2761 Ex. 1.— In (2760) let h = c=\, then
l-i + i-l + &c. = i + i \^ (2)-4/ a)} = log 2.
For ^ (2) = 1+4/ (1), by (2755) ; r|, (f) = 2 + 4^ (l)-21og 2, by (2754-5).
2762 Ex. 2.— In (2760) let & = 1, c = 2, then
l-\+\-\+&c. = 1 + 14^ (i)-!-]' (I) = f-
2763 V'(H-«) = r"-^rf.r+V(i)-
Jo .1' — 1
Proof. 4-(l + a) -^(l) + ^-^|xF^ + "%rl^.2!^^^"'^'^- ^^^ ^-^ ^^)
1_(1_9;)« a(a-l) . a(a-l)(a-2) 3 „
a; 1.2 1.2.0
therefore v|, (1 + a) = f ^"(l--'^')" dx + v|/ (1).
Jo
Substitute 1 — a; in the integral.
2764 i^(l+.0-V'(l+M = r^!— 4-%Ar.
,'0 .r — 1
[By (2768)
THE F UNCTION yj/ (.v) . 40 9
Ex. — Put h = —a; then
;/.(! + a) _;/.(!_ a) = - +^(a)-4^(l-a) (2756) =- -TrcotTra (2745).
ex 0/
2765 Therefore ■--_i_- tZ» = - -tt cot Tra.
i/j(a?) AS A DEFINITE INTEGRAL INDEPENDENT OF i/r(l).
2766 ^(,.) = -j;(j_L.+i^jrf«.
Proof. v|/A^)=W«-— ^-... ^^ — -with^ = oo (2743).
1,1., 1
-1 ^X + fl.-l
But - + -^ + ...+ , , ^ , dz,
X x + 1 X+fl — l J^ l-z
by actual division and integration.
Also iog^ = p(^^-g)cZ. (2367).
Put 2 = i/'^ in the first integi'al, therefore
Jo 1-'/ Jo 1-2/'^
Replace i/ by z, and suppress the term common with the second integral
of (i.), and we get »//(«.) = J | '-j^^ - fj-^ \ dz.
Put g'^ = u, and this becomes
But when ^ = oo the product /i(l — m^^) has — logtt for its limit (1584);
and w'^ = 1. Hence the result.
2767 ^(^*^)=ri'
1 ? da
Proof, r («) = j e-^z'^-'^dz; d^Y (x) = j e-'z'^-Hogzdz.
Jo Jo
But, by (2427), log2=[ ^~"~^~°% ^«,
3 G
410 TNTEGUAL CALCULUS.
C' C^ /< - "■ /) - at
.•.d^r(x) = i e-':f-'- dzda
J n J
= r[c-.[%----'&-rc-"+->-r'-'&] '^'
-wLich establishes the formula since
d,v{x) - r(.o = .z.iog r 0^) = ^ (x).
2768 logro.) ^£[G.-i)c-^^-^-^!=£f]^'
2769 =:fT!jl£l!_,,.+ill?£.
2770 ^(„.)=j;[^-j^,/f
Proof. — Integrate (2767) for x between the limits 1 and x, observing
that logr(l) = 0; thus
Jo ' log(l + «) > a
Subtract from this the equation obtained from it by making ;c = 2, and
multiplying the result by x—\. We thus obtain
log r (.) = i; [0.-1, (1+,.)- ^^""^";"""^1 s^-
Substitute ^ = log (! + «), and (2768) is the result. To obtain (2769), sub-
stitute z = (l+a)-\ Lastly, (2770) is the result of ditferentiating (2768)
for X.
NUMKRICAL CALCULATION OF \ogr{.v).
2771 The second member of (2768) can be divided into two parts, one of
which appears under a finite I'onn, and the other vanishes with x. If we pnt
1 \ '-i . , ,,_ I
1-
r = (.-i-. -,) -,.w« = ,-.£._,^,
=r
then Iogr(,«)=^ iP+Qe-i>-)d^ (i.)
If Q bo developed in ascending powers of l, the terms which contain
negative indices are -- + ^, = li .--a^-.
NUMEEIGAL CALCULATION OF log Fur). 411
Put F(x) = { {P + Re-^^jdi
=i:[((-^)-i^)--(M)Hf ^^^•)'
and .. (..) = [^ (Q-B)e-^-^-cU - j^^ (j^" j" 1) -^' f - ("^^
Then, by (i.), log T (x) =i^(.0 + w(a;) (iv.)
F (x) can now be calculated in a finite form, and ra- (.f) will liave zero for its
limit as x increases.
First, to sbow that F (^) and w (|) cfin be exactl}^ calculated.
and, by substituting -g-^,
Again, 2;)utting' « = 1 in (iii.), we have
. -«=i:(i^4-i)-^f ^^'-'^
and, by substituting A^,
"0)=j;(^,-,\-i)-«f (vii.)
The diffex-ence of (vi.) and (vii.) gives
-r(r^-^-^)-f ^-•>'
Jo ^
since 1 — = -. = ,- 1^>-
l — e^i 1 — e--5 1-e -«
Subtract (viii.) from (v.), thus
^ (1) = ^ r(^:^^ _ ^) ^^ = I _ l£^ (2429).
Also, by (iv.), -F(i) + w(^) =- /r(i) = i?7r, .-. Fil) =: ilog (27r)-i...(ix.)
i-'O^) may now be found by calculating Fi.r) — F{^) as follows: —
By (ii.), F(.)-Fii) = |^[(.-i)e-f+ (1 + |) (e-^-e-i^J |
Jo ^' ^ Jo ^
= i-a'+(^-i)loga: (2427-8),
.-. F(x) = ^\og{27r) + (x-l)\ogx-u:, by(ix.);
.-. by (iv.) logrOi;) = 1 log (2;r) 4-(a;-i) logx-x + xjr (x), (x.) ;
2772 ■•• r(*) = e--^-a;-r-iv/(27r)e-('-) (xi.)
412 INTEGRAL CALCULUS.
When X is very large, e^ W differs but little from unity. For vt(x) diminishes
■without limit as x increases, by the value (iii.)
Replacing xsr (x) in (x.) by its value (iii.)) and observing that
log r (a; -I-]) =loga; + logr (a:),
■we get log r (.c + l) =-i-log (Stt) -|- (aj + |) log x — x
+J„(n^^-T--2)^'^^ ("^-^
Now, by (1539),
\l-e-f i 21 k 1.2 1.2.3.4 1 ... 2;i 1 ... 2« + 2'
where ti is < 1 . Also
Jo ^"'^' Jo '^
So that equation (xii.) produces
2773 log r (.r+ 1) = i^^^ + (.r + i) log .v^a.^
, ^2 ^i_ I zr ^J^in + 2
This series is divergent, the terms increasing indefinitely. The comple-
mentary term, which increases with n and is very great when n is very great,
is, however, very small for considerahle values of n. For instance, when
a = 10, the values obtained for log r (11), by taking 3, 4, 5, or G terms of
the series, are respectively,
10-090820096, 16104415343, 16^104412565, 16-1041 12563.
CHANGE OF THE VARIABLES IN A DEFINITE
MULTIPLE INTEGRAL.
2774 Let ,7', ?/, ^ be connected with ^, v, I by three equations
Then, when tlie limits of the integral containing the new
variables can be assigned independently, we have
TRANSFORMATION OF A MULTIPLE INTEGRAL. 413
where <l> is what F becomes when the vakies of x, y, z, in terms
of ^, 7), I, obtained by solving the equations u, v, w, are sub-
stituted.
Proof.
F (x, y, z) cluhj dz = III f (^^ n, 1^ ff J ^^^ ^^ <^^-
To find Xt, consider rj aud ^ constant, and differentiate the three equations
ti, V, w for 4', as in (1723). To find y^, consider i^ and x constant, and differ-
entiate for r/. To find Z/-, consider x and y constant, and differentiate for ^.
We thus obtain
d (itviv) d (ttvw) d (uvw) d (uvtv)
dxdych^_ d {kyz) d inzj) d (46O ^ _ d (tnQ
dE, dt] dZ d (uvw) d {uvw) d (uviv) d. (uvtv) '
d {xyz) d {yzS) d {zkn) d {xyz)
observing' that two interchanges of columns in a determinant do not alter
its value or sign (559).
Similarly in the case of any number of independent variables.
When, however, the limits in the transformed integral
have to be discovered from the given equations, the process
is not so simple.
In the first place, we shall show how to change the order
of integration merely.
2*7*75 Taking a double integral in its most general form, we shall have
F(x,y)dydx (i.)
cb r4> (.1-) rp r* t
F(x,y)dxdy=^\
The right member will generally consist of more than one integral, and S
denotes their sum. The limits of the integration for x may be, one or both,
constants, or, one or both, functions of y. ^ is the inverse of the function \p,
and is obtained by solving the equation y = i//(.x'), so that x =■ ^(^). Simi-
larly with regard to (p and 4>.
An examination of the solid figure described in (1907), whose volume
this integral represents, will make the matter clearer. The integration, the
order of which has to be changed, extends over an area which is the projec-
tion of the solid upon the plane of xy, and which is bounded by the two
straight lines x = a, x = h, and the two curves y = '4^ (.^')> V = 'p(x)-
The summation of the elements PQ.qp extends from a to h, and includes
in the one integral on the left of equation (i.) the whole of the solid in
question.
But, on the right, the different integrals represent the summation of
elements like PQ^^rj), but all pai^allel to OX, between planes y = a, y = fi, &c.
drawn through points where the limits of x change their character on account
of the boundaries y = \p{x), y = cp (x) not being straight lines parallel
to OX.
414
INTEGRAL OALCULUS.
2776 Example. — Lot the figure represent the pro-
jected area on the xy plane, boanded by the curves
y = \p (x), y = (b (x), and the straiglit lines x = a, x = b.
Let y = cp (x) have a ruaxinium value when x = c.
The values of y at this point will be ^(c), and at the
points where the straight lines meet the curves the
values will be (p(a), <P\h), i// (a), v/'(fc).
According to the drawing, the right member of
equation (i.) will now stand as follows, U being written
for F{x,y),
Udyd,,
C't' {a) rb [<!> (b) Cb
Vdydx+\ Udydx
Uia)]a J* (a) J* (2/)
The four integrals represent the four areas into which the
by the dotted lines drawn parallel to the X axis. In the last
and ^iiy) are the two values of .r corresponding to one of y
the curve y = f (x) which is cut twice by any x coordinate.
divided
part of
2777 To change the order of integration in a triple integral,
from z, y, x to y, x, z, we shall have an equation of the form
F {x, y, z) dx dy dz = I.\ \ F (x, y, z) dz dx dy
ixii^i{x)}4>i(x,y) J 2^1 J ^1 U) J *i (c, a-) (ii.)
Here the most general form for the integrals whose sum is indicated b}^ 2
is that in which thf limits of y are functions of z and x, the limits of a; func-
tions of z, and the limits of z constant. Referring to the figure in (lOOG),
the total value of the integral is equivalent to the following. Every element
dxdydz of the solid described in (1907) is multiplied by F(xyz), a function
of the coordinates of the element, and the sum of the products is taken.
This process is indicated by one triple integral on the left of equation (ii ) ;
the limits of the integration for z being two unrestricted curved surfaces
2 = <^i (x, y), z = 02 (x, y) ; the limits for ?/, two cylindrical surfaces y = 4^i{x),
y = \p2 (^) ; ^■^d the limits for x, two planes x = a'j, x = x.^.
But, with the changed order of integration, several integrals may be
required. The most general form which any of them can take is that shown
on the right of equation (ii.) Solving the equation z = 0, (.'', y), let
?/, = <l>j (z, x), ?/,, = <t., (z. x) be two resulting values of y ; then the integra-
tion for y may be efiected between these limits over all parts of the solid
where tlie surface z = (pi (x, y) is cut twice by the same y coordinate.
The next integration is with respect to a;, and is limited by the cylindrical
surface, whose generating lines, parallel to OY, touch the surface z = fi{x, y).
At the ])oints of contact, x will have a maximum or minimum value for each
value of z ; therefore J^^j (a;, ;/) = 0. Eliminating y between this equation
and that of the surface, we get x = 4', (z), x = "ir^ (z) for the limits of .t.
Lastly, th.e result of the previous summations is integrated for z between
two parallel planes « = z,, z = Zj, drawn so as to include all that portion of
the solid over which the limits for x and y, already dotermiued, remain the
Bame.
TBANSFOBMATION OF A MULTTPLE INTEGRAL. 415
The reiuaiiiiiig integrations will take place between z = z^ and similar
successive parallel planes ; and, according to tbe portion of the solid which
any two of these planes intercept, the limits of x for that integral will be one
or other of the bounding surfaces, curved or plane, the limits of y, one or
other of the curved surfaces.
The general problem to change the variables in a multiple
integral,' and determine the limits from the given equations,
may now be solved.
2778 First, in the case of a double integral,
/•.r2 /-MS (j)
F{x,y)(Udy (m).,
to change from q^, y to S, v, having given the equations u = 0,
v = 0, involving the four variables.
To change y for rj, eliminate ^ between these equations ; thus y =f(-v, v)
and dy = /^ {x, ?/) dt]. Substituting these values, we shall have
F(x, y) dy = F{xJ(x, 7) }/„(»•, v) dn = F, (x, n) dn.
Also, if T)^ corresponds to i/^, the equations y^ = ^^ (f) and y^ =/(«, Vi) will
give ?7i = 4'i (^')- Similarly V2 = ^2 C^)-
Hence the integral (iii.) may now be written
F, (x, v) dx fZ^ = 2 F, (X, r,) dr, dx (iv.),
J.riJ>/'i(.r) JliJ^iC,,)
the form on the right being obtained by changing tJie order of integration, as
explained in (2775).
Next, to change x for £, eliminate y between the equations « = 0, v = ;
thus, x = g{t,, v) and dx — g^ (^, >?) dt,. Substituting as before, we shall have
F,(x,rj)dx = F^{t,-n)dl
Also, ^1 cori'esponding to x^, the equations x^ = "^^ (jj) and «i = f7 (^d ^) W°'
duce ^1 = m■^ (»?), and in the same way ^3 = m.^ (n)- Hence, finally,
F{x,y)dxdy = :z\ F,{^, ^) dii di (v.)
In the last transformation from x to ^, the most general form of tho
integrals which may be included under S has been chosen. When any of the
limits of* are constants, the process is simplified.
2779 Again, to change the variables from x,y,z to ^, r?,^,
in the triple integral,
F{x,y,z)dxclych (vi.),
■rJ>^iU-)Jxi(.r,2/)
having given the equations u = 0, 'y = 0, iv = between the
six variables x, y, z, ^, »?, t-
416 INTEGBAL CALCULUS.
First, to cliange from z to C, eliminate ^ and n between the three
equations, and let the resulting equation be z =/(*, y, 0- From this
dz = f^(x, y, 4) dZ] therefore
F(x, y, z) dz = F{x, y,f(x, y, 0}/^ (^. V, d'C = F,{x, y, dl
Also, if ^1 corresponds to the limit z^, the equations z^ = Xi (■"> !/) ^^*^
»i =/('^ 2/' ^1) give Ci = <^i (.e, 2/)- Similarly C^ = ^3(2?, ?/).
The integral (vi.) may therefore be written
F,(..,y,0fZ.^%c?^ = 2 F,(x,y,OdCdxdy
J .vi J M£) Ui(^-,y) J^i J*i(^) J*i(^,j-) (vii.),
the last form being the result of changing the order of integration, as ex-
plained in (2777). We have now to change from y to >) ; we therefore
eliminate z and ^ from the equations u, v, w, obtaining an equation of the
form y =/(f, x, j;), and proceed exactly as before. The result, as respects
the general foi'm of integral in (vii.), will be
F,(x,v,Od^dxdr, (vm.)
The order of x and t) has now to be changed by (2775). Since C is a
constant with respect to integrations for x and ?j, '^x(0' ^".(O "will also be
constants, while \^(ii,x), XjC'T, «) will be functions of the single variable x.
Suppose r] = \{ii,x) gives x = A^{i;,r]). Similarly, x = A^{!i,ri) maybe
the other limit.
At the point where x = '^i(i^) and ri = \(^,x), we shall obtain by
eliminating «, say, v = l^i(0- Similarly, from x='^i(Q and »? = A,, ( (T, :c)
suppose, we get r) = fA2(i^) for the next limit; then a general form for the
transformed integi'al will be
r^2 f/^2(^) rA2(^,r,)
F,(x,r,,Odi:dr,dx (ix.)
J^Jmi(6")JAi(^,7,)
It now remains to change from the variable x to ^. Eliminating y and z
between the equations u, v, iv, we have a result of the form x-=f {l, v, 0-
Substituting for x and dx as before, we arrive finally at the form
F,{^,V,0<-Udr,d^ (x.)
It should be noticed that the limits x = A^ii', »;), x = Ao(C, v), in (i^-), are
not necessarily different curves. They may, in some of tlie partial integrals,
be different portions of the same curve. This was exemplified in the last
integral of (2770).
MULTIPLE INTEGRALS.
The following theorems, (2825) to (2830), which are
given for three variables only, liold good for any number.
Let X, y, z be quantities which can take any positive values
MULTIPLE INTEGRALS. 417
subject to tlie condition tliat their sum is not greater tlian
unity; then
«o«r pfi--ri-^-^ I 1 m 1 n n 7 J T(l)T{m)r{n)
"^^"^"^ JoJo Jo -^ ^ r(/+m+« + l)
Here x+ij + z = I is the hmiting equation.
Proof.— Integrate for z ; then for y by (2308) ; finally for x by (2280),
and change to the gamma function by (2305).
2826 fflV>y-4--V/^/,r/^^^^g^^ ^f^ ^^^ ^"V
when ( — ) -}-(^) +( — j =r 1 is the hmiting equation.
Proof.— Substitute x = l—Y, y = (^-jY, z= (—)'', and apply (2825).
2827 When the hmiting equation is simply ^ + r) + ^ = A,
the value of the last integral becomes
.z^.n^n r{l)T(m)T{n) \
r(/+m+n + l)'
2828 The value of the same integral, taken between the
limits h and h-\-dh of the sum of the variables, is
.^..«.n-i r(/) T(m) T(n) „
Proof.— Let u be the value in (2827) ; then, by Taylor's theorem, the
value required is
which reduces to the above, by (2288).
2829 j]p-^r-'^'^'y('^'+^+-) f^'^'^^f^-
- T{l + m-\-n) X^ ^ '
3 H
418 INTEGRAL CALCULUS.
if x-\-}/-\-z = h and 11 varies from to c. In otlier words, the
variables must take all positive values allowed by tlio condi-
tion that their sum is not greater than c.
Proof.— For each value of 7t the integration with respect to x, y, z gives,
by (2828), fih) // — » r(l)r(rn)r(n)
the variations of .r, y, z not affecting h. This expression has then to be in-
tegrated as a function of h from to c.
2830 fij>->v»-'f-/[(l)''+(iy+(-^)''( didridC
'dh,
1"P' rl ' 1 '" I " \ J«
itli the hmiting equation ( — ) +(-^) +() =^-
Proof.— From (2820) by substituting x = f-^~ Y, &c.
2831 If X, y, z he n variables, taking all positive values
subject to the restriction cX'- + ?/- + r+ ... >1 ; then
CCC (lv(h/(h.&c. _ 7r'("+^)
But, if negative values of the variables are permitted, omit
the factor 2" in the denominator.
Proof.— In (2830) put l=im = &c. = l; a = j3 = &c.=l; ];i = q = &c. = 2;
f(^'^ ^^~7r\ )T' '' — I; ^'^^ t^iG expression on the right becomes
_ r(i»)r(i)
5).
The integral is = 17 (in, \) (2280) = t\ jy ' U; (2305
Hence the result. Iil(«+i)} .
But if negative values of the variables are allowed under the same re-
striction, .i"+ij^ + ~-+ ... :^ 1, each element of the integral -will occur 2" times
for once under the first hypdtliesis. Theivt'ore the former result must bo
multiplied by 2".
MULTIPLE INTEGRALS.
419
2832 If n positive variables, ,r, y, z, &c,, are limited by tbe
condition a;'^-|-?/"-^ + 2;'^-f-&c.> 1, then
<f) ( ax -\-hy-\-cz-\-kQ.) dx dy d.%
i(w-i)
_i(w-l) PI
2'-^r{i(«+i)}Jo^'
wliere F = a^ + K^ + c' + &c.
Proof. — Change the variables to £, rj, ^ by the
oi'thogonal transformation (171*9), so that
a^ + h^ + c^ + S^c. = k\ and ax + hy + cz + &o. = H.
The intearral then takes the form '
cp (JcO di ch] cU ,.. with k^ + >f + T" + &c. > 1 .
Now, integrate for >/, <r, &c., considering ^ con
stant, by adapting formula (282G). The limiting
equation is
aj
y
z ...
(I
b
c
^-
k
k '"
rt
V
c
»/
k
k
k '"
a"
h"
c"
<
k
k
iT '"
(- ^? V+ ( —-^ -] + &c. to 71-1 terms,
Therefore put I = m =^ &c. = l ; p = q = &c.==2 ; a = ft = &c. = v/(l -4').
The.esultis ^\ ^H) <±0 ^J^^^i^^
which is equivalent to the value above.
d^,
2833 With the same limiting equation for n variables and
the same value of k,
Proof. — Making the same orthogonal transformation as in (2832), the
. , , 1 , {[{ ipikDdldndi:...
integral changes to J J J . . . -^__^,-_,_^,
Considering ^ constant, the integration for the remaining variables is efiected
by (2830). Adapting the integral to that formula, we have
for the limiting equation.
420 INTEGRAL CALCULUS.
Here I = m = &c. = 1 ; p = q = &c.= 2; a = ft = &c. = 7(1-^) ;
f (h) = _ ; c = 1 : and the reductions are similar to those in (2832).
2834 If in (2832-3) negative values of tlie variables are
admitted (since the limiting equation is satisfied by such),
each element of the integral with respect to »?, t, &c. will then
occur 2''"^ times, and therefore the result in each case must
be multiplied by 2''"^, and the limits of the integration for ^
will be —1 and 1 instead of and 1.
EXPANSIONS OF FUNCTIONS IN CONVERGING
SERIES.
The expansion of a function by Maclaurin's theorem (1507)
can be at once effected if the ■n}^^ derivative of the function is
known, or if merely the value of the same, when the inde-
pendent variable vanishes, is known. Some ?z*^^ derivatives of
different functions, in addition to those given at (1461-71),
are therefore here collected. When the general value would
be too complicated, the value for the origin zero alone
is given.
DERIVATIVES OF THE /i*'^ ORDER.
The following is a general formula for calculating the n^^
derivative of a function of a function.
If 7/ be a function of z, and z a function of x,
w^iere r = 1, 2, 3, ... 7?. successively, and a is put =zm each
term of the expanded binomial, after differentiation.
Proof.— Assume ?/„^ = A,y:+ ~2/2z+ rf 2/3-- + ••• + ryf 2/'«-
To determine any coefficient A,., form r equations from this by making
y = z, 2^ z^, ... z'' in succession : nmltiply these r equations respectively by
yz-\ —C(r, 2) z-\ G(r, 3) z-\ ... ( — l)'*^'?;-'", and add the results. All the
coefficients excepting A^ disappear. This is shown by differentiating the
equation (1 -x)'" = 1-rx + G (r, 2) z'-C (r, 3) ;»»+... d= x*-
EXPANSIONS OF FUNCTIONS- 421
successively for x, and making x zero after each diflfereutiation. Thus, finally,
with a put = z, after expanding and differentiating the binonaial.
2853 Examples. — The formula may be applied to verify
equations (1416-19).
Jacobi's formula (1471) may also be obtained by it.
2854 (hrv+Dx siu-' .V
__ 1.3... (2/1-1) U n_y l.3C{n,2) ^,
"" 2''{l-.if(l-j:y^ I '^n-1 2/1-1. 2/i-3
l-3-5C(^^,3) Z-^+..,±Z"L wliere Z = i=^.
2/1-1. 2/i-3. 2/1-5 ^ ^ V l-\-x
Proof. (sin-^ cb)(„,i), = \{l-x)-^- {l + x)-^n. (1434).
Expand the right member by (1460).
2855 </„;r tan-^t-. This derivative is obtained in (1468).
The following is another method, which also includes the
result in (1469).
1 t ( 1 1
.(1).
tan" ic — , , o c, I
l«-l(-l)«f 1 1 )
••• by (1425) d„^ tan- . = — 1^ I ^^^^ - ^— y„ j •
Put a; = cote, therefore x±i= y(H-a;'') (cos ± z sin 0), which values
substituted in (1) convert the equation, by (767), into
(tan-^ x}„^ = (-1)"-' \n-l sin" sin nO.
2856 rf„;,{e^'°'^cos(c«'siua)} = e^«°^«cos(.^sina+n«).
Proof. — By Induction.
LAGRANGE'S METHOD.
2857 Lemma.— The ?^"' derivative of a function u=f(x)
will, by Taylor's theorem (1500), be equal to 1.2 ... % times
the coefficient of h"" in the expansion of fix-\-h) in powers of
h by any known method.
422 INTEGRAL CALCULUS.
Let u = a-\-hx-\-Ciir, and therefore ?/^ = i + 2c,c; tlien
d^_^{a-\-hx-\-cx^Y is equal to either of the following series,
with the notation of (2451-2).
2858 i(H,,n-.,,.[nS . n^^r'cu n^:r'c^n- ■ ].
^005 1 U U^ I 1 (r) + l(r-l) j^2 + • •• + l(r,)l[r-Z,) ,^o,, + •••]'
2859 or, putting 2?i = 7?i and Aac — lr = (f,
Pkoop. — Cbangitig x into x + /i in ?t", it becomes (7t + «^7i + c7(.')". Tlien,
by (2857), drxU" will be = | r times the coefficient of h^ in tlie expansion of
this trinomial. (2858) is the result, and it may be obtained by expanding
{(ic + ttj.h) + ch^}" as a binomial, and collecting the coefficients of A'' from the
subsequent expansions. The value (2869) is found by taking
expanding, collecting coefficients of h^, and multiplying by 1.2... r, as
before.
2860 Ex.— To find d„^(a' + x-)". Applying fornuila (2859), we have
u = a^ + x^, ti^ = 2x, q= 2a, r = n. Therefore
(a' + x'r = (2n)'."; I X-+ f^;'~^], aV-^
L 2)1 (2ii — l)
1 . 2 .2?i ... {2n~ 3) )
2861 rf«.t-'= e-'[a«(2.r)«+... + :^«-'-(2.r)''-^''+&<).|,
with 7- = 1, 2, 3, &c. in succession.
Proof.— By the method of (2857). Putting e<^(^*A)' = e"^ e'"^* e"''\ expnnd
the factors containing h by (150), and Irom the product of the two series
collect the coefficients of h".
ciocn 1 siu <, /o \„sin/ ., , mr\ ,
•••+1(Tt(^^^) eo^V + — 2 j"^
EXPANSIONS OF FUNCTIONS. 423
Proof.— i„^ (cos A'- +i sill ic") = (7„,,e''-^-'. Expand the right by (2861),
putting i"-'" = e' '"-''''", since, by (760), i^"" = * sin ~ = i. Also put
^^.iin-r^l^ = eos \s^+ (iiZ^I + ^sin [a'H ^^^^" | ,
and then equate real and imaginary parts.
2864 fh.r-^^ = i-^r [^"'■+ {n+l-2'^} c^"-'^^
+ {(7(^^ + 1, 2)-2«(« + l) + 3"[e^»-''^
+ \C(n + l, 3)-2«(7(H + l,2) + 3''(» + l)-4"}e'"-^>^ + &c.]
Proof. — Let u be the function. By differentiating u it is seen that
the A's being constants. To determine their values, expand u = (e^ + 1)"*,
and also (e^ + l)"'^S by the Binomial theorem; thus
(e- + l)«-i = e("'^ •)-+(«.+ 1) e'- + a(n + l,2) e^'^'^'^+C (n + 1, 3)e(«-^)^+&c.
From the product of the two expansions the coefficients A^, A^-x, &c. may
be selected.
'-1
2865 dr,,,iim-\v = (-1) -^ \n-l or zero,
according as n is odd or even.
Proof.— By Rale IV. (ISo-i)- The first and last differential equations
(see Example 1535) are, in this case,
(l + x')y,^ + 2xy, = (i.) ; 2/(». 2)0.0 + ^ («+ 1) 2/»xo = (ii.) ;
with y^ = 1 and f/2xo = 0-
Otherwise.— B J (1468), putting x = 0.
2867 ^.osin-^t^ = 1.3^o\.. {ii-2y or ^ero,
according as n is ocZtZ or eveii.
Proof. — By differentiating (1528).
Othenvise.—As in (2865) where equations (i.) and (ii.) will become in this
case (l-a!^)i/2^ = a:?/^ (i.) 2/(„.2)xo = ^^^2/»^o ... ..-(ii-)
2869 ^4.0 {sm-\vy = 2.2\4?.6' ... {n-2r or zero,
according as ^i- is even or odd.
Proof. — As in (2865) ; equations (i.) and (ii.) being identical with those
in (2867).
424 INTEGRAL CALCULUS.
2871 d„,,i-o^(mHm-\i')
= (-l)f m2(wr-2-)(m---4-) ... [/>i--(7i-2)-],
or zero ; according as n is even or odd and > if even.
2873 f^?„ ,0 sin (r>i siu \r)
= (_1)"-T m(m2-l)(m2-3^) ... [m--(n-2)-],
or zero ; according as n is odd or ei-e?^ and >1 if odd.
mir
2875 f4^o eos rm COS \v)
2876 or
according as n is 0(7(? and > 1, or even and >0.
2877 «/„,o siu (»i cos-^r)
— (_l)fm2(m'— 2-)(?n^— 4-) ... [m^—in—2f] siu JmTr,
2879 or ^^^
= ( — l)^'m(m2-l)(m2-3-) ... [m--{n — 2f] cos-imTr,
according as n is ei'^'^^ and > 0, or odd and > 1.
Observe that, in (2871-3), sin-^0 = 0, and in (2S75-9),
cos~^ = ^i are the only values admitted.
Pp^oof.— For (2871-9). As in (2865) ; equations (i.) and Cii.) now bo-
coming in each case
(!-»') 2/2x-a32/x+»'V = (i-) 2/(«.2);ro= (■«'-«i')2/r,xo (ii.)
Otherwise.— By the method of (1533).
2880 Let y = x cot ,r, then
wcos^i^ 7r = ?/oSiu — ir+...+//,^o^(^',>*)sin— 2" '^+
...+//(n-l):rO>iSill^,
with integral values of r, from to n — l inclusive.
EXPANSIONS OF FUNCTIONS. 425
2881 Thus, denoting y„-,Q shortly by ?/„, we find, by making
n= 1, 2, 3, &G. successively in the formula,
_ 2 _ _8_ _ 32 _ 128
Proof. — Take the nth derivative of the equation a; cos a; = y sinx by
(14G0), reducing the coefficients by (1461-2), and putting x finally = 0.
2882 The derivatives of an odd order all vanish. This may be shown
independently, as follows : —
Let y = ^ (x), then <p (x) is an even function of x (1401) ; thei-efore
9"-'(x) = -f"^'(~x);
... ^2„U(0)=:_^^-'(0); ... f-^^(0) = 0.
2883 d„,,{{l+.v-y siu(mtau-^.r)} = (-1)"'^ m^^l or zero,
according as n is odd or eve7i.
ni n-2
2885 (l,,,{{\-\-a)')'' cos(mtan-i.:i?)} = (-1) ' m^^^l or zero,
according as n is even or odd.
Proof.— As in (2865). Equations (i.) and (ii.), both for (2883) and
(2885), are now (l+«') 2/2^-2 (m-1) xy^ + m (m-1) y = (i.),
and 2/(«t2)a'0 = — {m — n) {m — n — 1) y^^ (ii.)
Formula (ii.) gives the factors in succession, starting with y^ = 0, y^ =■ m
in (2888) ; and with y, = 1, y^=0 in (2885).
2887 f/.xo{(l+'?")~^cos(mtan-^a7)} = ( — l)'m^'^^ or zero,
according as n is eveii or odd.
Pkoof. — Change the sign of m in (2885).
Note. — In formulaa (2883-7) zero is the only admitted value of tan"' 0.
2889 '^"-(^l) = (-1)' "'^'^ ^^ '''^'
according as 7? is even or odd; by (1539).
2891 When |> is a positive integer,
or zero, according as n is > or < j;.
Peoof. — Put y = e"^ cos hx and z = x^ in (1460), employing (1465).
3 I
426 INTEGRAL CALCULUS.
MISCELLANEOUS EXPANSIONS.
Tlic following series are placed here for the sake of
reference, many of tliem being of use in evaluating definite
integrals by Rule V. (2249). Otlier series and methods of
expansion will be found in Articles (125-129), (149-159),
(248-295), (756-817), (1460), (1471-1472), (1500-1573).
For tests of convergency, see (239-247).
Numerous expansions may be obtained by differentiating
or integrating known series or their logarithms. These and
other methods are exemplified below.
2911 cot.i = l--i- + -l---^-i-
a: IT — x TT+ci" Utt — ,v
+ _i ^—+-L &c.
^ 27r+.r 'dir^.v ^ aTT+A-
Proof. — By differentiating the logarithm of equation (815).
2912 ^cot7r.t=i--h--i-:+ ^ ■ ^
0? cV — i a'-\-l .r— 2
1 , 1 + 1 +&e.
Proof. — By changing x into irx in (2911).
2913 taD.r = ^-i 1__|_.^
iTT+.r- f TT— .r Itt+cI.^
&c.
Proof. — By changing x into ^■t — x in (2011).
2914 cosec .r = — '
V TT— .r 77+ .J' 27r— .r
' 27r+.t' t\7r—.r tin-^-.v 47r— .r
Proof. — By adding together equations (2911, 2913), and changing x
into 53!.
EXPANSIONS OF FUNCTIONS IN SERIES. 427
2915 -.J!— =l + -i -^
smmiT m 1 — m 1 + wi
1,1 1 ^
2—m 2-^m 3— m 3+m
Proof. — By putting x = m-n- in (2914),
2916 cot. = i_?^_^-^'-&c.
cV |_2_ L4_ [o_
For proof see (1545). The reference in that article (first edition) should
be to (1541) not (1540).
2917
\JL \Jl IA
2918 coseCci^ = i--f^i^pi^A.r
Peoof. — By (2916) and the relations
tan x=. cot » — 2 cot 2x, cosec a; = cot 4a; — cot «.
2919
-kz^ -, = l+2acos.r+2a'-cos2^^+2a'cos3<i^+&c.
1— 2acoStr4-a
cos cr
^^^^ i-2acos.tM-a^
= — i-_i-^(coScr+acos2cr4-a^cos3cr+«^cos4i'+&c...)
1— a- 1—a^^
2921
— _sinf ^ _ sin^-j-a sm2.v-^a' sin3ci + a^sm4i + &c.
1— 2a cosji+a
Proof. — By (784-6) making a=l3 = x and c = a.
When a is less tlian unity and either positive or negative,
428 INTEGRAL CALCULUS.
2922
1 a^ a^
--^log(l + 2acos.iH-«^) = ttcoscr— — cos2.:r+ — cosSci— &c.
2923 tan-i_- 1 — = asino? — sin2a7+--T-sin3ci'— &c.
l+«coSci' 2 ti
Proof. — Putting z = a (cos x + i sin a;), we have
log (1 + 2) = log (1 + a cos a; 4- m sin a;)
= |log(l + 2acosa^ + a'') + aan-' ''"'"•'' (2214),
1 + a cos .«
Z^ gS jgl
and also log(l + 2) = z o "^" "q t- + &c.
Substitute the value of z and equate real and imaginary parts.
2924 Otherwise.— Ho obtain (2922),
log (1 + 2a cos X + a?) = log ( 1 + ae'') + log ( 1 + ae"*^) .
Expanding by (154), the series is at once obtained by (768).
2925 Otherwise. — Integrate the equation in (786) with respect to a, after
changing a and /3 into x, and c into — a.
2926 When a is greater than unity, put
log(l-l-2acos.r+«^) = loga^-fiog (l+2«~^ coscr+w"^),
and the last term can be expanded in a converging series by
(2922).
2927
log 2 cosher = cosc^-— ^cos2cr+icos3^'— Jcos4r-|-&c,
2928
log 2 sin ^x = —cos .r— ^cos 2jg—^ cos 3.r— ^ cos4r— &c.
2929 2'*' = sin cv—\ sin 2x-\-\ sin 3.i'— J sin 4i -f &c.
2930 i (it— .r) = sin .v-^l sin 2x-\-\ sin 3cr+i sin 4r+&c.
Proof.— (2927-30) Make a =^ ±1 in (2922-3).
2931 4^ = sin cV+i sin 3<r+i sin OcV+&c.
2932 77= 2y2(l+i"i-|4-i+-A— &c.).
Proof.— Add together (2029-30), and put ,r = .|t.
"When 71 is less than unity, and (/, = 1 + \/(l + ^r'),
2933 log(l-f w coSct) = Iog(lH-2acosjr+a-)— log(l-h«'),
EXPANSIONS OF FUNCTIONS IN SERIES. 429
and is tlierefore equal to twice the series in (2922), minus
log (!-]-«-). But if a be greater than unity, expand, as in
(2926), by
2934 log(l + ;icos.iO
= log (l+2a~^ cos.^+a"-)+loga^— log (1+a^).
2935
(l+2« coscr)" = ^ + ^1 coScr+^2 cos2.v-\-A^ cos3.v-\-&g.,
where
A = l + C(n,2)2a'+... + C(n, 2p) C (2p, p) 0"^+ ...,
A, = 2a\n+C(n,S)Sa'-{-... + C(n, 2p + l) C (2p + l, p) a-P+ ...},
and A - ^r-^(n-r-]-l)a-rA,
(n+r+l)«
If n be a positive integer, the series terminates with the
n-\-V^' term, and the values of A and ^j are also finite.
Proof. — Differentiate the logarithm of the first equation ; multiply up
and equate coefficients of sin ra; after transforming by {GG6) ; thus ^^^1 is
obtained.
To find A and A^, expand (l + 2a cos re)" by the Binomial Theorem, and
the powers of cos a; afterwards by (772).
LEGENDRE'S FUNCTION" X^.
2936 (l-2«.r+«^)-^= l+X,«+AVr+...+XX+
with x„ = -J— 4^ {^' - 1 )^
Proof. — Expand by the Binomial Theorem, and in the numerical part
of each coefficient of a" express 1.3.5 ... 2/i — 1 as j 2/< -^ 2" | /i .
Consecutive functions are connected by the relation
2937 d^x,^^ = (2;i+i) x,+^,Z„_,.
Proof. — Difierentiate the factor once under the sign of difierentiation in
the values of X„^.i and X„_i given by the formula for X„ in (2936).
A differential equation for X„ is
2938 (1-^-) d^Jin-'^xd^X^^n (w + 1) X, = 0.
430 INTEGRAL CALOFLUS.
2939 When jp is any positive integer,
1^+2^+3"+... + (/i-l)P
_ 5ji^ n^ Bm^-^ B,n^-^ B,n^-' ^^
~'^? |/9 + l 2 p +|2 \p-l 1 4 \p-3'^\ii |7i-5
concluding, according as n is even or odd, with
(_1) _ or (1) i„_i|2^
Pkoop. = . . Expcand the left side by division, and
e-^— 1 X e^ — 1
each term subsequently by (150). Again, expand the first factor of the right
side by (150), and the second by (1539), and equate the coefficients of x" in
the two results.
See (276) for the values of the series when p is 1, 2, 3, or 4. But the
general forraula there is incorrectly printed.
Let the series (2940-4) be denoted by S^,,, S'.,,, So,,, sL+i5 as
under, n being any positive integer ; then
2940 S,,= l+±+^, + ^,+&c.... = ^^"B^-
2941 S^^^l-±+l--±+&c. ... = ^^ -'-"B,...
2942 s,„ = l + ± + l, + l:+&o....= ^^^'-"B,,.
Proof. — (i.) S^n is obtained in (1545)
(ii.) 8,,-s:,, = 2l^^^
(iii.) s,„ = ^(s,„-]-s:„)
(ii.) 8,,-s:, = 2 (_4 + J^ +^"-) = ^^^- ^^'' '^'^^ ^■•"••
2943 ^..^l+.p + y;: + ^+&c-==- i;,.|2„-l '
ird.,,,,. cot TTj;
2944 4...1-5i.+^-,-i.+&c.= lf^
Proof. — By differentiating equation (2912) successively, and putting
iT = J in the result. To compute d„x cot ttx, see (1625).
EXPANSIONS OF FUNCTIONS IN SERIES. 431
2945 The following values liave been calculated by formulaa
(2940-4).
77" o( TT o. TT
tt'
0' '~"90' ' 945' ' 9450'
^' ""12' ' ~ 720' ' ~ ^30240' ' 1209600 '
TT
tt'
g, -4-j^^, "^"900' ' 161280'
TT ' _ "n"
57r^ ' GItt'
3946 T^^-(^-5)[^-(^][^-(^^
L (47r-a)-J L (47r + aj-J
0Q4.7 cosa; + cosa T o;^! ^ £!_"! H ^]
^y^< l + cosa L (TT-aj^J L (7r + a)^J L (3T-a)^J
L (37r + a)'dL (57r-aj=J
Proof, ^osa^-^o^^ = sin |(a + g)sini(a-a;) Expand the sines by
1— cosa sin^ |-a
(815) . The two n + V^ factors of the numerator divided by the corresponding
ones of the denominator reduce to
/ 2ax + cc^ \ /■■ 2ax—x'' \
\ 2nn-a/ \ 2u.n + al
(2n7r-a)V \ (2»7r + a)2
Similarly with (2947) employing (816).
2948 cos. + tan|siu.= (1+-I-J (l--|-J (l+jl^J
x(l-.^)(l + ^)(l-r^)...&.,
2949 c„s.-c„t|sin.= (1-1) (1 + ^^J (1-^J
Wl+ 2^)(l_ 2^)(l+^
Proof, cos a; + tan — sin x = SSlil^L-^. Expand the cosines by (816),
2 cos ^a
and reduce. Similarly with (2949), employing 815.
432 INTEGRAL CALCULUS.
2950 ^=4^-('^)'][^+(^:r][^+(3^)'j-
2951 ^'= [i+(;01[i+(if)1[i+(s)]-
Proof.— Change 6 into ix in (815) and (81 G).
2952 e>-2e.s.+ .
2 (1 — cos a)
Oft CO fe^ + 2 cnsa + e"'^
2953 2(1+ COS a)
= Mi^S] M^S] Ms7r:n M^n ■•■-
Proof.— Change x into ix in (29'lG-7).
FORMULJE FOR THE EXPANSION OF FUNCTIONS
IN TRIGONOMETRICAL SERIES.
2955 When X has any value between I and —I,
*W = ij>W'^''+|C:J>('')eos'iI^rf«...(i.),
where n must have all positive integral values in succession
from 1 upwards
But, if x=l or —I, the left side becomes Z^ (/) + /(/>( — /).
Proof. — Bj (2919) we have, when h is < 1,
l—h^ l + 2/iCO3 + 2;i-cos29 + 2;i'cos3^ + &c....
l — 'J.koosd + lv
Put d = ''^^ — — ; multiply each side by <p (v), and integrate for v from
— Z to I; then make h = 1. The left side becomes, by substitutiing z = v — z,
n 0-h')<t,(v)dv _[•'-" jl-lr) f (z + z) dz ^
J-^ l-2^cos^^''~''^ + /r J-^-a-(l-/0H4/isiu^g
When ^ = 1 each element of the integral vanishes, excepting for values of v
which lie near to x. Therefore the only appreciable value of the integral
arises from such elements, and in these z will have values near to zero, both
positive and negative, since x has a fixed value between I and — 1. Let these
values of z range from — /3 to a. Then between these small limits we shall
, sin' irz ttV , / , N ^ , X
have ____ = _, and ip{a: + z) = <p{x),
EXPANSIONS OF FUNCTIONS IN SFPdES. 433
and the integral takes the form
when 7; is made equal to vmity, which establishes the formula.
In the case, however, in which x = I, siu" ^^- vanishes at both limits, that
is, when z = and when 2 = — 2Z. We have therefore to integrate for a
from -/5 to 0, and also from -21 to —2l + n, a and /3 being any sra^all
quantities. The first integration gives Icp (I) as above, putting a = U. The
second integration, by substituting y = z + 2l, produces a sin\ilar form with
limits to a, and with f (ji;-2l) in the place of (j> (x) giving lij> ( — 1) when
x = l. Thus the total value of the integral is Jf (l) + i<!' ( — 0- The result
is the same when x = — l.
That the right side of equation (i.) forms a converging series appears by
integrating the general terms by Parts; thus
^ (.) eos —j—^ dv = -- ) ^ (.) s:n _^— ^ _^
(v) sm ^- ^ dv,
which vanishes when n is infinite, provided f (v) is not infinite.
Hence the multiplication of such terms by h" when )b is infinite produces
no finite result when h is made = 1, although 1" is a factor of indeterminate
value.
2955« A function of the form ({>(:x) cosnx, with % infinitely
great, has been called '' a fluctuating function,'' for the reason
that between any two finite limits of the variable x, the func-
tion changes sign infinitely often, oscillating between the
values (^(03) and —<l>(x). The preceding demonstration shows
that the sum of all these values, as x varies continuously
between the assis^ned limits, is zero.
By similar reasoning, the two following equations are
obtained.
2956 If 3> ^^las any value between and Z,
.^(,,.,)= 1 j'V(.)rf.+isrfV(,Oeos!^^:ipr^rf. ... (2).
But if a; = 0, write -^ (0) on the left; and if x=l, write
If X has any value between and ?,
2957 = 1 fV(„)rf.+ l2r[V(.)cos^!:^i^)rf«...(3).
3 K
434 INTEGRAL CALCULUS.
But if ,i'=0, write ^(}>(0) on the left; and if x = l, write
im
2958 ^ (.<■) = 4 fV ('•) rf"+ 1 sr cosi^ (".■OS ^>(/-) rfr
This formuk is true for any vahie of x between and /,
both inclusive.
But if X be > /, write (^ (,/' ~ 2ml) instead of <^{x) on the
left, where %nl is that even multiple of I which is nearest to x
in value.
If the sign of ,/' be changed on the right, the left side of
the equation remains unaltered in every case,
2959 H^v) = |2rsm^fsm^>(r)<fo (5).
This formula holds for any value of x between and I
exclusive of those values.
If X be > I, write ■±(\>{x~ 2ml) instead of (p{x) on the left,
+ or — according as x is > or < 2nil, the even multiple of I
which is nearest to x in value.
But if X be or /, or any multiple of /, the left side of this
equation vanishes.
If the sign of x be changed on the right, the left side is
numerically the same in every case, but of opposite sign.
Proof. — For (2958-9). To obtain (4) take the sum, and to obtain (5)
take the difference, of cquatJDns (2) and (o). To determine the values of
the series when x is > /, put ,e = 2ml =b '/, so that x' is < Z.
Examples.
For all values of x, from to tt inclusive,
ftrt/>/\ TT 4 ^ , cos elr , cos 5.r , o ^ ?
2960 '^'= ^ cos.r+— ;^^ + -^^:^+&c.^.
PiiOOF. — In formula (4) put <j) (x) = x and I = tt, then
f^ 7 rvsinJiv , co?,nv~\'" 2 ^
V cos nv dv = h • ^r- ' = ~ "T °^ ^j
y L n W Jo n
according as n is odd or even.
Similarly, by foi-mula (5), equation (2929) is reproduced.
For lQI values of x, from — -^tt to ^tt inclusive.
on/^i 4 { . siiiJlr , siiio.r n )
2961 *•= —Unuv-—^j:r--}-—^, ^C.K
Pkoof.— Change ,*■ into Itt—x in (29G1).
EXPANSIONS OF FUNCTIONS IN SEBIES. 435
2962 ^':l^^ = ?^-l^ + ^l^-&c.
2 e'^ — c "^ a -\-l ir-\-z^ rr + ir
Pkoof. — In formula (5) put 6 (.c) = ^<'-v_(,-ax j^,^(| i — ■„■ -^ then
2963 If i>(^') be not a continuous function between x =
and r/' = /, let the function be <p {x) from x = to x = a, and
;//(<iO from c'K = rt. to c^=/; then, in formulse (4) and (5), we
shall have (pio') or i// (.c) respectively on the left side, according
to the situation of x betAveen and a, or betvveen a and I.
But, if x = a, we must write i (</>(«) +^(«)} foi' the left
member.
Proof. — In ascertaining tlic valae of the integral in the demonstration
of (2955), we are only cjiicerned witli the form of the function cl ise to the
value of aj in question. Hence the result is not affected by the di.^continuity
unless x=a. In this case the integration for z is from — /3 to with (p (x)
for the function, and from to a with 4' G^) for the function, producing
^<p(a) + l^ (a).
2964 Hence an expression involving x in an infinite
series of sines of consecutive multiples of --^ may be found,
such that, when x lies between any of the assigned limits
(0 and a, a and h, b and c, ... h and I), the serie3 sh dl be equal
respectively to the corresponding assigned functions
provided that the integrals
\\m^^)M.) ds, j'sm'^,/; W dr., ... j'.i.i(«f )/„(.) d.
can all be determined.
2965 The same is true, reading cosim for slii' throughout,
with the additional proviso [as appears from formuLi (4)] that
the integrals
[fMdx, ^f.^dx,... U\„{,e)dx
Jo J a Jk
can also be determined.
2966 Ex. 1. — To hud in the form of a series of cosines of multiples of
X a function of « which shall be equal to the constants a, (i, or y, according
as X lies between and a, a and b, or h and vr.
436 INTEGRAL CALCULUS.
Formula (4) produces, putting l = Tr,
2 » = x 1
H 2 — cos nx
< « cos ».i; (/./! + /3 cos vi.f ax + y cos /(.c Jx >
= 1 {« (a-/3) + Z.(/3-y) + -y}
2 jz = X 1
+ — S"^^^-y cos?;x- {(ct — ft) shi7ia+(fl—y)iimnh + y sin jitt},
2967 Ex. 2. — To find a function of x having the value c, when x lies
between and a, and the value zero when x lies between a and L
By formula (4), we shall have
f '"r*^ J z' \ 7 [" ^"^^' 7 '^^ • nir(Z
cos -— (v) ax = c cos — - ctv = — sin — ,
Jo Jo ^'■"" ^
since <p (v) = c from to a, and zero from a to Z.
rpi P , / N ft , 2c C . ira TTX 1 . 27r(i '2.TZX
Ihereiore ^ (.f) = — + - j sin - cos — + -- sm cos - -
■ 1 . oira o~X , n 7
+ -3-sm-pcos-^+&e.J.
When x = a, the vaUie is |- [0 (a) + 0] = ^c, by the rule in (2963). This
may be verified by putting a = — ] in (2923).
2968 Ex. 3. — To find a function of .r which becomes equal to Jcx when
X lies between and |/, and equal to k (l — x) when x lies between II and /.
By formula (4),
</) {v) cos — r- tZy = liv cos (Zi-'+ A' {l — i') cos - txiJ.
Jo ^ Jo ^ Ji/ ^
This reduces to ^/' f 2 cos — -cos viTr- 1) = - i'^'^\ or 0,
TT^II- \ 2 / TT-ltr
according as n is, or is not, of the form 4?/i + 2. Also
rl nil rl l.n
f (v) dv = Jc {' V dv + k (l-v) du = '4- ;
Jo Jo Ji/' ^
(p (,(■) = _ -] cos —- + — r COS -— + — 2 cos -— h &
1-2-^ '"' ^ -^ IF ''^^ T" "^ Fu^ '"^ "T- ^ ^^^- 3
APPROXIMATE INTEGRATION.
2991 Let I /(.?') ^?.c be tlie integral, and let tlie curve
7/ =_/'(.'■) be drawn. By sunirainp^ the areas of tlie trapezoids,
Avliose parallel sides are the )i-{-l eipiidistant ordiuates
AVl'BOXniATE INTEGRATION. 437
2/oj ?/i5 ■•• Vm we find, for a first approximation,
f/W ilv = tzIL (,y,^+2y,+2^,+ ...+2//,._,+//») (i.)
SIMPSON'S METHOD.
2992 If V\ be tlie ordinate intermediate between y^, —■ f {a)
and y,=f[h), then, approximately,
f/(.r)rf,c = *r;^0A+4//,+;y,) (ii.)
* a ^
Proof. — Take n = ?j in formula (i.) ; write y.^ for y^, and suppose two
intermediate ordinates each equal to //j. The area thus obtained is equal to
what it would be if the bounding curve were a parabola having for ordinates
?/o' //]' Vi parallel to its axis. Otherwise by Cotes's formula (2995).
2993 A closer approximation, in terms of 2/^ + 1 equi-
distant ordinates, is given by Simpson's formula,
Proof. — We have
\ f(x)dx= \y(,v)clc+\'\f(x)dx+... + { f(x)Jx.
Jo Jo ^ — J '-^
Apply formula (ii.) to eacli integral and add tlie results, denoting by y,. the
value of y corresponding to x ^ — .
2994 When the limits are a and b, the integral can be
changed into another having the limits and 1, by sub-
stituting X = a-\-{h — a) II .
COTES'S METHOD.
Let n equidistant ordinates, and the corresponding
abscissge, be
1 2 n — 1
Vo, Vi, y-i--. Vn-i, Vn and 0, -, — . . . -^, 1.
2995 A formula for approximation mil then be
]f{d^) cLv = A,y,+A,y,-\-... + A,i/,,+ ... + A,?j, (iv.),
wli(
(_i)»" r' (<«•)'_",'
.= -1> \^^!^^,U: (-2460)
r I n — r Jo nj; — r
438 INTECmAL CALCULUS.
Proof. — The method consists in substituting for / (.'^) the integral
function
r taking all integral values from to « inclusive. Wlieu x = r, we have
\P(^r)=yr; so that \l (x) has n+l values in conimou with /(a;). The
approximate value of the integral is therefore \p (x) dx, and nui}^ be written
tis in (iv.) •''*
By substituting 1—x, it appears that
(«)
joOiX — r ]^)HX — {)l — r)
and therefore A^ = A„_,.. Consequently it is only necessary to calculate
half the number of coefficients in (iv.)
2996 The coefficients corresponding to the values of 7/ from 1 to 10 ai'e
as follows. Every number has been carefully verified, and two mis]>rints in
Berti-and corrected ; namely, 2089 for 2'J89 in line 8, and 89500 tor 89600 in
line 11.
11 = 1:
A = A
2 '
n = 2:
A = ^2
_ 1
A = |.
n=3:
^0=A
_ J_
8 '
A = -.. = f
n = 4<:
A> = ^i
7
-^-^^-^'
-^-i-
11=- 5 :
A = ^h
_ 19
288'
0-.
^-^-f.-
n = (5:
^0 = ^6
~ 84u'
q
A, =A. = -^,
- * 280
1 - "^^
n = 7: A. = A, = ^^, A, = A, = ^^^ ,
" 7 1^280' ^ "^ 17280'
^ _ 4 _ 49 A - 1 - 2989
^^-'^^-040' ^^-'^-f/lBO-
989 , . 2944 . . 4G4
8
2«3o0' '^' ~ "^7 - 14175
14175'
< A ^-*8 . 454
''^ = ^ =14175' ^^ = -2835-
APPBOXIMATE INTEGnATION. 439
2851. ^^ = ,,,= 1^, A, = A,= ^,
8DGUU- ^ ^ b'JGUO ^ ^ 224U'
. _ . _ 12i">9 , _ . _ 2S89
in A A 10067 . , 26575 , , _ 1H175
^ _ 4 _ 5r>75 .i_^__4825 TTSO?
GAUSS'S METHOD.
2997 When f{x) is an integral algebraic function of degree
2n, or lower, Gauss's formula of approximation is
Cfi.v) = J,/(.r„) + ...+^./W + ...+^«/GrJ (v.),
wliere ^'o ... c^?^ ... cr,^ are the 7?-f-l roots of the equation
rPi.v) = d,,,,,.^U-+^{.r-ir-'}=0 (vi.),
and A^= r 0.-.r).. G.-.v_OGr-.v,O...G.-.g ^^^^,
Jo (.tv— ^t'o) . . . (ctv— cr,._i) (.r,— civ+i) . . . i'-Vr—^r,,) _ _ _ ^^->^
The formula is evidently applicable to a function of any
form which can be expanded in a converging algebraic series
not having a fractional index in the first 2)i terms. The result
will be the approximate value of those terms.
Proof. — Let '•P (■'-') = (.i' — .-Co)(x—a\) ... (.» — a-,,),
and let f (x) = Q-sly(x) + B (viii.),
where /(«) is of the -211^^ degree, Q of the n — V'\ and B of the n"\ since \p{x)
is of the 71+ I"' degree.
Then the method consists in choosing a function v^ (;)!) of the ?i + 1"' degree,
so that Q 4^ (x) dx shall vanish ; and a function B of the n^^ degree, which
shall coincide with/(.7;) when x is any one of the n + 1 roots of i// {x) = 0.
(i.) To ensure that Qi/{x)dx = (). We have, by Parts, successively,
writing N for i// {x), and with the notation of (2148),
\xm=x^^\N-,l(^x^-^\N)
= x^ [ N-pxP-' [ N+p (p-l){ l^x^'-' f -v)
= &c. &c.
= X'' [ N-px''-' [ N+p (p-1) X''-' { N-...zk\ji\ ^^^N (ix.)
Now Q\P (x) is made up of terms like x^ 4> (x) with integral values of jj from
to n — 1 inclusive. Hence, if the value (vi.) be assumed for ^(x), wo
440
TXTEGnAL CALCULUS.
see, by (ix.), that (2\p (x) dx will vanish at both limits, because the factors
X and .X — 1 will appear in every term.
(ii.) Let It be the function on the right of ef|nation (v.) Then, when
X = Xr, "we see, by (vii.)* that Ay = 1, and that the other coefficients all vanish.
Hence R becomes /(.r) whenever x is a root of xp Oc) =0.
The values of the constants corresponding to the first six values of n,
according to Bertrand, are as follows. The abscissas values, only, have been
recalculated by the author.
0:
1:
= 4
X, = •2113-2487, Jc
as = -7880751;].
Ai = '5, log
•6989700
w = 2: a-„ = •11270167, Ao = A, = -^?^, log = 9-44:36975 ;
x■^ = '5 ;
X., = •88729833, A, = f, log = 9-6478175.
71 = 3: a;.^ = •06943184, ^o = ^» = '1739274, log = 9-2403G81 ;
X, = -33000948, A, — A., = ^3260726, log = 9^5133143
X, = •66999052 ;
ar, = -93056816.
X, =-04691008,
A, = A, = -1184634,
x^ = -2307653i,
A, = J3 = -2393143,
Xi = "O,
A, = -2844444,
a'3 = •76923466;
X, = -95308992.
log = 9-0735834
log = 9-3789687
loix = 9-4539975
5 : x^ = -03376524,
flji = -16939531,
X, = -38069041,
x[ = -61930959 ;
x^ = -83060469 ;
x^ = -96623476.
A, = A, = •0856622,
A, = A, = -1803808,
A, = A, = •2339570,
log = 8-9327895
log = 9-2561903
W = 9-3691360
As a criterion of the relative degrees of approximation obtained by tho
foregoing methods, Bertrand gives the following values of
f Logii+iO dx = -'^ log 2 = -2721982613.
Method of Trapezoids,
n = 10,
-2712837.
Simpson's method,
n = 10,
-272-201'2.
Cotes's ,,
n = 5,
-272-2091.
Gauss's „
n= 4,
•2721980.
For other formulae of
approximation.
see also p. 3^
CALCULUS OF VARIATIONS.
FUNCTIONS OF ONE INDEPENDENT VAEIABLE.
3028 Let ?/ rr /(.?]), and let F be a known function of x, ij,
and a certain number of tlie derivatives t/.^, 7/2^, 7/3^, &c. The
chief object of the Calculus of Variations is to find the form
of the function /(a;) which will make
Vd^ (i.)
a maximum or minimum. See (3084).
Denote //.,., ?/o.r, ^3^,., &c. by j;, q, r, &c.
For a maximum or minimum value of U, ^U must vanish.
To find S?7, let Bij be the change in y caused by a change in
the form of the function y =f(x), and let dj^, dq^ &c. be the
consequent changes in j^, q, &c-
Now, 2^ = y,^.
Therefore the new value of p, when a change takes place in
the form of the function y, is
therefore Sp = (S//)., ; that is, g {jA = '^^.
Similarly, Sq = {^p)ri
^r={Sq)^, &c (ii.)
Now m=rWdx (1488). Expand by Taylor's theorem,
.ro
rejecting the squares of ^y, Bj), Sq, &c., and we find
m = r iVJy-^VJp+V,^^...) dx,
or, denoting F„, F,„ F„ ... by N, P, Q, ...,
BU= r {my + PSpi-Q^q-^ ...) dx. (iii.)
J .To
3 h
442 CALCULUS OF VABIATIOXS.
Integrate eacli term after the first by Parts, observing that by
(ii.) \^2^dx = Sij, &c., and repeat the process until the final
integrals involve hjdx. Thus
Nhjdx is unaltered,
^F^pdx = ny-^PJydx,
j" Q^pdx = Q^p-QJy + \Q,Jydx,
3029 Hence, collecting the coefficients of 8//, ^p, S^, &c.,
BU= ^'\N-P,+ Q,,-Rs^„+ ...) ^i/clx
+ Sg,(il-~8, + T,,-...X-Sr/o(il-^^. + T,.-...)o+&c. (iv.)
The terms affected by the suffixes 1 and must have x
made equal to x^ and x^ respectively after differentiation.
Observe that P^., Q^, &c. are here complete derivatives;
Vi P» ^'j ^'5 &c., which they involve, being fvmctions of x.
Equation (iv.) is written in the abbreviated form,
3030 ZU ={Khij8.v-\-Ih-H, (v.)
The condition for the vanishing of S?7, that is, for mini-
mum value of U, is
3031 K = iV-P.+Q,.~P3.+ &c. = K),
3032 and //i-//o= (vii.)
Proof, — For, if not, we must have
r
KSydx=:E,-n,
that is, the integral of an arbitrary function (since 1/ is arbitrary in form) can
be expressed in terms of the limits of ^ and its derivatives ; which is impos-
bible. Tlierefore II,— H^ = 0. Also K = ; for, if the integral could vanish
witliout K vanishing, the/or>Ji of the fuaction ^1/ would be restricted, which is
inadmissible.
FUNCTIONS OF ONE INDEPENDENT VARIABLE. 443
The order of K is twice that of the highest derivative contained in V. Let
n be the order of K, then there will be 2n constants in the solution of equa-
tion (vi.) and the same niimber of equations for determining them. For
there are 2)i terms in equation (vii.) involving o?/i, Bij^, dp^, &c. If any of
these quantities are arbitrary, their coefficients must vanish in order that
equation (vii.) may hold; and if any are not arbitrary, they will be fixed in
their values by given equations which, together with the equations furnished
by the coefficients which have to be equated to zero, will make up, in all, 2?i
equations.
PARTICULAR CASES.
3033 I- — When V does not involve x explicitly, a first
integral of the equation K = can always be found. Thus,
if, for example,
a first integral will be
+ QP.-Q.P
The order of this equation is less by one than that of (vi.)
Proof.— We have V^=Np-\-Pq+ Qr + Bs.
Substitute the value of iV from (vi.), and it will be found that each pair of
terms involving P, Q, E, &c. is an exact differential.
3034 II- — When V does not involve y, a first integral can
be found at once, for then N= 0, and therefore K= 0, and
we have -P«— Q2.c+^3a:~~<-^c. = 0;
and therefore P—Q^.-\-Roj. — &g. = A.
3035 III- — When V involves only y and p,
V=Fp-i-A, by Case I.
3036 IV. — When V involves only i? and q,
V = Qq+Ap+B. See also (3046).
Proof. K = —P^+Q2x = 0, giving, by integration, P = Q^ + A.
Also V, = Pq+Qr = Aq+ Q,q + Qr.
Integrating again, we find V = Qq + Ap + B,
a reduction from the fourth to the second order of differential equations.
444 CALCULUS OF VABTATIONS.
3037 Ex. — To find the bracliistoclirono, or curve of quickest descent,
from a point taken as origin to a point x^ij^, measuring the axis of y down-
wards.*
Velocity at a depth y = v^gij.
Therefore time of descent = I — -=i- dx.
Here V = J^-±^ = -^(-~- +A, by Case HI.
By reduction, y (l+j?^) = — ^ = 2a, an arbitrary constant.
That is, since p = tan 6, y = 2a cos^ 9, the defining property of a cycloid
having its vertex downwards and a cusp at the origin
H,-n, reduces to -~^_^{(ph\- (p^yX) = 0.
If the extreme points are fixed, Sy^ and hj^ both vanish.
The values x^, t/i, at the lower point, determine a.
Suppose «!, but not y^, is fixed. Then h/^ is arbitrary ; therefore
its coefficient in (3) (P— Qa!+&c.)i must vanish; that is, (Vp)^ = 0, or
) — ^ C = 0, therefore «, = 0, which means that the tangent at the
l^y(l+p^)^i
lower point is horizontal, and the curve is therefoi'e a complete half cycloid.
3038 Tn the example of the brachistochrone, it is useful to notice that —
(i.) If the extreme points are fixed, ^//„, cy^ both vanish.
(ii.) If the tangents at the extreme points have fixed directions, fj\„ ^Pi
hoth vanish.
(iii.) If the curvature at each extremity is fixed in value, ^p^, cq^, ^pi, cq^
all vanish.
(iv.) If the abscissfB x^, x-^ only have fixed values, hjf„ hji are then
arbitrary, and their coefficients in II^ — I1^, must vanish.
3039 Wlien the limits x^, x^ are variable, add to the value of
Win (3029) V^dx,-VJx,.
PkoOF. — The partial increment of U, due to changes in a\ and a-^, is
^dx,+ 4^ dx, = V,dx-Y,<U,. By (2253).
dx-^ dx,
3040 Wlicn r/^1 and 7/i, o\ and t/o J^^c connected by given
equations, y^ = ^ {,,\) , i/o = x {''o)-
EuLE. — Put
2.^1= W(^i'i)—Pi} fJr, and 8//,= IxM-^Po] ^^-^'o'
* The Calculus of Variations originated with this problem, proposed by John Bcrn.ulli
in 16i)6.
FU^X'TIONS OF OXE INDEPENDENT VARIABLE. 445
and afterwards equate to zero the coefficients of dxi and dx^^,
because the values of the hitter are arbitrary.
Proof. — iji + %, being a function of x^,
(2/1 + hi) + d., iVi + hi) d^-^h = ^ (•*■. + ^•^•1) = ^ («i) + "P' (-^i) '^^i 5
therefore cij^+j^dx^ = \p' (x^) dx^, neglecting Sjulx^.
Ex.— In the brachistochrone problem (3037), the result thus arrived at
signifies that the cycloid is at right angles to each of the given curves at its
extremities.
3041 If V involves the limits x^, x^, y^, jh, Ih, Ih^ &c., the
terms to be added to 8?7 in (3029), on account of the varia-
tion of any of these quantities, are
dx^\ [V..+ VyjH+V,/2,+ ...\dx
■JXo
+ r {^oSyo+T;,g//x+T;>o+T;,gpi+&c.} dx.
In the last integral, g//o, ^j/i, ^Jh, &c. may be placed outside
the symbol of integration since, they are not functions of x.
Hence, when F involves the limits x^, x^, y^, y^, i?q, Px, &c.,
and those limits are variable, the complete expression for
gifjis
3042 8t7= p{iV-^.+ Q2.-^3a+&c.] hyclv
J Xo
%J Xq
446 CALCULUS OF VARIATIONS.
3043 Also, if y, = xp{x,) and y, = x{^\) be equations re-
stricting the limits, put
^y, = {i.'{x,)-2h] dx^ and 8//0 = [x'(«'o)-B} dx,, (3040)
The relation 7^= is unaltered, and, by means of it, the
additional integrals which appear in the value of H^—B^
become definite functions of x.
3044 Ex. — To find the curve of quickest descent of a particle from some
point on the curve ?/o = xC^'o) ^o the curve y^ = ^ {x^).
p, and T/o- Equation (3042) now reduces to
^U = r(N-P^) ^ydx + V,dx,-{V,-\Vy,podx} dx,
+ P,cy,-{Po-\ Vy,dx]cy, (I).
Now E=0 gives N-F^ = 0; therefore V = Pp + A (3035) -
.1 f /"i^' - 7^' + A
therefore \ -77 wTTT^
Vl/-?/o v/(!/-2/o)(i+i')
Clearing of fractions, and putting A = ^^ , this becomes
(2/-2/o)(l+r) = 2a (2).
^^^° ^ = ^^ = y(.-.!ici+/;i = ^k) (^>-
Hence F=l±/^; F,„ = -7, = -^ = -P. (by i:= 0),
therefore ^JyA^ = ^""^^ = ^^ ^^^•
Substituting the values (2), (3), (4), in (1), the condition H.-Uo produces
(l+2^\)dx,-{l+Po2h)dxo+PiCy^-p,cyo = 0.
Next, put for By^ and By^ the values in (3040) ; thus the equation becomes
{l+P,^'(^i)]d.r-{l+p,xi'^-o)}dx, = (5);
dx^, dx^ being arbitrary, their coefficients must vanish ; therefore
p^^P'(x,)=-l and i'lX'C'^o) =-!•
That is, the tangents of the given curves \p and x ^^ ^^e points x^yo and x^i/i
are both perpendicular to the tangent of the brachistochrone at the point a-,?/,.
Equation (2) shews that the brachistochrone is a cycloid with a cusp at
the startiug-poiut, since there y = t/q, and thorclbro jjj = 00 .
FUNCTIONS OF ONE INDEPENDENT VARIABLE. 447
OTHER EXCEPTIONAL CASES.
(Continued from 303G.)
3045 V.--Denoting //, y,, yo. ••. Vn. "by y,Pi,2h --.Pn;
and T;, F,_, T; ... T;^ by N, P„ P, ... P,,;
let the first m of the quantities y,lh,p-i, &c. be wanting in the
function F; so that
Then ir= fZ,„.P,„-fV+i).P,„+i+... (-ir-"cZ„,P. = 0,
which equation, being integrated m times, becomes
p„-4P,«+i+^z2.p.+2-...(-ir-'"^?(.-.).-p.
= Co + Ci.i'4-...+c,„_iaj"'-^ (i.),
a differential equation of the order 2n — m.
3046 VI. — Let X also be wanting in F, so that
y = f{Vm,Pm^l ■■-Pn);
then K = is the same as before, and produces the same
differential equation (i.) From that equation take the value
of P,„, and substitute it in
V^ = P,nPm+l + Pm+lPm+2+--- + Pn Pn+1'
Each pair of terms, such as Pm+2P>m+3~ <^2gPm+2Pm+ij is an
exact differential ; and we thus find
F=C + P„, + iJ^,„4.1+(P«. + 2B« + 2-4^», + 2iWl)+...
+ (PnPn-d.PnPn-l + d,,P,p,.,) - . . . (" l)^^-'"-^^(.-.-l) .PnP>n^l
+ ^ {co-\-CiX+ ... +c,^_-,x'''-^) p)„,+idx.
The resulting equation will be of the order 2n—m — l, or
m + 1 degrees lower than the original equation.
3047 ^n. — If V. be a linear function of p^, that being the
highest derivative it contains, P,, will not then contain p^.
Therefore d^^P^ will be, at most, of the order 2n — l. In-
deed, in this case, the equation ^ = cannot be of an order
higher than 2?i— 2. (Jelletf, p. 44.)
448 CALCULUS OF V ART AT TONS.
3048 VIII. — Let 2^,„ be the lowest derivative whicli V in-
volves; tlien, if P,^ =f[x), and if only the limiting values of
X and of derivatives higher than the 7?^*'' be given, the problem
cannot generally be solved. (Jelletf, p. 49.)
3049 IX. — Let N= 0, and let the limiting values of x alone
be given ; then the equation K = becomes
or, by integration, P—Qj, + B.,,,—&c. = c,
and the two conditions furnished by equating to zero the co-
efficients of Sv/i, %o) ^iz-,
{P-Q, + &G.\ = 0, (P-Q. + &c.)o = 0,
are equivalent to the single equation c = 0, and therefore
Hi — Hq = supplies but 2n — l equations instead of 2n, and
the problem is indeterminate.
3050 Let U=rVdx-\-V', where
F= F{x, 7/, j9, q ...) and V =f{x,, x,, y„ '!h,p„Pu &c.)
The condition for a maximum or minimum value of U arising
from a variation in y, is, as before, K = ; and the terms to
be added to H^ — Hq are
r;r/.To+i';;s.'/o+F;„^i^o+ ... +F;d^+F;%i+&c.
If the order of V be n, and the number of increments cIxq, S//o,
&c. be greater than n-{-l, the number of independent incre-
ments will exceed the number of arbitrary constants in K, and
no maximum or minimum can be found.
Generally, U does not in this case admit of a maximum
or minimum if either V or V contains either of the limiting
values of a derivative of an order = or > than that of the
highest derivative found in V. (JcUetf, p. 72.)
FUNCTIONS OF TWO DEPENDENT VARIABLES.
3051 Let F be a function of two dc})eudent variables ij, z,
and their derivatives with I'cspcct to ,r; that is, let
V=f{A,y,iy,q,..z,p\q ,..) (1),
FUNCTIONS OF TWO DEPENDENT VARIABLES. 449
where p, q, ... ^ as before, are tlie successive derivatives of ^,
and p', q, ... those of z.
Then, if the forms of the functions y, z vary, the condition
P' .
for a maximum or minimum value of ?7 or Vdx is
W= r{KSf/-\-K'B:^) dj+H,^H,-^H;-H^=0 ... (2).
Here K', H' involve z, }'>'■> ?'» •••5 precisely as K, jff involve
y, p, q, ... ; the values of the latter being given in (3029).
3052 First, if y and ^z are independent, equation (2) ne-
cessitates the following conditions :
^ = 0, ^'=0, H,-H,+H,'-H,'=0 (3).
The equations K = 0, K' = give y and z in terms of a;,
and the constants which appear in the solution must be deter-
mined by equating to zero the coefficients of the arbitrary
quantities ^y^, %i, ^p^, Sp^ ... Szq, S^i, ^j)^, Sjh't ... »
which are found in the equation
H,-H, + H,'-H,'= (4).
3053 The number of equations so obtained is equal to the
number of constants to be determined.
Proof.— Let V = f{x, y, y„ y^ ... i/„^, z, z^, z^ ... 2^^),
K is of order 2/t in y, and .-. of form (p (a;, ?/, y^... 1/2,,^, «, Zj, ... «(«+„)x) ••• (i-)i
K' is of order 2w in z, and .'. of form ^ (a*, y,yt ... y{m*H)xi ^t ^x ••• ^jibx) ••• (ii-)
Differentiating (i.) 2m times, and (ii.) m + n times, 3wH-n+2 equations are
obtained, between which, if we eliminate z, jJj. ... 2(3,„+„)a., we get a resulting
equation in ?/, of order 2{m-^n), whose solution will therefore contain
2(m + n) arbitrary constants. The equations for finding these are also
2 {w,-\-7i) in number, viz., 2rt in H^ — Hf^ and 'Im in H[ — H'q.
3054 Note. — The numberof equations for determining the constants is not
generally affected by any auxiliary equations introduced by restricting the
limits. For every such equation either removes a terra from (4) by an-
nulling some variation (cy, ^p, &c.), or it makes two terms into one ; in each
case diminishing by one the number of equations, and adding one equation,
namely itself.
3055 Secondly, let y and z be connected by egjiie equation
3 II
450 CALCULUS OF VABLiTTOXS.
(j, (^xy:c) = 0. y and z are tlieii found by solving simultane-
ously tlie equations
(j) {a\ y,z) = and K : <l>y = K' : <^^.
Proof — (p (x, y, z) = 0, and therefore <}> (x, y + ^y, z + Sz) = 0, wten the
forms of y and z vary. Therefore (p^^y + cpJz = (1514), Also K^y + K'h = 0,
by (2). Hence the proportion.
3056 Thirdly, let tlie equation connecting 7j and z be of tlie
more general form
4>{^^>y>p>q ... ^,p',q' ...) = (5).
By differentiation, we obtain
If (wliich rarely happens) this equation can be integrated so
as to furnish a value of ^z in terms of %, then dj)', dq\ &c.
may be obtained, by simple differentiation, in terms of Sz/, ^_p.
Generally, we proceed as follows : —
^V=my+P^p-hQh + ...+N'h + F^P+Q'^q' + (7).
Multiply (6) by X, and add it to (7), thus
,„ + {N'-^\<t>,)^z + {P'+Xi>,)^jy + (8).
The expression for SZ7 will therefore be the same as in (2), if
we replace iV by iV+Xe^^, P by P + X<^^, &c., thus
3057 BU=:^\{{N+\<l>,)-(P+\ct>X-\-...}Si/
+ [{N'-\-\<l>,)-{P'^\<t>,)^+...} Bz](Lv
+ {P+X(^,-(Q+X(^,).+ ...},8^,
-{P+X(^,-(g+X<^J,+ ...}«8i/o
+ {g+X(^,-(i?4-X<^,.).,+ ...},87>,
-;g+X(^,-(/{+X(/>,),+ ...}o87>o
&c. &c.
+ similar terms in P, Q ... p\ q ... &c. ...(0).
3058 To render W independent of the variation h, we must
FUNCTIONS OF TWO DEPENDENT VARIABLES. 451
tlien equate to zero tlie coefficient of Sz under the sign of
integration; tlius
N'+\<l>,-iP'']-H,).+ {Q'+H,)2.-&G. = (10),
the equation for determining X.
3059 Ex. (i.)— Given F= Fix, 7j,p, q ... 2), where
z='\vdx and v = F(x,y,p, q ...). :^f^
Tlie equation f is now z—^vdx = or v—Zj.= 0,
0i/ = '"j/. 9p = '"p^ 'Pa= ^1' ^^-^
<p, = 0, (j>p,= —1, ^3'= 0, the rest vanishing.
Substituting these values in (9), we obtain
SU= r[{N+Xv,-{P + Xv,),+ (Q + Xv,),^- ...} ^y + {N'+X,\8:]dx
J J-u
+ \P+\v^-(Q + Xv,)^ + ...\Jij,-{P + \v^-(Q + \v,), + ...],Si/,
Tor the complete variation DU add V^dx^—VQdxQ. To reduce the above so
as to remove Sz, we must put N' + K= 0, and therefore \ =— j N'dx. Let
\ = M be the solution, u being a function of a;, y,p,q ... »• Substituting this
expression for A, the value of cU becomes independent of ^z.
Ex. (ii.) — Similarly, if z in the last example be = j^^'" (2148), (p becomes
v—Zjjj, = 0; and, to make N'+\^ vanish, we must put A = —y^^N'.
3061 Ex. (iii.)— Let Z7= f x/l + yl + zldx (1).
Here N = ; N' = ; P = — =^==^ ; P' = J---= =; Q=0;
Q'=0; and the equations K—0, K'—O become
P, = 0, P; = 0, or
Solving these equations, we get
y^ = ?n ; z-^ = n; or y = mx-\-A ; z = nx + B.
3082 First, if x„ y„ z^, x^, y^, z^ be given, there are four equations to
determine m, n, A, and B.
This solves the problem, to find a line of minimum length on a given
curved surface between two fixed points on the surface.
3063 Secondly, if the limits .Tj, x^ only are given, then the equations
{P\ = 0, (P)o=0, (P')i = 0, (P')o=0,
are only equivalent to the two equations m = 0, n — 0, and A and B remain
undetermined.
452 CALCULUS OF VABL-ITIOXS.
3064: Thirdly, let the limits be connected by the equatlous
"We shall have (^^.^ + ^^, jh + <Pz, 2'i) ''•^'i + <Pi^, ^V i + '•I'^-i ^'-i = ^•
Substitute <}>^^ = m^cpj.^, ((>2^=: 7i^<p^^, 2h=^'>^h p'\ =■ '>i ', thus
(1 + m???i + MHj) (ZXi + Wj ^i/i + 7(i ^Zj = 0.
Eliminate dx^ by this equation from
and equate to zero the coeflBcients of ^i/i and Sz■^ ; then
m^V^ = (P)i(l + ??z?i2j + 7mi) ; ?ijT^i = (P')i(l+mm-i + nn^).
Replacing Fj, Pj by their values, and solving these equations for m and n,
we find m = m^, n = Vi-
Similarly from the equation \p (.?•„, t/^, z^) = we derive m = m^, n = n^.
Eliminating x^, y^, z^, a^^, i/^, z^ between these equationp, and
y^ = mx^+A; z^ = vx^ + B', y^ = mxQ-\-A; z^=nx^-]rB;
(--^i, 2/i, «i) = ; y\> O^o, 2/o, 2o) = ;
four equations remain for determining vi, n, A, and B.
3065 On determining the constants in the solution o/ (8056).
Denoting j;, q,r ... by l^uP^jPs ••• j "^6 liave
and for tlie limiting equation,
<l>{^^\I/,Pl,Pl, ."Pn,^,P'l,P2, ".p'm) =0.
V is of the order n in y and 7?i in z.
^ is of the order oi in y and m' in ^.
3066 Rule I. — If m Z^c > m', and n r?7/it'r > or < n, the
order of the final differential equation will he the greater of the
two quantities 2(mH-n'), 2(m' + n); and there will be a
sufficient number of subordinate equations to determine the
arbitrary constants.
3067 Rule II. — 7/'in be < m', and n < n', the order of the
final equation ic ill generally be 2(m'+n'); and its solution
may contain any number of constants not greater than the least
of the two quantities 2(m' — m), 2 (n' — n).
For the investigation, see Jelleff, pp. 118 — 127.
3068 If V docs not involve x cxplicitl}^, a single integral of
order 2{nii-n) — l maybe found. The value of V is that
given in (303:^), with corresponding terms derived from z.
FUNCTIONS OF TWO DEPENDENT VARIABLES. 453
Proof.— dV = Ndij + P,dp, + ... +P„d2}„ + N'dz + P\dp\+ ...+P'^dp^.
Substitute for ^ and iV' from the equations K = 0, K' = 0, as in (3033),
and integrate for V,
RELATIVE MAXIMA AND MINIMA.
3069 In tliis class of problems, a maximum or minimum
value of an integral, Ui = Vich, is required, subject to the
condition that another integral, U.^ = V.^^dx, involving the
J Xq
same variables, has a constant value.
Rule. — Find the maximum or minimum value of the func-
tion Ui + alJg; that is, take V^Yi + aV.;,, and afterwards
determine the constant a hy equating Ug to its given value.
For examples, see (3074), (3082).
GEOMETRICAL APPLICATIONS.
3070 Proposition I. — To find a curve s which will make
F [x, y) ds a maximum or minimum, F being a given function
of the coordinates x, y.
The equation (5), in (3056), here becomes
where p = Xg, p' = y^, x and y being the dependent variables, and s the in-
dependent variable.
In (3057), we have now, writing u for F(x, y),
N = u^, N' = My, fp = 2p, fp, = 2p';
the rest zero. The equations of condition are therefore
^ -M^— cZs(X.Cs) = and Uy — dg{\yj) = (1).
Multiplying by Xg, yg respectively, adding and integrating, the result is
\ = u,
the constant being zero.*
Substituting this value in equations (1), differentiating tta;^ and uyg, and
putting Wj = u^Xg + u^yg, we get
Vsiu^Vs-UyXg) = uxog (2),
Xg(u^Xg—U:,yg) = uyog (3).
* Bee Todhunter's "History," p. 406.
454 CALCULUS OF V ABLATIONS.
Multiplying (2) by y , and (3) by x„ and subtracting, we obtain finally
« (2/» a'2» — «» 2/2«) = «x 2/» — ^h ^s, or
3071 «_^dud^_dud. (4)^
*^"* p (Lv ds dy ds
p being the radius of curvature.
To integrate this equation, the form of u must bo known,
and, by assigning different forms, various geometrical theorems
are obtained.
3072 Proposition II.— To find the curve which will make
\F{x,y)ds^-\f{x,y)dx (1)
a maximum or minimum, the functions F and / being of given
form.
Let F{x, 7/) = n and /(x, y) = v.
Equation (1) is equivalent to \{u-\-vx^ ds.
In (3057) we now bave V=u + vp; and for (p, p'^+p^ = 1, as in (3070).
Tberefore N = u^ +pu^ ; P = Vp = v; </)j, = 2p-,
N' = u,^+pv^ ; (pp' = 2i> ; the rest zero.
Therefore, equating to zero the coefficients of ^x and cy, the result is the two
equations ^x + P^x — (v + ^p)s = 0,
n^+pvy—{\p)s = 0;
or d, (Xx,) +v, = Ux + ^>>'^x,
dsl^y,) =n^ + x^v^.
Multiplying by a;,, y, respectively, adding, and integrating, we obtain, as in
(3070), X = «, and ultimately,
Qn»7Q 1_ 1 /du_(ly_du. d^ dv\
dU/d p - ~ w \d.v ds dy ds ^ dyl'
3074 Ex. — To find a curve s of given length, such that the volume of the
solid of revolution which it generates about a given line may be a maximum.
Here \{ifx,—a})ds must be a maximum, by (3069), a? being the arbitrary
constant. The problem is a case of (3072),
u = a\ u^ = 0, v; =0, V = y\ v^ = 2y.
1 2ij
Hence equation (3073) becomes — = - "i •
Givinrr D its value, - ^^ "^^^"-'^ f where p = -f^], and integrating, the result
PP,, V dxy
— = ■' ' : from which x = \ — y-- „ — ^— .
FUNCTIONS OF TWO INDEPENDENT VARIABLES. 455
FUNCTIONS OF TWO INDEPENDENT YAEIABLES.
3075 Let V = f(:x,y,z,p,q,r,s,t),
in whicli x, y are tlie independent variables, and p, q, r, 5, t
stand for z^, Zy, z.2^, z^y, z-^y respectively (1815), z being an in-
determinate function of x and y.
Let U=\ \ Vdojdy,
and let tlie equation connecting x and y at the limits be
^ {x. y) = 0. The complete variation of Z7, arising solely from
an infinitesimal change in the form of the function z, is as
follows : —
Let F„ Fp, &c. be denoted by Z, P, Q, B, S, T.
Let ^ = {P-B,-\8y) ^z-^\8^+mp,
^ = (Q-Ty-^S,)^z+^S^p + nq,
X={Z-Pr-Qy + R2. + S,y + T,y) h.
The variation in question is tlien
3076 81/ =£(v.,=, -V',=..+<^,=,.g -^'-'S) ''"
Proof.— ^ Ydxdy=\ hVdxdy
J Xo J yo }xo]ya
J Xo J 2/0
as appears by differentiating the values of ^ and »//. But
Jj,„ d.^ -^ cZa; Jj,^ ^' "' dx ^^ -^ dx
by (2257), and T"!^ '^^ ^ '/'iz-^'i-^y-V
Hence the result.
3077 The conditions for a maximum or minimum value of
U are, by similar reasoning to that employed in (3032),
</» = 0, ^ = 0, X = ^-
456 CALCULUS OF VAmATTONS.
GEOMETRICAL APPLICATIONS.
3078 Proposition I.— To find the surface, S, which will
make [I F{x., y, z) dS a maximum or minimum, i^ being a given
function of the coordinates x, y, z. [Jelhtt, p. 276.
Here, putting F (x, y, z) = «, V=u^l +/ +
2';
\i^. F - "^ • Q=-y^J^
and V„ Vg, Vt are all zero.
dP_ V (du ^ du \ ^ ^^ a + q') r-pq.^ ^
dx ^i^f^(^^\dx ^ dzl (i-|-^/ + 22)S '
dQ^ q (du du\ ^^ (l-\-p')t-pqs
dy v/l+/ + 2' ^ d'j dz I (H-/ + 2^)t
The equation x = or Z-P^-Q, = gives
(1 + r/^) r-2pqs+ (1 +r) t 1 ( du^ du_du\Q^
(l+/ + 22)i u^/i+p' + q:'^ dx dy dzl
If E, W be the principal radii of curvature, and Z, m, n
the direction cosines of the normal, this equation may be
written
3079 i + i.+l(.|^ + .J + n3 = 0,
and according to the nature of the function u different
geometrical theorems may be deduced.
3080 Proposition II.— To find the surface S which will make
\\F{x,y,^S)dS+\\f{x,y,z)dxdy
a maximum or minimum ; F and / being given functions of
the coordinates a3, y^ z.
Let F (a;, y, z) = lo and / (x, y, z) = v. Proceeding throughout as in
(3078), we have V = u^/Y^^fT^+v,
Z = s/l+p^ + q*U, + V„
and the remaining equations the same as in that article if wo add to the
V,
reBultiiig ditloroutial equation the term — * on the left
u
APPENDIX. 457
This equation may then be put in the form
R R u\ cLv dy dz dz /
where I, m, n are the direction cosines of the normal to the
surface.
3082 Ex. — To find a surface of given area such that the volume contained
by it shall be a maximum.
By (3069), the integral | | (z-a-yT+pT^) dxdy
must take a maximum or minimum value. The problem is a case of (3080).
We have u-=—a, v = z, m^. = 0, 14^ = 0, t/^ = 0, «^ = 1 ;
and the differential equation of the surface (3081) reduces to
3083 or \^^,-\.
APPENDIX.
ON THE GENERAL OBJECT OF THE CALCULUS OF
VARIATIONS.
3084 Definitions. — A function whose form is invariable is
called determinate^ and one whose form is variable, indeter-
minate.
Let du be the increment of a function u due to a change
in the magnitude of the independent variable, hi that due to a
change in the form of the function, Du the total increment
from both causes ; then
Dv = du-^hL.
Thus, in (3042), the terms iavolving dx^ and dx^ constitute du,
and the remaining terms 8// ; the whole variation being Du.
hi is called the variation of the function u.
3085 A primitive indeterminate function, u, of any number
of variables is a function whose variation is of arbitrary but
constant form ; in other words, ^hi = 0.
3 N
458 CALCULUS OF VARIATIONS.
3086 Let V =^ F.u be a derived function, — tliat is, a func-
tion derived by some process from the function u ; F denoting
a relation between the forms, but not between the magnitudes,
of u and v.
The general object of the Calculus of Variations is to
determine the change in a derived function v, caused by a
change in the form of its primitive ?l
The particular derived functions considered are those
whose symbols are d and , denoting operations of differ-
entiation and integi^ation respectively.
SUCCESSIVE VARIATION.
3087 Let the variation of the variation, or second variation
of V due to a change in the form of the involved function,
y =f(x), be denoted by S (§F) or ^'F; the third variation by
B^V, and so on.
By definition (3085), y being a primitive indeterminate
function, and ^?/ its variation, ^-y = (1).
3088 The second variation of any derivative of y is also
zero, i.e., ^"^p, ^^q, &c. all vanish.
• Proof.- P(y„,) = B (c^,,,) = {S (hj)},^ = (chj)„, = by (1).
3089 If V = f{x, y, 2^, q, r, &c. ...), where ?/ is a primitive
indeterminate function of x, then
where, in the formal expansion by the multinomial theorem,
hj, ^j), &c. follow the law of involution, but the indices of r/^,
dp, &c. indicate repetition of the operation dy, d^, &c. upon F.
Proof.— First, ^V=(^yd^ + dp dp + ^d^+...) V.
In finding PV, each product, such as hj dyV, is differentiated again as a
function of i/, f, q, &c. ; but, since the variations of ^?/, cp, &c. vanish by (2),
it is the same in effect as tlioiigh ey, ^j), &c. were not operated upon at all.
They accordingly rank as algebraic quantities merely, and therefore
PV= (^yd, + Spdp + ^qd,+ ...yV.
Similarly for a third differentiation ; and so on.
IMMEDIATE INTEGRABILITY OP THE FUNCTION F.
3090 Def.— When the function F (3028) is intcgrable
without assigning the value of y in terms of x, and therefore
APPENDIX. 459
integrable whatever tlie form of the function y may be, it is
said to be immediately integrahle, or integrable per se.
3091 The requisite condition for V to be immediately
integrable is that K= ^ shall be identically true.
Peoof. — Vclx must be expressible in tlie form
■where (p is independent of the form of y. Hence, a change in the form of y,
which leaves the values at the limits unaltered, will leave
I 'Vdx = ; that is, i ^K^y = 0.
But the last equation necessitates 7v = 0, since cy is arbitrary. And K=0
must be identically true, otherwise it would determine y as a function of x.
DIFFERENTIAL EQUATIONS.
GENERATION OF DIFFERENTIAL EQUATIONS.
3050 By differentiating ordinary algebraic equations, and
eliminating constants or functions, differential equations are
produced. Some methods are illustrated in tlie following
examples.
3051 From an equation between two variables and n
arbitrary constants, to eliminate the constants.
Rule. — Differentiate r times (r<n), and from the r + 1
equations any r constants may he eliminated, and thus C (n, r)
different equations of the r*^ order (3060) obtained, involving
-3-^, -r — ^, &c. Only r-|-l, however, of these equations will he
dx'" dx''"^
independent. By differentiating n times and eliminating the
constants, a single final differential equation of the n*^ order
free from constants may he obtained.
3052 Ex. — To eliminate the constants a and h from the equation
y = ax^ + hx (i.)
Differentiating, we find - "^ = 2ax + h (ii.)
Eliminating a and b in turn, we get
a. 4^ +hx = 2y, a:^ = ax- + y (iii., iv.)
dx ax
Now, differentiating (iii.) and eliminating h produces the final equation of
the second order,
^^ || -2x^ +2y = (v.)
The same equation is obtained by differentiating (iv.) and eliminating a.
3053 To eliminate tbe function <p from tbe equation z = (p (r),
where z? is a function of x and //. We have
Therefore '^. i
^v r V /
GENERATION OF DIFFERENTIAL EQUATIONS. 461
3054 To eliminate </> from 'ii = (p {v), where ti and v are func-
tions of X, y, z.
Consider x and y the independent variables, and differ-
entiate for each separately, thus
^*x + ^^z ^^:c = 1>'{v) {I'x + ^z ^x) ,
Uy + U,Zy = <}>'{v) {Vy+V,Z^),
and, by division, <{>' (v) is eliminated.
3055 To eliminate (p^, <p2, ... <pn from the equation
F {x, y, z, «^i(ai), faW, ... i>nM] = 0,
where a^, a^, ... a^ are known functions of x, y, z.
Rule. — Differentiate for x and y as independent variahles,
forming the derivatives of ¥ of each order ^ up to the (2n — 1)*^
in every possible ivay ; that is, F; F^, ¥y; Fg^., F.^^, Fj^,; Sfc.
There will be 2n^ unhnoivn functions, consisting of (pi, (p^, ... ^«
and their derivatives, and 2n'^ + n equations for eliminating
them.
3056 To eliminate ^, <^i(g, (p.,{^), ... <p,,{l) between the
equations
F [x,y,z,l, .p,{^), U^) ... <PM] = ^^
f{x,y,z,K,i>,(^),<p,i^.)...<p^{'.)\=0.
Rule. — Consider z a,nd ^ functions of the independent
variables x, y, and form the derivatives of F and f up to the
2n — 1*^ order in the manner described, in (3055). There ivill
be 4n^+n functions, and 4n^ + 2n equations for eliminating
them.
3057 To ehminate ^ from the equation
F{x,y,z,w,<p{a,^)} = 0,
where a, j3 are known functions of x, y, z, u\
Rule. — Consider x, y, z the independent variables. Dif-
ferentiate for each, and eliminate <p, (p„_, (p^ between the four
equations.
462 DIFFERENTIAL EQUATIONS.
DEFINITIONS AND RULES.
3058 Ordinary differential equations involve the derivatives
of a single independent variable.
3059 Partial differential equations involve partial deriva-
tives, and therefore two or more independent variables are
concerned.
3060 The order of a differential equation is the order of the
highest derivative which it contains.
3061 The degree of a differential equation is the power to
which the highest derivative is raised.
3062 A Linear differential equation is one in which the
derivatives are all involved in the first degree.
3063 The complete primitive of a differential equation is
that equation between the primitive variables from which the
differential equation may be obtained by the process of differ-
entiation.
3064 The general solution is the name given to the complete
primitive when it has been obtained by solving the given
differential equation.
Thus, reverting to the example in (3051), equation (i.) is the complete
priviifive of (v.) which is obtained from (i.) by diilerentiation and elimination.
The differential equation (v.) being given, the process is reversed.
Equations (iii.) and (iv.) are called the first integrals of (v.), and equation
(i.) the final integral or general sohifion.
3065 ^ particular solution, or particular integral^ of a
differential equation is obtained by giving particular values to
the arbitrary constants in the general solution.
For the definition of a singular solution, see (3068).
3036 To find when two differential equations of the first
oi'dor have a common primitive.
Rule. — ViffrrentiatG each rguatioii, and eliminate its
arbitrary constant. The tivo results will agree if there is a
common primitive jivhichy in thai case, will be found by elimi-
nating y, between the given equations.
Ex, — Apply the rule to equations (iii.) and (iv.) in (3052).
SINGULAR SOLUTIONS. 463
3067 To find when two solutions of a differential equation,
each, involving an arbitrary constant, are equivalent.
Rule. — Eliminate 07ie of the variables. The other will also
disappear, and a relation betiveen the arbitrary constants ivill
remain.
Otherwise, if V=G, v = c he the two solutions : Fand v
being functions of the variables, and G and c constants ; then
dV dv__dV du_
cLv dy dy d.v
is the required condition.
Proof. — V must be a function of v. Let V= (p(v); therefore Fj = ^^v^
and Fj,= 'pv'^'y't then eliminate <p„.
Ex. — tan"' (aj + 7/) +tan"' {x — y) = a and x^-\-2hx- = 2/^ + 1 are both solu-
tions of 2xyy^ =■ x^ + y^ + ^' Eliminating y, x disappears, and the resulting
equation is h tan a. = 1 .
SINGULAR SOLUTIONS.
3068 Definition. — "A singular solution of a differential
equation is a relation between x and y which satisfies the
equation by means of the values which it gives to the differ-
ential coefficients t/.^, y.^j., &c., but is not included in the com-
plete primitive." See examples (3132-3).
3069 To find a singular solution from the complete primitive
^{x,y,c) = 0.
Rule I. — From the complete primitive determine c as a
function ofx, by solving the equation j^ = 0, or else by solving
Xc = 0, and substitute this value of c in the ptrimitive. The
result is a singular solution, unless it can also be obtained by
giving to c a constant value in the primitive.
3070 If the singular solution hivolves j only, it results from
the equation jc=0 only, and if it involves x only, it results
from Xc = only. If it involves both x and y, the two equa-
tions Xc = 0, yc = give the rame result,
OF TKB
TTNI^ERSITT
464 DIFFERENTIAL EQUATIONS.
3071 WJien the primitive equation (p (xyc) = is a rational
integral function, ^c = '^(^1/ ^^ used instead o/ Xg = or
ye = 0.
Proof. — Let <p (x, y, c) = be expressed in the form
y = fChc) (1).
Then, if c be constant, yx=fx (2);
and, if c varies, yx=fx+fcCx (3)-
When c is constant, the differential equation of which (1) is the primitive is
satisfied by the vahie of ?/^ in (2). But it will also be satisfied b}' the same
value of y^ when c is variable, provided that either fc = or /^ = co , and in
either case a solution is obtained which is not the result of giving to c a
constant value in the complete primitive; that is, it is a singular solution.
But f^ = is equivalent to y^ = 0, and /j. = oo makes y^ = (Xi , and therefore
X = constant.
GEOMETRICAL MEANING OP A SINGULAR SOLUTION.
3072 Since tlie process in Rule I. is identical with that
employed in finding the envelope of the series of curves
obtained by varying the parameter c in the equation
(j) (x, y, c) -— ; the singular solution so obtained is the equa-
tion of the envelope itself.
An exception occurs when the envelope coincides with one
of the curves of the system.
3073 Ex. — Let the complete primitive be
w = ca;+ v^l— c", therefore yc = x ; yc=^^ gives c = — -
Substituting this in the primitive gives y = vl + x"^, a singular solution. P
is the equation of the envelope of all the lines that are obtained by varying
the parameter c in the primitive; for it is the equation of a circle, and the
pinmitive, by varying c, represents all lines which touch the circle. See also
(3132-3).
3074 "The determination of c as a function of aj by the solution of the
equation y^ = 0, is equivalent to determining what particular primitive has
contact with the envelop at that point of the latter which cori-esponds to a
given value of x.
"The elimination of c between a primitive y=f(x,c) and the derived
equation y^ = 0, docs not necessarily lead to a singular solution in the sense
above explained.
" Por it is possible that the derived equation ?/p = may neither, on the
one hand, enable us to determine c as a function of x, so leading to a singular
solution ; nor, on the other hand, as an absolute constant, so leading to a
particular primitive.
SINGULAR SOLUTIONS. 465
" Thus the particular pi-imitive y = e" being given, the condition yc=
gives e"^ = 0, whence c is +co if a; be negative, and — oo if « be positive.
It is a dependent constant. The resulting solution J/ = does not then
represent an envelope of the curves of particular primitives, nor strictly one
of those curves. It repi^esents a curve formed of branches from two of them.
It is most fitly chai^acterised as a particular primitive marked by a singularity
in the mode of its derivation from the complete primitive."
l^Booles '■'' Differential Eqiiations,'^ Supplement, p. 13.
DETERMINATION OF A SINGULAR SOLUTION FROM THE
DIFFERENTIAL EQUATION.
3075 Rule II, — Any relation is a singular solution luhich,
ivhile it satisfies the differential equation, either involves j and
makes Py iifinite, or involves x aiid makes ( — \ infinite.
3076 " One negative feature marks all the cases in which a solution
involving y satisfies the condition p^ = ao . It is, that the solution, while
expressed by a single equation, is not connected with the complete primitive
by a single and absolutely constant value of c.
" The relation which makes p,, infinite satisfies the differential equation
only because it satisfies the condition ?/c = 0, and this implies a connexion
between c and x, which is the ground of a real, though it may be unimportant,
singularity in the solution itself.
" In the first, or, as it might be termed, the envelope species of singular
solutions, c receives an infinite number of different values connected with the
value of a; by a law. In the second, it receives a finite nnmber of values also
connected with the values of x by a law. In the third species, it receives a
finite number of values, determinate, but not connected with the values of x."
Hence the general inclusive definition —
3077 "^ singular solution of a differential equation of the
first order is a solution the connexion of which tuith the com-
plete primitive does not consist in giving to c a single constant
value absolutely independent of the value of x."
[Boole's " Differ ential Equations," p. 163, and Supplement, p. 19.
RULES FOR DISCRIMINATING A SINGULAR SOLUTION OF
THE ENVELOPE SPECIES.
3078 Rule III. — ]Y]icn py or ( — j is made infinite hy
equating to zero a factor having a negative index, the solution
^' may be considered to belong to the envelope species''
3079 "In other cases, the solution is deducible from the
3
466 DIFFERENTIAL EQUATIONS.
complete primitive by regarding t; as a constant of multiple
value, — its particular values being eitlier, 1st, dependent in
some way on the value of x, or, 2ndly, independent of x, but
still sucli as to render the property a singular one."
\_Boole's " Bifferential Equations," p. 164.
3080 Rule IV. — A solution which, ivliile it makes py infinite
and satisfies the differential equation of the first order, does not
satisfy all the higher differential equations obtained from it, is
a singular solution of the envelope species.
m-\
Ex. : i/x = my '"■ lias the singular solution y = when m is >1.
m -r
Now 2/ra; = "i O'i— 1) ••• O't— r + 1) y '" ,
and, when ?• is > m, the value y = makes y^-x infinite. The solution is,
thei-efore, by the rule of the envelope species.
3081 Rule V. — " The proposed solution heing represented hy
u = 0, let the differential equation, transformed hy making u
and X the variables, be u^-l-f (x, u) = 0. Determine the in-
f^ dn
tcgral -^- as a function of x and u, in ivhlch U is either
Jo U
equal to f (x, u) or to f (x, u) deprived of any factor ivhlch
neither vanishes nor becomes infinite when u = 0. If that
integral tends to zero ivlth u, the solution is singular" and of
the envelope species. [Boole, Sttjyplement, p. 30.
3082 Ex. — To determine whether ^ = is a singular solution or par-
ticular integral of y^ =z y (log ?/)^.
Here w = y, and I — , •' ^., = — , .
^ J 2/ (log yy log'^
As this tends to zero with y, the solution is singular.
Verification. — The complete primitive is ?/ = c"--', and no constant value
assigned to c will produce the result y =: 0.
3083 Professor De Morgan has shown that any relation
involving both x and y, which satisfies the conditions j>y = oo ,
p^=: OD , will satisfy the differential equation when it does not
make 7/2.r> as derived from it, infinite ; that it may satisfy it
even if it makes y^.^ infinite ; and that, if it does not satisfy the
differential equation, the curve it represents is a locus of
points of infinite curvature, usually cusps, in the curves of
complete primitives. [Boole, Sxpiilcmmf, p. 35.
FIEST ORDER LINEAR EQUATIONS. 4G7
FIRST ORDER LINEAR EQUATIONS.
3084 M+N-^ = 0, or Md.v-\-Ndi/ = 0,
M and N being either functions of x and. y or constants.
SOLUTION BY SEPARATION OF THE VARIABLES.
3085 This method of solution, when practicable, is the
simplest, and is frequently involved in other methods.
Ex. xy(l + x')dy=z(l + ,/) dx,
therefore /, ' ., = — rr- — jr,
l + y^ x(l + x^)
and each member can be at once integrated.
HOMOGENEOUS EQUATIONS.
3086 Here if and N, in (3084), are homogeneous functions
of X and y, and the solution is affected as follows : —
Rule. — Put y = vx, and therefore dy = vdx + xdv, and
then separate the variables. For an example , see (3108).
EXACT DIFFERENTIAL EQUATIONS.
3087 Mdx-\-Ndy = is an exact differential when
M, = N,,
and the solution is then obtained by the formula
SMd.v-\-^{N-d,{^Mda^)] dij = C.
Proof. — If F=0 be the primitive, we must have Vj. = M, Vy = N;
therefore V^y = My = JV^,. Also V ==\ Mdx + ij) (y), (p (y) being a constant
with respect to x.
Therefore N = V^ = dyj Mdx + (p' (y),
therefore <}> (t/) = ^ {N-dyJMdx} dy+G.
3088 Ex. (x'-Sx'y)dx + Of-x')dy = 0.
Here 3Iy = —Sx^ = N^. Therefore the solution is
C=.^-xhj + ^[f-o:^-dy (I -a) I dy
468 DIFFERENTIAL EQUATIONS.
3089 Observe that, if M(Jx-{-Ndij can be separated into two
parts, so that one of them is an exact differential, the other
part must also be an exact differential in order that the whole
may be such.
3090 Also, if a function of x and y can be expressed as the
product of two factors, one of which is a function of the
integral of the other, the original function is an exact differ-
ential.
ijnyi JiiX. — — cos — dx ,- cos - dy = cos — . ; — ^ = 0.
y y r y y r
Here — is the integral of the second factor. Hence the solution is
y
sin -^ = G.
y
INTEGRATINa FACTOR FOR Mdx-\-Ndij = 0.
When this equation is not an exact differential, a factor
which will make it such can be found in the following cases.
3092 I. — When one only of the functions Mx + Ny or
Mx — Ny vanishes identically, the reciprocal of the other is an
integrating factor.
3093 n. — If, ivhen Mx + Ny = identically, the equation
is at the same time homogeneous, then x~^°"^^^ is also an in-
tegrating factor.
3094 III.— If neither Mx + Ny nor Mx-Ny vanishes identic
cally, then, when the equation is homogeneous, — 1^ is an
integrating factor ; and ivhen the equation can be put in the
form <f>{xy)xdy-\-x{xj)jdK = 0, - ^.^^_-^ ^*« an integrating
factor.
Proof.— I. and IH. — From the identity
M dx + Ndy = i f (Mx + Nij) d log x>j + (Mx - Ny) d log - | ,
FIRST OBDEB LINEAB EQUATIONS. 469
assuming the integrating factor in each case, and deducing the required
forms for 31 and N, employing (3090),
11, — Put v= -^, M = a?"^ (v), N = x'"\p (v), and cly = zdv + vdx in
Mdx + N(hj and 3Ix + Ny.
3095 The general form for an integrating factor of
Mdx+Ndy = is
wliere v is some chosen function of x and y ; and the condition
for the existence of an integrating factor under that hypothesis
is that
M —N
3096 XT-^ — TT— 'iniist be a function of v.
Nv^—MVy
Proof. — The condition for an exact differential of M/jdx + NfJtdy = is
(Mii)y-= (Nn) J. (3087), Assume f^ = <!> (v), and differentiate out; we thus
obtam 2— = —-^ — -f- ,
The following are cases of importance.
3097 !• — If an integrating factor is required which is a
function of x only, we put i^i = <p {x), that is, v = x; and the
necessary condition becomes
— ^^— — - must he a function of x only.
3098 II- — If the integrating factor is to be a function of xy,
the condition becomes, by putting xy = v,
M —N
- miist be a f miction of wy only.
Ny—Mx
3099 III- — If the integrating factor is to be a function of
-^, the condition is
x
— i^ — — -^ must he a function of -^.
Mx+Ny -^ -^ X
If Mx-\-Ny vanishes, (3092) must be resorted to.
470 DIFFERENTIAL EQUATIONS.
In this and similar cases, the expression found will be a
function of v = -^ if it takes the form F (v) when y is re-
X
placed by vx.
3100 IV. — Theorem. — The condition that the equation
Mdx-\-Ndy = may have a homogeneous function of x and y
of the n^^ degree for an integrating factor, is
^^K,.-M,)+nKv^j,^^^ where t. = i^.
3101 The integrating factor will then be obtained from
Proof.— Put fx—v = x^xpiu) in (3097), thus
V Nvj.—Mv,j'
Perform tlie differentiations, and, by reduction, we get
■>pIu) Mx + Ny
The right member must be a function of u in order that 4> (u) may be found
by integration.
3102 Ex. — To ascertain whether an integrating factor, which is a homo-
geneous function of x and y, exists for the equation
(y'^ + axy^) dy — ay^dx + (x + y)(x(ly—ydx) = 0.
Here M = -(ay^ + xy + y-), N = (y^' + ax/' + xy + x-).
Substituting in the formula of (3100), we fiud that, by choosing n = -3,
the fraction reduces to ^^V-^^^ , and, by putting y = ux, it becomes -"": ^
y ^'
a function of u.
fi = x-'e ^^ -^=y-'e
the integrating factor required. It is homogeneous, and of the degree - 3
in X and y, as is seen by expanding the second factor by (150).
3103 If by means of the integrating factor ^it the equation
fLMd.e^l^Ndy = is found to have V=G for its complete
primitive, tl'ien the form for all other integrating factors will
be /"/(^)j wlicre/ is any arbitrary function.
FIRST ORDER LINEAR EQUATIONS. 471
Proof. — The equation becomes
fiMf (V) dx+fxNf(V) chj = 0.
Applying the test of integrability (3087), we have
{i^^ifiy)}, = {i^Nf{v)u.
Differentiate out, remembering that
and the equality is established.
3104 General Rule. — Ascertain by the determmcdion of an
integrating factor that an equation is solvaUe, and then seeh to
effect the solution in some more direct waij.
SOME PARTICULAR EQUATIONS.
3105 (acr+%+c) d.v-]-{a\v+b'y+c') dij = 0.
This equation may be solved in three ways.
I. — Substitute X = ^ — a, y = r] — ^,
and determine a and j3 so that the constant terms in the new
equation in E and n may vanish.
11. — Or substitute ax-\-hy-{-c = ^, a'x + b'y -{-c =v.
3106 But if a:a' =h:b\ the methods I. and II. faiL The
equation may then be written as a function of ax-\-hy.
Put z = ax -{-by, and substitute hdy = dz — adx, and after-
wards separate the variables x and z.
3107 III. — A third method consists in assuming
(Ayi + C) d^-h{A'^ + G') ch = 0,
and equating coefficients with the original equation after sub-
stituting ^=zx-\-7n-^y, V = ^-\- m.2y.
m^^, ?% are the roots of the quadratic
anv' + ih + a)m + h'=0.
The solution then takes the form
{(ami— a')(.r+m,?/) + cw*i—c' }"'"'""'
{ (am2 — a'){.v-\-m.2i/)-\-cnh—c' } «'«^-«'
472 DIFFERENTIAL EQUATIONS.
3108 Ex. (Sij-7x + 7)dx + (7ij-Sx + S)chj = 0.
PiTfc X = ^ — (1, 2/ = >? — (3, thus
(Sj,-?^) c^+(7v-3^i,) dn = (i.),
with equations for a and /3, 7a —3/3 + 7 = ; 3a — 7(3 + 3 = 0;
therefore a = — 1, /3 = (ii.)
(i.) being homogeneous, put r] = v^, and therefore drj = vd^ + ^do (3086) ;
.-. (7v'-7)Uii-\-(7v-3)^ulv = 0, or ^+pfldv = 0.
c, iV—7
The second member is integrated, as in (2080), with & = 0, and, after
reduction, we find 5 log (»j + ^) + 2 log (»/ — ^) = G.
Putting I, = x — \ and n=- y, by (ii.) the complete solution is
{y + x-Yf{y-x + \y=G.
3109 When P and Q are functions of x only, the solution
of the equation
^+Py = is y=Ce-^'^ (i.)
by merely separating the variables.
3110 Secondly, the solution of
%^-Py = (i is y = c-^"'' { c+Jeci"%/..} .
This result is obtained by the method of variation of
imrameters.
EuLE. — Assume equation (i.) to he the form of the solution^
considering the parameter C a function of x. Differentiate (i.)
on this hypothesis, and put the value of j^ so obtained in the
proposed equation to determine C.
Thus, differentiating (i.), we get ?/j. = (7j.e J ' —Fy,
therefore Q ^= CxG
J''"", therefore C = [ qJ'^"" dx + C.
Then substitute this expression for C in equation (i.).
Otherwise, writing the equation in the form (ry—Q)dx + dy = 0, the
integrating factor J '^ may be found by (3097).
3111 y^J^Py=Qf
is reduced to the last case by dividing by //" and substitutiug
^ — y •
FIEST OEBEB LINEAR EQUATIONS. 473
*3212 P,(Lv+P,d^+Q {^itIi/-^/(lv) = 0.
Pj, P3 being liomogeneoiis fimctions of ,*; and 1/ of the |/''
degree, and Q homogeneous and of the q^^"- degree, is solved
by assuming
Put y = ox, and change the variables to ,»j and v. The result
may be reduced to
which is identical in form with (3211), and may be solved
accordingly.
3213 (A, + Brr + G,y)(xdy-ydx)-(A, + B,x + G,y)dy
-(iA, + B,:c^G,y)dx = 0.
To solve this equation, put x = t, + a, y = v + ft,
and determine a and /3 so that the coefficients may become homogeneous,
and the form of (3212) will be obtained.
RICCATI'S EQUATION.
3214 U, + bu' = C.V^'^ (A).
Substitute // = ilv, and this equation is reduced to the
form of the following one, with n = Jii + 2 and a = 1. It is
solvable whenever m (2^+1) = —4t, t being or a positive
integer.
3215 d'i/,-ai/-]-bt/ = c.e (B).
I. — This equation is solvable, when n = 2a, by substituting
y = vjf, dividing by x-"", and separating the variables. We
thus obtain r— :, = af~'^dx.
c — ov
Integrating by (1937) or (1935), according as h and c in
equation (B) have the same or different signs, and eliminating
1; by y = vsf, we obtain the solution
3216 , = ^|,,.C£^^ (1).
* The preceding articles of this section are wrongly numhered. Each number and
reference to it, up to this point, should be increased by 100. The sheets were printed off
before the error was discovered.
3 p
474 DIFFERENTIAL EQUATIONS.
3217 01- ,/ = ^[- -^),..«tan I C- -iVtzM I (2),
3218 II- — Equation (B) may also be solved whenever
n — 2a , •-• • .
— =r f a positive integer.
Rule. — Write z for j in equation (B), and nt+a for a
m the second term, and transpose b and c if t he odd.
Thus, we shall have
xz^— {nt -{- a) z -{-hz' = c-x" (when t is even) (3),
xz^^ —{nt-Y a) z + cz^ = haf (when t is odd) (4) .
Either of these equations can be solved as in case (I.), when
11 = 2 {nt-\-a), that is, when — - — ' -- t. .^' having been de-
termined by such a solution, the complete primitive of (B)
will be the continued fraction
y = ■
--^ + — ^— +...+ j: +— ...ioj,
where h stands for b or c according as t is odd or even.
3219 III- — Equation (B) can also be solved whenever
71 -\~ 2 a
—- — = t a positive inte2:er. The method and result mil be
271 ^ °
the same as in Case II., if the sign of sl he changed throughout
and the first fraction omitted from the value ofj. Thus
y =
71 — a , 2n — a , , (t—l)n — a , ^v
— h— r— + -+^ T + -— (6).
Proof. — Case II. — In equation (B), substitute y = A+ - — , and equate
a ^'
the absolute term to zero. This gives ^ = , or 0.
Taking the first value, the transformed equation becomes
dx
n + a
x^-(n + a)y, + cy\=lx''.
Next, put )/, = r ^—, and so on. In this way the /"' transformed
equation (3) or (4) is obtained with ,■;; written for the ^"' substituted variable yt-
FIB ST OEBEB NON-LINE AB EQUATIONS. 475
Case III. — Taking the second value, ^ = 0, the first transformed equa-
tion differs from the above only in the sign of a ; and consequently the same
series of subsequent transformations arises, with —a in the place of a. The
successive substitutions produce (5) and (6) in the respective cases for the
values of y.
3220 Ex. u, + i<.' = cx-^. (3214)
Putting -ii = -^, -r-— 2 >
^ X dx x^
and the equation is reduced to xy^—y^if = cxi of the form (B). Here
a =1, 5 = 1, 7^ = 1, and ^-^±^=2, Case III. By the rule in (3218),
changing the sign of a for Case III., equation (3) becomes
Solving as in Case I., we put z = vx^, &c. ; or, employing formula (1)
directly,
, = ye.* ^^^^ ; and then, by (6), y = -^ - ,
3c +
the final solution.
FIRST ORDER NON-LINEAR EQUATIONS.
3221 Type
where the coefficients P^Pzi ... Pn i^ay be functions of x
and y.
SOLUTION BY FACTORS.
3222 If (1) can be resolved into n equations,
and if the complete primitives of these are
V, = c„ V, = c,, ... n = c„ (3),
then the complete primitive of the original equation will be
(V-c)iV-e) ... (F,-c) = (4).
476 DIFFERENTIAL EQUATIONS.
Proof. — Taking n = 3, assume tlie last equation. Differentiate and
eliminate c. The result is
(r,-r,y- (v-v,y {v-v,rdv,dr,dv, = (5).
By (2), dV^ = f<i {ij^—i\)dx, &c., where ft, is an integrating factor. Sub-
stitute these values in (5), rejecting the factors which do not contain differ-
ential coefficients, and the result is
which is the differential equation (1).
3223 Ex.— Given yl + ^>j,, + 2 = 0.
The component equations are ^/, + l = and 7/, + 2 = 0, giving for the
complete primitive
(y + ,r-c)(y+2x-c)=0.
SOLUTION WITHOUT RESOLVING INTO FACTORS.
3224 Class I.—Type <l> (.r, jy) = 0.
When aj only is involved with jj, and it is easier to solve
the equation for x than fov ji, proceed as follows.
liuLE. — Obtain x = f (p). Differentiate and eliminate dx
hy means of dy = pdx. Integrate and eliminate p hy means
of the original equation.
Similarly, when y =f(p), eliminate dy, &c.
3225 Ex.— Given x = ay., + hy'i, i.e., x = ap + hf (1),
dx = adp-\- 2hp dp, therefore dy = p dx = apdp + 2hp-dp,
therefore !/ = o ^ — ^ r ^ •
Eliminating jfj between this equation and (l),the result is the complete primitive
(ax + 6hy - hcf = (6a7/ - 4i'" - ac) (a' + -ihx) .
3226 Class 11.-%;.
■<<^ (;>)+.# 0^) = x(/>)-
EuLE. — Biff'erentiate and eliminate y if necessary. Inte-
grate and eliminate p by means of the original equation.
If the equation be first divided by ^{p), the form is
simphfied into
3227 //-.*<^(/>)+x(/>)-
Differentiate, and an equation is obtained of the form
»r^,+ P.r = Q, wliere P and Q are functions of p.
This may be solved by (3210), and j) afterwards eliminated.
FIEST ORDER NON-LINEAR EQUATIONS. 477
3228 Otherwise, a differential equation may be formed
between y and jj>, instead of between (c and |7.
3229 Or, more generally, a differential equation may be
formed between x or y and t, any proposed function of ^, after
which t must be eliminated to obtain the complete primitive.
3230 Glairaufs equation, which belongs to this class, is of
the form i/ =pc€+f{p).
Rule. — Differentiate, and two equations are obtained —
(1) Rr = 0, and .-. p = e; (2) s^+f(p) = 0.
Eliminate -p from the original equation hy means of (I), and
again hy means o/(2). The first elimination gives y = cx-f-f (c),
the complete primitive. The second gives a singular solution.
Proof. — For, if Rule I. (3169) be applied to the primitive y = ex +f (c),
we have x+f{c) = 0; and to eliminate c between these equations is the
elimination directed above, c being merely written for p in the two equations.
3231 Ex.1. y=px + x^/T+f.
This is of the form ?/ = i>'(p (p), and therefore falls under (3227). Differ-
entiating, we obtain xdp + dx\^l+p^-\ ''' ■ ^'^ ■ ^ ^ = 0,
\/(1+F)
since dy ^=^ pO.x; thus
(70^) + !^?)'''"+^"='''
in which the variables are separated.
Integrating by (1928), and eliminating p^ we find for the complete
primitive x^+y'^ =■ Cx.
3232 Ex. 2. y =px+V¥^'p\
This is Clairant's form (3230). DiflTerentiating, we have
^ I.. ^ 1 = 0.
dx L \/{lr — d-p-)
The complete primitive is y ■= cx+ \/(h^— ctV) ;
and the elimination of p by the other equation gives for the singular solution
a^y' — b'x" = a-F', an hyperbola and the envelope of the lines obtained by
varying c in the complete primitive, which is the equation of a tangent.
3233 Ex. 3. — To find a curve having the tangent intercepted between
the coordinate axes of constant length.
478 DIFFERENTIAL EQUATIONS.
The differential equation which expresses this property is
= «./• 4-
2/=i-+-7= (!)•
Differentiating crives -/- j ;»; H 7 [ =0 (2).
^ " dx L (l+/)t3
Eliminating p between (1) and (2) gives,
1st, the primitive ij = cx-\ — (3);
V 1 + c
2nd, the singular solution x^ + y^ = a? (4) .
(3) is the equation of a straight line ; (4) is the envelope of the lines
obtained by varying the parameter c in equation (3).
3234 Class III. — Homogeneous in x and j.
Type a^^<l>(^^,p^=.Q.
Rule. — Put j = vx, and divide by x". Solve for p, and
eliminate p by differentiating y = vx ; or solve for v, and
eliminate v by putting y = ^; and in either case separate the
variables.
3235 Ex. y=px + xyi+p\
Substitute y = vx, and therefore p = v + xv^. This gives u = p + -/_! +i^*.
Eliminate p between the last two equations, and then separate the variables.
, , . (IX 2vdv _ r.
The result is h T~r~^^ — '^'
X 1 + v
from which a; C^'H^) = (7 or a;- + r = Cc
The same equation is solved in (3131) in another way.
SOLUTION BY DIFFERENTIATION.
3236 To solve an equation of the form
Rule. — Equate the functions (p and 1// respectively to arbi-
trary constants a andh. Differentiate each equation, and
elimindte the CDiislanfs. If the results aqrce, there is a common
EIGEEB OBDEB LINEAB EQUATIONS. 479
primitive (8166), which may he found by eliminating j^ hetiueen
the equations ^ = a, ^ = b, and siibser[iiently eliminating one
of the constants hy means of the relation F (a, b) = 0.
Ex. ^-yy.+f{f-fyl) = ^'
Here the two equations x—yjj^. = a, f (y^—y^yl;) = ^>
on applying the test, are found to have a common primitive. Therefore,
eHminating y^, we obtain
Also, by the given equation, a + h = 0.
Hence the solution is f {y^ — {x + hf} = h.
HIGHER ORDER LINEAR EQUATIONS.
3237 Type g+p/J3 + ...+P„..,,.-^+/',,; = Q,
where Pi ... P„ and Q are either functions of x or constants.
Lemma. — If y^, ?/2, ... y,, be n different values of y in terms
of X, which satisfy (3237), when Q = 0, the solution in that
case ^\-ill be t/ = C^i/, + C.2ij.,-{- . . . + C„?/,,.
Pkoof. — Substitute y^, y^, ... ?/„ in turn in the given equation. Multiply
the resulting equations by arbitrary constants, G^, C^, ... C« respectively;
add, and equate coefficients of F^, P^, ... Pn ""-ith those in the original
equation.
LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS.
3238 ;A.v+«i?/(«-i):f+...+«(«-i)«/..+««3/= Q (i)-
3239 Casel.—mien Q = 0.
The roots of the auxiliary equation
m*^+aim"-i+...+«.-im+«. = (2)
being ruj, hlj, ... m,„ the complete primitive of the differential
equation will be
3240 p = Cie-^-^-+C,e--+... + C\e-«-^- (3).
If the auxiliary equation (2) has a pair of imaginary roots
480 DIFFERENTIAL EQUATIONS.
(a^ih), tliere will be in tlie value of y the coiTespondiug
terms
3241 Ae"'' cos kv-\-B(f'' s'mkv (4) .
If any real root m' of equation (2) is repeated /• times, the
corresponding part of the value of // will be
3242 {A,+A,.v+A,a;'-^...+A,_,.v^-') e^>'\
And if a pair of imaginary roots occurs r times, substitute
for A and B in (3241) similar polynomials of the r — V^^ degree
in X.
Pkoof. — (i.) Substituting y = Ce'"^' iu (1) as a particular solution, and
dividing by Ge'"^, the auxiliary equation is produced, the roots of whicli
furnish n particular solutions, y = Cje"'"^', y = C'oe'"^^, &c., and therefore, by
the preceding lemma, the general solution will be equation (2).
(ii.) The imaginary roots a ± ib give rise to the terms Ce"^''''^-'''+ C'e"^''''^,
which, by the Exp. values (766), reduce to
(G+C) e"^ cos hx + i{G- C) e"^ sin fe.c.
(iii.) If there are two equal roots m<^ = m^, put at first ???., = 1n^-\-^l. The
two terms (7ie"''^+ ae""'""'-^- become e'"-^' (C^i + G/'-O- Expand e*^ by (150),
and put G^-\-G.j = A, CJi = B in the limit when k = 0, G-^ = cc, G^ = —oo.
By repeating this process, in the case of r equal roots, we arrive at the form
(A, + A,x + A,x' + ...+A,.,x'-')e"'^^;
and similarly in the case of repeated pairs of imaginary roots.
3243 Case Il.—When Q in (3238) is a function of x.
First method. — By variation of parameters.
Putting Q = 0, as in Case I., let the complete primitive be
y = Aa-]-B(3 -h Cy + &c. to n terms (6),
a, j3, y being functions of x of the form c"'-\ The values of
the parameters A, B, G, .,., when Q has its proper value
assigned, are determined by the ;/- equations
3244 A^a +B,.(3 +to n terms = 0,
yl,«, +J9",i3,,. 4- „ -0,
^%v +y|,i3,,, + „ = 0,
A^, B,., &c. being found from these equations, their integrals
must be substituted in (6) to form the complete primitive.
EIGEEB OBBEB LINEAR EQUATIONS. 481
Proof.— Differentiate (6) on the hypothesis that A, B, Gy&a. are func-
tions of X ; thus
;y,= (^a, + i?/3,+ ...) + (^.a + B,/3+...).
Now, in addition to equation (1), n — 1 relations may be assumed between
the n arbitrary parameters. Equate then the last term in brackets to zero,
and differentiate y, in all, n — 1 times, equating to zero the second part of
each differentiation ; thus we obtain
y^ =Aa^ +BI3^ +&C. and Aj,a +B^l3 -j-&c. = 0,
y,^ =Aa,^ +B(3,^ +&C. and A^a^ +B,ft^ +&c. = 0,
y{n-l)x= ^«(«-ljx + -BA«-I)x+&C. and ^x"(»-2)x + -5x/3(H-2)a;+&C. = 0.
The n quantities A^., B^., &c. are now determined by the n — 1 equations on
the right and equation (1). For, differentiating the value of ^(,j.i)xj we have
y,^ = {Aa„^ + B(3„^ + &C.} + {^^a(,_i)x + 5^/3(„_i).. + &c.},
and if these values of ^^, y>x, ... y,ix be substituted in (1), it reduces to
^x«(»-l)x + -B,.A»-lia-+&C. = Q,
for the other part vanishes by the hypothetical equation
ynx+(hyin-i)x+---+ci>iy = o,
since the values of y^, ... y(H-i)x, and the first part of y^x: are the true values
in this equation.
3245 Case 11. — Second ifef/iot?.— Differentiate and eliminate
Q. The resulting equation can be solved as in Case I. Being
of a higher order, there will be additional constants which
may be ehminated by substituting the result in the given
equation.
3246 Ex.— Given y,,-7y. + V2y =:^ x (1).
1st Method.— Pxxiimg x = 0, the auxiliary equation is m^—7m + 12 = 0;
therefore m = 3 and 4. Hence the complete primitive o{tju—7y^+12y =
is y = Ae'''+Be"' (2).
The corrected values of A and B for the primitive of equation (1) are found
from
A,e"+ 5,6*^=0") .-. A,^-xe-'^- and A= ^i^g-^^+a.
SA,e''+4iB,e'' = xy
B,= xe-'^ and B =-^^ e-'^'+h.
Substituting these values of ^ and B in (2), we find for its complete primi-
tive y = ae\+le + -J^-
3247 2nd Method. y,,-7y.+ 12y = x (1).
S Q
482 DIFFERENTIAL EQUATIONS.
Differentiating to eliminate the term on the I'iglit, we get
The aux. equation is m'^—7m^ + l2m^ = ; therefore m = 4, 3, 0, 0,
Therefore y = Ae'^' + Be'^+Cx + D (2) ;
7/, = 4>Ae"' + dBe''+G', y,. = 16^e^ + 95e^
1 7
Substitute these values in (1) ; thus C = —; D = y— r ;
therefore, substituting in (2), y = Ae'^ + Be^+ ^ + ^44^^ before.
3248 When a particular integral of the linear equation
(3238) is known in the form ij =f{x), the complete primitive
may be obtained by adding to y that value which it would
take if Q were zero.
7
Thus, in Ex. (3247), 2/ = ?i^ + TT7 ^^ a particular integral of (1) ; and
iZ 144;
the complementary part Ae^'^ + Be^'' is the value of y when the dexter is zero.
3249 The order of the linear equation (3238) may always
be depressed by unity if a particular integral of the same
equation, when Q = 0, be known.
Thus, if y-i.+Piy-2.+P2y..+P3y= Q (!)>
and ii y = z be a particular solution when Q = ; let y = vz be the solution
of (1). Therefore, substituting in (1),
(z^, + P^z,, + P,z, + P,z)v + &c. = Q,
the unwritten terms containing i\, v^x, and v^,-.
The coefficient of v vanishes, by hypothesis ; therefore, if we put v^ = u,
we have an equation of the second order for determining w. it being found,
V = yudx-\-G.
3250 The linear equation
where A,B^ ... L are constants, and Q is a function of x, is
solved by substituting a-\-hx = e\ changing the variable by
formula (1770), and in the complete primitive putting
t = log {a + hv) .
Otherwise, reduce to the form in (3446) by putting
(lA-bd- = X^, and solve as in that article.
HIGEEB OBBEB NON-LINEAR EQUATIONS. 488
HIGHEE ORDER NON-LINEAR EQUATIONS.
3251 Ty,e K-....g.^.-g)-0-
SPECIAL FORMS.
3252 F{.v, ?/,,., ?/(,.+a).^. ... ?/„,.) = 0.
When tlie dependent variable y is absent, and 7/,.,. is the
derivative of lowest order present, the equation may be de-
pressed to the order n—r by putting 7/,,,. = z. If the equation
in z can be solved, the complete primitive will then be
y
= f 2 + f (2149).
3253 F{i/, ?/,,, 7/(,^i), ... I/,,) = 0.
If X be absent instead of y, change the independent vari-
able from Q3 to y, and proceed as before.
Otherwise, change the independent variable to y, and
make j^ (= y^) the dependent variable.
For example, let the equation be of the form
3254 F(y, 7/„ y,^, 2/3.) = (1).
(i.) This may be changed into the form
F(y, av, x,„ x,,) = by (1761, '63, and '66) ■
and the order may then be depressed to the 2nd by (3252). The solution
will thus give x in terms of y.
3255 (ii.) Otherwise, equation (1) may be changed at once into one of
the form F (y, p, py, p,^) = 0, by (1764 and '61),
the order being here depressed from the 3rd to the 2nd. If the solution of
this equation be ^ = (?/, c^, c.^), then, since dy —pdx, we get, for the com-
plete primitive of (1), x=\ ^-^ +C3.
J {y, Ci, ^2)
3256 fc=i^W.
Integrate n times successively, thus
jnx
484 DIFFERENTIAL EQUATIONS.
3257 ?hv=F{yy
Multiply by 2ij^^ and integrate, tlius
KchJ J ^-^^ ^^^' ^^{2 F{y)dy + c,]
3258 ?/«x. = i^{!/(«-i)x},
an equation between two successive derivatives.
Put y(n-i)x = ^j tben z^. = F{z), from which
Hm+_^ «•
If, after integrating, this equation can be solved for z so
that z = (l>{x, c), we have V(«_i)a- = ^ (a?, ^)5 which falls under
(3256).
3259 But if z cannot be expressed in terms of x^ proceed as
follows : —
_ ^ dz r dz
2/(n-3).-j^ j^^;
T^. „ f <i;2 f (^2; f dz
Fmally, j, = J ^^ j _... J _.;
the number of integrations and arbitrary constants introduced
being n—1.
3260 ?/«. = ^{y(«-2)4-
Put ^(«_2)^ = ^; then 2^2^. = i^(/), which is (3257), the
solution giving x in terms of z and two constants. If z can be
found from this in terms of x and the two constants, we get
Z or 2/(„_2).^. ==<^(^',^l, ^2)5
which may be solved by (3256).
3261 But if z cannot be expressed in terms of x, proceed as
in (3259).
DEPRESSION OF ORDER BY UNITY.
3262 When F {.V, ;/, 7/,, ;/,,, . . . ) =
is rendered homogeneous by considering
a\ y, ?/,., ?/2,., ?/3,,., &c.
HIGHER OBBEB NON-LINEAB EQUATIONS. 485
to be of the respective dimensions 1, 1, 0, —1, —2, &c. ; put
,v = e\ y = ze\ &c.
The transformed equation will contain the same power of e in
every term, and will reduce to the form
F{z,ze, z.,,, ...) = 0,
the order of which is depressed by unity by putting Zq = u.
3263 When the original equation is of the 2nd order, the
transformed equation in u and z may be obtained at once by
changing a\ y, ?/.,., y.,^ into 1, z, if+^j u-\-uUg, respectively.
The solution is then completed, as in example (3264).
Peoof. — We have a? = e* ; y ■=. ze^ ;
y^= e, + z; y^^^ e"" (zo, + z,) ; ys^ = e"-" (^39— «») ; and so en.
The dimensions of x, ij, y^, &c. with respect to e^ are 1, 1, 0, —1, —2, &c.
Therefore the same power of e' will occur in every term of the liomogeneous
equation.
3264 Ex.: x'y,,= {y-xy,)\
Making the above substitutions for x, y, y^, and 1/2^., the equation becomes
^20 + ^9 = — ^e-
Put Zg =: ii; thus
71^ + u = — «« = —uuz, therefore tt^ + 1 = — «r, ^ = —dz,
therefore t?ar^u=a—z (1935), therefore z^ = « = tan (a— 2),
therefore dd = cot (a— z) dz, therefore = — log & sin (a — 2) (1941).
But 6 = log X and z = -^,
X
therefore bx = cosec la — —), or bx = sec f c + — | ,
by altering the arbitrary constant.
3265 When F {.V, y, ?/,, ij,,, . . . ) =
is made homogeneous by considering cV, y, y^, y^^i, &c. to be of
the respective dimensions 1, ?i, n— 1, w— 2, &c. ; put
x = e^, 2/ = ze'"'^
and depress the order by putting Zq = u, as in (3262).
3266 When the original equation is of the 2nd order, the
486 DIFFERENTIAL EQUATIONS.
final equation between u and z may be obtained at once by
changing
.r, ;/,?/.,, ?/2.r iiito 1, z, u+nz, m«^+(2n— 1) u+n{n — l)zy
respectively.
3267 Ex.: y2,y. = ^y (l)-
With the view of applying (3265), the assumed dimensions of each
member of this equation, being equated, give
n—2 + n—l = l-\-n, therefore n =■ 4.
Thus x=:e'; y = ze''', y^ = e'' {z, + 4^z); y,,= e'' {z,, + '?z, + l'lz).
Substituting in (1), e disappears ; and by putting z, = u, z., = uu„ the equa-
tion is reduced to
0tH4«z)cZtt+(7ttH40H2 + 482--2) dz = 0,
which is linear and of the 1st order. This equation is also obtained at once
by the rule in (3266).
3268 When F(cl^ ?y, ?/.,, y,:,, ...) =
• 1 j_ \n(lx
is homogeneous with respect to ?/, y,^, y.^,-,, (tec, put y — e' ,
and remove e as before by division. The equation between u
and X will have its order less by unity than the order of F.
3269 Ex.: 2/2.+P2/x+Q2/ = (1),
P and Q being functions of x.
Here ?/ = eJ ; y^ = uy; y.^^ — {u^ + ti ) y.
Substituting, the equation becomes ?t^ + «^ + P«+Q = 0, an equation of the
1st order. If the solution gives u = (p(x,c), then | ^ (re, c) J.« = log ?/ is
the complete primitive of (1).
EXACT DIFFERENTIAL EQUATIONS.
3270 Let dU=<l> {x, y, y„ y,,, . . . ?/,,,) dx =
be an exact differential equation of the n*'' order. The highest
derivative involved will be of the 1st degree.
3271 Rule for the Solution {Sarrus). — Integrate the term
Inculcing y,,^ nnt]b resjyect to J(n-i)x only, and call the result Ui.
Find dUi, considering both x and y as variables. dU — dUi
ivill be an exact differential of the n — I*'' order.
MISCELLANEOUS METHODS. 487
Integrate this with respect to J{n-2)x only, calling the resiilt
U2, a?i(i so on.
The first integral of the proposed equation ivill he
3272 Ex. : Let dU = {y' + (2x7j-l) yl + xy2. + ^y3x} d.v = 0.
Here U^ = x%^, dU^= (2xyi^ + x'ys^ dx ;
dU-dU^ = {7/+ (2xy — l) y^-xyo^) dx = 0;
.'. TJ.i = —xy„ dU2 = —{y;^+xy2x)dx, dU—dUi—dU2—(y^ + 2xyy^T.)dx',
tlierefore JJs = xy^, and JJ =■ xSjix — xy^ + xr/ =■ C
is the first integral.
3273 Denoting equation (3270) by clJJ —■ Vdx, the series of
steps in Rule (3171) involve and amount to the single condi-
tion that the equation
N-P,-{-Q,,-R,,-\-&G.= 0,
with the notation in (3028), shall be identically true. This
then is the condition that V shall be an exact 1st differential.
3274 Similarly, the condition that V shall be an exact 2nd
differential is
P-2Q,+3i?,,-483,+&c. = 0.
3275 The condition that V shall be an exact 3rd differential
is g-3i?,+ ^ s^-^-j^ T3.+&C. = 0,
and so on. [Euler, Comm. Petrop., Vol. viii.
MISCELLANEOUS METHODS.
3276 ih.-\-Pi/.+ Qtjl=0 (1),
where P and Q are functions of x only.
The solution is y = ^e'^'''-' {2^Qe~'^'^''\b^-hIx.
Proof. — Put e^ "' =■ z, and multiply (1) by 2; then, since z^ = Pz,
'^y:r+Qzijl^O. Put zy, = u-\ .-.im^-Qz--, .-. 71 = V(2 ^ Qz-dx), &c.
3277 z/2.+ Qz/i+^ = (1),
where Q and R are functions of y only.
488 DIFFEBENTIAL EQUATIONS.
The solution
is x = lJ'"(2\Be'^'""dy)-hl:,.
Proof. — Put e^ " = 2, and multiply (1) by z.
3278 ih^'+Pi/..-+Ql/'^ = ^>
where P, Q involve x only.
Put y^., = z, and the form (3211) is arrived at.
3279 y-2.+P!/l-{-Qi/: = 0-
This reduces to the last case by changing the variable from x
to 7/ by (1763).
3280 For a few cases in which the equation
y.+Py' + Qy+B =
can be integrated, see De Morgan's " Differential and Integral
Calculus," p. 690.
3281 2/2. = ci.v+hij.
Put ax + hy = t (1762-3). Eesult U, = bt. Solve by (3239)
or (3257).
3282 (1 -^^') ih-^^'y.+q'y = 0.
Put sin~^rtf = t, and obtain y-zt + c/y = 0.
Solution, y = A cos {q sin~^ x) + B sin (q sin~^ x).
3283 (l + «ct-'0 yz,-\-axy,±Ti) ■= 0.
Put I ,'' '*' — ^= t, and obtain y.H±fv = ^ ^^ above.
i^{l+ax')
3284 LlouviUo's equation, yi.v-\-f{x)y^-\-F(y)yl = {).
Suppress the last term. Obtain a first integral by (3209), and
vary the parameter. The complete primitive is
3285 Jacohrs thcuron. — If one of the first integrals of the
equation ?/.., =/0r, y) is ?/., = <^ (.r, y, c) (i., ii.).
APPROXIMATE SOLUTION. 489
tlie complete primitive will be
Proof. — Differentiating (ii.), we obtain <p:,-\-<p<p,j =/(«, !/), and differen-
tiating this for c, f:,c + (t>c<py + (p<p,,c = 0. But, by (3i87), this is the condition
for ensuring that (p,d}j — (i>,<pdx = shall be an exact differential ; therefore fc
is an integrating factor for equation (ii.), l/z—f (j^, V, c) = ^^
Equations involving the arc s, having given
3286 ds' = (M + chf or s^ = \/l-{-yl.
3287 s = cLv-\-btj.
Here \/l -\-yl = a-^-hy^,.. Find y^^ from the quadratic equation.
3288 ci'2, = a.
Change from s to <« by (1763) ; /. — s'^s.^^. = a, .'. s~^ = 2ax-\-Ci
or
1+7/2 = — i — . ^. ^r /( 1 l)clx+i
^" 2ax-\-c' ^ ]\\2ax-\-c )
APPROXIMATE SOLUTION OF DIFFERENTIAL EQUATIONS
BY TAYLOR'S THEOREM.
3289 The following example will illustrate the method : —
Given y,, = xy,-]ry, .'. y-s., = {x'-]-2) y^,+xy.
GeneraUj, let y,,. = A,,y^,+Bjj ; //(„+i)., = A,^,y^,-\rB,,+,y.
But, by differentiation,
y(H+i).«= {A,,x-{-A',,+B,,) y^ + {A,,+B:,) y,
.' . ^,,+1 = A,,x-\- yi; + B,, and ^„+i = A,, + B'^.
But A,= x, B,= l, .'. A,= x' + 2, B, = x;
A, = x'-{-bx, B, = x' + S, &c.
Now, when x = a, let y = b and y^ = ir, then, by Taylor's
theorem (1500),
y = a-}-p {x-a) + (A,2Ji-BJj) (l^ + iA,p+B,h)^-^^^
+ &C.,
which converges when x—a is small. [De Morgan, p. 692.
3 R
DIFFERENTIAL EQUATIONS.
SINGULAR SOLUTIONS OF HIGHER ORDER
EQUATIONS.
DERIVATION FROM THE COMPLETE PRIMITIVE.
3301 Let v/,,, = <i>{'>',y,y.;y2. •■■ ^(^-dJ 0)
be the differential equation, and let its complete primitive be
^7=/(»''«j^'^ ••• ^) (^)'
containing n arbitrary constants.
3302 Rule. — Tu find the general singular suliitloii of (1),
eliminate abc ... s between the equations
y = U jx = fx> y-ix = fax • • • J y(n-i) - = f
til tax Ia2x ••• t;i(n-l)x
and
tlj Ibx tb2x ••• lb (11-1
(11-1) X
=
(n-l)x VV
(4).
Is Isx ls2x ••• Is(n-l):
The result is a differential equation of the n — V^' order, and
the integral of it, containing n — 1 arhitrarij constants, is the
sing^Uar solution.
Proof. — Differentiate (2), cousidering the parameters a, b ... s variable,
thus y^ = fx+fa 't.r + • •• +/,Sx. Therefore, as in (3171),
2/2X = /■>. if fax a. +fi,x b.v +...J\xS. = 0, as well ;
and so on up to y„^ = f„x- Eliminating a^, h^, ... 6\ between the n equations
on the right, the detei-minaut equation (4) is produced with the rows and
columns interchanged.
3303 Ex. : y-^y^ + ^^y^^-yl- (y^-Xy,^' =
The complete primitive is y =■ -j- +h,c + c(r + h'^
From which y^.=:ax + h
and the determinant equation is
pn2a, x\ ^
X +2Z/, 1 I
or
+ 2Z^.v
.(2).
.(3),
(4).
SINGULAR SOLUTIONS OF HIGEEB OUDBB EQUATIONS. 491
Eliminating a and b from (2), (3), and (4), we get the differential equation
M!, + (2x + f)cU ^ ^ , ^
the integral of -whicli, and the singular solution of (1), is
^/(iey + 4^x' + x') = a;v/(l + rc*)+log{^+ y(l + x')} + G.
[Boole, Sup., p. 49.
3304 Either of the two * first integrals' (3064) of a second
order differential equation leads to the same singular solution
of that equation.
3305 The complete primitive of a singular first integral of
a differential equation of the second order is itself a singular
solution of that equation ; but a singular solution of a singular
first integral is not generally a solution of the original equa-
tion.
Thus the singular first integral (5) of equation (1) in the
last example has the singular solution 16y-\-4<x^-{-x'^ = 0i which
is not a solution of equation (1).
DERIVATION OF THE SINGULAR SOLUTION FROM THE
DIFFERENTIAL EQUATION.
3306 Rule. — Assuming the same form (3173), a singular
solution of the first order of a differential ecptation of the n^^
order will make LYnx; infinite ; a singular solution of the
d (y(n-l)x)
second order ivill mahe ^ , ^-^"^^ , , ^ , ^•^"'^-^ , hoth infinite ;
d(y ,,_,),)' d(y („.,),) -^
and so on. [Boole, Sup., p. 51.
3307 Ex. — Taking the differential equation (3303) again,
y-^y^+i^%z-yl- (y^-xy.^y = (1),
and differentiating for y^ and ?/2j, only,
{^x' + 2x (y.-xy.J-2y.J d(y,;)-{x + 2(y^-xy^)} d(y,) = 0.
The condition ,; / = ^ requires
^x' + 2x {y..-xy,,)-2y,, = 0.
Substituting the value of yo^^ obtained from this in equation (1), and rejecting
the factor (;^- + l), the same singular integral as before is produced (3303,
equation 5).
493 DIFFERENTIAL EQUATIONS.
EQUATIONS WITH MORE THAN TWO VARIABLES.
3320 Pd.v+Qdii-^Rdz = {S (1).
P, Q, B being here functions of x, y, 2, the condition that this
equation may be an exact differential of a single complete
primitive is
3321 P (Q-R.) + Q {R-P^R (P-Q.) = 0.
Proof. — Let f^ be an integrating factor of Pdx + Qcly + Bdz = 0. Then
—fiPcIx = ijQdi/ + fiIlc1z, and, by (3187), for an exact differential, we must
have (i^Q)~ = (At-R)v Write this symmetrically for P, Q, and B, differentiate
out, and add the three equations after multiplying them respectively by P,
Q, and B.
To find the single complete primitive of equation (1),
3322 Rule. — Consider one of the variables z constant, and
therefore dz = 0. Integrate, and add (j> (z) for the constant of
integration. Then differentiate for x, y, and z, and compare
with the given equation (1). If a primitive exists, ^(z) will be
determined in terms of z only by means of preceding equations.
The complete primitive so obtained is the equation of a
system of surfaces, all of the same species, varying in position
according to the value assigned to the arbitrary constant.
3323 Ex.: (x-Sy-z)dx + (2y-dx)dy + (z-x)dz = (1).
Condition (3321) is satisfied ; therefore, putting dz = 0, we have
(x -Sy-z) dx + (2 // - 3.1') dy = 0.
Applying (3187), M,, = — 3 = iV„ and integration gives
^x"' - Sxy—zx + y- + (l> (z) = 0.
Difierentiating now for .t, y, and z,
(x-Sy- z) dx +(2y- Sx) dy + { f (,v) - x } dz = 0.
Equating coefficients with (1), f'(z)=z, therefore (f>(z) = ^z-+C.
Hence the single complete primitive is
x' + 2y^ + z- — 6xy — 2zx = C,
the equation of a system of surfaces obtained by varying the constant G.
3324 When the equation Pdx-\-Qdy-\-Rdz = is homo-
geneous, put x = 2iz, y = vz. The result, when the coefficient
of dz vanishes, is of the form
3325 Mdu-\-Ndv = i),
EQUATIONS WITH MORE THAN TWO VARIABLES. 493
solvable by (3184). Otherwise it is of the form
3336 ~ = Mdu-{-Ndv,
z
and the right will be an exact differential if a complete primi-
tive exists.
3327 Ex.: y%ilx^-zxdij-^',mjih=^^ (1).
Condition (3321) is satisfied. Patting
X = %{jZ, y = vz, dx = tidz-\--4 du, dy = vdz + z dv,
(1) becomes ^Zf + f^ + f = 0,
z oil 6v
and the solution is log {zu'v^) = G or xyz = G.
When the equation
Pdxi-Qdy + B(h = (1)
has no single primitive :
3328 Rule. — Asswne (p{x, y,z) = (2)
and differentiate ; thus
(^,dxH-<^ydy + .^,dz==:0 (3).
The form of <p being given, eliminate z and dz from (1) hy
(2) and (3). The result, being of the form
Mdx + Ndy = 0,
can be integrated, and the solution taken lolth (2) co7istitutes a
solution of equation (1), and represents a system of lines {by
varying the constant of integration) drawn on the surface
^ (x, y, z) = 0.
3329 Ex.: {l + 2m)xdx+(l-x)ydy-\-zdz = 0.
Tiie condition (3321) not being satisfied, assume «- + t/^ + z^ = r^ as the
function <p, therefore xdx + ydy + zdz = 0; and by eliminating z and dz,
^mdx — ydy = 0, the integration of which gives y^ — 4imx =. G, a cylindrical
surface intersecting the spherical surface in a system of curves (by varying
G), whose projections on the plane of xy are parabolas.
The condition that
3330 Xd.v-{- Ydi/+Zdz+rdt = 0,
where X, Y, Z, T are functions of x^ y, z, t, may be an exact
494 niFFJEIiENTIAL EQUATIONS.
differential, may be sliewn, in a manner similar to tliat of (3321) ,
to be expressed by any three of the equations
3331 Y{Z,-T,)+Z{T,-Y,) + T{Y,-Z^)=0,
T {X,- r,) +x(r, - r, ) + y{t,, -x,) = o,
X(Y,-Z,) + Y{Z,,,-X,)+Z{X^-Y,.) = 0,
the fourth being always deducible from the other equations.
If this condition is fulfilled, the solution of equation (3330)
is analogous to (3322).
Integrate as if z and t were constant, and therefore dz and
(It zero, adding for the constant of integration ^ {z, t).
Differentiate next for all the variables, and determine (jt hy
comparison with the original equation.
3332 If a single primitive does not exist, the solution must
be expressed by simultaneous equations in a manner similar
to that of (3328).
SIMULTANEOUS EQUATIONS WITH ONE
INDEPENDENT YARIABLE.
GENERAL THEORY.
3340 Let the first of n equations between n + 1 variables be
Pdd+P.du+P.ih-]- ...+P,,dw = (1),
where P, Pi ... P« may bo functions of all the variables.
Let X be the independent variable. The solution depends
upon a single differential equation of the n^^^ order between
two variables.
Solving the n equations for the ratios dv : dy : dz : &c., let
dx d.ii dz (/?/'
. dy _Q^ dz_ _ Q2. ^ _ Oil
" dx Q ' dx ~ Q' dx Q'
Differentiate the first of these equations ii — \ times, substi-
tutino: from the others the values of r;,. ... ii\,, and the result
SIMULTANEOUS EQUATIONS. 495
is u equatioDS in ,f/^,, t/^.i- ••• 1J,lv> and the primitive variables
33, ?/, 2 ... W.
Eliminate all the variables but x and y, and let the differ-
ential equation obtained be
F{x,y,y_,..,y,,) == 0.
Find the n first integrals of this, each of the form
F{x, y, ?/,. ... y(n-i)x) = G, and substitute in them the values of
2/.1-) ?/2.rj ••• i/».i-5 ill terms of x, y, z ... w, found by solving the n
equations last mentioned. Thus a system of n primitives
is obtained, each of the form F{x,, y,z ... lu) = G.
3341 The same in the case of three variables.
Here n = 2. Let the given equations be
P.,(Lv-\-Q,di/+R,dz = 0.
3342 Therefore q^^/^^q^^^ = B,pf-P.,B, = P^-F^^^
From these let ?/^, = (|) {x, y, z), z,, = ^p {x, y,c).
Therefore t/o., = «^., + <l>, y,, + 1>z -.v
Substitute the value of ^.,, and eliminate - by means of
y^, = <p (x, y, z). An equation of the 2nd order in x, _?/, //.„ //o,,.
is the result. Let the complete primitive of this be
y = X ('«j (h ^)- Then we also have (p (x, y, ^) = (/.,x {>''> «? ^^)-
These two equations form the complete solution.
FIRST ORDER LINEAR SIMULTANEOUS EQUATIONS WITH
CONSTANT COEFFICIENTS.
3343 In equations of this class, the coefficients of the
dependent variables are constants, but any function of the
independent variable may exist in a separate term.
Such equations may be solved by the method of (o34U),
but more practically by indeterminate multipliers.
3344 Ex. (1): I +7.-2/ = 0, JL+2x+5y = 0.
Multiply the second equation by m and add. The result may be written
;^>a.) +(„„ + ,) (..+ |1=-1,|=0 (.).
496
DIFFERENTIAL EQUATIONS.
5*y;-l
To mako the whole expression an exact diftercntial, put :^-— -= = ^»- This
(2);
gives
^, ^n'==J^
(1) now becomes "^ (^ + ^»?/) + (2m + 7)ix + my) = 0,
and the solution is x + my = ce- -'"^'''^ and x + vi'y = c'e--'"'*"'-.
Solving these equations, and substituting the values (2),
iy =zce-^'''^^-c'e'^'''^^ = e-'''\{c-c')cost-i(c + c) sin^j,
^. = .--|(^ + ^^^)cos^+f^-^^)sinfl,
1 \ 2 ' " 2 / V 2
or 7/ rre-^'CC^cosi-C^'sinf), •'« = le-'mC+C) cost + (0-0') sin t}.
3345 Ex. 2 : a-, + 5.7; + 2/ = e*, ^y ^ + Sy - a; = e".
Multiply the second equation by vi, and add to the first
cl(x±my)_^^._^^^^ ( _^_^ l±3m | ^ ^.^^^^^.._
Put "*" = iji, thus determining two values of vi, and put x-\-m.y =z; thus
5 — m
2^ + (5 _ „i) 2 = e' + «ie-^ This is of the form (3210) .
]vfoTE. — The equations of this example, written in the symmetrical form
of (3342), would be
dx _ dy _
e^ — hx — y e-* + x — ^y
dt.
3346 General solution hy indeterminate rnultijjliers.
dx __ dy_ _ dz
Let
be given with
^2 = a.,x-^h.y-^c.^z^d.2.
suc'h that
(1).
Assume a third variable t and indeterminate multipliers J, m, )i
dt _ Idx + mdy+ndz _ Jdx-\-m dy + ndz
T "" ZPi + mP.^ + rtPs ~\{lv-\- my + nz + r)
The last fraction is an exact dift'ereiitial, and, ia determine \, /, )/?, «, r,
wc have
r/, / + (/.,(;;+((..( /t = A/, : (/j — A a. a^
b^ I + b.^ III. + h-^ n = Xin,
Ci I + c'a m + Cj Jt = X»,
dil + ditn + d^n = A?-,
^1
b,-K b.
SIMULTANEOUS EQUATIONS. 497
The detei'minant is the eliminant of the first three equations in I, m, n.
The roots of this cubic in \ furnish three sets of values of I, in, n, r, which,
being substituted in the integral of (1), give rise to three equations involving
three arbitrary constants ; thus,
c^t = {l^x + m^y -f n^z + r^) ^s.
Eliminating t, we find for the solution two equations involving two
arbitrary constants.
A similar solution may be obtained when there are more than three
variables.
3347 To solve ^ = ^ = ^ = &e. ...(1),
where P == ax -\- by + c, P^ = a^c + &iy + Ci , &c .
Assume ji = ai,-\-h] + cZ, ^j = aj^ + tj?? + Ci<^, &c.,
and take — = = — (2),
Pi P-2 P
the solution of which is known by (3346). Substitute i, = xi^, n = y^, and
these equations become
xd^ + ^dx ^ ydi: + (dy ^ cU_
Pi P2 P'
and therefore '— = — = — .
p,-xjp p.-yp ]J
Dividing numerators and denominators by C, the first equation in (1) is pro-
duced, and therefore its solution is obtained by changing ^, rj in the solution
of (2) into x^ and y^.
Certain simultaneous equations in which the coefficients
are not constants may be solved by the method of multiphers.
Thus,
3348 Ex. (1): Xt-^P{ax + ly) = Q, y^ + P (ax + h'y) = E,
P, Q, B being functions of t. Multiply the second equation by in, add, and
determine m as in (3344), The solution is obtained from
^ + my = e-'""""'^''|a+f e^'^""''''-(^''(Q + .;iE) dt\, (3210)
with two values of m.
334:9 Ex.(2): x,+ ^{x-y) = 1, y,+ ] (x + 5y) = t
are equations solvable in a similar mannei*, and the results ai-e
[Boole, p. 307.
3 s
498 DIFFERENTIAL EQUATIONS.
REDUCTION OF ORDER IN SIMULTANEOUS EQUATIONS.
3350 TiiEOKEM. — n simultaneous equations of any orders
between n dependent variables and 1 independent variable are
reducible to a system of equations of the first order by sub-
stituting a new variable for every derivative except the
highest.
3351 The number of equations and dependent variables in
the transformed system will be equal to the sum of the indices
of order of the highest derivatives. This will, therefore, in
general be the number of constants introduced in integrating
those equations. If, after integrating, all the new variables
be eliminated, there will remain n equations in the original
variables and the above-named constants. These equations
form the complete solution.
In practice, such reduction is unnecessary. The following
are methods frequently adopted. : —
3352 Rule I. — Differentiate until by elimination of a vari-
able and its derivatives an equation of a higher order in one
dependent variable only is obtained.
3353 Rule II. — Employ indeterminate multipliers.
3354 Ex. (1): Xit = ax + by, y2t= a'x + b'y.
By Rule I., differentiating twice for t and eliminating y and y.,t, we obtain
■^'4i:~ (^ + &') *2!;+ {ah' — ah) x = 0,
which may be solved by (3239).
Otherwise by Rule II., exactly as in (3344), we find
am" + (a — h')m—h = 0,
and for the exact differential
(_x-\-iny).^t = (a + ma) (x-\-viy),
the solution of which, by (3239), is
X + my = Cye ^'^^ ' »'«'' * + C^e " ^^^ ^ '"«'' *
in duplicate with the two values of m.
3355 Ex. (2): x.i-2ayt-\-hx=i0, y2t + 2a.Vt + hy = 0.
Diilerentiate, and eliminate y, y^, y^t ', thus
Xu + 2 (2a:- + h)x2t+b-x = 0,
and solve by (3239). Otherwise assume
;<;=:£ cos at + V sin at, y '= i) (^^s at — l sin at,
and the given equations reduce to
^,, = -{a' + h)l, ^„,= _(aH?')r;,
■which are solved in (3257). [Boole, p. 311.
PARTIAL DIFFERENTIAL EQUATIONS. 499
3356 Ex. (3). — Let u = 0, V =z 0, iv = be three equations in x, y, z, t,
involving derivatives of t up to x^t, ijco ^u-
To obtain an equation between x and t. Differentiate each equation.
6+7 = 13 times, producing 3 + 13 X 3 = 42 equations involving derivatives
of t up to aJie;, 2/i9i, Zos,f Between these 42 equations eliminate ?/, ?/<, ... 3/19^,
2!, 2!^, ... 020^5 in all 41 quantities, and an equation of the 16th order in x and t
is the result. [De Morgan.
3357 If a number of equations involve the quantities x, X2t,
x.if, &c., iff, i/sf, y5t, &c., all in the first degree, these quantities
may be eliminated by assuming
X ■=■ L ^mpt, y = M cos, pt.
3358 If there be n linear homogeneous equations in n vari-
ables x,y,z, ... and their derivatives of the 2nd order only,
the equations may be solved by putting
cV = L sin j)t, y = Msiii^^, z = N smpt, &c.
3359 Ex. : Xot = ax + hy, y.^ — gx +fy.
Putting X = L sin 2yt, y = Msinpt,
(a+p') L + hM=01 .1 a+p\ b I _ q
9L+(f+p')M = 0y --Ig, fi-f\ '
p and the ratios L : M are thus found.
Suppose L = —Ich and M = l-(p^-\-a),
then X = —Jcb sin pt, y = ]c(p^ + a) sinj^iJ,
and Jc and t are arbitrary constants.
PARTIAL DIFFERENTIAL EQUATIONS.
3380 An equation is termed a general primitive or a com-
plete primitive of a partial differential equation, according as
the latter is obtained from it by eliminating arbitrary functions
or arbitrary constants, as illustrated in (3150-7).
LINEAR FIRST ORDER P. D. EQUATIONS.
3381 To form the P. D. equation from the primitive
u-= <\> (?'), where u and v are functions of x, y, z.
500 DIFFER EN TIAL EQUATIONS.
Rule. — Differentiate for x and j in turn^ and eliminate
^'(v). See (3054).
Otherwise. — Differentiate the equations u = a, v = b; thus
Uidx-hUydy+Ugdz = 0,
Vj^dx+Vydy+v^dz = 0.
Therefore -^ = -^ = ^, where P = ^|'^Jj^, ^x.
Tlien the P. D. equation loill he
Proof. — Since 2 is a function of x and y, z^dx + Zydy = dz. But dx = hP,
dij = kQ, dz = kB, therefore IPz^ + lQzy = hB.
3382 Ex. — The general equation of a conical surface drawn through the
. , / 7 V . II — h ^ / z — c\
point (a, 0, c) IS '^ = (t> ,
x — a \x—al
the form of ^ being arbitrary.
Considering z as a function of two independent variables x and y, differ-
entiate for X and y in turn, and eliminate <p' as in (3154). The result is the
partial differential equation
{x—a) z^ + (y—b)Zj,-]-z—c = 0.
3383 To obtain the complete primitive; that is, to solve
the P. D. equation, Pzj.-\-Qzy = E,
P, Q, B being either functions of x, //, z or constants.
Rule. — Solve the equations
dx _ dy _ dz
"P ~ Q ~U'
Let the two integrals obtained he u = a, v = b ;
then u =z <l> (i^)
ivill he the comiilete jprimitive.
Propositions (3381) and (3383) extended to any number
of variables.
3384 To form the partial diiferential equation from the
primitive ^ («, i?, ... iv) = (1),
where ?6, v, ... w are n given functions of n independent vari-
ables a.', i/i ... z and one dependent t.
PARTIAL VIFFEBENTIAL EQUATIONS. 501
KuLE. — Differentiate for all the variables thus,
<^,du + (/),dv+ ... +<p,A^ = (2).
Therefore, since (f) is arbitrary, du,dv...dw must separately
vanish, giving rise to the n equations
du = u^dx + Uy dy 4- . . . + Ut dt = 0,
dv = Vx dx + Vy dy + . . . + Vt dt = 0,
dw = Wjdx + Wydy + . . . + Wtdt = 0.
Solving these for the ratios, by (583), tve get
^_4L- il^ii (3),
P, Q ... R, S being functions of the variables or else constants.
Noiv, t being a function of all the rest,
t^dx + tydy+...+tzdz = dt (4),
therefore, by (3) and (4), the partial differential equation
required is
3385 Pt,-^Q%-^...^-Rt, = S,
3386 If ^^ V ...whQ n functions of n variables, x, y ... t, the
condition of interdependence of the functions or existence of
some relation expressed by equation (1) is J{u, v ... w) =
(see 1606) ; that is, the eliminant of equations (2) must vanish.
3387 Conversely, to integrate the partial differential equa-
tion p^,+g^,+ ...+iiL= s (1).
Rule. — Solve the system of ordinary equations
| = |=.c. = J = | m,
and let the integrals obtained &e u = a, v = b, ... w = k ;
then ^ (u, V, ... w) = tvill be the complete primitive.
I/" P, Q ... R, S are linear functions of the variables, the
integrals of equations (2) can always be found by the method of
(3346).
502 DIFFERENTIAL EQUATIONS.
Note. — Suppose, in equation (1), tliat any coefficients P, Q
vanish; then, by (2), dx = 0, dij = 0, and therefore the cor-
responding integrals are x = a, y = h. The complete primi-
tive thus becomes
(\>{x, y, u, V ... w) = 0.
3389 When only one independent variable occurs in the
derivatives of the partial differential equation, the equation
may be integrated as though the others were constant, adding
functions of the remaining variables for the constants of
integration.
3390 (^^- 1) • ~~ — ^ Integrating for x as though y were con-
dx ^y^ — x^
stant, the complete primitive is
2 = ?/sin-^— +0(2/).
y
Some equations are reducible to the above class by a transformation.
Thus :
3391 Ex. (2): K,, = x' + y\ Put z, = «,
therefore ii„ = x^ + if, therefore ?4 = ^^ = x^'y + i^/* + (^) >
therefore z = ^x'^y + ^xy'^ + j (p (x) dx + \P (y) ,
or z = \ (a;V + xy"") +x(^) + 'P(y)'
3392 Ex. (3): (:x-a)z^+(y-h)z_„ = c-z.
Solving by (3283), -^ = "^ = — ■
^ •' ^ ^' x-a y—h z — c
The integrals are
log(2/-Z>)-log(a;-a) = loga ) ^^, lLl± = c, ^^^=C',
\og(z — c)—\og(x—a)=logC'j x — a ' x—a
therefore ^~ = o ( ^~'^ ) is the complete primitive.
x—a \x—a/
For the converse process in respect of the same equation, see (3382).
3393 Ex. (4). — To find the surface which cuts orthogonally all the
spheres whose equations (varying a) are
x- + y' + ^'-2ax = (1).
Let (j) (x, y,z) =■() be the surface. Then
(.f-a)0^ + //f„+~0,- = O
by the condition of normals at right angles. Substitute the value of a from
(1), and divide by 0-; thus,
{x^-y''-z'')z, + 1xyz^ = 2zx.
PARTIAL BWFEEENTIAL EQUATIONS. 603
By (3383),
dx _ dy _ dz_
^ = — gives ^=c for one integral.
y X z
Substituting y = cz, we then have
dx _ dz
x'-{c' + l)z'~2zx
which, being a homogeneous equation in x and z, may be solved by putting
z = vx (3186). Tlie resulting integral is '^ ■' = G. Hence the com-
plete primitive is ^ "^^ ~'"~' = ^ f ^ J and the equation of the surface sought.
3394: Ex. (5). — To find an integrating factor of the equation
(a;hj-2y*) dx+(x7f-2x') dy = (1).
Assuming z for that factor, the condition (Mz), = (Nz)^ (3087) pro-
duces the P. D. equation
(xf-2x')z,+ (2f-xhj)z, = 9(x^-y-^)z (2).
The system of ordinary equations (3283) is
dx _ dy _ dz
xy^—2x^ ~~ 2y^—xhj ~ 9 {x^—y^) z
The first of these equations is identical with (1) (and such an agreement
always occurs). Its integral is — ^ -f -y = c.
Al=o ydx + xdy ^ dz
^^'° xy'-2xSj + 2xy'-xSj 9 (a^-y') z'
which reduces to 1 ^ -f — = ;
x y z
and thus the second integral is x^y^z = c.
Hence the complete primitive and integrating factor is
Any linear P. D. equation may be written as a homogeneous
equation with one additional variable; thus, equation (3387)
may be written
3395 l\. + Quy + . • • + R^^'. = ^^h-
SIMULTANEOUS LINEAR FIRST ORDER P. 13. EQUATIONS.
3396 Pkoi'. I. — The solution of sneU equations may he made
to depend upon a sijstem of ordinary 1st order differential
S04 DIFFERENTIAL EQUATIONS.
equations having a nnviber of variahles exceediiir/ hij niore than
one the number of equations.
Let there be n equations reduced to the homogeneous form
(3395) involving one dependent variable P and n-\-ni inde-
pendent. Select n of the latter, x,y ... ^, and let the remaining
??t be ^,r]...l. From the n equations find P,,, Py ... P^ in
terms of P^, P,, ... P^, and arrange the results as under :
P.. + ci,P, + h,P^...+hP, = 0'
Py + a2P^ + h,P^...-j-h,P^ = 0[ (1).
P. + c''nF,-\-h.P^...+KP^ = o,
Multiply these equations by A^, X25 ... X,^ respectively, and add ;
thus,
A1P.. + X2P, ... +X.P. + 2 (Xa) P, + S {U) P„ ... +2 (XZ-) P, =
(2).
From this, as in (3387), we have the auxiliary system
dx___ dy_ _ dz_ _ d^ _ dn _ dl /on
Y^~ \ "'~ K ~2(Xa)~2(X6) "'~^{U) ^^^'
and, by eliminating X^, X2 ... X,„
d^ — a^dx — a-idy ...—andz =
dn — h-^dx — h^dy ... — Jj^dz = 01 r,^\
dl - 1\ dx - h,dy ... - /.•„ d^. = 0.
Then, if u = a, v = I), &c. be the integrals of (4), they
will be values of P satisfying the equivalent system (1), and
the integral of that system will be F(u, v, ...) = 0.
3397 Prop. II. — To integrate a system of linear 1st order
P. D, equations.
Let A = ad^^-\-hdy ... -\-kd,,
so that AP = represents a homogeneous linear P. D. equa-
tion of the 1st order.
Rule. — ^'Reduce the equations to the homogeneous form (1);
express the result symholically hy
AiP = 0, A.P = 0, ...A,P = 0,
PARTIAL DIFFERENTIAL EQUATIONS. 505
and examine ichether the condition
is identically satisfied for every pair of equations of the system.
If it he so, the ec[iiations of the auxiliary system {Prop. I.) will
he reducible to the form of exact differential efiuations, and
their integrals being ii = a, v = b, w = c, ..., the complete
value of P will he F (u, v, w, ...)j l^^^ farni of F being
arbitrary.
" If the condition be not identically satisfied, its ap2)lication
will give rise to one or more new partial differential equations.
Gonibine any one of these with the previous reduced system, and
again reduce in the same toay.
" With the neio reduced system proceed as before, and continue
this method of reduction and derivation until either a system
of P. D. equations arises, hetiveen every two of lohich the above
condition is identically satisfied, or, wliicli is the only possible
alternative, the system P^ = 0, Py = 0, ... appears. In the
former case, the system of ordinary equations corresponding to
the final system of P. D. equations, will admit of reduction to
the exact form, and the general value of P ivill emerge from
their integrals as above. In the latter case, the given system
can only be satisfied by supposing P a constant.''
3398 "Ex.: P,+ (/+.-v/ + a-r)P,+ (7/ + .— 3,OP, = 0,
P,+ (a^„-; + // -.r//) P, + (./ -y) Ft = 0.
Representing these in the form A^P = 0, A.P = 0, it will be found that
(AiAj — zi,.,Ai)P = becomes, after rejecting an algebraic factor, xP^ + Pt = 0,
and the three equations prepared in the manner explained in the Rule will
be found to be
P, + (3.r + OP. = 0, P, + yP, = 0, P, + .7;P„, = 0.
No other equations are derivable from these. We conclude that there is but
one final integral.
" To obtain it, eliminate P^, P^, P^ fi'oixi the above system combined with
Pjx + P,jdy + P,dz + Ptdt=0,
and equate to zero the coeflBcient of P, in the result. We find
dz—{t + 3x-) dx -ydy-xdt = 0,
the integral of which is % — xt — x^ — lif = c.
" An arbitrary function of the first member of this equation is the general
value of P." [Boole, Sup., Ch. xxv.
For Jacobi's researches in the same subject, see Crelles Jonrnal, Vol. Ix.
3 T
606 DIFFERENTIAL EQUATIONS.
NON-LINEAR FIRST ORDER PARTIAL DIFFERENTIAL
EQUATIONS.
3399 Tupe F{x,j,z,z.,y^,) = ^ (!)•
Chaepits's Solutioj^. — Writing p, q instead of z^ and z,.,
assume ilic ecjnations
A^^dy = -Al- = -4B- (2).
Find a value of p from these by integration, and the corres-
ponding value of q from the given equation, and substitute in
the equation
dz = p(U-\-qdi/ ( 3 ) ,
and integrate by (3322) to obtain tJic final integral.
Proof. — Since dz=pclx + qdy, we have, by the coudition of integrabihty,
j/y = q^. Express p„ and q^^ on the hypothesis that z is a function of x, y ;
jj a function of x, y,z ; q a function of a;, y, z,p ; considering x constant when
finding j5,/, and y as constant when finding q^. Equating the values of p^ and
(/,, so obtained, the result is the equation
Ap, + Bpy + Cp, = I),
in which A, B, C. D stand for —q^,, 1, q—pqi„ q^+pq-.
Hence, to solve this equation, we have, by (3387), the system of ordinary
equations (2).
3400 Note. — More than one value of p obtained from equations (2) may
give rise to more than one complete primitive.
The first two of equations (2) taken together involve equation (3).
DERIVATION OF THE GENERAL PRIMITIVE AND SINGULAR
SOLUTION FROM THE COMPLETE PRIMITIVE.
EuLE. — Let the complete primitive of a F. D. equation of
the \st order be
z = f(x,y,a,b) (1).
3401 ^i^^t-G general prunitlre is obtained hy clluiliiating ;i
bet'lveen z = f {x, y, a, «^ (a) } and f., = (2) ,
the form of (p being specified at pleasure.
3402 ^-I'lie singular solution is obtained by eUtninating a and
h between the coinplete 'primitive and the equations
t-0, f,. = (3).
PABTIAL DIFFEBENTIAL EQUATIONS. 507
Proof. — By varying a and h in (1),
p = f^+faa^+f,h, q = f,+faa,+f,Ar
Therefore, reasoning as in (3171), we must have
faa^+fiK^O and f,,a,,+f,h,, = (3),
therefore either /„ = 0, /,, = 0, leading to the singular solution ; or, elimi-
nating fa, ft,, «.. ^y - a, J h, = 0,
and therefore, by (3167), h = (p(a). Multiply equations (3) by dx, dy re-
spectively, and add, thus fada+fi,db = 0. Substitute l = (j> (a) in this and
in (1), and the equations "(2) are the result.
SINGULAR SOLUTION DERIVED FROM THE DIFFERENTIAL
EQUATION.
3403 Rule. — Eliminate p and q from the differential equa-
tion by means of the equations
Zp = 0, z,j = 0.
Proof.— Let the D. E. be z =/(«, y,p, q), and the C. P. 2 = Fix, y, a, h).
Now p and q being implicit functions of a and h, we have, from the first
equation, z^ = z^pa + z^ qa, z,, = z^po + ^a ?Zi.
Hence the conditions z„ = 0, z,, = in (3) involve, and are equivalent to,
z, = 0, z, = 0.
3404 All possible solutions of a P. D. equation of tlie 1st
order are represented by the complete primitive, tbe general
primitive, and tlie singular solution. [Boole, p. 343.
3405 To connect any given solution witli tlie complete
primitive.
Let z = F{x, y, «, h) be the complete primitive, and
^ r= ^ (^x, y) some other solution.
Determine the values of a and h which satisfy the three
equations F = (}>, F^. = (^^., F^ = (p^.
If these values are constant, the solution is a particular
case of the complete primitive; if they are variable so that
one is a function of the other, the solution is a particular case
of the general primitive ; if they are variable and unconnected,
the solution is a singular solution.
3406 CoE. — Any two solutions springing from different
complete primitives are equivalent.
508 DIFFEBENTIAL EQUATIONS.
3407 Ex.: z=px + qy+pq (1).
By (3299), ^ = .7.^ = |. = .| (2).
T 1 z — px . xy + z
and we have q = — ^— ; A = —q„ = 7-^- — n? ;
p+y (p+yy
He.ce (2) becomes i^ ,. = ., = ,-MiL, = f ;
.-. dp = 0, p = a ; .'. q=: '-. Substituting in dz = pdx + qdy,
a + y
dz = (idx+^'^ dy (3).
a + y
By (3322), making z constant, -^ + -^ = 0,
therefore — log (2 — oaj)+log (a + ^z) = (^) (^)-
Differentiate for x,y,z, and equate with (3), thus (p'{z)={), therefore
<p {z) = constant (say —log 6); therefore, by (4), z = ax + by + al), the C. P.
of (1).
3408 To find a singular solution by (3402), we must eliminate a and h
between z^^ = 0, 2;,, = ; that is, x + h = and y + a = 0,
therefore z = —xy—xy + xy = —xy
is the singular sokition.
To find the general primitive by (3401), eliminate a between the two
equations z = ax + (y + a) <p (a) and x + {y + a) ^' (a) + (a) = 0.
NON-LINEAR FIRST ORDER P. D. EQUATIONS WITH MORE
THAN TWO INDEPENDENT VARIABLES.
3409 Pkoi'. — To find the complete primitive of tlie differ-
ential equation
F{d\,a^, ... a\„z,j)„pi ...p,) = (1),
3410 Rule. — Form the linear P. D. equation in $ denoted by
the summation extending from r = l to r = n. From the
auxiliary system (3387) n — 1 integrals
^l = «l, ^2 = rt2, ... ^„-i = an-i
''• I Vdx, "^ ^^- dz ) dpr dpr Vdx, ^ ^'
SECOND OBDEB P. D. EQUATIONS. 509
onay he obtained. From these equations, together with {l),find
Pi J P2 ••• Pn ^''^ terms of Xi, x,, ... x,,, substitute the values in
dz = pidxi + padxg ... +PnClXn,
and the integral of this last equation will furnish the solution
re([uired in the form.
f (xi, X2 . . . x„, z, ai, aa . . . a^) = 0.
[Boole, Biff. Eq., Ch. xiv., and Snp., Ch. xxvii.
SECOND OEDEE P. D. EQUATIONS.
3420 Type F{.v,i/,z,z,,.,^„z,,,z,,„z.,,) = 0.
The derivatives 2.,., z^, z^^, z^y, z.,y are briefly denoted by
_p, q, r, s, t respectively.
z being a function of tlie two independent variables x and
y, the following values are of freqnent use
3421 dz = pcU-\-qdy ; dp = rdx-\-sdij ; dq = sdd?-\-tdi/.
If u be any function of x, y, and z, the complete deriva-
tives of u are indicated by brackets, thus
3422 M = u, +pu,, (Uy) := Uy + qu,.
A linear 2nd order P. D. equation is of the type
3423 Rr-^Ss+Tt= V (1),
in which _B, S, T, Fare functions of x, //, z,p, q.
Pkoposition. — Any P. D. equation of the 2nd order which
has a first integral of the form u =f{v), where u and v involve
'C, y, 2, p, q, is of the form
3424 Rr+Ss + Tt+U{rt-s') = V (2),
where B, S, T, U, Fare functions of x, ij, z,p, q, and
3425 U=u,,v,'-u,Vp (3).
Proof. — Differentiate u = / (v) for x and y separately, considering x, y, z,
p, q all involved in n and v, and eliminate f'{v). The result is equation (2),
■with the values
510 niFFEBENTIAL EQUATIONS.
3426 f^l^ = n^ {Vy) - (Uy) Vp, ^iT = v^ (u,) - {v,) u,^,
■with the notation (3422), ^ being an undetermined constant.
3427 Cor. — The condition to be fulfilled in order that
equation (1) may have a first integral of the form n =f{r) is
SOLUTION BY MONGE'S METHOD OF
3428 Rr-^Ss+Tt=V.
Rule. — Wri'tr the tiro equations
Rihf-S(iidij^-r(ii^ = o (1),
Rdp(Ii/-Vclvdi/-\-Tdqclv = (2).
Besolve (1) into its factors, producing the tnv equations
dy — m^ dx = and dy — mgdx = ,
From dy = m^dx and equation (2) combined, if necessary,
with dz = pdx + qdy, find two 1st integrals u = a, v = b;
then u = f (v) will he one 1st integral of the given equation.
Similarly from dy = modx fnd another 1st integral.
3429 The final 2nd integral may he found from one of the
1st integrals hy Lagrange's method (3383).
3430 Otherwise, determine p and q in terms of x, y, z from
the two 1st integrals; substitute in dz =pdx-\-qdy, and then
integrate by (3322) to obtain the final integral.
3431 If equation (1) is a perfect square, there will be only
one 1st integral, and Lagrange's method only is applicable.
Proof. — By (3427) we may put n^ = mi(,^„ v,^ = mvp ; and also
dz = pdx + qdy (3321) in the complete derivatives
(du) = u,,dx + u,idy + %(,ulz-\-Updp + u^dq = 0, (dr) = &:c. = 0;
.-.by (3422) (71,) dx + (?/.„) dy + u, (dp + m dq) = | ,3.
(v,)dx+(v,)dy + v,(dp + mdq) = 0^
Solving these equations for the ratios dx : dy : dp + nuhj, we obtain at once
dx _ dif + mdx _ mdij _ dj> -f hi dq /,>.
ir~ s ~ T ~ V ^■^'
with the values of J?, S, T, V in (342G).
SECOND OEDEB P. D. EQUATIONS. 511
Equations (1) and (2) are the result of eliminating »i from (4). These
two equations with dz = pdx + qdy suffice to determine a first integral of
(3428) when it exists in the form « =/(i),
3432 Ex.(i.): q{l+q)r-{p + q-^2pq)s^p{l-]-p)t = 0.
Solving the quadratic equation (1), we hud
2nlx + qdy = 0, or (1+2^) dx + (l + q) dij = i) (5).
First, dz = x>dx-\-qdy = 0, .•. z = A.
Monge's equation (2) is 2 (l + s) dpdy+jp (l+p) dqdx = 0,
which, by pdx = —qdii, gives -^£- = --^ ; and, integrating, ——L—B.
Hence a first integral is — '- = (p{z) (6).
Next, taking the second equation of (5) with
dz = p dx + q dy, dx + dy + dz = 0, .-. x + y + z = C.
Also, by (5), equation (2) now reduces to qdj) = pdq, and by integration,
p = qD ; therefore the other first integral is p = qyp {x+y + z).
For the final integral integrate j; — ^^' = 0; i.e., z^~\pz,, = 0, by (3383) ;
, dy dz All dx + dy + dz
.-. dx = — T-. ^ = -TT' •"• - = A, and dx = —-^ — .
^ (x + y + z) 0' l-xP{x + y + z)
C d(x + y + z) ^j,(,,^^,) + j,>.
]l-^(x + y + z) ^ ^-^^ ^
Hence the second integral is ;«— /(;^' + 2/ + ~) =^ F {:).
3433 Ex. (ii.) : ^,,.-a'^,, = 0.
(i.) Here, in (3428), E = 1, S = 0, T=-a\ F=0; therefore (1)
and (2) become dij^—a^dx- = 0, dpdy — ahJqdx = 0.
From (1) dy + adx = 0, giving y + ax = c, and converting (2) into
dp + adq = 0, which gives p+aq = c ; therefore a first integral is
p + aq = f(y + ax) (3).
Similarly, from (1), dy — adx = gives rise to another first integral
2) — aq = yp(y — ax) (4).
Eliminating j>» and q by means of (3) and (4) from dz = pdx + qdy,
dz = (2a)-' {(/. (y + ax)(dy + adx)-xl^(y-ax)(dy-adx)},
therefore, by integrating, z = ^ (y + ax) +^ (i/ — «'*-')•
For the symbolic solution of the same equation, see (3-5G6).
find
SOLUTION OF THE P. D. EQUATION.
3434 Ri'^Ss-^rt+U{rf-s')= V (1).
Let iii^, iiiz be the roots of the qLiaclratic equation
3435 7rr-Sm-\-RT-\-UV=0 (2).
512 BIFFEBENTIAL EQUATIONS.
Let »! = a, L\ = b, and ^i., = a', r.y = b' be respectively
the solutions of the two systems of ordinary differential
equations.
3436 Vdp = riuthj- Tih ^ ITdp = m,dij-Tdx ^
Udq = mJx-Bdi/ (3), Udq = m,dx -Edy [ (4).
th = jj dx + q dij ) d:i — j; dx + q dy )
Then the first integrals of (1) will be
To obtain a second integral :
3437 1st. — When m^, m.^^ are unequal, assign any particular
forms to /i and /,, then substitute the values of p and q, found
from these equations in terms of x and y, in dz = pdx-\-qdy,
which integrate. Otherwise, assign the form of one only of
the functions f\, f.i, involving an arbitrary constant C, solve
for jj and q, and integrate dz =pdx-\-qdy, adding an arbitrary
function of C for the constant of integration.
3438 2nclly. — When ')%, m.2, are equal, and therefore, by (2),
S-' = 4.{BT^-JJV) (5).
Equations (o) and (4) coincide, and, since m = ^S,
reduce to
3439 mp=iSdy-Tdx (6),
Udq = iSdx-Bdy (7),
dz = p dx-\- qdy (8) .
Here py = 7.,., and therefore the last equation is integrable
if the values oi p and q, obtained by integrating (6) and (7),
be substituted in it. Let n = a, v = b be the integrals of (6)
and (7) ; and let z = (}> {x, //, a, b, r) (0)
be the integral obtained from (8).
The general integral is found by making the parameters
(/, b, c vary subject to tAVO conditions b = f{a), c = F (a) ;
that is, by differentiating
z = <l>{x,y,a,f{<i), F(n)}
for (i, and oHmhiatino' ti.
3440 The general integral therefore represents the envelope
uf the surface whose equation is (l>).
SECOND ORDER P. D. EQUATIONS. 513
Proof. — (Boole, Sup., p. 147.) Assuming a Isfc integral of the form
u =f(v), eliminate /j, and v from equations (o426) by multiplying (i.) by
(H,J?f^, (ii.) by («„)«,„ (iv.) by (».r)0'v/)> (/Ob'^^'vz, andadding. Again,
eliminate /x and v by multiplying (i.) by («.,,)"> (ii-) by («v)^ (iii.) by (Ux)(Sh)i
(v.) by (^iij.) 7ip + (Uy) Uq, and adding. The two resulting equations are
B (u,) H, + T («,) u, - U (u,) («,,) + Vu,, n,^ = I
E,(n,y+soQM + TOhy+v{(n,r) »,+k)«j = o)
Multiply the 2nd of these by m, divide by V, and add to the 1st equation ;
the result is expressible in two factors either as (11) or (12),
{BOtd + ^^hM + V<h,}{'>^hM+TOh,) + Vu,^} = (11),
{B(u.)+vi,(u,) + J%}{m,M + TM + ru,} = (12),
m,, vi., being the roots of the quadratic (2). By equating to zero one factor
of (11) and one of (12), we have four systems of two linear 1st order P. D.
equations. Taking each system in turn with the equations
(ti^)+stip + tUq = 0,
and eliminating (uj), (u^), u^, u,^, we have the de-
terminant annexed for the case in which the 1st
factor of (11) and the 2nd of (12) are equated to
zero. In this case, and also when the 2nd factor
of (11) and the 1st of (12) are chosen, trans-
posing mj, Wj in the determinant, the eliminant is equivalent to
V{Br+Ss + Tt^U{rt-s')-V} = 0,
B
iil^
V
'h
T
V
1
r
s
1
s
t
= 0,
having regard to the values of m^vio and m-^ + m.^ from (2).
When the 1st factor of both (11) and (12) is taken, the 2nd order P. D.
equation produced by the elimination is
Vt-B(rt-s') = 0,
and when the 2nd factor of each is taken, the elimination produces
rr-T(:rt-s') =0.
Hence the hypothesis of a 1st integral of (1), of the form n =f(v),
involves the satisfying one or other of the systems of two simultaneous equa-
tions, (13) or (14), below :
Now multiply the 2nd equation of (13) by \ and add it to the 1st.
In the result, collect the coefficients of lo^, Uy, Up, u^, u.. The Lagrangean
system of auxiliary equations (3387) will then be found to be
dx, __ di/ _ f?p _ d^ 'l^ _. 'l^
B + Xnii ~ m, + \T~ V~ XV Bp + vi,q + \(Tq + m-ip) '
Eliminating X, equations (3) are produced. Treating equations (14) in
the same way, equations (4) are produced.
3 u
514 DIFFEltENTIAL EQUATIONS.
POISSON'S EQUATION.
3441 P = {rf-srQ,
where P is a function of p, q, r, s, f, and homogeneous in r, k,
t ; Qis a function of a3, v/, 2;, and derivatives of ?:, which does
not become infinite when rt — s^ vanishes, and n is positive.
Rule. — Assume q = ^ (p) and express s and t in terms ofq^
and r; thuSi rt — s"^ vanishes, and the left side becomes a func-
tion 0/ p, q, and qp.
Solve for a Isf integral in terms o/p and q, and integrate
again for the final solution.
Proof.— s = 5, = q^p^ = (/,, r ; f = q, = q.rpy = qlr\
therefore rt — s- = 0. Also P is of the form (r, s, f)'" = r"' (1, q^,, cj],)'".
Hence the equation takes the form (1, g^, ql)'"' = 0.
LAPLACE'S REDUCTION OF THE EQUATION.
3442 Rr+Ss+Tt+Pp + Qq+Zz= U (1),
where B, S, T, P, Q, Z, U are functions of x and // only.
Let two integrals of Monge's equation (3428)
Edy'-Sdxdy + Tdir =
be (p {x, u) = a, ^ (;>j, //) = h.
Assume ^ = ^ {■>', [/), v = 4^ {^v, y).
3443 To change the variables in equation (1) to i and j?,
we have
r = ;:,,. = %e + 2;v^,^,7,., + ^,^r,H;:^L, + r,.„,, ; (1701)
t = Z,^ = Z2^ti + 2^^^t,^V, + ^2A+-A!j + -r,V,y ;
The transformed equation is of the form
,,^^Lz, + ]\P, + Nx=V (2),
where L, ilf, N, V are functions of ^ and >;. This equation
may be written in the form
{d, + M){d^ + L), + {N-LM-L^::=V (3).
If N-LM-L^==i) (4),
we shall have
(d.-^M )::'=¥ with (J„ + L).v = c',
LA W OF BEGIPBOGITY. 515
and the solution by a double application of (3210) is obtained
from
By symmetry, equation (1) is also solvable, if
y-LM-M^ = (5).
But if neither of these conditions is found to hold, find z
in terms of z' from (3). It will be of the form
. = ^4 + 7?.' + ^',
where A, B, G contain ^ and v. Substitute this for z in
(c/, + i) z = z', and the result is of the form
The same conditions of integrability, if fulfilled for this equa-
tion, will lead to a solution of (1), and, if not fulfilled, the
transformation may be repeated imtil one of the equations,
similar to (4) or (5), is satisfied.
3444 CoK. — The solution of the equation
z^,-\-az^ + hz^ + ahz = V
3445 For the solution of equation (2), wbeu L, M, V contain also z, see
Prof. Tanner, Proc. Lond. Math. Soc, Vol. viii., p. 159.
LAW OF RECIPROCITY. [Booh, ch. xy.
3446 Let a differential equation of the 1st order be
<l>{^v,ij, ^,p, q) = (1).
Let the result of interchanging cV and ]), y and ^, and of
changing z into 2hi-\-qy—z, be
^(p, q,px-\-qy-z,x,y) == 0... (2);
then, ii z = -^ (*, ij) be the solution of either (1) or (2), the
516 DIFFERENTIAL EQUATIONS.
solution of the other will be obtained by eliminating ^ and »?
between the equations
a; = d^xfi {^, f}), ij = (l^xlf {^, rj), z= iv-^rji/-^ {$, rj).
3447 Ex.— Let zz=pq (1), 2^^^ + r2y-z = xy (2),
be tbe two reciprocal equations.
The integral of (2) is z = xy+xf {^^\ .-.»// (bj) = bi + t>i ( y ) •
i,, T) have now to be eliminated between
»^=v-|/'(|)+/(|), !/ = £+/'(!). '-f" (3).
Each form assigned to/ gives a particular integral of (1). If / ( ^ ) = '^ y + Z^,
the equations (3) become x=^rj-\-l, y = ^ + a, z = tn,
and the elimination produces z = (x — l)')(y — a).
3448 In an equation of the 2nd order, the reciprocal equa-
tion is formed by making the changes in (3446), and, in
addition, changing
r into — — -, s into ^„ t into
rt—s rt—s" rf—s-
then, if the 2nd integral of either equation he z = \p (.r, //), that
of the other will be found by the same rule.
3449 The above transformation makes any equation of the
form <^ Q), q) r+^ {p, q) .v+x ilh q)t =
dependent for solution upon one of the form
X (■^% «/) »'-^ G*'^ 1/)^+^ (^^ y) t = 0.
3450 And, in the same way, an equation of the form
is dependent for solution upon one of the form
Rr^-Ss+Tt= V.
See Be Morgan, Camb. Phil. Trans., Vol. VIII.
SYMBOLIC METHODS.
FUNDAMENTAL FORMULA.
Q denoting a function of 0,
3470 {do-m)-'Q = e"''j6'-'"^ Qd0.
8 YMBOLIG METHODS. 517
Proof. — The right is the value of 1/ in fcbe solution of d,y — my = Q by
(3210). But this equation is expressed symbolically by (dg — ra) y = Q (see
1492), therefore y = (cZ^— 7?i.)'^Q.
Let X = e\ therefore d^ = xd^ and xdQ = dx. Hence
(3470) may be written
3471 {.v(h-m)-^ Q = cr"^ Ja-»^-i Q(Lv.
3472 Cor.— {de-m)-^0 = Ce"'\
3473 or (.vd.,-m)-'0= av'\
Let F(m) denote a rational integral function of m; then,
since dee"^' = one"'\ f^o, e'"' = mV"^ &c., the operation f/, is
always replaced by the operation mX . Hence, in all cases,
3474 F{de)e"'' = e'"'F(m).
3475 F(de) e''"Q = e'>''F(de+m) Q.
Formula (2161) is a particular case of this theorem.
3476 e^''F(de) Q = F {de-m) e'»^ Q.
Also, by (3474-6),
3477 F(m) = e-'«^F(fye'"^
3478 F (f/,+m) Q = e-'F {d,) e'«^ Q.
3479 F (de) Q = e-"''F{de-m) &''' Q.
To the last six formulce correspond
3480 F ixd^ .r"^ = x"^F{iii).
3481 F{a:d,) .v"'Q = .v"'F{M,-\-m) Q.
3482 ^v'^'F (ctY/,) Q= F {.vd,, - m) a^'^Q.
3483 F (m) = .v-"'F (erf/,) .v'".
3484 F{a^d^-]-m) Q = .v-"'F{a'd,) .v"'Q.
3485 F i^d,) Q = x-"'F {.vd,-m) .r'"Q.
If U=a-{-hx-\-cx^ + &c., then, by (3480),
3486 F{.vd,) U= F(0) a+F(l) bjv-{-F{2) c.r+&c.
518 DIFFERENTIAL EQUATIONS.
3487 F-'(.vd,) U= F-'{0) a-\-F-'{l) Kv-\-F-\2) cx^^kc.
3488 F(.v(]_,,7/(Iy, z(L, ...) <r'"ifz\.. = F{m., n,p ...) .r'y«^..
3489 ^v^'Un,. = (h{cU-\) {do-2) ... {(h-tl+l) U,
or, more succinctly, writing D for dg,
D{n-l) ... {D-n-\-l)n or D'"in (2452).
3490 Otherwise ,r"«,„,. =: .vd^{.rd,, — l) ... (.?y/,-/< + 1) /^
Proof. — As in (1770). Otherwise, by Induction, differentiating again,
and remembering that a-^ = x.
Note. — In the symbolic solution of differential equations,
we may either employ the operator xd,, directly, or the
operator d^ after substituting e^ for x. Formulae (3480-5) or
(3474-9) will be required accordingly.
3491 {(^(/))e'-^j"g
= ct>{D) ^{D-r) (t>{D-2r) ... <l>{D-(n-l) r\ e"'-'Q.
Proof. — By repeated apjilication of (3475) or (3476).
For ready reference, formulas (1520, '21) are reprinted
here.
3492 /(.*' + /0 = e^'"^-f{.r).
3493 fi'V+h, 7/+ A-) = e'"'..--H./(.r, ;y).
Let cIq + «! X -\- CLuCr ...-{- cr„ r/;" = /' {,r) ,
then, denoting d^ by D,
3494 /( D) in- = uf{D) v^uof (D) r+ ^/" (i>) r + &c.,
where / (D) means that D is to be written for x after differ-
entiating f{x).
Proof. — Expand uv, D.nv, I)- .uv ... B'^.uv by Leibnitz's theorem (1400);
multiply the equations respectively by a^, a,, a^ ... a„, and add the results.
3495 uf(D) r ^f{D) . uv-f" (D) u,v+ ^f" D.u,^ i—&c.
SYMBOLIC METHODS. 519
Proof. — Expand uv^, uv-i^i uv-m ••• ''f''^'-.!) by theorem (1472), and proceed as
in the last.
3496 Fid,) ^^" m.v = F(-m') ^"^ m.v.
A more general theorem is
3497 F (7r^)(7.,„+ »,_„„) =z/^ (->H^)(»,,^^+ «_,,),
where ii and tt have the meanings assigned beloAV (3499),
and i = \/ — 1 .
Theorem. — If ^ and x^ denote any algebraic functions of x
and //, it may be shown, by (3474) and (3475), that
3498 ^ {(!.+!/) H^r) = <!> {(h^^v) xl^Qf).
3499 Let u, or, more definitely, n,^ = (x, y, z, ...)", represent
a homogeneous function of the n^^^ degree in severable vari-
ables, and let
3500 ^ = ^4+K+-^.~ + &c.
Then, by (3480),
3501 TTW = nu, TT^H = 7ru, irhi = nhi, &c.
3502 Hence F {'n)u = F{n)u.
REDUCTION OF F{Tr,) TO /{it).
3503 Let u be any implicit function of the variables, and
let TT = TTi + TT.,, where tt^ operates only upon x as contained in
u, and TTg only upon x as contained in ttil, &c. after repetitions
of the operation tt. Then
3504 '"'i" = '^ti, '^[ft =^ (tt— l)7r?/,
3506 <« = {rr — r-\-l) ... (tt— 2)(7r— 1) nn.
Proof. — tt^u = (^r — tt,,) u = ttk,
since tt^ has here no subject to operate upon.
TTj (( = (tt — ■n-.J TTH = (tt— 1) 7r«,
for, Ttu being of the 1st degree, tTo and 1 are equivalent as operators. In the
next step, tt^ and 2 are equivalent, and so on.
Cor. — "When u is a homogeneous function, we have, by
520 DIFFEEENTIAL EQUATIONS.
(3501), TT^'a = n^'ii, therefore ir and n are equivalent operators
upon u. Hence (3506) may be written
3507 7T[i( = (»->• + !) ... {n-2){n-l)nu = n^:lu,
which is Euler's theorem of homogeneous functions (1G25),
since in that theorem the operator is confined to v.
3508 As an illustration, let ttu = (xd^ + ydy) u = ttjM,
then Trjit = (.t;^ J-j.^ + 2xy cl^y + y'doy) u, ir^u = {tt^ + tt.,) ttu = tTj u + ir, ttu.
Here Tr^mi = (xd^ + ydy) (xd^ + ydy) u,
the operation being confined to x and y in the second factor (.3503), and there-
fore producing {xd^-\-ydy)i(, merely.
Hence tt^m = {x'd.,:r-^'lxyd^,i-\-y'^do,j-\-xd^-\-ydy) v, which proves (3505).
li U = Uo-{-Ui-{-U2-\- ..., a series of homogeneous functions
of dimensions 0, 1, 2, ..., then, by (3502),
3509 F {n)U= F (0) u,-\-F{l) u,+F(2) u,-^...,
3510 F-'(7r)U= F-'{0) u,+F-'{l)u,-^F-\2)u,+ ...
3511 Ex. 1 : a" Z7 = ^(o + ^i^.ti + ahi^ + . . . ,
3512 a-''U = u^ + a-^u,-{-a~hL2-\-...
Ex. 2 : Ji u have the meaning in (3499),
3513
and simikrly for the inverse operation F~^ (tt).
Proof. — By (8502) applied to the expansion of the subject by (150).
3514 __ ^ _______ u,^,
where j)-\-q-\-r-\- .... = m, and pi = 1.2 ... i).
Proof. — Equate coefficients of a"' in the expansion of
(l + rt)'^f7=(l + a)'''"-^(Ua)^"''(l + a)--"'... tr,
reducing by (3490).
3515 The general symbolic solution of the equation
F{de)a = Q is
u = F-' ((/,) Q-^F-' ((/J 0, by (1488-90).
SYMBOLIC METHODS. 521
3516 The solution of the equation (:^)238), viz.,
where Q is a function of x, is most readily obtained by the
symbolic method. Thus 7%, 7iu, ... m,, being the roots of the
auxiliary equation in (3239), and A,B,G .,. N the numerators
of the partial fractions into which (7;i"-|-fti^)i"~^+ ••• +««)~^ can
be resolved, the complete primitive will be
3517
where (4-''0~'Q = e""-je-'"'^-QcZa-, (3470)
and the whole operation upon zero produces, by (3472), for
the complementary term,
3518 (7,e'""'-+a,e'"^''-... +C,e'v.
Proof. — Equation (1) may be written
or (4-w^i) (d^—ini) ... {dx-mn) y = Q,
.-. by (3515), y = \(d,-ind(d,-m,) ... id,-m,)\-' (Q+0),
whicb, by partial fractions, is converted into the formula above.
If r of the roots m^, m,, ... are each =.- ni, those roots give
rise in (3517) to a single term of the form
3519 {A + 54 + Cch. • • • -h Rd,,) e-'- [ 6-"^'^' Q.
Proof. — By (1918), the r roots equal to in will produce
\A\d,-my + B'{d,-vi)-'-\.. -^R'(d,-m)-'\ Q,
or (A + Bd,+ Cd,,... +Bd,.,?)id.-m)-'-Q.
3520 But, by (3470), (d^-vi)-' Q= (d^-ni)-'e'"^^e-"'^ Qd^
= e'"' I J e-™^e»'^ L-"" Qdx | dx = e'"4 e"'"^ Qdx, and so on.
3521 Ex. (1) : y^^-y.^-5y^-3 = Q.
Here 7n^—vi" — om- 3 = {m — '3)(m + iy,
. 1 ^ 1 1 L_^
{m-3)(m + lf 16(Hi-3) 16(w + l) 4(m + l)-^'
therefore y = yV (d.~S)-'Q~^\ (4 + 1)"' Q-i O^. + l)"' Q
= i^e'^ J e-^'- Qf?"-— tV'^^ J e- ^ Qr/,.' - ie-'j^e" QdxK (3520)
3 X
59.2 DIFFERENTIAL EQUATIONS.
3522 Ex. (2): u,,,.Jrfrf( = Q,
thei-efore /' = ('/.., + «-) '(^.
Here (vi^ + cr)-^ = (2m) -' {()» -/a) -'-(m + wt)"'},
therefore u = (2ia)'^ {(dj. — ia)'^ Q — (Jj.4-m)~' Q\
= (2ia)-' {e'"^ Je-'"^ Qdx-e-'"=' je'"^ Qdx} (3470)
= a" ' sin cw j cos o,f Q (?.« — a " ' cos a.e j siu ux Q dx,
by the exponential values (766-7).
3523 CoE. 1.— The solution of iu,-^a^u = is
u = A cos ax-\-B sin ac«.
3524 Cor. 2. — The solution of jf^,.— rt^jt = is
Change a into ia in the fifth line of (3522), and put Q = 0.
3525 When Q is a function whose derivatives of the 7i^^ and
higher orders vanish, proceed as in the following example.
Ex. (3): Uo,: + a'^u = (1+x)-,
therefore n = (d., + a") "' (1 + xy + (d.,, + a^) " '
= (a-'-a-%, + a-%,-&c.)(l + 2x + x') + (d,,-\-aY^0
= a'^ (l + xy—2a'^ + A coaax + B sina.i",
the last two terms by (3523).
Exceptional Cat<e of the Inverse Process.
3526 Ex. (4) : M2.r + «"« = cos JW,
.-. 'II = (d,, + aY' (cosux + 0) = 1 (d,, + a^)'' (e'"-'- + e-'"-'' + 0)
= I (e'"^ + e-''")(-w- + (r)'' + ^cosaa; + 2?sinaa; by (3474) and (3523)
= coswa; (a''— «^)"^ + &c.
Now, if n = a, the first term becomes infinite. In such cases proceed
as follows : —
Put A = A'-(a'-n')-\ and find the value pf ^"^ ".r-cosa;?; ^ ^^^^
n = a. By (1580) it is = iL^Mf . Thus the solution is
X sin ax , ,, , ti •
u = — \-A cos ax + 1) sina.t;.
2 a
The same result is obtained by making Q = cos aa; in the solution of (3522),
For another example, see (355'.').
SYMBOLIC METHODS. 523
3527 Ex. (5) : y,,-9>u + 20y = .^V%
therefore y = {(d,-4^)(cL-5)}-' x'e'"^ + {(ih-4)((l-5)}-'
= e''-{((l-l'){cl.-2)}-'x' + Ae'^- + Be'\ (3475, 3517, 3472)
Now 0,^^-3m + 2)- = \ (l-^J^)''
C^^ 3,.-m- ^/3m^y^^^
Hence the solution becomes
y = e'"- { i + f 4 + H:^ + &c- } ^' + ^6*' + ^^'
= ^"" { Y + T + T 1 +^^''+-^^''-
3528 Ex. (6) : {d^-aYu = e--\
therefore n = {d^-ay'e"- = e^'^' (d,)-^ (3476) = e^^^^ + f o). (2149)
3529 Ex. (7): (d, + ayy = sin mx,
there fore y = (' K + «) " " sin »ia) + (fZ^ + a) ' ^
= (4-a)' ('k-«')"' sin ma; + e-"^' (c/,.)'' [by (3478) with ^ = 0]
= (-vr-a^)-- (d,—ay sinonx + e-'"' (Ax + B) (by 3496)
= (ui' + a")"- ( — VI' smmx — 2amcos7nx + a^ sin ma;) + e'"''^ (Ax-\-B).
REDUCTION OF AN INTEGRAL OP THE w"' ORDER.
3530 ^ Q = ;^3Y] { ^"'' J Q^^^^'- (« - 1) ^^"~' J Q^^'^'
+ C {n, 2) .r"-« j V''^/^'' . . . ± \Qd''-' cLv,
where n — l\ = 1.2 ... n.
Proof.— Bj^ (3489) d,M = e-'"(d-n + l)(d,-n + 2) ...d,Q (1),
therefore d.,M = {{d~n+l){d,-n + 2) ... d,]-'e^''Q
'^-1- +C'(«,2)(t7,-« + 3)-^-&c.}e"^(3 (3517)
= -J_{e(»-i)VJ,)-ie'-(7i-l)e("--'^(cZ,)-ie=" + &c.}Q.
w— 1 !
Then replace e' by a*.
The equation
3531 «<y..+6ct^'^//..,+ &c. = ^+J5.v+ at'^+&c. = q
524 DIFFERENTIAL EQUATIONS.
may, by (3480), be transformed into
U' W,-)'';' + ^> (.v/...)^'; + &c.} !j = (2 or F{xd,) ij = Q.
The solution is then obtained from
3532 y = F-' (.''4) Q + ^-' (j^'l,) 0.
The vahie of the 1st part is given in (3487).
3533 If a, i3, y, &c. are the roots of F{m) = 0, the second
part gives rise to the arbitrary terms
3534 If a root a is repeated r times, the corresponding
terms are
.1- { t\ (log aY-'-\- C, (log .r)'-^+ . . . + C,} .
Proof. — The partial fractions into which F'^{xd:,) can be resolved, as
in (3517), are of the type G (a-d^ — ony^ 0, m being a root of F (jc) = 0. Bnt
(xd^.—m)"^ = Gx"^ (3473), G being an arbitrary constant.
For a root in repeated r times, the typical fraction is C (xd^.—m)''^, p
being less than. r. Now
(xAl-my Gx>" (log.r)^'-' --= {de-my Ge'"' iP'' = e"" (d,y CT^"' (3475) = 0,
therefore (xd,.-m)-P = Gx'" (log xy-\
The equation
3535 ai/,,e-{-bi/ne+&G. =f{e% siu 6, cos 6)
is reducible to the form of (3531) by x = e^ ; or, substituting
from (768), it may be written
Fide)y = :^{A„,e'"%
and the solution will take the form
3536 ^ = %A,,,e-'F-'{m)-^F-^{(h) 0,
for the last term of which the forms in (3533-4) are to be
substituted with .v changed to e\
3537 Ex. (1 ) : a;V = ax'" + hx"
xd^ (xd^-l)(xd^-2) y = ax"'+hx",
.-. u = {xd, (xd,-l)(xd,.-2)}-' (ax"' + h.r") + {xd, (xd,,-l)(xd,-2)}-'0
= ^ + ^ + .1 + Ih + Gx;
m {m-l) (7U-2) 71 (7i- l)(ji-2)
by (3180) and (3533). A result evident by direct integration.
(tjni--
SYMBOLIC METHOb'S^c:. , 525
3538 Ex. (2) : x%, + 3xy,, + y= {l-x)-\ By (3490)
{ xd^ (xd^ - 1 ) + 3xd^ + 1 } !/ = (-^4 + 1 )' y = 1 + 2.^ + 3a;2 + &c. ,
.-. 2/=GiY7, + l)-^(l + 2,« + 3cuH) = (0+l)-^ + 2(l + l)-^a) + 3(2 + l)-V +
(3480) =l+|.+^+&c.+ ^-^^^ + -^ =-llog(l-..) + &c.
<i o a' .'(' a;
3539 Ex. (3) : ,y,^+(4.7J-l) 2/^+(4aj'^-2.i;4-2) 7/ = 0.
Let TT = cZ, + 2a". Then the equation may be Avxntten
7r(7r-l)^ = 0, .-. 7/ = {7r(7r-])}-'0= (7r-l)->0-tr->0.
Let (7r-l)-^0 = ?s .-. (tt — 1)?< = 0, ov u,+ {2x — l)ii — (}, .-. « = i4e^'-^
3540 The solution of a P. D. equation of the type
where 7/1, u.,, &c. are homogeneous functions of the 1st, 2nd
degrees, &c. in x, y, and Tr^ = xd^,-\-yd,j (3503), is analogous to
(3531), and is obtained from that solution by substituting
7ti, U2, &c. for Bx, Gx?, &c. ; and, for such terms as Gx"-, an
arbitrary homogeneous function of x and y of the same degree,
3541 Solution of F{tt)u = Q,
where F(7r) = 7r" + ^i7r"~^H-yl.27r""-+^„,
and Q=- u^^-{-Ui-\-Uo-\-kc.,
a series of homogeneous functions of x, y, z, ... of the respec-
tive dimensions 0, 1, 2, &c.
Here u = F-'{7r) Q + F-\7r)0.
3542 The value of the 1st term is given in (3510). For the
general value of the last term (see Proof of 3533), let
F {711) = have r roots = m ; then
3543 C{w-m)-PO=C{a{\og.vy-'+v{\og.vy-\..-^w},
where u, v, ... w are arbitrary functions of the variables all of
the degree m.
3544 CoE.— (7r-m)-i = (cr, 1/, ... )'«,
that is, a sing^le homogeneous function of the variables of the
degree m (1620).
526 DIFFERENTIAL EQUATIONS.
3545 Ex. : x-zox + 2xyzj.,, + y'eo,,—a {xz^ + tjz^^ + az = n.,„ + «„,
u,„, u„ being homogeneous functions of the m^^ and /t"' degrees. The equation
may be written (tt- — OTTj-f a) z = ";« + '?'« 5
or, by (3505), (7r-rt)(7r-l) z = «,„ + «,„
therefore z = {(7r-a)(7r-l)}-' (u,, + n„) + {(7r-a)(7r-l)}-'
= !!i» \ ^*»_ + cr„-i-[7i.
{m — a)(m—l) (n — a)(n — l)
The first two terms by formula (3502) ; the last two terms are arbitrary
functions of the degrees a and 1 I'espectively, and result from formula (3543)
by taking ^ = 1 and m = a and 1.
3546 To reduce a P. D. equation, wlien possible, to the
symbolic form
{WJ^A,n"-^+A,W^-\..~\-A,)n= Q (1),
where n = Md^ + Nd^ + &c . ,
and Q, M, N, &c. are any functions of the independent
variables.
Consider the case of two independent variables,
{Md,-\-Nd,Y u = ]\Pu,, + 2MNu,y + N'u,,
+ {MM, + NM,) 71, + {MN, + Ny^ n^, . . . (2) .
Here the form of n is obtainable from the right by con-
sidering the terms involving the highest derivatives only, for
these terms are algebraically equivalent to (Md^-^Ndy)'^.
The reduction being effected, and the equation being
brought to the form of (1) ; then, if the auxiliary equation
3547 w''+A,m''-'-^A,in''--...-{-A, = (3)
have its roots a, h, ... all unequal, the solution of (1) will be
of the form
3548 u = {n-ay Q,+ {u-h)-U}+&c (4).
The terms on the right involve the solution of a series of linear
first order P. D. equations, the first of which is
3549 Mn, + Nil,, J^ ...-au = Q,
and the rest involve h, c, &c.
If equal or imaginary roots occur in the auxiliary equation,
we may proceed as in the following example.
SYMBOLIC METHODS. 527
3550 Ex.:
(l + xyz,^-^xy (l+x') z,.,,-\-4.x'i/%, + 2x (1 +x') z^ + 2y (x'-l) z, + a'z = 0.
Here IT = (1 + x") cl^. — 2xijd„, and the equation becomes (11^ + a') z = 0.
Let the variables x, y be now changed to ^, t], so that 11 = d^. Therefore,
since n (4) = 1, n (4) = (l + .r) ia.-2,r^£, = 1.
Therefore, by (3383), -^ = -^ = J£,
1+a- -2xy
from which, by separating the variables and integrating, we obtain
x'y + y = A (1),
and, by (1430), ^ = tan-^x + B (2).
Also, since 11 (?/) = 7/^ = 0, (l + x"') t]^—2xyn,, = 0.
Therefore -^, = ^^ = -^,
l + .y- -2xy 0'
the solution of which is equation (1). Thus
^ = tan~^ X and rj = x'-y + y.
The transformed equation is now (_d2^ + a-) z = 0,
and the solution, by (3523), is
z = <p(r]) cos ak + \p (ji) sin a£,
arbitrary functions of the variable, which is not explicitly involved, being
substituted for the constants (3389). Therefore finally,
z = <l> (x-y + y) cos (a ta,n' ^ x) + \p (x-y + y) sin (a tan' ^ a').
MISCELLANEOUS EXAMPLES.
3551 «2.v + W2.+ M2.-=0.
Put d2y + d.2z = a'. Thus tt-.^ + a^u = 0, the solution of which, by (3523)}
is ii = <p (y, z) cosax + tp (y, z) sina.i;,
arbitrary functions of y and a being put for the constants A and B, Expand
the sine and cosine by (764-5) ; replace a^ by its operative equivalent, and,
in the expansion of sin ax, put a\p (y, z) =■ x (y, ") ; thus
u = f (y, z)- 1^ 0?,, + fU cp (y, z) + 1^ (cZ,, + c?,,) ^ (y, z)-&c.
o ; o ;
[See (3626) for another solution.
3552 u, + ti^ + ?/_, = ciT/^-.
Here a = (rZ, + d, + d,)~' (xyz -f 0)
= {d.,-d_,, {d, + d,) + d.,, id, + d,y-...]{xyz + 0).
Operating upon xyz, we get
u = ix'^yz—^x^ (2 + 2/) + Ta^*,
528 DIFFERENTIAL EQUATIONS.
the rest vanishing. For symmetry, take ^rcl of the sum of three such
expressions ; thus
Operating upon zero, we have, in the first place, d_j.Q = <p (yz)
instead of a constant, therefore d.^.^O = xf (yz), &c.
The result is
{1-x (d„ + d,)+hy (d. + d.y- ...}<!> (yz) = e-^^"^^"--'cp (yz) = 6 (y-x, z-x)
(3493) the complementary term.
3553 Otherwise, putting cZ^ + tZ^ = 5), we have, by (3478),
(d, + ^)-'xyz = e-^-^d.,e^-^xyz = e--*-3^cZ.,{.. (y + x) (z + x)], (3493)
= e-^^{Wyz + :Lx'(y + z)+}x^}
=^ ^x' (y-x)(z-x)+ix' (y + z-2x)+^x\
which agrees with the former solution.
3554 « Ux + bUy -\- cu^ = cvyz.
Substitute x=. a^, y = hri, z = c(, and the equation becomes
which is solved in (3552).
The same methods fm^nish the solution of
3555 ttu,. -\-bUy'\- cu. = x'"y''z^\
3556 .^'5!.,+/p, = 2ajij\/a^-z\
Put z z^ a sin y,
.-. TT^ = a cos f . TTi", .'. TTV = 2,13?/, .". z-=.as\n(xy-{-c).
3557 aa;u^-{-hyUy-\- czu.^nu = 0.
Put X = i,", 7/ = T]'', z = C ;
.-. ^H^ + y]i(^ + ^u^—nu = 0, .-. by (3544) u = (x" , y'', z'' )".
3558 The solution of any P. D. equation of the type
F{.vd,, ijdy, zd„ ...)« = XA.v"y^'' •••
is, by (3488) and (3557),
W = ^ 177 "^ T + 777—1 7 -, 7 <'•
SYMBOLIC METHODS. 529
3559 Ex. : xu^^-^yUy — au = Q,,^,
where Q,,= {x,yy^ (1620).
Here ti = (ni—a)'^Q„^+ TJa- When a = m, this solution becomes inde-
terminate. In that case, as in (3526), assume
m — a m — a
Differentiate for a, by (1580), putting Q„ first in the form
thus u = iQ,„ (log X + log y) + F,„.
Similarly, the solution of
3560 xu^.-\-yUy-\-xu-—mu — Q,^
is n = ^Q,„ (log X + log y + log ^) + V,„.
3561 ct w,+i/i*^+5:w^ = c.
The solution, by (3560), is
u = ic (log x + \ogy + log 2) + Fo-
3562 ^2x—^a^xy+tt%y = or {d,—adyyz = 0.
2 = (d^-ad^)-'0 = (fZ.,-a(^^)-' (y) e"^"y (3472)
by putting ot^j, for m, and ^ (y) for C. The second operation produces, by
(3476), z = e"''"v{x<p(y) + ^P(y)}=x<p(y + ax) + ^P(y + ax). (3492)
3563 ^v%^, —if^-iy + 'J'%- —y^u = 0.
This reduces to (xd^ + yd^) {;xd^ — yd,J) 3 = 0.
Here tt = xd^^+yd^, and m = in (3544),
therefore ^ = («, i/)"+ ( i^', — j ,
the second term being obtained by substituting y'^ = y', and so converting
the second factor into (xd^ + y'd^,). The above may also be written
F and/ being integral algebi'aic functions.
3564 x,,,-a%y-\-2aM,-{-2a%!:^y = 0.
Putting y = at], this equation is equivalent to
(d,-d., + -2ah) (d^ + dr,) z = ;
3 Y
530 DIFFERENTIAL EQUATIONS.
putting X =■ log .7/ and ?; = loy- ;;', this gives, by (354-i),
= e-'^"'-F(,j + ax)+f{y-ax),
the functions being algebraic and integral.
3565 U2^—ahi.2y = <!> (.V, y).
.-. u = {cl,,-a'ch,)-' {x, y) (3515)
= {2ad,Y' [ (,l-ad„)-'-{(l + ad,)-' } (x, y) (3470)
= (2ad^)-^ { e"^"v fe""^";' (^ (x, y) dx-e-"-'"'.' (e"^"«' (x, y) dx ] (3470)
= (2a)"' j I *i {x, y + ax) — ^., (,r, y — ax) j- c7//,
since e"^";/ (a-, ?/) = ./. (a;, 1/ + ax) (3492).
Here *, (x, y) = | ^; (a^, 9/-«,r) (Z.t; + v/. (?/),
*2 (■'■, y) = J0 (■■'■, 2/ + «.'■) fZ.i' + x (!/)•
3566 If «/> 0^', //) = 0, the solution therefore becomes
H = xj/^ 0/4-«^?')H~Xi (Z/~"'0 [-BooZe, ch. 16.
For the solution in this case by Monge's method, see
(3433).
3567 %^^.—aZy = e""' cos tii/.
z= (d^— ady) -^e"'^ cos ny — e"^".'/ [ e ' '"'"i' e'"' cos ny dx (3470)
= e«^''i/ [e'«^cosn (y-ax) dx (3492), and this by Parts, or by (1999), is
_ f^axiiy^mx I j^^ pQg ^^ (?/ — aa?) — awsinn (y — ax) ] (Hr4-«V)"' + e"^''.v0 (y)
.-. 2 = e'"-'^ m cosny — an sinvy \ (m- + a'ii')'^ + (]> (y + ax), by (3492).
3568 z-a::,,= i).
■, = (J,_ar/,,) -^ = e'""^'-^</> (0), by (3472),
^ («) taking the place of the constant G.
Therefore s = (a!) + ai^2.-l-iaV^4^ + &c. (3492)
Otherwise, to obtain z in powers of a;, we have, putting Ir = a~\
j;o.-62»i = 0,
.-. z = {(d, + ldh(d.+ hd])}-'0 = e'''-"'(t> (t) + e-'"-">x}. (0 (3518) ;
then expand by (150).
3569 «2a -f^2// = ^os 1UV COS mi/.
SYMBOLIC METHODS. 531
z = (t?>^. + c?2;,)~^ COS nx cos raij.
Treating (h, and cosmy as constants, we have, by (3526), putting d.,y for a^
s = cos nx (diy — n-) '' cos my + A cos ax + B sin a,^, or by (3496),
= cosnx cosmy ( — m--n^)-' + (p (ij) cos (xd,,) + \p (y) sin (*f7^),
A and J5 becoming ^ (y) and vp (//)•
3570 ^,^+2zh-\---24'+(''- = COS (w4+w^).
Therefore (tt + ia) (tt - ia) z = i {e'C'^-'^^' + e-f'"**"^'},
where tt = r7, + f?^. Therefore, by (3510), with x = e*, (/ = e^
Ca^— (j^i + w) a^ — (m + ?i) -^
a^—^m + ny
3571 Prop. I.— To transform a linear differential equation
of the form
into the symbolical form
MD) u^-f,{D)eUi+MD)e^'u^kc. = T (2),
where Q is a function of .v, T a function of 6, x = e^ and
Multiply the equation by Qf ; then the 1st term on the left
becomes, by (3489),
(,, + le' + cr'-' +...)D{D-1) ...{D-n-i-l) u.
This reduces, by the repeated appUcation of formula (3476)
with the notation of (2451), to
3572 aD^"Ui-\-h {D-iy^^ eUi^-c {D-^Y"^ e-'u-^ka.
The other terms admit of similar reductions.
3573 Conversely, to bring back an equation from the sym-
bohc form (2) to the ordinary form (1), employ formula (3475)
so as to transfer e"" to the left of the operative symbol.
3574 Ex. : x' {xhi,,-\- 7xu^.+ hu) = g''{I) (i»-l) + 7D + 5} «
= e-' (D- + 6 J> + 5) « = e-^ (D + 1) (D + 5) u
= (X»-l)(D + 3)e"tt (3476).
632 DIFFERENTIAL EQUATIONS.
For tlie converse reduction, the steps must be retraced, employing (3475).
See also example (3578).
3575 Prop. II. — To solve tlie equation
u+a^<l> (D) e'u+a,<f> (D) ^ (D-l) e'' u ...
wliere U is a function of 0.
By (3491)
Putting p"it for this, the equation becomes
3576 Therefore
where qi, q.2 ... q„ are the roots of the equation
and A,. =
q'l.-^
iq,'-qi){q>--q-^ ••• (qr-qn)'
The solution will then be expressed by
U = A^Ui-\-A.2Ui ... 4-^nW„,
where ?/^ is given by the solution of the equation
3577 u,-q4iD)e'u,.= U.
3578 Ex.:
(x' + 5x' + G.v') «o. + (4x + 25x' + S6x') u, + (2 + 20a; + 3G.r'0 u = 20.v».
Putting X = e*, and transforming by (3489),
(l + Se' + Ge^O i>(-D-l) H + (4 + 25e'' + 3Gc-') D(( + (2 + 20e'' + 36e-^) it = 20e^
The first term = D (D-1) « + 5 (D-l)(D-2) e'^ + G (D-2)(D-3) e'"«
by applying (347G). The other terms similarly ; thus, after rearrangement,
(D + l)(i' + 2)« + 5(D + l)-e'« + GD(i) + l)e'-'« = 20e^
Operating upon this with {(D + l)(D + 2)}-', we get
or (1 + .V + Cp^)« = e^ if p = (i)+l)(D + 2)->e^
therefore « = {3 (l + 3p)-'-2 (1 +2,))'} <>'' = 3;/ -22,
if 2/ = (1 + 3p) - ' e="' and .- = (1 + 2p) "' e^*.
SYMBOLIC METHODS. 533
Hence (l + 3p)y = e'' or y + S (D+l)(D + 2)-' e'y = e'' ;
therefore (D + 2) y + 3 (D + 1) e'y = e"> (S-\-2), by (3474),
or (-D + 2) y + Se' (D + 2) y = 5e^ by (3475) ;
that is, (x + Sx')y, + 2(l + Sx) y = 5x'.
Similarly (x + 2x') 2, + 2 (1 + 2.v) z = 5x\
Solve these by (3210), and substitute in m = 3y — 2z.
3579 Prop. III. — To transform tlie equation
u+(f>{D) e''u = U into v-\-<f>{D^n) e'v = V,
put u = e'^^v and U — e^^ V.
Proof.— By (8474), because (D) e'^^'-^'tJ = e»> (D + n) e'^v.
3580 Prop. IV. — To transform the equation
?f+(^(i>) e"-' u = t/ into v+V'(i>) e^^'v = F,
put
3581
where P HI>) - HD) ^JD-r) <l>{B-2r) ...
Proof. — Pnt u =f(D)v in the 1st equation, and e'''f{B)v =f{D—r)e'''v
(3476). After operating with f'^ (D) it becomes
v + <l>(D)f(D-r)f-'(D)e'-^v=f''iD)U,
therefore (p (D) f(D-r) /"' (D) = 4^ (D) by hypothesis ;
and so in inf. Also U =^ f (D) V.
3582 To make any elementary factor x(^) ^^ 'Pi^) ^®'
come, in the transformed equation, x (-^ i ^^'0' where r is an
integer; take ^ (D) = xi^ + nr) Xi(-^)- See example (3589).
3583 To make any factor of (p {D) of the form — ^y /^
disappear in the transformed equation, take '^(-D) = xi{D),
where Xi(-^)' ^^ ®^^^ case, denotes the remaining factors of
<^(X>). See example (3591).
534 DIFFERENTIAL EQUATIONS.
3584 III tlie application of Proposition IV., differentia-
tion or integration will be tlie last operation according as
■^j- /^7^ (3581) lias its factors, after reduction, in tlie
numerator or denominator, and therefore according as \p (D)
is formed by algebraically diminishing or increasing tlie several
factors of (p (D). However, by first employing Proposition III.,
the given equation may frequently be so prepared that the
final operation with Prop. IV. shall l^e differentiation only.
See example (1).
For further investigation, see Boole's Dif. Eq., Ch. 17, and Supplement,
p. 187.
3585 To reduce an equation of the homogeneous class
(3531) to a binomial equation of the same order of the form
The general theory of such solutions is as follows. Let
the given equation be
it+q {(i>+«i)(i> + «2) ... {D+a,)}-\^'Ui = U ... (1),
%, rtg, ... ctn being in descending order of magnitude. Putting
w =r e'^'^v, by Prop. III.,
'V + q{D{D-7i:;^,) ... (Z>-^^7=^J}-^^«^• = e"^'U... (2).
To transform these factors, regarded as ^ (D), by Prop. IV.
into ■^{D)=D{D — l)...{D-n + l), we convert D into D + rn
(3582), r being an integer.
Hence for the j/^^ factor we must have
D-\-rn — ai-{-aj, = D— j; + 1,
3586 and therefore a^ — iip = rn -\-p — 1 (3) .
If this relation holds for each of the constants a^ ... rt„,
equation (1) is reducible to the form
3587 u^-q{D{D-l)...(D-n-\-l)]-\"'i,=^ Y (4),
which, by (3489), is equivalent to !/„.,. + </'/ = 1,,.,. = X.
1/ being found in terms of x from the last equation, and,
V being = P, / ^ (^ ij (3580), the solution will result from
3588 „ = !-...»P,. (/;-i)(_^>-^)y(/^-"+i) ,;
while U and Y are connected by the same relation as u and y.
SYMBOLIC METHODS. 635
3589 I^x- 1 : Given ./•■>«3., + 1 8a;-«,, + 84to. + 96a + Say'u = 0.
Putting X = e" and employing (3489), this becomes
{D (D-l)(D-2) + 18D (D-l) + 84D + 96} u + 3e'ho = 0,
or (D + 8) (D + 4) (D + 3) « + Se'' tc = 0,
therefore « + 3 {(D + 8}(D + 4)(D + 3)}-^e^^^« = (1).
Employing Prop. III., put u = e~^"v,
therefore (3476) v + 3 {D (D-4>)(D-5)}-' e''v = (2).
To transform this by Prop. IV. into
y + S{D(D-l)(D-2)}-'e'^y = (3),
we have
P tm = ■ D(P-l)(J-2)(D-3)(J-4)(J}-5) ... ^ .„_! wr,_9>)
'4,{D) Z)(D-4)(D-3)(I»-3)(D-7)(Z>-8) ... ^ '^ "^'
.-. i;= (D-l)(D-2) ;/, .-. ii = e-''(D-l)iD-2)7j (4),
and the solution is obtained by differentiation only, performed on the value
of 2/ as obtained by the solution of (3), that equation being equivalent to
D(D-l)(D-2)y + -Be''y = 0, or, by (3489), y,, + Sy = 0.
If, however. Prop. IV. were used to pass directly from (1) to (3), we
should have
<p(D) ^ I)(D-l)(D-2)(D-S)(D-4)(D-b) ...
'4^(D) (i) + 8)(D + 4)(X» + 3)(D + 5)(D + I)D...
1
(i' + 8)(D + 6)(D + 4)(D + 8)(D + 2)(D + l)'
and equation (4) would involve integrations of y as high as D'^^y.
3590 Note. — By the literal application of Rule IV., the right side of
equation (3) ought to be F = {(!> -1)(Z) — 2)}"^ ; but no such term is
required when the original and transformed equations are of the same order,
for in such cases the arbitrary constants introduced by the operation upon
zero disappear with the terms containing them in the final differentiation >
The result is the same as if the operation upon zero had not been performed.
In the following example, V has to be retained.
3591 Ex.2: (x-x')u,,+ (2-l2x')u,-'30xu = (1).
Multiply by x, transform by (3489), and remove e"' to the right of each
function of B by (3476), thus
u—
(J + 4)(J + 3)„,,,_
M = (2).
D^D + l)
Transform this by Prop. IV. into
v-^±^e-v = V (8).
We have ?i == P, ^^ u == (-D + 4) (D + 2) n,
7= {(D+4)(D + 2)}-iO = Ae--' + Be-'' (3518).
The operation upon zero is required in this example (see 3590), because (8)
536 DIFFERENTIAL EQUATIONS.
is of a lower order tliun (2) ; but only one term of the result need be retained,
because only one additional constant is wanted. Hence (3) becomes
(D + l)v-(D + S) e-% = (D + 1) Ae-'-' = -Ae-'\
Changing again to x, this equation becomes
(x'-x') v,-4:xh- + A = 0.
The value of v obtained from this by (3210) will contain two arbitrary con-
stants. The solution of (1) will then be given by
u= (D + 4)(Z> + 2)r.
3592 Ex. 3 : W2«-« (h + 1) x-'u-qho = 0, [Boole, p. 424.
n being an integer.
Multiplying by x" and employing (3489), this becomes
u-cf {(D + n)(D-n-l)}-^ e-'u = 0.
This is changed by Prop. III. into
v-q'{D(I)-2n-l)}-'e''v = 0, with m = e'-'V,
and this, by Prop. IV., into
y-q'{D(D-l)}-'e''y = or t/,.-5V = (3489).
y being found from this by (3524), we then have
u = e-'-P. ^-'^ . y = e-'-(D-V)':,y = x-"(xd^-my.
JJ — Zn — 1
But, by (3484), F (xd,-m) = x"'F (xd,) x-'\
.'. w = X-" X (xd,) x-\x' (a;4) x-' ... x'"-' (xd^^) x'^'^'y,
or u-=x-^"''^(x'd,yx-"'*'y
= X-'"' (x'd^y x--"-' (Ae'^ + Be-"') (3525).
This may be evaluated by substituting z = x'-. (See Educ. Times Reprint,
Vol. XVII., p. 77.)
3593 Ex. 4 : n,,-a'u,,-n (n + 1) x'^ u = 0.
The solution is derived from that of Example (2), by putting q = ad^^,
and arbitrary functions of y after the exponentials instead of A and B ; thus
to = X-'"' (xM^y X-""' {e"^"." ./) (y) + e "■'" -4 (y) }
= X-"-' (x'd^y X--" " {0 (2/ + ax) + >//(?/ + ax) }, by (3492).
[Boole, p. 425.
3594 (l+ttd-) u.,,'\-axu,±nUi = 0.
To solve this equation or its symbolical equivalent obtained
by (3489), viz.,
3595 u^.^liJ^=g^e-u = il
Substitute / = [-tt^ — ^ i" the soluticm of u.t^n-n = 0, by (3523-4).
J v/(l + aa;^)
SYMBOLIC METHOBS. 537
3596 Similarly, to solve the equation
or, the same in its symbolical form,
[ dx •
Substitute t = — 7—^ r in the solution of w^, ± ?t*i( = 0.
Jx^(x^ + a)
(3596) is obtainable from (3593) by changing 6 into —0.
3598 Pfaff's equation,
When Q = 0, the symbolical form becomes
b(D-n)(D-n-l) + e(D-n) + c,
"^ aD{D-l) + cD+f ' "-^ ^^^'
If w be not = 2, substitute 20'= nd, and therefore 2cfe = nd^..
3599 Thus „+^i|5^1(|5|le-. = (2).
where Oj, aj are the roots of the equation
h (^na-v)(^na-n-l) +e (^na-n) -{-g = (3),
and /3i, /3j are the roots of
a |n/3 (|7i/3-l)+c i"/3+/ = 0.
Four cases occur —
3600 I. — If Oi— Qj and /3i— /Bj are odd integers, (2) can be reduced by
Prop. IV. (3581) to the form v + ^ ^^~"'^ ,^^~"^~^^ e'^'v = 0,
and then resolved into two equations of the first order.
3601 II. — If any one of the four quantities Oi— /Jj, aj— /Pj, «2 — /3i, fj—Z^a
is an even integer, (2) can be reduced by Prop. IV. to an equation of the
first order.
3602 III. — If Pi—Ij-j and aj + uj — /3j — /jj are both odd integers, then, by
Props. III. and IV., (2) is reducible to (3595).
3603 IV. — If aj-oj and Oi + Qj— /3i— /Sj are both odd integers, (2) is
reducible in like manner to (3597). [Boole, p. 428.
Note. — The integers may be either positive or negative, and when even
may be zero.
3 z
538 DIFFERENTIAL EQUATIONS.
SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS
BY SERIES.
3604 Case I. — Solution of the linear differential equation
/o (D) u-f, (D) e^' u = or f, {.vtQ u-f, {.vcQ .v^ u = 0,
in which fo{D), fi{D) are polynomial expressions of the form
ao + chD-\-cuD\.:-\-a,D'^ and f,{D) = {D-a){D-h){D-c) ....
3605 Let </>(!))=/, (D)-f-/o(D), and let
3606 ^ («) = 1 + <^ (rt + r) X' + <!> {a + 27-) <p{a + r) x'^
-{-<p (rt + 3r) <p {a + 2r) cp {a + r) x^''-{-&c.
Then the solution will be
3607 u = A.v'<P{a)+B.v'^{b) + av'^ic) + &c.
Proof. — Operating with f~^ (D) and writing p for ^ (D) e''",
u -pu = f-' (D) = Ae"' + Be''' + &c. (3518)
Therefore, by (3515), « = (l-p)"' (Ae''' + Be'"+ ..,)
= (l+P + p'+...)AG"' + (l+p-^p'+...)Be'' + &c.
Now in each terra substitute for p" the value in (3491), and remove D by
formula (3474).
Case II. — Solution of
3608 MD) u+MD) e'u+MD) e~'u ... +/,(D) c^^'u =
where A(D) = {D-ct){D-h){D-c) ...
Let ^ (a) := 1 + F, (a + l) x-\-F^ {a + 2) x' + Slc,
where the coefficients F^{a-{-l) or v^, i^2(« + 2) or v^, &c. are
determined in succession by the formula
3609 /o (m) v,„+f, (m) r„,_: . . . +/. (m) i'„,_, = and v, = 1
The solution will then bo expressed by
3610 u = ^.r«*(a) + /^.f'*(«')-f CV*(c) + &c.
Proof. — From (1)
«= {l + 0.(/J)e"...+0,(D)e'-}-7o-'(^)-O (3),
where l'r(D) = fr{D) ---fo{D).
Hero fo-'(r))0 = {(D-a)(D-h) ...}-' = Ae"" + Be^' + . . . (3518);
and {l + ^,(D)e"..,+./.„(D)e-j-' = l + F,(D)e« + i^,(X»)e^^+...
SOLUTION OP LINEAR DIFFERENTIAL EQUATIONS. 539
To determine F^, F^, &c., operate upon each side with {l+0 (D) e' + ctc.},
and equate coefficients of powers of e ; thus formula. (2) is obtained. (3)
now becomes
u= {l + F,(D)e' + F,(D)e''+...}(Ae'"^+Be"^+...) (4).
Multiply out; apply (3-i74), and put x for e".
3611 Ex. : xht2x—(a + h — l) xio^+ahio — qxhi = 0,
or, by (3489), {B-a){B-h) ti-qe^'u = 0.
Here f,(D) = (D-a)(D-b), A (D) = 0, f,(D)=-q.
Therefore (2) becomes (m—a)(m—b) v,„ = qVin-2,
therefore F^, F^, &c. vanish, and ^o («) = 1»
F,(a + 2) =
qFg (g) _ q
(a + 2-a)(a + 2-Z>) 2(a + 2-5)'
F,^a + ^) = --^M^
(a+4-a)(a + 4-6) 4.2 (a + 4-&)(a + 2-Z>)
Therefore ^ (a) = 1 +
qr
-.2,„4
2(a + 6-2) 4.2(a-6 + 4)(a-6 + 2)
Similarly we find F^(h-\-2), F^{h + 2), &c., and thence ^(Z^); and, sub-
stituting in (3610), we have
_ A a I ^?-^"'' I ^^'■^°'' I
u- ^-^ +2(a4-6-2)"^4.2(a-6 + 4)(a-i + 2) '"
^ ^2(6-a + 2)^4.2(6-a + 4)(6-a + 2)
3612 The solution is arrived at more quickly by formula (3607). We
have {!>) —
±
(D-aXB-hy
<^(a + 4) = -
producing the same series by the value of * (a). Similarly with $ (6),
(»+2) = 20^^^' *('' + *) = 4-(^;^^' *=■
3613 When /o(-D) lias r factors each =: B—a, the corres-
ponding part of the value of u in equation (4) will produce
3614 ^o+"4,log.^^+^2(log.^f ... +^._x(logcr)-S
where the coefficients A^, A-^, ... are each of the form
3615 But if any one of the quantities F, (a + r) = (3608),
then Gr = also.
540 DIFFERENTIAL EQUATIONS.
Proof.—/"' (D) now contains a term of the form
e''"(co+Ci0...+c,e'-O = t^'v, say.
The corresponding part of ii in (4) is
{l + ^i(D)e''+...} e'"'v
= [e''' + e^''*'^'F,(D + a + l)+c"''"'F,{D + a + 2) + ...]v by (3475).
Expand each function F by Taylor's theorem in powers of D, operate upon v,
and arrange the result according to powers of 0.
In practice, proceed as in the following example.
3616 Ex. : a;u2;,-{-u^+cf^u = 0.
Multiplying by z and changing by (3489), this becomes
DV + s'e'" u = 0. I>-u = gives u = A + Bd.
Substitute this value and operate with I), considering A and B as variables,
and equate to zero the coefficients of the powers of ; thus
B'A + q'e^'A + WB = 0, B'B + q'e'^B = 0.
Then change D into m, and e'^J. into a„_r, to obtain the relations
m^a„,-^q^a^.i+2'mb^ = ; m''b„+ q^b^.^ = 0,
which determine the constants successively in terms of a^ and l^ (which are
arbitrary) in the equation
u = ao + aiX^ + atX*+ ...+\ogx (bQ + b2x'' + h^x*+ ...),
which thus becomes the solution sought. [J5ooZe, Biff. Eq., p. 439.
SOLUTION BY DEFINITE INTEGRALS.
3617 La2)lace^s method. — The solution of the equation
^(^((/,)w+^(c/,)i. = (1)
is u=C^{e'''^S'^'\cf>t)-'}dt (2),
the limits being determined by
/'*i«^'=0 (3).
Proof.— Assume t* = e e'^Tdf, and substitute in (1), putting f (cZ,) c''
= ^(0e^' (3474), thus
{xe^'f (0 Tdt+ [ c"xP (0 Tdt = 0.
• This mothod of solution is meruly indicated here, and the reader is referred to Boole's
jD//. Hq., Ch. xviii., for a comploto investigation.
SOLUTION OF DEFINITE INTEGRALS. 541
Integrating the first term by parts, this becomes
e'^cp (0 r- je-' [cZ, [<|>(t)T}-^p (t) T^t^O (4),
an equation which is satisfied by equating each term to zero. The second
term thus produces a value of T by integration by (3209), and this value
substituted in the first term, and in the value of u, g-ives the results (3)
and (2).
3618 Ex. (1): a^u,:,+au^'-q\vu=0* (4).
Here <}> (4) = c^a,— j^ ;// (d^) - ad^, <p (t) = f^-q", ;// (0 = at. Hence
(2) and (8) become
u = G{e''(f-q')^''dt; e''' (t' - q')^ = 0;
a being positive, the limits are t = Joq, and, putting t = qcos d, we find
u=z C I V"'^sin"-'etZ0 (5).
3619 The solution in series by (3608) is as follows. Equations (1) and
(2) of that article are in this case
D(D + a— 1)m— gV^M = and m (m + a — l)vn—q^Vn-2 = 0.
Thus, a in (3608) =0, and I = 1-a. Therefore (3610) becomes
" = ^{^+2^) + 2.4(a + li(a + 3) +^^-
^^-^-1^+2|g^ + 2.4(3-ai(5-.) -^^-} ^'^'
Both series are convergent by (239 ii.).
The results deduced by Boole are these —
3620 (5) is equivalent to the particular integral represented by the first
series of (6).
3621 A second particular integral, by assuming u = e'^'^'^t;, is found to
be, when 2 — a is positive,
u = C^x'-" I e'^^'^sin'-^flcZfl (7).
Jo
3622 When a lies between and 2, the complete integral is
tt= (7ijV^«"'«sin"-^0(Z9+C2a;^"'[V*'='"«sin^-"^(^6l (8).
* The method by definite integrals is elucidated by Boole chiefly in the solution of this
important equation.
542 DIFFERENTIAL EQUATIONS.
3623 But, if a = l, the solution becomes
u= fV*'°«« {A + Blog(xsm''d)] dO (9).
3624 If a does not lie between and 2, tlieu, if a be negative, put
a = a—2n, and replace the fix'st term of (8) by
(7, (xd, + a-l)(ixd, + a-S) ... (xd, + a-27i + l) [V"^^sin<^'-ic^0 ... (10),
the transformation being eflFected by (3580).
3625 And if a be positive and > 2, put u = e^'^-''''''v = x'^'''v. This con-
verts a into 2— a, a negative quantity, and the case is reduced to the last
one.
3626 Ex. (2). — To solve by tliis metliod tlie P. D. equation
ihx+i(2y + U2z= (1)
wlien r = \/{x'^ + i/)'
Eliminating x and y, (1) becomes
rU2r + l('r + '>'^''2z = (2).
Now the solution of this equation is number (9) of Example (1), if w^e
change x into r, q into id^, and A and B into arbitrary functions of z. We
thus obtain
n= [V<=°^^^'^={./)(2) + ;//(2)log(rsiu=e)} dd (3),
or, by (3492),
M= ^j j^ + tVcoseJ fZ9+ ;p(2 4-tVcos0) log(rBin-^) c?^ (4).
See (3551) for another solution.
3627 If « be the potential of an attracting mass at an external point,
and if it, = F {z) when r = ; then, since log r = oo , ^ (z) must vanish ;
therefore F(c) = <\, (s) dd = tt^ (z).
Hence (4) reduces to « = — I JT" [ -.fiV cos Q \ dO.
ParsevaVs Theorem.
3628 If, for all values of n,
and A'-\-B'ii-^^G'u-^^ ... = ^p{u) (1),
DIFFERENTIAL RESOLVENTS. 543
then
AA'+ BB + ... = L^^^[<l> (e^-^) tA (e'-0 + <^ (e'^ 4 (e"''^)] dO.
Proof. — Form the product of equations (1), and in it put u = e'" and e''"
separately, and add the results. Multiply by dd, integrate from to tt, and
divide by 27r.
P. D. EQUATIONS WITH MORE THAN TWO
INDEPENDENT VARIABLES.
3629 By means of Fourier's theorem (2742), the sohition
of the equation
may be deduced by a general method in the form
ti = (1 + f/,) (((((( e'^^'-'^^^'^ xjj {a, b, c) dadbdcdXdiidp,
the limits of each integration being — oo to go , and the func-
tion ^ being arbitrary and different in the two terms arising
from the operator {l-\-df).
Boole, Ch. xviii., and more fully in Cauchy's Exercice di'Anahjse MathS'
matique, Tom. I., pp. 53 et 178.
3630 Poisson's solution of the same equation in the form of
a double integral is
7i=(l + dt)\ I ^sin.^4' O'^ + ^^'^siulsinT?, ?/ + /iLsin^cos /?, z + ht coa ^) d^dr/
with the same latitude in the interpretation of xp.
[Gregory's Examjyles, p. 504.
DIFFERENTIAL RESOLVENTS OF ALGEBRAIC
EQUATIONS.
3631 Theoeem I. (Boole). — ''If y^, y^ .-.Vn are the n roots
of the equation
y^-air'-^l = () (1),
and if the m^^ power of any one of these roots be represented
644 DIFFERENTIAL EQUATIONS.
by u, and if a = g\ then u as a function of B satisfies the
differential equation
„-(!izix)+i!i-iy"-'Y^_»_i)rz)<»)-i-.e-„ = o,
\ n n / \n n / •- J
and the complete integral of the same will be
3632 " Cor. I.— If m=-l and if n be > 2, the differential
equation
/)(»-.) „_i (!i^ i>- ± _iy""'e.»„ =
n \ n n I
has for its general integral
y iV ••• Vn-i being any n — \ roots of (1).
" If be changed into —0, and therefore D into — D, the
above results are modified as follows : —
3633 " CoE. II.— The differential equation
has for its complete integral
u = C,y'^-^ C^yl . . . + C^yZ,
Vii 2/2 ... Vn being the roots of the equation
ay'-?/"-^ + a = (2).
3634 " CoK. Ill,— The differential equation
„-„(/>-2).»-..[(!i^i>+±)'"-"]-V«„ = 0,
supposing 7i > 2 has for its com})lete integral
2/i>?/2 ••• 2/n-i being any n—\ roots of (2),
BIFFEUENTIAL RESOLVENTS. 545
" Theorem 11. {Ilarley). —
3635 " The differential equation
^ -"^ \ n n I \n n I
is satisfied by the ??i^'' power of any root of the equation
if—xif-'^-a = 0,
u beino; considered as a function of x.
3636 " Cor.— The differential equation
is satisfied by the Wi^""^ power of any root of the equation
?/'*— ////"-'■ + (« — !) X = 0."
[Boole, Biff. Eq., Sup. 191—199.
3637 See also Boole, Phil. Trans., 1864; Harley, Froc. of the Lit. and
Phil. Soc. of Manchester, Vol.11.; Bawson, Proc. of the Bond. Math. Sac,
Vol. 9.
4 A
CALCULUS OF FINITE
DIFFERENCES.
INTRODUCTION.
3701 III this branch of pure mathematics a fuuction (/> (,')
is denoted by %., and (j){x-\-h) consequently by ■u_^+,,. The
increment h is commonly unity . If Ax denotes the increment
h, and A^^^. the consequent increase in the value of ?;^.5 we have
3702 A«., = «,+^,,-w,,.
3703 When Ax diminishes without limit, the value of
Ai/,
or
U, + A,—U,
is
du.
K
Ax
cLv
3704 The repetition of the operation A is indicated as
follows :
A At*,. = A-u^, AA^», = ^^«.f5 ^^^ so on.
3705 Ex. : Let Uj. = x^,
X =1 2 3 4 5
'i
1 4 9 16 25
Ax'=S 5 7 9
AV= 2 2 2 ...
FORMULAE FOR FIRST AND n''^ DIFFERENCES.
3706 A'X,. = (in (n-l) ... («-'>•+!) .r"-'- + (V»'""'"' + &c.,
3707 A^'u,.= an{n-}) ... ;i.2.1.
FOBMULJE FOB FIB-ST AND 7;*" DIFFFBENCES. 547
3708 Hence the 71th. difference of a rational integral func-
tion of tlie nth deo'ree is constant.
3709 So also A".r" = 1.2.3...7i.
3710 Notation.— Factorial terms are denoted as follows : —
3711 ^ =u\r-l
3712 Thus x{x-l) ... {x-m + l) = x^'-\
3713 1 -...i-m)
Hence \m, ml, and m^'"^ are equivalent symbols.
3714 According to (2452), x''"' would here be denoted by a;'.'"'. The
suffix, however, being omitted, it may be understood that the common differ-
ence of the factors is always —1.
and, if m<Cn, A^i-^'"^ = 0, since Ac = if c = constant.
3718 A.v^-'") = -ma.^^-"-'\ A\v^->"^ = (-m)^"^^^-'"-"^
3720 AmI;"^ = {u..i-u.-,n^^) "i'"-^^
3722 Ex.:
A(ax + by'"'> = am(ax + by"'-^\ ^(ax + hy""> = -am (ax + hy-'"-'^\
3724 Alogw,=:logh + ^^(, Alog<-i) = logiiH±2_.
3726 Art-^- = {a — 1) «■'', A'Vr"-'' = {a"' — iya"'-'\
owoo AM^iu/ I 7\ /o • «\"siii ^ , / , n(a-\-'rr))
3728 A" («ct"+^) = 2sm— - ^ rtci'+64- — ^^-r^ — ^f.
**^^ cos^ ' ^ V 2/ cos I ' 2 )
Pr.oOF. — A sin (art; + ?>) = sin (ux + a + b) — sin (ax + b)
= 2 sm ~ sm \ax-\-b^ — j .
That is, A is equivalent to adding — — — to the angle and multiplying the
2
sine by 2 sin -.
^ 2
548 CALCULUS OF FINITE DIFFERENCES.
3729 Conversely, tlie same formula holds if tlie sign of n
be changed throughout.
EXPANSION BY FACTORIALS.
3730 If A"'/'(0) denote the value of AX-v) when x = 0,
then (^ GO = <j> (0) + A(/) (0) .r+ ^^^ .v^'^ + ^^^ ^'^) + &c.
3731 If AcV=/^ instead of unity, the same expansion holds
good if for A"<^(0) we write (A"<^(<(')-r/^")^^o; that is, making
x = after reduction.
Proof. — Assume (a?) = aQ + a^x+a^x^-^ + a^x'^'-^&c.
Compute A0 (a;), A^^ (a')? &c., and put x=:0 to determine fto, ai, aj, &c.
GENERATING FUNCTIONS.
3732 If '?f.r^'" he the general term in the expansion of (p{t),
then (i)(t) is called the generating function of n^. or (p{t) = Gi',^.
Ex.: 0—t)-'=G(x + l), for a! + l is tbe coefficient of t'' in the
expansion.
3733 Gu., = <i>{t), G».,«=M), ... Gh,.,„ = M.
3734 GA»,, = (l-l)<^W, ... GA"»,, = (l-l)V(0.
Proof. — G^u^= Gu^^^—Qu^, &c.
THE OPERATIONS 27, A, AND d^.
3735 -E* denotes the operation of increasing k by unity,
Eu^, = H.,+1 = jf.,+Ait., = (1 +A) t/..,.
The symbols E' and A both follow the laws of distribution,
commutation, and re;petition (1488-90).
3736 E^ 1 + A = c-^- or e^.*
Proof. — Eu^ = a^.x = u^ + (lv^ix+h^2x^''^+ :^-^<-^ix^'x + &'C-
2.3
= (l + '7,,+H.,+ ^jh. + &o.) u, = e"'u,.
By (1520), A.r being unity.
* The letter d is reserved as a symbol of differentiation only, and the suffix attached to
it indicates the independent variable. Sec (1187).
FOBMULzE FOB FIRST AND u"' DIFFERENGBS. 549
3737 Hence A = 6'^^! and D = \ogE.
3739 Consistently with (3735) E"^ denotes the diminisliing
X by unity ; thus E~'^n,. = u^^^i.
For Eti^.i = u^, .". w^.i = E'^tCj;.
Ux+n "2^'^^ terms o/u^ and successive differences.
3740 «^.+« = u,-\-n^u, + C (»., 2) A2?f,+C (w, 3) A%,+&c.
Proof. — (i.) By induction, or (ii.) by generating functions, or (iii.) by
the symbolic law :
(ii.) Gu,,, = (i)'V (0 = [ 1+ ( j -i) I 'V(o-
Expand by the Binomial theorem, and ajiply (3734).
(iii.) «,,„ = E««^=(1 + A)«tv
Apply the laws in (3735) by expanding the binomial and distributing the
operation upon %.
Conversely to express /y'u^. in terms of u^,,, 7?.,+i, v^^.+c,, &c.
3741 A"Jf^. = tf,^^^^--nff, ,+„_!+ C (71, 2) w.,+«_2 . . . (-1)" u,.
Proof.— AX- = (^-l)""a; (3736).
Expand, and apply (3735) as before. Putting .»; = 0, we also have
3742 AX = ^K-nu„-,+ C,,,u,_, ... (-1)" ii,.
3743 A"cv"^ = (,t'+?j)'"-« (.v+n-l)'"
-\-C (71,2) {.v-\-n-2y"-&c.
3744 A'^O'" = M™-n (7i-l)"^+C(w, 2)(w-2)'«
-C{n,3){n-3y"-^&G.
3745 Ex. : By (3717) A"0" = n ! Hence a proof of theorem (285) is
obtained.
3746 A'%,v,,= {EE'^iyu,v„
where E operates only upon if^ and E' only upon i\^,.
Proof. A?(,_,,v ^ = Uj, + i i\, , i — u^^ i\ = Euj. . E'v^ — n^r^ = (EE' — 1 ) u^ v^..
ApiMcations of (3746).
3747 Ex. (1): ^''u,v,= {~iy{\-EE'Yu,v,.
Expand the binomial, and operate upon tlie subjects u-^, v^ ; thus
550 CALCULUS OF FINITE BIFFERENGE8.
374:9 Kx. (2) : To expand cfsmx by successive differences of sin a-.
A" a"" sin a) = | _E? (1 + A') — 1 1 " a"" sin a; = | ^ + ^^' } ""^ sin x
= { A" + «A"-'J?A'+(70^2) A"--E-A'H&c. } a^sina;
= A" a\ sin .c + /iA"-^ a^ '^ A sin a? + C (ii, 2) A''--a^"- A^ sina; + &c.
= n"'! (a — l)''sina' + ?i(a — l)"''«Asina)+6'(ii, 2)(a — l)"'-o^A^sina! + &c. | ,
by (3727), wliile A'' sin « is known from (3728).
3750 Ex. (3) : To expand A"Mj,?;j, in differences of u^ and v^ alone : put
E = l + ^, E'=l-^^' in (3746), thus
A«M^'y^= (A + A' + AA')"m^.v^,
wliicli must be expanded.
A"Ux in differential coefficients of \\.
3751 A«w., = C%+^i^/r'+^2C'?r,-+&c.
Proof. — A"»^. = (e"-" — 1)"?^,. (3737).
Expand by (150) and (125) as if cZ^ were a quantitative symbol. See also
(3701).
-r-Yi i^'>^ successive differences of u.
3752 g={log(l+A)}"«,
The expansion by (155) and (125) will present a series of
ascending differences of u.
Pkoof.— e""=:l+A, .-. J, = log(l + A).
3753 Ex.:if% = i, ^ = A^^ — 2^'Y~~¥'^^'''
If G be a constant,
3754 i>{n)C=cl>{A) €=<!>({))€ and <PiE)C = <t>{l)C,
Since every term of (1>(D), or of (j) (A) C, operating iipon C, produces 0;
and every term of (E) operating upon produces G.
IIERSCHEL'S THEOREM.
3757 <p{c') = <l>{E)c^^-^
3758 =ci>0)+ct>(i£)i)j-j-ci>{E)iy^.^+&c.
tNTEEFOLATION. 551
Proof.— Let ^ (e^ = A^^-A^e* ... +^„e"*
= A,e'-' + A,Ee'-' ... +AnE"e'-^ = (A, + A,E ... +A„E„) e"-'
= <^(iJ) e°-^ = (E) { 1 + O.f + j*^ +&C. } ,
and 0(i;)l = 0(l) by (3756).
A THEOREM CONJUGATE TO MACLAURIN'S (1507).
3759 <l>{f) = <l>{D)e'-^
3760 =^{0)+<l>{(QO.t+<j>{(h)0\^-^&G.
PiiooF.— <p{t)=<l> (log eO = ^ (log i') e°' (3757)
= </.(D)e°-' (3738) = ^ (-D) (1 + O./+ +&c.|,
and ,/;(D)l:=</)(0) (3754).
n being a positive integer,
3761
" ~ ^At'" "^ 1 . 2 . . . (yi+ 1) f/a'"+^ "^ 1.2... («-}-2) ^/ci""^-^ "^
Proof.— By (3758), putting (e') = (e'-l)",
(e'-l)"=(i;-l)"0./+(:EJ-l)"0-.-^'-+&c. =:A"0./ + A"0^^ + &c.
Put t — d^, and employ (3736) and (3737).
INTERPOLATION.
Aiyproximate value of u^ in terms of n particidar equi'
distant values.
3762 If u^ is an integral algebraic function of the degree
n—1, ^''u,. vanishes, and therefore by making x = 0, and
writing x for n in (3740),
3763 "... =^ u,-]-.rAu,-\-C,^,A'u, ... +C,,,_iA'-^w,.
This is formula (265). The given values are u^,, Aii^, A^ii^,
&c., corresponding to «, h, c, ...
3764 For an application of the formula to the problem of interpolation,
see (267), in whicli example x = 1-54 and u^ = log 72-54.
652 CALCULUS OF FINITE DIFFERENCES.
3765 When tlie term to be interpolated is one of a set of
equidistant terms, employ (3741). A"?(^; = 0, as in (37G2) ;
tliercforc
3766 '^-""«-i+^.,2*^.-2-C\3'^.-3 ... + (-1)"'^ = 0.
3767 Ex.: From siuO, sin SO'', sin 45°, and sin 60', to deduce the value
of sine \h\
The formula gives sinO— 4 sin 15° + 6 sin 30°— 4 sin 45" + sin GO^ = 0,
or -4sinl5°+3-2y2 + iN/3 = 0,
from which sin 15" = 1(6-4/2 + 73) = '2594.
The true value is "2588 ; the error "0006.
LAGRANGE'S INTERPOLATION FORxMULA.
3768 Let ft, b, c, ...h be r^ values of x, not equidistant, for
wliicli the values of ?/.^, are known ; then generally
3769
"- ^^'-(a-b){a-c) ... {a-k)'^^^'lb-a)(b-c) ... [b-k)
'{k-a){k-b){k-c)..:
Proof. — Assume i(,„., = A (x — h)(x — c)... (x — Ji)
+ B(x--a){x-c) ... (x-k) + C(x-a)(x-b)(x-il) ... (x-Jc) + &c.,
and determine A, B, C, &c. by making x = a, h, c, &c., in turn.
If the values of a,b,c, ... Jc are 0,1,2, ... n—l, (3769)
reduces to
3770 u - u ■^-Gr-l)...(.r~/i+2)
3770 K^ - */«-! 1.2.8... Oi~l)
_ ,rOr-l)...(.r->i+a)Gr-n+l)
1.1.2.3... (;i-2)
.,, a>(,r-l) ■■■ (.r-»+4)(.t-» + 2)(.r-;^ +l) ,
+ "«-^ 2.1.1.2.3...(>,^ '^^•' ""'
3771
MEGBANICAL QUADBATUBE. 553
MECHANICAL QUADRATURE.
The area of a curve whose equation is y = z/.,. in terms of
n-\-l equidistant ordinates u,Uy, ... n,^, is approximately
3772 nu+!^..+{l^-f)^+il^-.,+,.^^
\6 4 ,3 / o !
+iy -— +1^'^ — i- +-3- -^^N iry-
Proof. — The area is = I u^-dx. Take the vakic of n.^- in terms of
Jo
i6(„ 16, ... ?«„_! from (376o) and integrate.
3773 When n = 2, {\ijLv = ^^-^^[h+ik
Jo t>
3775 « = 4, f »,fe = l4(«+»,)+64.(».+-0+2t»,_
Jo ■ 4o
3776
n = 6, ) f^,,^/cr = ^ {u-\- J^2 + "i + '^u+') ('^1+ ''5) +6/rjj .
Jo iU
In the last formula, which is due to Mr. Weddle, the co-
efficient of A^u is taken as -^-o instead of 1-4-0, its true value.
These results are obtained from (3772) by substituting for
each A its value from (3742).
COTES'S AND GAUSS'S FORMULA. •
3777 These give the area of the curve directly in terms of
fixed abscissae.
They are obtained by integrating Lagrange's value of //.,.
(3769-71), and arc fully discussed in articles (2995-7).
4 B
55-4 CALCULUS OF FINITE DIFFERENCES.
LAPLACE'S FORMULA.
3778 ^"ujLv = l^^-t- wi+wo ••• + 1^
the coefficients being those in the expansion of
^ {log (1 + 0}"-
•^1^2 12 24 72U 5 "^
Hence, putting ?f j = A^y^.,
f' , Mn + «i A- Wo , Adrift o^
and so on ; then add together the n equations.
3779 Formula (3778) contains A?f,„ ^\„ &c., which cannot
be found from 7(o5 ''i ••• ^hr
The following formula does not involve differences higher
than A?/.„_i.
3780 ^\Ar = '-^^-ui+u, ... +1^
- ^ (Af/,_i- A^^,)- ^ (A-?/„_i-A-/^,)-&c.
Proof.— In the proof of (3778), change ,., , ,. into E — -- — t^jttk'
log(l + A) log(l — Aii )
and put E'hv_, = iv^.i (3739) after expansion, and proceed as before.
SUMMATION OF SERIES.
3781 Definition: ^m., = //„4-Mv,+i+'^.+2- •• + "..•-!•
3782 Theorem : 2//, = A-^m,+ C,
where G is constant for all the assigned values of x.
SUMMATION OF SERIES. 555
Proof. — Let <l>(x) he such that Af (x) = ^i^, then (p (x) = A'^u^,
therefore «„ = (a + l)—(j) (a). Write thus, and add together the values of
^*a) «a+i> ••• "x-i- Therefore, by (3781), S?*^, = («)— <^ («) = ^"'"o- — ?> ("■)>
and ^ (o) is constant with respect to x.
Taken between the limits x = a, x = h — l, we have the
notation,
3783 ^iifl or tl-\,. = tih-tu^, = ^-'ih-^-'u„.
Functions integrable in finite terms :
3784 Class I. ^.r(-)=lL_. + C\
m+1
3785
%{fLV-\-hY"'
a{m-\-l)
fO.
3786
Glass
11.
X-,A-n.
) — '^ 4- r
I
and n
3y (3718),
-_,,,+! +^-
otation (3711).
3787
%iaa
,+by-^
a {m-
(-W + 1)
-1) '
Formula? (3785) and
(269) and (271). They
(3782).
(3786) are equivalent
are the direct results
to the rules
of theorem
3788
Xa-' = :r.
a — 1
[By (3720).
Class III. — If u^. be a rational integral function,
3789 ^r'^'"V. = {c*^+C,,A+€'.,3A-^+ ... }«.•
Proof.— By (3735) and (3736),
7^-r — 1 ('14-AF — 1
«„ + «,.! ... +ua..-i = a + E + E'^ ... +E^-^) .„ = ^-^«. = ^-^ ^
= the expansion above.
3790 The formula has been given at (266) and an example of its appli-
cation. The series there summed is 1 + 5 + 15 + 35 + 70 + 126 + to 100 terms.
The function u^ which gives rise to these terms is found by (3763) to be
u^ = (x* + lO^H 85a;'- + bOx + 24) -^ 24.
556 CALCULUS OF FINITE DIFFERENCES.
3791 ir t;]iis function bo presented as «,., and ^l'^ u_, be required, we
first find 71^= 1, ?t,= 5, «.,= 15, &c. ; then the differences A?*,,, A^«u, ... A*«^ =
1, 4, G, 4, 1, and then, by (3780), the required sum, as in the example re-
ferred to.
3792 For another example, let S".7;' = 1 + 2^.. +n^ be required.
Here ^x' = 3x' + 5x + l, A\c^ = 6x + 6, AV = 6,
therefore AO^ = 1, ^'0' = 6, A'O' = 6 (1),
x^ may now be expressed in factorials, and the summation may then bo
effected by (3784). First, by (3730),
x^zzz x + Sx(x~l)+x(x-l)(x-2);
therefore, by (3784), SlV = 2 (n + lf (3783),
n 3 _ n(n + l) S(n + l)n(n-l) (n + l) n (n-l)(n-2) _ n\n + lf
'^~2'^ 3 4 -4"
3793 Otherwise, by (3789), taking a = 0, we have
7(^ = x^, u„ = 0, A?<„ = 1, A-?t,| = 6, A'«u = 6, as above.
Therefore
y,-i ^ n(n-l) 6n(n- l) (n-2) 6n (n-l)(n-2)(n-S) _ n'jn-lf
"' ''' 2 "•" 1.2.3 1.2.3.4 4 '
therefore, changing n into ?i + l, S"'?fjr = — — — — .
4
3794 Class IV. — When the general term of a series is a
rational fraction of the form
— ' !— , where 7/ ^. = ax-\-b,
and the degree of the numerator is not higher than cc+m — 2 ;
resolve the numerator into
by (3730). The fraction then separates into a series of frac-
tions with constant numerators which can be summed by
(3787).
3795 If the factors u.^.... ?/.,.+,„ are not consecutive, introduce
the missing ones in the denominator and numerator, and then
resolve the fraction as in the foregoing rule.
3796 Fx. : To sum the series --— + -— + - - + to n terms.
1.4 2.5 3.0
SUMMATION OF SERIES. 557
Tbe ..^ncrna is — ^— = 0^+^)0^ + ^)
n (n + l)+2n + 2 _ 1 ^
«(n + l)(% + 2)(n + 3) 0i + 2)(« + 3) (ri + l)(u + 2)(« + 3)
2
The sum of w terms is, therefore, by the rule (271),
/!_ J_Ulf^ ^ U^f-^ ^
\3 « + 3y 2 \2 3 (w + 2)(« + 3)/ 3 \2.3 (ri + l)(^j+2)(7i + 3)
^ 11 ^7t? + 12n + ll
~ 18 3Oi + l)(n + 2)0i + 3)"
If the form in (3787) is used, the total constant part G is determined finally
by making n = 0, which gives G =—.
3797 Theorem. f{E) a^j> {x) = a^f{aE) <i> {x),
f being an algebraic function.
Proof. — Let a = e'", then the left
= f{E)e'''^<l>{x)=f{e")e'^^^<p{_x) (3736) = e'»V(e'' "")«/> C^O (3475)
^a^f{aE)<p{i,).
Class V. — If <p (x) be a rational integral function,
3798
The upper limit is understood to be x—1, and a constant is to
be added, (3781-2).
Proof.— Sa^^ (a-) = ^-\i^f(x) = (i;-l)-'a^0 (x) = a^ (aE-l)-'<p (x)
(3797) = a- { a (1 + A)-l } "V (x) = j£^l^l+^^y'cp (x).
Then expand the binomial.
2a'''^(x) in successive derivatives of ^(x).
3799 2«-^-<^GiO = ^ [ i+^if Oi')+ ^ <^"Gr)+&c. ( ,
w.ere .„ = (S^iyV = (l + ^yV.
Proof.— By (3757), \P (e^) = xj^ (E) e"-^; therefore (see last proof)
a^{aE-l)-u\x) (putting ^ = e^) = ft^ (aE-l)"' e"-^*^ (a;)
558 CALCULUS OF FINITE DIFFERENCES.
3800 Ex.: To sum the series 2.1+4.8 + 8.27 + 10.04+ to n terms.
AVo require 2V + 22'.«* = 2^ + 2^ (a;*-2A.«' + 2-A-.i;''-2'AV)
= 2^-.f* + 2^ { x^ - 2 (3a;'^ + 3.« + 1) + 4 (O.f + 0) - 8 . }
= 2^-{2.i!»-0.r + 18.T-20}.
3801 If ^~"''.r be known for all integral A^aliies of n, and if
^\. be rational and integral,
Proof. 2«^.t;^. = {EE' -lyUi^v,^ = (A^' + A')-^ «,,v^ (3740) and (3730).
Expand tbe binomial operator, observing (3738).
3802
Pkoof. ^IX-^x = i^ + ^^y^^v^, as in (3801),
= A-" v^u^-n (A-"-^E) v^^u^+ (7„,.> (A-«--J5;-) v^Ahi,,-&c.,
producing tbe above by (3735) and (3782).
Observe that, in (3801) and (3802), two foi'ms ai'e obtainable in each case
by expanding the binomial operator from either end of the series.
3803 Ex. : To sum the series sin a + 2" sin2rt + 3^ sin 3a + to a- terms.
The sum is = x' sin ax + '^x' sin ax. Taking u_^ = sin ax and v^ = x^, we
know A~"sinfta3, by (3729) ; therefore (3801) gives
2a;^ sin ax = (2 sin ^a)~' sin I ax — I (a + tt) j (x — 1)"
— (2 sin|n)~-sin , (cc — (a + 7r) j (2.i! — 3) + (2 sin |a)'^sin | a.« — f (a + 7r) | 2.
APPEOXIMATE SUMMATION.
3820 The most useful formula is the following
1 (Pif,. , 1 f/'-'j/'"
Pkoof.— 2vA^ = ((-"-l)-'»^. E.xpand by (l.'')3;t) wiMi D in the place
of a;.
AFPBOXIMATE SUMMATION. 559
Ex. 1 : The value of %v^' at (2939) is given at once by the formula.
3821 Ex. 2 : To sum the series 1+ — + —... H approximately,
- +2- = - +C + loga;-f - -i-, + -l^-&c.
X X x 2x r2x' 120a;*
Put X =■ 10 to determine the constant ; thus
from which C = "577215, and the required sum is
.577215 + log.+ ^-^-|^ + ^-^,-&c.
3822 Ex.3: i+^ + i^ + i^+&c.,
x^ 2x' 2x^ 2x' 2x' 2x^ 2x"''
' x^ 2 2 4 12 12 20 12'
The convergent part of this series, consisting of the first five terms, is an
approximation to the sum of all the terms.
3823 A much nearer approximation is obtained in this and analogous
cases by starting with the summation formula at a more advanced term.
E.,.: 1+i + i + i+^^^i^
^2035 J^ _L 3^_M.+&e
1728 ^2.5'^ ^2.5^ ^2.5* 2.5«^
^ 2035 111 1
1728 50 250 2500 187500 "^ '''■
The converging part now consists of a far greater number of terms than before,
and the convergence at first is much more rapid.
3824 Ex. 4: The series for logr(ie + l) at (2773) can be obtained by
the above formula when x is an integer. For, in that case,
logT (x + 1) = logl + log2 + log3 ... +log.<; = log.u + S logs;,
and (3820) gives the expansion in question, the constant being determined
by making x infinite.
3825 Formula (3820) may also be used to find [ uj.c by the
process of summation, and thus answers the purpose of
Laplace's formula (3778).
560 CALCULUS OF FINITE DIFFERENCES.
2"Uj in a scries of derivatives of u^.
3826 Lriwma.—
...W + l){6''-l]-\
Proof.— Put v„ for n-l\ (e/-l)-". Then
Vn.i =—(dt + n) i\ = {dt + n){dt + n — l) i'„.i.
S%j. may now be developed.
3827 Ex.— To develope 2^*., {Booh, p. 97)
with jLy = 0, and
^..1 = (-l)"A,..i - (2r + 2) ! = I - I + I +(2.1 + 3.1,-1)
+ 2r{(r + 2)(r+l)J,.2 + 3 ('• + !) ^..1 + 2^1, }r.
Therefore, changing t into f?^., we get
2V = |||«.^/.«-|||«.^/.^+j«.c7..-|«.+ ^^'^-&c.
3828 2"Uj m a series of derivatives of Uj_n.
Let 33" cosec'* 3; = 1 — C^'k^ -H O^rc* — &c. , then
iLoole, p. 98.
3829 2
^(0)-^(l)+,^(2)_&c. = iJl-| + ^-&e.(^(0).
By tins formula, a scries of the given type may often be trans-
formed into one much more convergent.
p.ooK.-Tbo loft = ^m = 2+^*'(0) = i rTp:*^.
the expansion of which is the series on the right.
3830 -'^-^' — ^^ s^^^^ ■'^ ~" "o^ "*" "^ ~ T "^^^^ •^"Q^i^iiug ^'^c fi^'st six terms,
it becomes ^^ + I - ^ + &c. Takiug (0) = (0 + 7)-\ '
00 7 o
1 1,„ 1(1,1, 2 ^ 2.3 ^o
y- Q +&C. = - I y + ^-7-3 + ^--^-^g-^ + 8.7.8.9.10+'^'-
The sum after six terms converges rapidly by this formula, and more rapidly
than if the formula had been applied to the scries from its commencement.
PLANE COORDINATE GEOMETRY.
SYSTEMS OF COORDINATES.
CARTESIAN COORDINATES.
4001 In tliis system (Fig. 1)* the position of a point P in a
plane is determined by its distances from two fixed straight
lines OX, OY, called axes of coordinates. These distances
are measnred parallel to the axes. They are the abscissa PM
or ON denoted by x, and the ordinate PN denoted by //. The
axes may be rectangular or oblique. The abscissa ,*' is
reckoned positive or negative according to the position of P
to the right or left of the y axis, and the ordinate // is positive
or negative according as- P lies above or below the x axis
conformably to the rules (607, '8).
4003 These coordinates are called recMngular or ohlique
according as the axes of reference are or are not at right
POLAR COORDINATES.
4003 The polar coordinates of P (Fig. 1) are r, the radius
vector, and 9, the inchnation of r to OX, the initial line,
measured as in Plane Trigonometry (609).
4004 To change rectangular into polar coordinates, employ
the equations ,r = r cos 6, y = r sin 6.
4005 To change polar into rectangular coordinates, employ
r = \/c^+/, e = taii-1 nty
* See the end of the volume.
4
662 PLANE COORDINATE GEOMETRY.
TRILINEAR COORDINATES.
4006 The trilinear coordinates of a point P (Fig. 2) arc
o, |3, y, its perpendicular distances from three fixed lines which
form the triangle of refeiryiice, ABC, hereafter called the trigon.
These coordinates are always connected by the relation
4007 aa+by8+cy=2,
4008 or a sill A +y8 sin B-\-y sin C = constant,
where a, b, c are the sides of the trigon, and % is twice its
area.
4009 If <'«j y are the Cartesian coordinates of the point
afty, the equations connecting them with the trilinear
coordinates are, by (4094),
a = oV cos a-f // sin a—p^,
fi = ,1' cos )8+// sill ^—pi,
y = OS cos y-\-y sin y—pz-
4010 Here « has two significations. On the left, it is the
length of the perpendicular from the point in question upon
the side AB of the trigon. On the right, it is the inclination
of tliat perpendicular to the x axis of Cartesian coordinates.
Similarly /3 and y.
4011 The angles a, [3, y are connected with the angles
Af B, C by the equations
y— 13= IT— A, a—y = 7r^B, a—fi = 'rr-\-C,
only two of which are independent.
4012 7^1, P-i, l>:i are the perpendiculars from the origin upon
the sides of the triangle ABC.
ARKAL COORDINATES.
If A, B, C (Fig. 2) be the trigon as before, the areal co-
ordinates a', /3', y' of the point F are
4013 a-±-!U!!.' B-^-!l£A y=x=£ii^.
SYSTEMS OF GOOBBINATES. 563
Tlie equation connecting the coordinates is now
4014 a+/S-fy'-l.
4015 To convert any homogeneous trilinear equation into
the corresponding areal equation.
4016 Substitute aa = 2a', byS = 2y8', Cy = 2y'.
Also any relation between the coefficients /, m, n in the
equation of a right line in trihnears will be adapted to areals
by substituting /a, rii{\ M for /, m, n. Similarly for a, b, c,
f, g, h, in the general equation of a conic (4656), substitute
aa\ bh^, cc^ /be, gca, hah.
In either the trilinear or areal systems, a point is deter-
mined if the ratios only of the coordinates are known.
Thus, if a : (^ : y = P : Q : B, then, with trilinear co-
ordinates,
/>2 P
4017 a = —p: — J— ^; and, with areal, a -^
aP-^hQ+cR' ' ' P+Q-^R
TANGENTIAL COORDINATES.
4019 111 this system the position of a straight line is deter-
mined by coordinates, and the position of a point by an
equation. If /o + )»/3 + y/y z= be the trilinear equation of a
straight line EDF (Fig. 3); then, making a, (5, y constant,
and I, m, n variable, the equation becomes the tangential
equation of the point (a, /3, y) ; whilst /, m,n are the co-
ordinates of some right line passing through that point.
Let X, ^, V (Fig. 3) be the perpendiculars from A, B, G
upon EDF, and let pi, p^, ih be the perpendiculars from A, B,
G upon the opposite sides of the trigon ; then, by (4624), we
have
4020 R\ = l2h, RiM = nip,, Rv = np,,
where B = ^{I^-\-'m^-\-n^ — 27nn cos A — 2nl cos B — 2lm cos C).
Hence the equation of the point becomes
4021
V a , /8 , Y ., X sin 6^ . sm 0. . sin 6. ^
X — -\-aJ—-\-v^-=0 or X i+/x =+v ^' := 0,
Pi Jh Ih Pi Pi P-
where p, = OA, 6, = ABOG, &g., and 2AJUJG = p,p,^me.
564 FLANE COORDINATE GEOMETRY.
Eormiila (4021 ) sliows tliat, when tlie perpendiculars X, ^, v
are taken for tlie coordinates of tlie line, the coefficients be-
come the areal coordinates of the point referred to the same
trigon.
4023 Any homogeneous equation in I, m, n as tangential
coordinates is expressed in terms of X, m, v by substituting for
/, m, n, — , — , — respectively. By (4020).
Ih Ih Ih
4024 An equation in X,^t, v of a degree higher than the first
represents a curve such that X, ^tt, v are always the perpen-
diculars upon the tangent. The curve must therefore be the
envelope of the line (X, ^t, v).
TWO-POINT INTERCEPT COORDINATES.
Let X = AD, ^i = BE (Fig. 4) be variable distances from
two fixed points A, B measured along two fixed parallel lines,
then
4025 a\-^rhii-^c = {)
is the equation of a fixed point through which the line DE
always passes. This may easily be proved directly, but we
shall show that it is a particular case of the system of three-
point tangential coordinates.
Let one of the vertices (0) of the trigon in that system be at infinity
(Fig. 3). Then equation (4022) becomes
X sin 0, , // sin 0., , . nni:^ • a A
1 -j. ti 1 -I- sin COE sin 6*3 = 0.
Pi P'2
For I' : p3 = sin COE always. Divide by sin COE ■ then X -^ sin COE = AD,
&c., and the equation becomes
5iEii^D+ 515^^1^ + sin ^3 = 0.
Pi P2
The only variables are AD and AE. Calling these X and ^, the equation
may be written n\ + h^-\-c = Q,
the form taken by ^/X + ?)^/ + c'j' = when v = 00 and c' vanishes.
ONE-POINT INTERCEPT COORDINATES.
4026 Tjct (I, h be the Cartesian coordinates of the point
(Fig. 5) ; and let the reciprocals of the intercepts on the axes
^^YSTEMS OF COORDINATES. 565
of any line DOE passing tlirougli he ^ = -j^,, v = -j^.
Then, by (4053),
4027 f^^+h = 1
is the equation of the point 0, the variables being t„ rj.
This is a case of the system of three-point tangential coordinates in which
two of the vertices (B, G) of the trigon are at infinity. Equation (4022)
now becomes ^l?]B-^ + sin BOB sin d, + sin COE sin 63 = 0,
Pi
sin 0, , sin 0., , sin Q^ — r\
-p^^-AD^-lE-^'
which is of the form a^-^-hr] = \.
TANGENTIAL RECTANGULAR COORDINATES.
4028 This name has been given to the system last described
when the two fixed lines are at right angles (Fig. 6).
The coordinates S, r/, which are defined as the reciprocals
of the intercepts of the line they determine, have now also the
following values.
4029 Let X, y be the rectangular coordinates of the pole of
the line in question with respect to a circle whose centre is
the origin and whose radius is h ; then
f=| and , = i,
since x.OM=y.ON= ¥; for M, N are the poles of y = 0,
■x = 0.
4030 The equation of a point P on NM whose rectangular
coordinates are OB = a, OS = h, is
a^^br, = l, by (4053),
this equation being satisfied by the coordinates of all lines
passing through that point.
4031 In all these systems an equation of a higher degree
in K, V represents a curve the coordinates of whose tangents
satisfy the equation.
ANALYTICAL CONICS
IN
CARTESIAN COORDINATES.
LENGTHS AND AREAS.
Coordinates of the point dividing in tlie ratio n : n the
right line which joins the two points ;/'//, x ij' .
4032 f^'-i^^^i^. r, = ^UL±2LJL.
n-\-n n-\-n
Proof. — (Fig. 7.) I = x + AG = x-\ '—, (.«'—.<;). Similavly for ?/.
4033 If« = »', f='^, v = ^-
4034 Lengtli of the line joining the points xi/, x'y'
= y(,r-y)^+(//-//7.
Tlie same with oblicjuo axes
4035 V(^v-^v'y-{-{t/-t/Y-\-2(.v-.v')[f/-i/)i'OS<o.
Proof.— By (Fig. 7), Euc. I. 47, and (702).
Area J. of a triangle in terms of the coordinates of its
angular points xYi/i, x.^y.,, r^y/g.
4036 A = I {.*'i//2-.t\.//i + .*"2;/3-'^V/2 + '^'3.'A— '^'i.^/^}-
PudOF. — (Fig. H.) By considering the tliree trapezoids formed by i/,, y^, y^
and the sides of the triangle, we have
A = i (//, +://,)(.r,-.'',) + i (!/o + f/3 ) Of, -■''•:) -^ (.'/» + . '/,)0''3--«i)-
LENGTHS AND AREAS. 567
Area of the triangle contained by the \ axis and the lines
1/ = miX-{-Ci, y = m.2,x + Co, (4052)
4037 A = J^^ill^ = .Jjlfllf'j^y («5G)
Proof. — (Fig. 9.) Area = | (.'^'i — ^dl^y a.nd p is found from
pm^—pm^ = Ci — Co. The sign of the area is not regarded.
Cob. — Area of the triangle contained by the lines
4038 4 = i|(£i=£5)!+(£j=^' + fe=^
(. }}ii—m
4039
_ { Ci (ma — mg) + c-i (^% — ^^^1) + ^' i (>>< 1 — m-^ ] '
2 (mi — m.y) {m.2 — nis) {ni-^ — nii)
4040 = iBi^--B,C,Y ^
^"*" 2B,B, {A,B,-A,B,) ^
ACiA'i — Square of Determinant (^4iR,ty
^"^^ ~ 2{A,B,-A,B,){A,B,-A,B.^{A,B-A,B,y
Pkoof.— (Fig. 10.) ABC = AEF+CDE-BED. Employ (4087).
_____^ E
Area of Polygon of n sides.
First in terms of the coordinates of the angidar points
4042 2A = (criy.,---.r.,^?/0 + (<^'2y3-^%?A') + --. + G*^.yi-^^'i?/J
Secondly, when the equations to the sides are given, as in
(4037).
4043 2A=i^l^^^i^^^^+...+i^^^^.
4044 Also three values similar to (4039, '40, '41).
Proof. — By (4367), adding the component triangles.
4047 Each expression for the area of a triangle or polygon
will be adapted to oblique axes by multiplying by sin w.
568 CARTESIAN ANALYTICAL GONICS.
TEANSFORMATION OF COORDINATES.
4048 To transform tlic origin to tlie point Id-
Put .i> = .i'+A, y = ij'-^k.
To transform to rectangular axes inclined at an angle B
to the original axes.
4049 Put
.r = .!>' Qo^e-y' siii^, // = //' cos^+.i' siii^. (Fig. 11.)
Pkoof. — Consider a;' as cos0 and t/' as sin^^. Then .<; = cos (^ -|- 9) and
i/ = sm(./. + ^) (627, '9).
Generally (Fig. 12), let w be the angle between tlie original
axes ; and let the new axes of x and // make angles a and /3
respectively ^vith the old axis of x.
4050 Put .r siu o) = x' sin (w — a) +//' sin (w — ^)
and y sin w = .v' sin a-\-y' sin ^.
Proof. — (Fig. !-•) The coordinates of F referred to the old axes being
OG = X, PC = y, and referred to the new axes, OM = x, I'M = y', we have,
by projecting OCP and 02IP at light angles first to CP and then to OC,
CD = MF- 31 E, FN = ML + PK,
which are equivalent to the above equations.
To change Rectangular coordinates into Polar, hk being
the pole 0, a the inclination of the initial line to tlie x axis
(Fig. lo), and xy the point P.
4051 Put .V = A + r cos (^+a), // = k+r sin (^+a).
THE RIGHT LINE.
EQUATIONS OF THE RIGHT LINE.
4052 yy=->/M+c (1),
4053 - + f =1 (2),
(I
THE BIGHT LINE. 569
4054 A' COS a-\-i/ sin a = j) (3).
4055 Aa'-\-Btj-\-C = (4).
Peoof. — (Fig. 14.) Let AB be the lino. Take any point P upon it,
coordinates OiV = .7;, PN = y. Tlien, in (1),' m = tanO, where 6 = BAX,
the inclination to the X axis ; therefoi^e mx = — OG, and c is the intercept
OB. In (2), a, h are the intercepts OA, OB. In (3), p = OS, the per-
pendicular from upon the line ; a = Z AOS.
p = OB + LP = X cos a + y sin a .
(4) is the general equation.
4056
m = tan^ ==- 4- = - — == -cot a.
B a
4060
sin^— cos^ — — ^
\/A~-\-B' x/A'+B'
4062
2)— c sin a— ^ — —
Vl+m- v/v4^+/i^
Oblique Axes.
Equations (4052, '53, '55) hold for oblique axes, but
(4054) must be written
4065 ^v cos a-\-i/ cos /3 = p. (Fig. 14)
4066 tan e = ^^^"^"^ ::= __ii!iifi_.,
l+mcosw A cos (o — lJ
(o being the angle between the axes.
Proof. — From vi = sin -^ sin (w — 9).
4068 w = c sin w C sin w
\/l+2m cos w+m' v^^'+jB^— 2.1/i cosw
Proof. — From p = c sin (w — 9) and (4 K56).
The equations of two lines being given in the forms (4052)
or (4055), the angle, (p, between them is given by
Af\y^r\ i / m—m AB' — A'B
4070 tau (^ = — , or .
1+mm A A -\-BB
Proof.— (Fig. 15) tan ^ = tan {i)-0'). Expand by (G32).
4 I)
670 CARTESIAN ANALYTICAL CONIC S.
To oblique axes :
4072 tau 4> = , , / , — 7v^ j 7
Proof. — As in the last, employing (40GG).
Equation of a line passing through x'y':
4073 y-y = »^ {^^ -^^'1> ^^'s- 8)
4074 or y — tn.v = y' — wicv' ,
4075 or .4.2 +% = A.v' + By'.
Proof. — From Figure (13), m being = tan 0.
Condition of parallelism of two lines :
4076 m = 7)i', or AB=AB.
Hence the equations differ by a constant.
Condition of perpendicularity :
4078 mm=~l or AA^BB = {i. (4070)
The same to oblique axes :
4080 1 + {m-\-m') cos 6>4-mm' = 0. (4072)
4081 or AA -^BB'= {AB'-^rA'B) cos co,
Af\ork ' l + «<eoscu
4082 or m = '— .
Wl + COSW
A line passing through the points x^y^, avyo '■
4083 iaii = Ilii:]li = ,n.
Proof.— (Fig. IG.) By the simihir right-angled triangles PCA, ABB.
4084 Or ,y = ,,M4-'M2ZJMj,
4085 or (^v-^r,)[ii~!r;) = {.v-.v,){y-n,).
Proof. — This equation represents a straight line because it is of the first
degree; and the coordinates of each of the given points satisfy the equation.
THE BIGHT LINE. 571
A line passing tlirougli x'y and perpendicular to a given
line (m) :
4086 y-y = - ^i^-^)' (4073, '78)
m
4087 or Bx—Ay= Bx — Ay
The two lines passing through xy' and making an angle
j3 (= tan'M^^o) with a given line {m^) :
4088 ?rz]L = p:zl!h. and I!h±2!h.. (4073,70)
x — X 1 + m^m.2 1 — m^tn.2
A line passing through hh and dividing the line which
joins Xii/i and ^2//2 ii^ the ratio 1^ : ??2 :
4089 ^ = ^jrfiZkl^-) (4073, '32)
Coordinates of the point of intersection of two lines :
4090 .^.= Jl=^ = ff^:f|.
4092 y = ^j!n2=££L. = _ 4#=4|- ("i*^)
"^ m.2—m^ A^B^—A.B^
Length of the perpendicular from a point x'y' upon a
given line
4094 = ^^' cos a-\-i/' sin a— p.
Proof. — Let AB (Fig. 14) be the line, and Q the point x'l/'. Then, by
(4054), a;' cos a + 2/' sin u = OT, the perpendicular from upon a parallel line
through Q, and j) = 08.
Otherwise, the same perpendicular
4095 = Aa,'-]-Bi/ -f g ^^QgQ^ ,gj^ ,94^
A/yl' + jy^
The same with oblique axes
Qfi _ {Ax'-\-By'
^^ ~ ^/{A-^+B-^-
obtained in a similar way from (4065-69).
4nqfi - (Ax -\-Bi/-\-C) sin (o
- ^^-i_^B'-2ABcos(oy
672 CARTESIAN ANALYTICAL CONIGS.
Condition of three lines intersecting in one point :
4097 CiWJo— rjomi+Ca/Wg— CaWia+Csmi — CiWig = 0.
The area in 1009 must vanish.
4098 Otherwise. — If certain values of the constants /, ??i, n
make the expression
vanish identically, the three lines indicated intersect in one
point.
Proof. — By (4099), for then values of x and y which make (1) and (2)
vanish also make (3) vanish.
A line passing through the point of intersection of the
lines Ax + Biji-C = and A'x-{-B'i/-\-G' = is
4099 A.v-\-B?/^C= k {A\v-\-B'i/+C'),
4100 or l{A.v-\-By-\-C)-m{A',v-]-B'i/-\-C') = 0,
hj I, and m being any constants.
4101 Rule. — If the equation of a right line contains a third
variable k in the first degree, the line altvays ])asses through a
iixed looint.
Proof. — For the values of x and y, which satisfy simultaneously the given
equations, also satisfy (4099), whatever k may be. See (4604).
4102 If in the equation of a line Ax-\-By-^G = 0, the co-
efficients A, B, G involve x, y\ the coordinates of a point
which moves along a fixed right line, then the first hne passes
through some fixed point.
Proof. — By means of the equation of the fi.xed line, y' may be eliminated,
and X then remains a third variable in the first degree (4101).
4103 To find the point in Avhich the line Ax + Bij + G inter-
sects the line joining the points xy, xy; substitute
A,v-\-Bij-\-C iovn, and y4.j'+%' + C for // in (4032).
Proof. — By (4095), since the segments intercepted are in the ratio of the
perpendiculars from xy^ xy upon the lino -4a; + J5y + C
THE BIGHT LINE. 573
'- /
- m '
7 _ sin (m — B)
sin w
sine
m = -. ,
smw
(Oblique)
I = cos 9,
m = sin 6.
(Rectangular)
Equations of the line with I, m for direction-ratios, lih a
fixed point on tlie line, and r the distance of the variable
point xy from /i/j.
.... x — h y — k
4104 " = ' —
4105 where I
4106 or
Polar Equation of a Straight Line.
4107 »* cos {d — a)= p.
(Fig. 17.) Here j:) is the perpendicular to the line from the
pole 0, and a is the inclination of 7; to the initial line OA.
When the line passes through the pole, the equation is
4108 ^ = constant.
A line passing through the two points r-^Q^, r.^O^.
4109 rr, sin ie-e,)-\-nr, sin (^i-^,) + »v sin (6,-6) = 0.
PROOF.-(Fig. 18.) APOA + AOB-POB = 0. Then by (707).
EQUATIONS OF TWO OR MORE RIGHT LINES.
The homogeneous equation of the n^^ degree,
4110 ,i^^-\-p^"-h/+p^'-Y+ . . . +p,^f = 0,
represents n right lines, real or imaginary, passing through
the origin.
For it is resolvable iuto n factors of the form (x — ay), by (405).
For the case of two right lines represented by the general equation of the
second degi'ee, see (4469).
Equation of two right lines through the origin :
4111 aa^'+2hx7j+bi/' = 0.
574 CARTESIAN ANALYTICAL CONICS.
If (j) be the angle between the lines,
yntn 4- JL \/(lr — ab) 2 siiio) v/(/r — «^)
4112 tan (^= ^ ^ ^ or — , , ^.,\ \
according as the axes are rectangular or oblique.
Proof. — Assume {ij — m.^a'^{y—m^x) =0, and apply (4088).
Equation of the bisectors of the angle <p :
4113 ka:^-^a-h)mj-hif = ^.
Proof. — Let y = ixx be a bisector (/i = tan i//) ; then, since 2J/ = (^, + l
-^^^^ = ^^h±J!h_ = -2^, by (4111) ; and ^ = I
1 — fj.^ 1 — ?;Ji«2.2 a — b X
The roots of tliis equation are always real.
GENERAL METHODS.
APPLICABLE TO ALL EQUATIONS OF PLANE CURVES.
4114 Let F(.^^;/) = (i.) ^nd f{.v,ij) = (ii.)
be the equations of two curves of any degree.
4115 To find the intercepts on the x and y axes.
Put y = in (i.), then x becomes the intercept on the x axis.
Similarly, 'put x = for the intercept on the j axis.
4116 To find the points of intersection of (i.) and (ii.).
Solve as simultaneous equations. Each pair of values of x
and J so obtained gives a point of intersection. Imaginanj
values give an hnaginary point.
4117 To determine equation (i.) so that the line may pass
through certain fixed points, x^y^, x^y^ &c.
Substitute x^j^, Xayg, ^'c. for xj successively, so forming
as many equations as there are points. From these equations
the constants iii (i.) must be determined in terms of Xi,ji, x.,,y..,
4118 The number of arbitrary points cannot exceed the
number of constants in the equation.
GENERAL METHODS. 575
4119 Condition tliat (i.) and (ii.) may touch.
At a point of contact tiuo or more points of intersection
must coincide, and therefore the equation for x or y, obtained as
in (4116), must have tiuo or more equal roots for each point of
contact. The contact is said to he of the second order when
there are three coincident points; of the third order when
there are four, and so on.
4120 To find the equation of the tangent at a point x'y' on
the curve f(x,y) = 0.
Form the equation to the secant through two adjacent points
Xiyi, x,j2 (4083), and determine the limiting value of -^^ '^'^
Xi X2
ivhen the points coincide by means of the equations f (x^, yi) = 0,
f(x3,y2)-o.
4121 Otherwise m=^, % (5101).
4122 For the equation of the normal, change m of the
tanp^ent into (4086).
m
4123 To express the equation of the tangent, or normal, in
terms of m and the constants of the curve.
From the equation of the tangent or normal, the equation to
the curve, and the equation furnished by the value ofm, eliminate
x' y', the coordinates of the point of contact of the tangent.
THEORY OF POLES AND POLARS.
4124 Let F(x, y, x, y) = represent the equation to the
tangent of a curve at the point x'y'.
Then F{x, y, x, y') = 0, the equation obtained by inter-
changing the constants x',y' with tlie variables ;«, 7/, represents
the polar of any fixed point x'y' not on the curve.
Let Xyiji, x^y^ (Fig. 19) be points A, B on the curve, and let the tangents
at those points intersect in x'y'. Consider the equations
Fix,.y,,x,y)=Q. ..(}), F(x„y„ x,y) = ... (2), F(x,7j, x',y') = ... (3).
Here (1), (2) are the tangents, and (3) is some straight line or curve
according to the dimensions of x and y. Also (3) passes through the points
576 CARTESIAN ANALYTICAL CONICS.
of contact .^'j//,, r/'o'/o, and raay therefore be called the cicrve of contact-; or, if a
right line, the chi)nl of contact of tangents drawn from x\ y\ i.e., the polar.
4125 Heuco the coordinates of the points of contact of
tangents from an external point ,i"'v/' will be determined b}^
solving (3) and the equation of the curve simultaneously.
4126 Again, let x'y (Figs. 20 and 21) be any point P not on the curve.
Then, from the equations
F(x,y\x,y)=0...i4), F (x,y,x.„y,) = ...(o), F (x,y, .r,,y,) = ... (G),
we see that (4) is some straight line ; that, if x.j,y^ and x^y^ are any two points
upon it, (5) and (6) are the curves of contact of tangents from those points ;
and that these curves of contact pass through the point x'y'.
4127 If the points x.^j/^, x^y^ are taken at A, B, where (4) intersects the
curve, (5) and (6) then become curves touching the given curve at A and B,
and passing through x'y'. We may call these lines the curve tangents
from x'y'.
4128 Lastly, let xy' in (o) be a point within the given curve (Fig. 22),
then the equations
F(x,y,x',y')=0...(7), F (x„y,,x,y) = ... (8), F (x,,y„x,y) = ... (9)
show that (7) is the locus of a. point, the curve tangents from which have
their chord of contact always passing through a fixed point. When x'y' is
without the curve, as in Fig. (19), the same definition applies to every part
of the hicus (3) from which tangents can be drawn.
4129 If the given curve be of a degree higher than the
second, the line of contact of the tangents from a point is a
curve, and the line of contact of the curve tangents from a
point is a straight line (Figs. 19 and 20). A similar converse
relation is exhibited in Figures (21) and (22).
If the curve be of the second degree, equations (o) and
(4) become identical. The line of contact or the polar is
always in this case a straight line, and so is the locus (7).
Figures (19) and (20) now become identical, as also (21)
and (22).
4130 The polar of the point of intersection of two right
lines with regard to a conic passes through their poles.
Proof. — As in (4124). Let (1) and (2) be the two lines, (,t,.Vi), (^VJi)
their poles, and x'y' their point of intersection.
GENERAL METHODS. 577
4131 To find the ratio in which the line joining two given
points xy, x'y' is cut by the curve /(x, y) = 0.
Substitute for x and y, the supposed coordinates of the
point of intersection i the values
n-\-n n-\-n
and determine the ratio n : n' from the resulting equation.
The real roots of this equation correspond to the real points of
intersection.
4132 To form the equation of all the tangents that can be
drawn to the curve from a point x'y' .
Express the condition for equal roots of the equation in
(4131), and consider xy a variable point.
4133 To form the equation of the lines drawn from x'y' to
all the points of intersection of two curves.
Substitute 7ix-\-n'x, ny' -\-n'y for x and y in both curves,
and eliminate the ratio n : n .
Proof. — Take any other point xy on the line through xij and a point of
intersection. The ratio n : n (4131) is the same for each curve, and there-
fore may be eliminated.
4134 To find the length, r = AP or AP' (Fig. 23), of the
segment intercepted between the point A or x'y' and the curve
/ {x, y) = on a straight line drawn from A at an inclination
9 to the X axis. That is, to form the polar equation with
x'y' for the pole and the initial line parallel to the x axis.
Substitute for X. and y, the assumed coordinates of the point
of intersection, the values x = ON or ON', y = PN or P'N',
that is, X = .r'+r cos^, j =: y'-\-r ^m.6,
and determine r from the resulting equation. That is, put
a = in (4051).
The real values of r are the distances of the points of inter-
section from x'y'.
4135 When an equation has been obtained for determining
X the length of a line, important results may frequently be
arrived at by applying theorem (406) respecting the sum and
product of the roots.
4 E
578 CARTESIAN ANALYTICAL CONIC S.
THE CIRCLE.
Equation with the centre for origin.
4136 a?^+i/^=r\ (Fig. 24.)
Equations of the tangent at the point P or x'y'.
4137 y-l/ =-jr (.r-ciO. (4120)
4138 ^v^v'+m' = r\
Also, by (4124), the polar of x'y', any point not on the curve.
4139 ?/ = ma^-^r Vl-^ni' ; m = — ■^. (4123)
4140 -V cos a-\-i/ siu a = r,
a being the inclination to the x axis of the radius to the point
'<^'y'-
Equation of the circle with a, h for the coordinates of the
centre Q. (Fig. 24.)
4141 (.,,_«)'^4.(,^_6y^=,.^
Tangent at x'y', or Polar,
4142 Op -«)(y -«)+(// -6)0/ -ft) = r\ (4138)
4143 or (cP — a) cos a+ (//— b) sin a = r,
a being the inclination of the radius to the point x'y'.
General equation of the circle :
4144 .^--|-/+2^>^^•+2^/+c = 0.
4145 Centre {-^g, -/). Eadius y{g^-\-f^-c).
Proof. — By equating coefficients with (4141).
Equation of the circle with oblique axes : (Fig. 25.)
4146 {.v-aY + 0/-0y-\-'2 {a'-^a){,/-b) cos <o = r~, (7u2)
4147 or .(••^+ 2d'i/ cos 0) +//" — 2((i-\-b cos co) .v
— 2{b-\-a cos6>)//
-|- cr + 2ab cos (D-\-b- = r~.
THE GIBGLE.
579
General Equation.
4148 ci'^+2.r// cos (o-{-7f-\-2g\v+2fi/-\-e = 0.
The coordinates of the centre are
fcOSCU — o- , 2- COS CO — f
4149
4150 Ptadius =
\i>'- — 2fi>' cos (o-\-f- — c siir <o
SlUOt)
Proof. — By equating coefficients with (4147).
Polar Equation.
4151 r'+P-2rl cos (^-a) = c\ (Fig. 2G)
4152 or r^ — 2l cos a r cos 6—21 siu a r sin 6-\-l^—r — 0.
Proof. — By (702), the coordinates of P being r and d.
General form of the polar equation : —
4153 r''-^2gr cos ^+2/r sin d+c = 0.
4154 tan a = ^. / = V^+f-
Proof. — By equating coefficients witli (4152).
4156 Equation of the circle passing through the three
points rt'i?/i, .Co//.,, x^y.^.
w y 1
<^i III 1
Proof. — Eliminate g,f, and c from (4144) by (4117).
Equation of the chord joining x^ij-^^ xjj., two points on the
circle x^ + y' = v" :
4157 ^r Grx+^r,) +// (y.+y^) = ^r^2+Ih!h+r', (4083, 4136)
4158 or .rcosi(6'i+6',) + 7/sini(6'i+6',) = rcosi(^,-^.,),
where rcos^i = r^i, 7'sin0i = y^, &c.
'*'l Ih
1
x^y-2
1
oi\yz
1
x^iVi
1
•^3 Ih
1
.V y
1
w
y
1
X,
y^
1
a\
y^
1
580 CAETESIAN ANALTTIOAL G0NIC8.
4159 Note. — The coordinates x, y of a point on the circle x- -\- y- = ''^^ may
often be expressed advantageously in this way in terms of 0, a single variable.
4160 Let S = {.v-ay-\-{?j-hy-'r' =
be any circle (Fig. 27). Then, if xij be a point P outside the
circle, S becomes tbe square of the tangent from P. If xy be
a point P' within the circle, S becomes minus the square of
the ordinate drawn through P' at right angles to the radius
throu2:h P'.
CO-AXAL CIRCLES.
(See also 984 and 1021.)
4161 If -S = .v'+ir+2g.v +2/// +c = 0,
be two circles, the equation to the radical axis is
If x = be taken for the radical axis, the equation to any
circle (radius r) of the system of coaxal circles (1021) is
4162 .v'-{-f-2K\v ± 82 = and k'-r^ = ± S\
+ in Figure (1), — in Figure (2). Here d = IB ii constant,
and h=. 10 a variable.
4164 The polar of xy' for any circle of the system passes
through the intersection of
.vx -\-i/i/ ± S^ = and .r + .r' = 0.
Proof.— Its equation is xx' + yTj'-h (x + x')±P = (4121). Then by
(4099).
4165 When Jc = ^, then h = ID = ID'. D and D' are
Poncelet's limiting points.
4166 The polar of D with respect to any of the circles
passes through D\ and vice versa, by (41 G4).
4167 Tangents from any point on the radical axis to all
circles of the system are equal (41(30, '01).
THE GIBGLE. 581
4168 The radical axes of three circles, S^, S^, S.^, meet in a
point called their radical centre.
4169 The reciprocals with respect to the origin D or D' of
the system of co-axal circles are all confocal conies (4558).
The equation of the circle, centre Q, cutting the system of
circles orthogonally is, putting IQ, = h,
4170 .r'+/— 2%— 8'^ = 0. (1230, 1236)
This circle passes through D and D'.
The common tangents to the two circles
(x-aY + (y-hy = r' and (x-ay+{y-hy = r'\
(See also 1037.)
The equation for a in (4143) is
4171 {a— a) cosa+(6— 6') sina+r=Fj''= 0.
Pboof. — Assume (4143) in a, h, r, a, and also in a', b', r, a as coinciding
lines. Then tana = tana'; therefore a' = a or tt + u. Take the difference
of the two equations.
The chords of contact are
4172 ia-a'){^-a) + {b-b){i/-b)-\-r{rzfr') = 0,
4173 {a-a'){ai-a')-\-{b-b'){i/^b')+r(r^r') = 0,
with — for exterior tangents, + for transverse.
Pkoof. — For these are straight lines, and they pass through the points of
contact of each pair of tangents respectively, by (4171).
The centres of similitude 0, Q are the intersections of the
external and transverse tangents respectively.
a'r—ar' b'r—br'
4174 Coordinates of 0,
r—r 7 — r
4175 Coordinates of Q, 'il±^, Vr+br
r-\-r r-\-r
Proof. — By equating coeflBcients in (4172) and (4142), the polar of
or Q.
4176 The six centres of similitude of three circles lie on
582 CARTESIAN ANALYTICAL CONICS.
four straight lines called axes of similitude. Sec the figure
of (1046).
Proof. — The coordinates of the three centres of the forms (4174, '75)
will in each case satisfy equation (4083).
4177 The equation of the external axis of similitude is, in
determinant notation (554),
(IrA) a3-(l?v/3) y + {nh(h) = 0.
Peoof. — By forming the equation of the right line passing through two
of the centres of similitude whose coordinates are as in (4174).
4178 The remaining three axes are found by changing in
turn the signs of i\, r^ ; r^, r^ ; rg, r^.
4179 If one of the circles touches the other two, one axis of
similitude passes through the points of contact.
4180 The angle 6, at which the circle F [x, y) = 0, radius
r (Fig. 29), intersects the circle whose centre is lih, and
radius U is given by the equation
iJ2-2iJrcos^ = jP(/i, A').
Proof : 6 = OQP, B'- 2.Br cos 6 + r- = FT' + /•- ^ F (Ji, 1-) + r\
(702) and (4160).
4181 Cor. 1. — If the circles are given by the equations
x^ + y'' + 2g'oj-{-2f'y-{-c' = 0, ;,^-{-y' + 2i/x + 2j'y + e = 0,
the equation for cos becomes, since h=—g, k= — /,
2Rr cos 6 = 2g'g'-^ 2ff'-c-c. (4145)
4182 CoE. 2. — The condition that the two circles may cut
orthogonally is
4183 CoR. 3. — By solving three such equations, we can
find the circle cutting three given circles orthogonally (4186).
4184 CoR. 4. — The condition that four
circles may have a common orthogonal
circle is the determinant equation
^3 g^ Jl 1
TEE FABABOLA.
583
4185 Cor. 5.— If the circle x- + if + iax-^2Fy^G = cuts
three other circles at the same angle Q, we have, by (4081),
three equations to determine G, F, G.
minant equation may be written
^v'+if -.V -7J 1
-{-2R cosp
The resulting deter-
4186 The first determinant, put =
-w -11 1
r. g\ A 1
r-2 g-z fz 1
rs g, fz 1
= 0.
= 0, is the orthoQ'onal
circle (4183),
similitude.
and the second, expanded, is the axis of
4187 The locus of the centre of a circle cutting three given
circles at equal angles is a perpendicular from their radical
centre on any of the four axes of similitude.
Proof. — By eliminating R and cos a between three equations, like (4180).
4188 Each of these four perpendiculars contains the centres
of two circles touchino- the three o-iven circles.
Proof.— Consider a = or 180', in (4180).
To draw the eight circles which touch three given circles,
see (946) and (1049).
4189 The equation of the fourth degree of two of the
touching circles is
23v'S,±Sl.yS,±T2^Ss = 0,
where 23 signifies the length of the common tangent of the
second and third circles, &c.
Proof. — By first showing that, if four circles are all touched by another
circle, the relation
4190 12.34± 14.23 ±31.24 =
will subsist, and then supposing the fourth circle to reduce to a point.
THE PARABOLA.
4200 Def.^ — A conic is the locus of a point which moves in
one plane so that its distance from a fixed point S, the focus ,
584
CARTESIAN ANALYTICAL CONICS.
is in a constant ratio (a) to its distance from a fixed riglit line
XM (the directrki).
When e = unity, the curve is a parabola. (See also p. 248,
et seq.)
Equation of the Parabola with origin of coordinates at the
vertex A.
4201 !r = 4«ci.
Here a = AS, x = AN,
y = FN.
Proof.— Geometrically, at (1229).
Analytically, from
PS'' = if + ix- af = PM" = (x + af.
The equations with the r f^ ^
origin at S and X respectively
are
4202 t/ = 4^a{a^-\-a),
y = 4«(cV-«). (4048)
Equations
of the
tangent
at x'y' :
4204
y-i/ =
f<-
.1'')
4205
mi =
■2a (,.•+,
v').
4206
.'/
= mdH-
JL, ,n
m
=
2a
y
(4120)
(4123)
4207 (4204) is also the polar of any point x\j , by (4124).
Its intercepts are —x and \]j .
Equations of the normal at x\j :
4208 ;/-;/= -|_(,r-.r'). (41^-^2)
4209 //'ci- + 2(11/ - (//'.*■ + '2(u/) = 0.
4210 y = ^njc - 2am - am\ (4123)
THE PARABOLA. 585
Equation of the parabola
with a diameter and tangent for
axes of coordinates.
4211 f = ^acc,
where 'r-
4212 a = acoBeG^e = SP;
x = PV; y = QV.
Proof.— Geometrically, at (1239). Otherwise, let VQ = -VQ'hc equal
roots of opposite signs of the quadratic (4221), V being the point xy,
therefore ?/^ or r^ = (y'"'—4^ax') cosec^ = 4a cosec" d.x,
since y'^ = 4a x abscissa of P.
4213 Equations (4204-10) hold good for these axes, with a
written for a in each.
For the polar equation of the parabola, see (4336).
4214 Quadratic for n^ : n.2, the ratio of the segments into
which the line joining two given points x^yi, a'o^a is divided by
the parabola i/—4ax = 0,
(4131)
Equation of a pair of tangents from any point x'y' :
4215 {y"-4^a.v'){t/-^tt^') = W-2« (^4-.^0}' = 0.
The condition for equal roots in (4214).
Quadratics for the coordinates of the points of contact of
tangents from x'y' :
4216 aj^^-(y'^-2ax') x-aw" = 0.
4217 /-2?/y+4a.r' = 0.
Proof. — Solve simultaneously the equations of the curve and the polar
(4205) and (4125).
Coordinates of the point of intersection of tangents at Xj^yi
and x^y.^ :
4218 a'=m y=y^.
4a 2
4 F
686 GAETE8IAN ANALYTICAL CONIGS.
Quadratic for m of the tangent from xy' :
4220 m\v — my + « = 0. (4206)
4221 General polar equation of the parabola, or quadratic
for r, the segment intercepted between a point, x]j\ and the
curve on a line drawn from that point at an inclination to
the X axis (4134),
r' sin2^+2r {ij sin ^-2a cos ^) +?/''— 4ay = 0.
Quadratics for the coordinates of the points of intersection
of the line Ax + By + G and the parabola y^ = 4<ax : (4116)
4222 A\v^-2 {2B'a^AC) .v+C = 0.
4223 Ai/'-^4<Bai/-\-4^Ca = 0.
Length of intercepted chord,
4224 W{{Bhi'-ACa){A'-^B')}-^A\ (4oa4)
Equation of the secant through x^i/i, x^y^, two points on
the parabola :
4225 !/ {yi+y2) = ^jiVz-^^ax, (4083)
4226 or y{m^-\-m2) = 2a-\-2m-^mo^oc.
4227 The subtangent NT = 2x. Fig. of (4201)
4228 The subnormal NG = 2a.
Pkoof.— Put y = in (4205) and (4208).
4229 The tangent PT' = 4^.v{a-^.v).
4230 The normal PG' = ki (a^.v).
The perpendicular p from the focus upon the tangent at xy :
4231 p = ^/aia^+a) = Vu^- (4212), (4095)
THE ELLIPSE AND EYPEBBOLA. 587
The part of the normal intercepted by the curve is equal to
4233 4«(l + m')t ^ _ 4« (4221), (4135)
m^ siu- ^ cos d
4234 The minimum normal = 6rt\/o and 9?i = v/2.
Length of a chord thi^ough the focus
4235 = -^ = 4a'. (4212)
Coordinates of its extremities, with the focus for origin :
- rtrt w 2« cos 6 '2a sin 6
4237 * = - ^-^j^^i, «/ = - ^^^em-
Coordinates of its centre :
4239 ^=^-|S^' y = 2a.oie.
THE ELLIPSE AND HYPEEBOLA. '
(See also p. 233, et seq.)
4250 Referring to the definition (4200) ; when e is less than
unity, the conic is an eUii^se ; when greater than unity, an
hyperhola.
Equation of the ellipse with the origin of coordinates at X
and SX=p.
4251 y^-^{.v^py = e\v\
Proof. — Bj the definition in (4200).
Abscissae of vertices : (Supply A' in the following figure.)
4252 XA = ^, XA' = j^. (4115)
588
OABTESIAN ANALYTICAL 00NIG8.
m'
M
P
/
^
X
m
\
\
c
T.'A\
SN
4254 Focal distances of vertices
4256
4260
SA
_ ep
SA' =
ep
SL = l = ep = a {1-e') = -^.
b'=a'{l^e');
(4251)
(4251)
4262
4264
1—e^
1—e"
4266
CS = /J\^=ae. (4252)
1—e^
4268 If h -.
= a tan a, tlien e = sec a in the hyperbola.
Equation with the origin at A :
4269 if=^(2a.v-a;')
Ell.
4270 f = -^(2a.r+.r^) Hyp.
Proof.— By (4200), i/ + {x-SAy = e' (x + AXf, &c.
Equations with the origin at the centre C :
4271 f = -^ («^-^i""0 = (1-^-) {ir-^).
(4269)
THE ELLIPSE AND HYPERBOLA.
689
4273
^3 .2
«2 ^ Ki
Proof.— By (4200), 7/+ {x + GSf = e^(a; + OX)^ &c.
4274
PiV: QN :: 6 : a.
(4271)
4275 Def. — QGNis the eccentric angle, (^, of the point P.
X and y in terms of the eccentric angle :
4276 a^ = acos<t>, y = h siu <ji. (Ell.)
4278 X = a sec <^, y = b tan <^. (Hyp.)
Five forms of the equation of the tangent or polar of the
point xy' :
4280
4281
4282
4283
4284
4285
y—ti = 7-;(<l?— 0?).
oox_,inL— 1
„2 "T 12 ■*•
a7C0S(^ ?/sin(^ ^ 1 (Ell )
aft
^sec(^ ytaii(^ _-^ (Hyp.)
(4120)
(4123)
(4276)
(4278)
X COS y-\-y siu y = \/ d^ cos'^y+ft'^ siu^ y,
•y being the inclination of ^.
(4054) & (4372)
690 CARTESIAN ANALYTICAL CONICS.
Five forms of the equation of the normal at x'y' :
4286 2/-y = g; (---')• ^''''^
4287 ^-^=a'-h\ or ha^-h/ = a'-b\
oc y
where h and h are the intercepts of the tangent.
4289 »=m.r-i! Lig!=^ . («23)
4290 «.^' sec j>—hy cosec <^ = a^—fy'. (4276)
4291 .*\^-^x2/ = {^x^'-Viy'). (^^^2)
where a^iyi is the extremity of the conjugate diameter.
Intercepts of the tangent or polar on the axes :
4292 -^ and — . (4115), (4281)
00 y
Intercepts of the normal : (4287)
,,2 _ 1^2
4294 On the X axis, — 5— x or e^x.
4296 On the 7/ axis, — " ~ y or — y37.//-
Focal distances r, r of a point <r?/ on the curve :
4298 {a±cx) in Ell.
4299 (ca7±a) in Hyp.
Proof, — From r' = {ae±x)--\-if, and (4272).
Perpendiculars from the foci upon the tangent :
4300 p = byj^, p' = byj'-. (4095,4282)
THE ELLIPSE AND HYPEBBOLA.
591
4302
•. (p. 588) sin SPT - P =. K = L_ =
r r y/rr
b
(4365)
4306
b' = pp\
(4300)
Segments of tangent and normal :
^
FG = — vrr = —
(4292)
PG' = ^s/r/='^. (4294)
o b
Bight Line and Ellipse.
Quadratic for the ratio n^ : n.,, in wliicli the line joining
two given points x-^y-^^, x.^y.^, is cut by the ellipse (4131).
4310
4(4+||_i)+2^Am.+.m^^i)4-(4+.||_i) = o.
n% \a^ ' b^ / 71.2, \ u b^ / \a 6 /
Equation of the two tangents drawn from xy :
^311 (S+S-0(5H-S-^)-(f+f-0="-
Proof. — By the condition for equal roots of (4310).
592 CARTESIAN ANALYTICAL C0NIG8.
Quadratic for abscissae of points of contact of the tangent
from xy' :
4312 cr^ {bV-^aY^)-2aWxiv'-^a' ih^-y^) = 0. (4282, 4125)
Quadratic for m of the tangent from xij :
4313 m' {x'--a^)-2mxy-\-y'-h'' = 0. (4282)
General polar equation of the ellipse, or quadratic for r,
the segment intercepted between the point q/i/ and the curve
on the right line drawn from that point at an inclination to
the major axis and x axis of coordinates.
4314 («' sin^^+i^ cos-^) r^
+2r (ay sin ^+6V cos^) + {aY+bV-a'b') = 0.
4315 Length of intercej)ted chord = difference of roots.
4316 Distance to middle point of chord = half sum of roots.
4317 Rectangle under segmc7its ^products of roots.
CoE. — If two chords be drawn to a conic at two constant
inclinations to the major axis, the ratio of the rectangles under
their segments is invariable.
For, if xij be their point of intersection, the ratio in ques-
tion becomes a^sin^0-j-Z>^ cos-0 : a^sin-0'-f-^'cos''O^', which is
constant if B and B' are constant.
Locus of centres of parallel chords :
4318 a^y sin e-^-h'x cos ^ = 0. (4314)
Quadratic for abscissae of points of intersection of the line
Ax-\-By-\-G = and the ellipse Wx^-\-a"y^—crlr' = 0. (411G)
4319 {A^a'-\-B-}r) x''-{-2ACd\v-\-Chi"-B-a-¥ = 0.
>iQOA _ -ACa''±Bahx/ A'a'-i-n-fr-C'
A-a-\-B-b^
4321 For the ordinates transpose A, B and a, h.
THE ELLIPSE AND HYPERBOLA. 693
Length of intercepted chord :
Hence the condition that the hne may touch the ellipse is
4323 A'a'-{-B%' = 0\
The chord through two points Xiij^, x^y.i, is
*^'^* ^? "^ P ~ a" "^ b' ^ '
or, denoting the points by their eccentric angles a, /3, the
chord joining a/3 is
4325 ±cos^ + |-sin5+^ = cos^.
The coordinates of the pole of the chord or intersection of
tangents at x-^y^, x.^y^ (or a/3 as above).
4326 V — ^^y^'^^^-^y^ ^ ^' iVi—y^^ -a ^^^^ («+^)
Vx-Vy-i oi\yx-x^y^i cosi(a-^)"
4329 V = ""^y^'^'^^-'y^ = ^'(-^^^"-^-J = b ^i^i(^+^) .
■^ ^\+^2 ^^^^2-^*^1 cosi(a-/8)
The following relations also subsist
>1 QQo "^^^ — «^ sin g sin ^ _ 6^ cos a cos ^
__ b (sina+sin/S) __ a (cosa+cosff)
- 2y - 2.V
" which are of use in finding the locus of {x, y) when a, /3 are
connected by some fixed equation."
(Wolstenholme's Problems, p. 116.)
4 G
694 CARTESIAN ANALYTICAL OONIGS.
4334 If a, i3, 7, S are tlie eccentric angles of tlie feet of the
four normals drawn to an ellipse from a point xy, tlien
a+)8+y+8 = 37r or 5ir.
Proof. — Equation (4290) gives the following biquadratic in z = tan }^(j),
by/ + 2(ax + a'-h') z' + 2 (ax-a' + h') z-by = 0.
Let a, h, c, d be the roots. Eliminate d from ah + ac + &c. = and ahcd = — 1
(406). Thus ab + hc + ca= — + -=^ + —r; from which, since a = tan|a,
^ ^ be ca ab
&c., we get sin(/3 + y) + sin(y + a) + sin(a + /3) =0;
and, since l — (ab + ac + &c.)+ahcd = 0,
tani(a+/3-f y + o) = 00, .-. a + /3 + y + 2 = Stt or Stt.
4335 The points on the curve where it is met by the
normals drawn from a fixed point xy' are determined by the
intersections of the curve and the hyperbola
a^'cc'y-y'yx = c'xij. (4287)
POLAR EQUATIONS OF THE CONIC.
The focus S being the pole (Fig. of 4201), the equation of
any conic is
4336 r(l+ecos^) = /,
being measured from A, the nearest vertex.
For the parabola, put e = 1.
Proof. —
r=SP; d = A8F; l = SL; r = e{8X + SN) (4200) = l + er cob 6.
The secant through two points, P, F, on the curve, whose
angular coordinates are a+/3 and a— /3 (Fig. 28), is
4337 r {ecos^4-seci8cos(a-^)} = /.
Proof.— Let ASQ = a, I'SQ = FSQ = ft.
Analytically. Take (4109) for the equation of PP'. Eliminate r, and r,
by (433G), and substitute 2a for e, + d^ and 2/3 for 0,-0,.
Geometrically. Let PP' cut the directrix in Z ; then QSZ is a right angle,
by (IIGG). Take C any point in PP' ; SC = r ; ASC = 0. Draw CD, CE,
CF, CG parallel to SL, SP, SQ, SX, and Dll parallel to XL. Then
l=SL= SH+HL.
arr SL or, n ^'^ ^P _ SL _ IlL _ BL
^^=^^^ = "''°^^- GG=PM-^-lJX-GG'
.-. IlLz= CE = r sin CSF sec /? = r cos (a — 0) sec /3,
.•. Z = er cos/(3 + >'sec/3 cos(u— 0).
THE ELLIPSE AND HYPERBOLA.
595
The equation of tlie tangent at the point a is, conse-
quently,
4338 r {e cos ^+cos (a-^)} = I.
4339 Length
A Focal Chord.
_ 2/
"~ 1— e^cos^^'
(4336)
Coordinates of the extremities, the centre G being the
origin :
_ a(e±cos6) _ / sin 6
l±ecos6 ' ^"Idzecos^*
4340
a^
4342 The lines joining the extremities of two focal chords
meet in the directrix. [By (4337)
Polar equation with vertex for pole :
4343 r^ (1 -e^ cos^ 6) = 21 cos d. (4200)
Polar equation with the centre for pole :
4344 r^ («' sin^ e+b' cos^- 6) = a'b\
4345 or r ^/(l-e' cos^ 6) = b.
Proof.— By (4273). Otlierwise, by (4314), with x' = y' = 0.
CONJUGATE DIAMETERS.
dr
T A A'
Equation of the ellipse referred to conjugate diameters for
coordinate axes :
4346
596 CARTESIAN ANALYTICAL C0NIC8.
where
4347 a" = »'^' h" = — .
a-sm'a+6'cos'a' a- sin' /8 + 6 ' cos^ /3
Here a = CD, h' = GP, a is the angle DCB, and jS the angle
FOB.
Proof. — Apply (4050) to the equation (4273), putting ^ = -^
= -^ and
tan a tan /3 = ^, by (4351)
When a = h', a+/3 = tt, and equation (4346) becomes
4349 ^-^ir = « ' = i {(I'+b').
Let the coordinates of D be x, y, and those of P x,y; the
equation of the diameter GP conjugate to GD is
4350 f +-f:
= 0.
4351 tan a tan ^ or
mm = -.
(4318)
a??/ in terms of xy\ &c.
4352 ^^ = -T^''
"=1-
Ell.
4354 .r = ±j/,
,,=±„.
Hyp.
Proof.— Solve (4350) with (4273).
4356 a;=dR, .
aj'=pN.
(4274, 4352)
4358 .r^+o?'^ = u\ ?/"+.?/ = 61 Ell. (4352)
4360 w^-w"' = n\ l/'-i/ ^ b\ Hyp. (4354)
4362 d' + h- = n'-hb'K Ell. (4358)
4363 (r — h'= a' — b'\ Hyp. (4360)
4364 a' = 6'H-eV. (4271, '61)
4365 b"' = a^—e'-d'^ = rr. (4298)
TEE ELLIPSE AND HYPERBOLA. 597
The perpendicular from the centre upon the tangent at
xy is given by
4366 7 = -^+^- (4281,4064)
The area of the parallelogram PCDL (Fig. of 4307) is
4367 pa' = ab = a'h' sin <u,
where p = PF, a' = GD, b' = CP, i^ = A PCD.
Proof.— From (4366), and (4352), and a'=^x' + y'\
Other values of p^ :
4369 P'^-i^-^W^r (^362)
4371 p^ = a" sin^ e-\- b' cos^ 0.
Proof. — From (4844, '67), putting r = a.
4372 p^ = a^ cos' y +6' sin' y, (4371)
y being the inclination of j9.
4373 p^ = a" (1 -e' sin' y). (4372, 4260)
Equations to the tangents at P and P\ the coordinates of
D being x, y :
4374 ocy'-yx =±ab (4073) m = -^.
DETERMINATION OP VARIOUS ANGLES.
4375 j)Cd=-^. Fig. p. 595. (4356)
4377 tan PCD = - ^, (4070, 4352-3)
where c = ^{a'-h') = G8.
698 CARTESIAN ANALYTICAL CONICS.
4378 tan (SPT) = — = 1+gcosg ^^^^^^ ^^SG, 4336)
^ c^ e sm ^
where = P8T. [See figure on page 588.
If i// be tlie inclination of the tangent to tlie x axis,
4380 ianxlf = -^ = £+^. (4280)
^ ai/ sm ^
Proof: ^p = + /SPr. Then by (631) and (4379).
4382 tan SPS' = tP^. (652, 4378)
2¥
4383 tan ^P^' = — ^, tan CPG = ^.
If OP^ OP' are tangents to an ellipse,
Proof.— By figure and construction of (1180), TOF' = ITOS'. Therefore
i.no' OS'+OS"-S^ 2(70H20/S^-4a^ _ „ ,
If »', y are the coordinates of 0,
4386 tanPOP' = 2v£(*!£!+£2:!z^.
Proof. — By (4311), taking terms of the second degree for the two parallel
lines through the origin and tan^ from (4112).
It is worthy of remark that the substitutions (4276-8)
may also be usefully employed when the axes of reference are
conjugate diameters : though, in that case, the geometrical
signification of ^ no longer exists.
I aN TV- f'sittI
TEE HYPERBOLA.
599
THE HYPERBOLA
REFERRED TO THE ASYMPTOTES.
4387 .ri/=JK+6^).
Proof.— By (4273) and (4050). Here x = CK, y = PK.
Equations of the tangent at P, {x, ?/).
4388 ciy+y?/ = i(«'+^')- (4120)
4389 y = m.v-{- v/m(«^+6'0. (4123)
4390
4391 Intercepts on the axes CI = 2a?' , CL = 2?/.
THE RECTANGULAR HYPERBOLA.
4392 Here a = h, e = \/2 ; and the equation with the
ordinary axes is
4393 ^'-1/ = «'• (4273)
4394 Tangent xx'-yy'^a''. (4281)
600 OABTESIAN ANALYTICAL GONIGS.
Equation with tlie asymptotes for axes :
4395 2ay = a\ (4387)
4396 Tangent a:y-\-xij = a\ (4388)
THE GENERAL EQUATION.
The general equation of the second degree is
4400 ci9^ + ^hxij + hf -H 2gx + 2/7/ + c = 0,
4401 or ax^ + hif + cz^+2fyz + 2gzx-{-2hxy = 0, with z = l.
The equation will be denoted hj u ov <\>{x,y) = 0.
THE ELLIPSE AND HYPERBOLA.
When the general equation (4400), taken to rectangular
axes of coordinates, represents a central conic, the coordinates
of the centre, 0' (Fig. 30), are
440^ ^1 - ^^_j^ - c' y ab-h' - C
Proof. — By changing the origin to the point xt/ and equating the new
g and /each to zero (4048).
For the case in which ab = /t^, see (4430).
4404 The transformed equation is ax^-\-2hjcij-\-hy^-\-c' = 0,
4405 where c' = aa^''-\-2Lv)/-\'hir+2g.v-^2fi/'-\-c.
4406 =g*<^'+/y+^.
^407 - «ft^+2fyA-^(f-^fA''^-^/^'^ _ A. (4466)
The inclination B of the principal axis of the conic to the
X axis is ffiven by
4408 tan 2^ = ^.
Proof.— (Fig. 30.) By turning the axes in (4404) through the angle 6
(4049) and equating the new h to zero.
TEE GENERAL EQUATION. 601
Tlie transformed equation now becomes
4409 «V+6y+c' = 0,
4410 iu which ti =^{a-\-b+ \/m^+(«-6)^ } ,
4411 b' = i {a-\-b-v'4sh'-\-{a-by} ,
a and b' are found from the two equations
4412 a-\-b' = (i-^b, ab' = ab — h\ [See (4418).
The semi-axes aiid excentricity are
4414 V--^, V-T' ^^'"^ e=^'(^l-|^). (4273) (4261)
For the coordinates of the foci, see (5008).
4416 Note. — If B be the acute angle determined by equation
(4408), we have to choose between 9 and 0-\-^ for the inclina-
tion in question, since tan 20 is also equal to tan (20 + 7r).
Rule.* — For the ellipse, the inclination of the major axis
to the X axis of coordinates will be the acute angle 6 or + ^7r,
according as h and c' have the same or different signs. For the
hyperbola, read " dfferent or the same.''
Proof. — Let the transformed equation (4409) be written in terms of the
semi-axes p, 2 ; thus cfx'^+p'y'^ ^ jfcf, representing an ellipse. Now turn
the axes back again through the angle —d, and we get
{q^ cos^ 6 +f sin^ d) X'- (p'^-c/) sin 2$ xy + {(f sin^ d-\-p^ cos' 0) t/^ = p\\
Comparing this with the identical equation (4404), ax^ + 21ixy-\-hi/ ■= —g\
we have ip'^ — 'i) sin 29 = — 2/i, p-q^ = — c';
sin20 = ^.^^„. Hence ^ is < -J
c p' — q" 2
when h and c' have the same sign, p being >q. A similar investigation
applies to the hyperbola by changing the sign of g'.
* This rule and the demonstration of it are due to Mr. George Heppel, M.A., of
Hammersmith.
4 H
602 CARTESIAN ANALYTICAL C0NIG8.
INVARIANTS OF THE CONIC.
4417 Transformation of the origin of coordinates alone
does not alter the values of a, h, or h, whether the axes, be
rectangular or oblique. This is seen in (4404).
When the axes are rectangular, turning each through an
angle 9 does not affect the values of
4418 ab-h', «+6, ^^-4-/^ or c.
When the axes are oblique (inclination w), transformation
in any manner does not affect the values of the expressions
4422 . , and —^ — ^^
These theoi'ems may be proved by actual transformation by the formulae
in (4048-50) . For other methods and additional invariants of the conic, see
(4951).
4424 If the axes of coordinates are oblique, equation (4400)
is transformed to the centre in the same way, and equations
(4402-6) still hold good. If the final equation referred to axes
coinciding with those of the conic be
4425 a\v'-^h'ii'-\-c' = 0,
and the inchnation of the new axis of ,i' to the old one, we
shall have c unaltered,
4426 tan 20= 2/,sm^-„sm2<«
2/t coso)— -a cos2a* — /> '
4427
, _ CT+fe— 2Acos6j+yQ , J, _ (i-\-h—2hQO^<o — ^Q ^
2 sin- 0} ' 2 sin- w '
where Q = a^-\-'U^-\-2ah cos 2w + 4A {a-\-h) cos a)-{-4^^.
Proof. — (4404) is now transformed by the substitutions in (4050),
putting /3 = + 9O°, and equating the new h to zero to determine tan 20.
a' and // are most readily found from the invariants in (4422). Thus, putting
the new /t = and the new w = dO\
. 7/ a + 6 — 2/i cosw n /w ah — h^
a +0 = —~ — - — and ab = -r-5 — ,
sm u> sin u)
equations which determine a and ?/.
THE GENERAL EQUATION. 603
The eccentricity of the general conic (4400) is given by
the equation
4429 __£!_ _ (a + b-2li cos coy _^
Proof. — By (4415), and the invariants in (4422).
THE PARABOLA.
4430 When ab — Jr = 0, the general equation (4400) repre-
sents a parabola.
For X, y' in (4402) then become infinite and the cui^ve has
no centre, or the centre may be considered to recede to
infinity.
Tui^n the* axes of coordinates at once through an angle %
(4049), and in the transformed equation let the new coeffi-
cients be a, 111, h\ 2c/, 2/ , c'. Equate li to zero; this gives
2/?
(4408) again, tan 26 = — -. If 6 be the acute angle deter-
mined by this equation, we can decide whether 9 or 9-\-^Tr is
the angle between the x axis and the axis of the parabola by
the following rule.
4431 KuLE. — The inclination of the axis of the imvahola to
the X axis of coordinates will be tJie acute angle 9 fh. has the
opposite sign to that o/a or b, and 9-\-\Tr if it has the same
sign.
Proof. — Since ah — h^ = 0, a and h have the same sign. Let that sign
be positive, changing signs throughout if it is not. Then, for a point at
infinity on the curve, x and y will take the same sign when the inclination is
the acute angle 9, and opposite signs when it is O + ^tt. But, since
ax^^lif = +00 , we must have 2hxij = — co, the terms of the first degree
vanishing in comparison. Hence the sign of h determines the angle as stated
in the rule.
4432 -"<'=V46' •^°«'' = V^-
Proof.— From the value of tan 20 above, d being the acute angle obtained,
and from /t^ = ab.
4434 Also a = and h' = a + b.
For a'b' = ab — lr = 0, and we ensure that a' and not 6' vanishes by
(4431). Also a+b' = a+b (4412).
604 CARTESIAN ANALYTICAL CONICS.
4436 ff = g COS e+f sin e = "^^ff^* -
4438 r = .cos^-^si„<»=^-:^i=^*.
4440 But if /i lias tlie same sign as a and b, change 6 into
O+Itt. (4431)
Pkoof.— By (4418, 4432-3).
The coordinates of the vertex are
4441 ''^-W^' y=--y-
Obtained by changing the origin to the point x'lj and equating to zero
the coefficient of y and the absolute term. The coefficient of x then gives
the latus rectum of the parabola ; viz. :
4443 L - - ^ - - 2 ^J>±/V^. (4437)
METHOD WITHOUT TRANSFORMATION OF THE AXES.
4445 Let the general equation (4400) be solved as a quad-
ratic in y. The result may be exhibited in either of the forms
4446 // = aa^+/3± \/ft {x'-1px^rq)^
4447 y = arJrfi±Vf^ {(^^-pY+iq-f)}^
4448 1/ = a.'+,8± v> (.r-y)(.r-S),
4449 where a=z —j, fi = -j, (^ = ^." .
4454 , ,. - >> („b c+2fkh-ar-hf,--''lr) _ bA
{(ib—lr)- C'-
4456 y and S = ;> ± V{f—q).
4458 Hero v/ = aa'4-/3 is the equation to the diameter DD
THE GENERAL EQUATION. 605
(Fig. 31), 7 and g are the absciss93 of D and D\ its extremities,
the tangents at those points being parallel to the y axis. The
surd = PN = FN when x — OM. The axes may be rect-
angular or oblique.
When ah — ]r = 0, equation (4446) becomes
4459 1/ = aa +/3 d= J ^q'-2p^v,
4460 where p = bg—hf, q=p—bc.
4462 In this case, ^, is the abscissa of the extremity of
the diameter whose equation is y = ax-}-^ and the curve has
infinite branches.
RULES FOR THE ANALYSIS OF THE GENERAL EQUATION.
First examine the value of ab— h^ and, if this is not zero,
calculate the numerical value of c (4407), and j^roceed as in
(4400) et seq. If ab— h^ is zero, find the values ofip' and q'
(4459). The following are the cases that arise.
4464 ab—h- positive — Locus an ellipse.
Particular Gases.
4465 ^ = — Locus the point xy'.
See (4402). For, by (4404), the conjugate axes vanish.
4466 6A positive — No locus.
By (4447-54), since q—p^ is then positive.
4467 /i = and a = b — Locus a circle.
By (4144). In other cases proceed as in (4400-14),
4468 ab—h^ negative — Locus an hyperbola.
Particular Gases.
4469 A = — Locus two right lines intersecting in the
point xy'.
By (4447), since g—p^ then vanishes. In this case solve as in (4447).
606 CARTESIAN ANyiLYTICAL CONICS.
4470 bA negative — Locus the conjugate hyperbola.
4471 a-\-b = — Locus the rectangular hyperbola.
By (4414), since a'= —h'.
4472 a = b = — Locus an hyperbola, with its asymptotes
parallel to the coordinate axes. The coordinates of the centre
are now —^ and — -^, by (4402). Transfer the origin to
the centre, and the equation becomes
4473 ■''^=T-
In other cases proceed as in (4400-14).
4474 ah — h^ = — Locus a parabola.
Particular Gases.
4475 |/ = — Locus two parallel right lines. By (4459).
4476 J)' = q = — Locus two coinciding right lines.
By (4459).
4477 Ji' = and (/ negative — No locus.
^ By (4459). In other cases proceed as in (4430-43).
Ex. 1.: 2x'^-2xy+y' + 3x-y-l = 0.
3 1
Here the values of a, li, h, (j, f, c are respectively 2, —1, 1, --, — — , —1,
G ah — h- 4
The locus is therefore an ellipse, none of the exceptions (44G5-7) occurring
here. The coordinates of the centre, by (4402), are
' _ kf-bg _ _i „' _ f/^'-" / _ _ 1
ah — h' ab — k '1
Hence the equation transformed to the centre is
2x'-2xy + y'-^ = 0.
Turning the axes of coordinates through an angle so that tan29=— 2
(4408), we find the new a and h from
a' + 6' = 3, ah' = \; (4412)
THE GENERAL EQUATION. 607
therefore a' = i (3-^/5), b' = ^ (3+ v/5),
and the final equation becomes 2 (3— v/5) a;^ + 2 (3+ ^/5) ?/* = 9.
The inclination of the major axis to the original x axis of coordinates is
the acute angle ^tan"^ ( — 2), by the rule in (4416).
Ex. (2): 12x' + 60xy + 75y'-12x-8y-6 = 0.
The values of a, h, b, g,f, c are respectively 12, 30, 75, —6, —4, —6,
a6-7i^ = 0; p' = bg-hf= -330; q'=f-bc. (4460)
Since jj' does not vanish (4475-7), the locus is a parabola. Proceeding,
therefore, by (4430-43), we have
ta.29 = ^^ = -|; sm« = ^, cos e = -l^. (4432)
By the rule (4431), we must take O+^tt for the angle, instead of 6. There-
fore g=-gsme+/cosg= ^^^J^^ = -^.
y = -/sine-, cos 0= %43 =729-
and 6' = a + 6 = 87 (4435).
Consequently the transformed equation is
Q^7 2 I 44 ,64 ^ ^
^^^ +"729"+ 729^-^ -^ = ^-
The coordinates of the vertex are computed by (4441), and the final
44
87x/29*'
Ex. (3) : x' + 6xy+9y' + bx + 15y + 6 = 0.
The values of a, h, b, g,f, c are respectively 1, 3, 9, — , — , 6,
ab — W = and p = bg — kf = 0,
therefore, by (4475-7), if there is a locus at all, it consists of two parallel or
coinciding lines. Solving the equation therefore as a quadratic in y, we
obtain it in the form (x + oy + 2) (x + Sy + S) = 0,
the equation of two parallel right lines.
The equation of the tangent or polar of x'l/ is
4478 u,,'<.v-\-Uy,i/-\-u,,z = or u,.a:'-\-Uyt/-]-u.z =0 ;
(4401, 1405) obtained by (4120) in the form
4479 («.!-'+%'+§•) .r+ (Ay +%'+/) y-\-g^'-^fy+c = 0,
4480 or {ax+hy-]-g) a;'+ {Kv^hy+f) y'J^^cc^fy^c = 0,
608 CARTESIAN ANALYTICAL CONICS.
4481 or
AVlien the curve passes through the origin, tlie tangent at
the origin is
4482 ga)-\-fy = 0. (4479)
And tlie normal at the same point is
4483 nv-g\i/ = 0.
4484 Intercepts of the curve on the axes, — — , — -/.
a b
4486 Length of normal intercepted between the origin and
the chord
= ^^ V ■ (4483-4)
a-\-h
Rigid Line and Gunic with the general Equation.
4487 Quadratic for n : n\ the ratio in which the line joining
xy, x'lj is cut by the curve.
Let the equation of the curve (4400) be denoted by
^ (aj, ?/) =0, and the equation of the tangent (4479) by
t// (a3, y, c-^^', ^') = ; then the quadratic required will be found,
by the method of (4131), to be
n^^ {x\ y) -i-2mi\f/ (.r, y, od\ y) + n^^ {x, y) = 0.
The equation of the tangents from .r'//' is
4488 <t> {'V\ y) i> Gr, //) = {^ (.r, //, a>', y')}\
Proof. — By the couditiou for equal roots iu (4487).
CoR. — The equation of two tangents tlirougli the origin is
4489 B.v^-'llliy^r Cy- = i). (46G5)
Tlie equation of the asymptotes of u (4400) is
4490 aul + 2/mj(, + bill = 0.
THE GENERAL EQUATION. 609
The equation of the eqiii - conjugates of the conic
ax^-\-21ixy-\-hif = 1 is
4491 (a+&)(«-r+2/u7/-f 6//-) = 2 ((ib-1i^(^-\-if).
Proof. — When the conic is ax^ + bif = 1, the similar equation is
(a + h) (ax' + hif) = 2ah (x' + ^f) or (ax'-hf) = 0,
given by the intersections of the conic and a circle. Transformation of the
axes then produces the above by tbe invariants in (4418).
4492 When the coordinate axes are oblique, the equation
becomes
{a-h){ax'-hif) + 2x (hx + hy) (h-a cos w) + 2ij (ax + hj)(]i-h cos w) = 0.
Greneral polar equation :
4493 (« cos^ 0-\-2h siu ^ cos e-\-b siir 6) r'
+ 2 fe cos ^+/siii 6) r-\-c = 0.
Polar equation with {x, ij) for the pole : (4134)
4494 {a cos' e^~ 2/i siu 6 cos e-\-h sin ' 6) r-
+2 {{ax^hy+g) cos^+(%+Act'+/) siu^} r^F{wy) = 0.
Equation of the hue through xif parallel to the conjugate
diameter :
4495 {^—^v){a.v'-^luj^g)^{y-y'){luv'^hy'^f) = 0.
Proof. — By the condition for equal roots of opposite signs (44'. »4).
Equation of the conic with the origin at the extremity of
the major axis, L being the latus rectum.
4496 if = L.v-{l-e') .v\ (4269, '59)
Equation when the point ah is the focus and
Ax -\-Iju-\-G = the directrix :
4497 ^{(•'-")'+0/-'')'5 =''7n§Jf- (^20"'«»^)
610 CARTESIAN ANALYTICAL CONICS.
INTERCEPT EQUATION OF A CONIC.
Tlie equation of a conic passing through four points whose
intercepts on oblique axes of coordinates are s, «' and t, t\ is
4498 ^+2/..,+ |:-.(± + J.)-,(l + |)+l =0.
Equation of a conic touching oblique axes in the points
wliose intercepts are s and t :
4499 ■^+2/u^//+^-^-^+l = 0,
4500 or (^ + |-^iy+nr// = 0.
Comparing with the general equation (4400), we have
4501 'S' = — , * = --p^ v = 2h-— = 2 ^•^ .
g f St c
Perpendicular p from wi/, any point on the curve, to the
chord of contact :
AKOP. W2 _ vs'f^l/f^co (4096, 4500)
*^"^ ^' -~ s'+t'-26'tQOS<^'
Equation of the tangent at xy' :
4507 2(^+^-l)(i^ + |--l)+.-(.<.'/'+.'» = 0-
4508 The equation of the director- circle is
(l + lstc)(x^ + y' + 2xy cos w)— /i (x + y cosw) — k (y + x cos uj) + hk cos w = 0.
The parabola with the same coordinate axes as in (4109) :
4509 (^ + f-l) = ^ o.. V7 + Vf=l-
Proof. — From (•l-'300), putting h= — ~ (4174), and tliorefbro v = -.
st St
TEE GENERAL EQUATION. 611
Equation of the tangent at x', y' :
4511 or ,; = ,„.r+_-j-^, » = -Vi?-
Equation of the normal at xy' :
4512 .'/ = »-+(^;^7+;)F. '« = n/-^-
4513 Normal through the origin xVs = yVt.
(4123)
(4122)
The equations of two diameters are, with any axes,
4514 iL_X = l and ^-f = -l-
S t St
Proof.— Diameter tbrotigh Of, -^^ = - by tte property OB = i?Q, iu
the figure of (4211). ^ '
Coordinates of the focus :
451 6 cV = "^^ y = — . (5009)
4010 s'-\-f-\-2stGOSto' ^ s'+f+2stcos<o
Equation of the directrix :
4518 cV{s-^t GOS(o)-\-1/{t-\-S COS 6)) = St COSG).
Proof. — Expand (4509), and form the equation of the polar of the focus
by (4479) and (4516).
When the axes are also rectangular, the latus rectum
4519 i>= ^fi-^. (4095,4516-8)
4520 Locus of the centre of the conic which touches the
axes at the points sO, 0^ :
t,v = Si/. (4500, 4402)
4521 To make the conic pass through a point xy' ; substi-
tute xy in (4500), and determine v.
612 CARTESIAN ANALYTICAL GONICS.
SIMILAR CONICS.
4522 Definition. — If two radii, drawn from two fixed
points, maintain a constant ratio and a constant mutual
inclination, tliey will describe similar curves.
4523 If the proportional radii be always parallel, the curves
are also similarly situated.
If tliere be two conies (1) and (2), with equations of the
form (4400), then—
The condition of their being similar and similarly
situated is
4524 ^ = -^ = t
a h b
Pi^OOF. — By (4404), changing to polar coordinates, r : r' ■= constant.
The condition of similarity only is
4525 M!=(^:; (4418-9)
h^—ab Ir — ab
or, with oblique axes,
^^gg (^^ + 6-2Aeosa))- ^ {u'^b'--2h'Q0f^0iy (4422_3)
h^ — ab li- — ab'
CIRCLE OF CURVATURE.
CONTACT OF CONICS.
4527 Def. — When two points of intersection of two curves
coincide on a common tangent, the curves have a contact of
ihQ first order; when three such points coincide, a contact of
the second order ; and so on. To osculate, is to have a con-
tact higher than the first.
4528 The two conies (Fig, ^2) whose equations are
ax^ +2//r*'// + />/r + 2j/<i! = (1),
a'x' + 21i'xii-\-h'tf + 2g'x = (2),
GIBGLE OF GUBVATUBE. 613
touch the y axis at the origin, 0, by (4482). EUminate the
third terms from (1) and (2), and we obtain x = 0, the line
through two coincident points, and
4529 {ah'-a'h) a'+2 {hh'-h'h) 1/-2 fe -?>» = 0,
the equation to LM, the line passing through the two remain-
ing points of intersection of (1) and (2). (4099)
Again, eliminate the last terms from (1) and (2), and we
obtain
4530 «- « g") ^'+2 {hg'-h'g') .vy+ (bg'-b'g) 2/ = 0,
the equation of the two lines OL, OM. [By (4111) and (4099)
4531 If the points L, M coincide, the conies have contact
of the first order. The condition for this is that (4530) must
have equal roots ; therefore
4533 {ag' - ag) (bg' - b'g) = {lig - h'g)\
4533 If the conies (1) and (2) are to osculate, M must
coincide with 0. Therefore, in (4529), hg' = Vg.
If in (4532) hg' = h'g, the conies have a contact of the
third order.
CIRCLE OF CURVATURE.
(See also 1254 et seq.)
The radius of curvatui^e at the origin for tlie conic
(Lv^-^2JLVt/-\-by^-\-2gx = 0,
the axes of coordinates including an angle w, is
4534 P = -
h siu (si
Proof. — The circle touching the curve at the origin is
x^-\-2xy co?,b)-\-if—2rx sin w = 0,
by (4148), and the geometry of the figure, 2rsinw being the intercept on
the X axis. The condition of osculating (4533) gives the value of p.
p is positive when the convexity of the curve is towards the y axis.
Eadius of curvature for a central conic at the extremity
P of a semi-diameter a, the conjugate being h'.
614 CARTESIAN ANALYTICAL CONICS.
4535 p = ^^=*l=if!*: = -*;. (4367)
a sm<y j) jr no
Proof. — Take the equation and figure of (434G) (a = CF). Transform
to parallel axes through P. Then by (4534).
The same in terms of x, y, the coordinates of the point P.
4539 , = (*V:My)!.
Pkoof.— By (5138), or from (4538) and the value of h at (4365).
The coordinates of the centre of curvature for P, the
point xijf are
4540 i= -~, V = — ~^^ where 0^= a^—b\
P,..00P.-(.ig. 33.) ..o. 5| = t^ = i, ana lj = ^^^=f^.
with the values of p, PG, and PG' at (4535) and (4309).
Radius of curvature for the parabola.
Taking the diameter and tangent through the point for
axes,
4542 p = ^=J^= ^^. (Fig. of 4201)
' sin u sm^ ff Si
By (4534), and equation (4211).
Coordinates of the centre of curvature at xij (rectangular
axes) :
4545 ^ = 3^+2«, ^ = -|7.-
Proof. — From i/ — i}:p = y:PG and p = 2a cosec" 0, PG' = 2acosec0
and y = 2a cot 6.
The evolute of a central conic (Fig. 33) :
4547 (aa^^-^{bf/)^={fr-Ir)K
4548 or {a\v^-^by-c'Y-\-27(rhh^\2nf = 0,
where c" = (r — lr.
CON FOCAL CONIC 8. 615
Proof. — Substitute for x, y in the equation of the conic (4273) their
values in terms of ^, t} from (4540). Otherwise as in (4958), or by the
method of (5157).
The curve has cnsps at L, H, M, and K.
The evolute of the parabola :
Proof. — As in (4548), from the equations (4201) and (4545).
CONFOCAL CONICS.
4550 ^ + |l=:l and 4 + 1^=1
are confocal conies, if
a^-a" = ¥-h'\
or the sign of h'''^ may be changed.
For the confocal of the general conic, see (5007).
4551 Confocal conies intersect, if at all, at right angles.
Proof. — If «, «' are the two conies in (4550), changing the sign of h"" to
make the second conic an hyperbola, « — «'=0 will be satisfied at their point
of intersection ; and {hj a' — a"^ = If' -\-h'') this proves the tangents at that
point to be at right angles (4078, 4280).
Otherwise geometrically by (1168).
4552 Tangents from a point P on one conic to a confocal
conie make equal angles with the tangent at P. [Proof at (1291)
4553 The locus of the pole of the line Ax+By + G with
respect to a series of confocal conies in which d^ — h" r= A, is
the right line perpendicular to the given one,
BCA-ACy-\-AB\ = {S.
Proof. — The pole of the line for any of the conies being xy ; Aa? = — Cx
and I?6^= —Cy (4292) ; also d- — lr' = A. Eliminate a" and W.
4554 CoR. — If the given line touch one of the conies, the
locus is the normal at the point of contact.
616 CARTESIAN ANALYTICAL CONIC S.
4555 Grai-cs' Theorem. — The two tangents drawn to an
ellipse from a point on a confocal ellipse together exceed the
intercepted arc by a constant quantity.
Proof. — (Fig. 132.) Let P, P' be consecutive points on the confocal from
which the tangents are drawn. Let fall the perpendiculars FN, P'N'. From
(1291), it follows that /.FP'N = F'FN', and therefore P'jV = P^". The
increment in the sum of the tangents in passing from P to P' is
BB' - QQ' + P'N- FN' = BB' - Q Q'.
But this is also the increment in the arc QB, which proves the theorem.
4556 M the tangents are drawn from a confocal hyperbola,
as in (Fig. 133), the difference of the tangents PQ, FB is
equal to the difference of the arcs QT, BT.
•The proof is quite similar to the foregoing.
4557 ^t the intersection of two confocal conies, the centre
of curvature of eitlier is the pole of its tangent with respect
to the other.
Pkoof. — Take -^ + ^ = 1 (i.) and -^ — ^, = 1 (ii.) for confocal conies.
a- h' a' '
At the point of intersection, x"- = ^^ and y"-= — (where c" = (r—h'-);
loy a' — a"- = h- + h"\ The coordinates of the centre of curvature of x'y' in
(i.) arc x" = ^\ y" = -^(^ (4540-1). The polar of this point with
respect to (ii.) will be ^^ + -Kr = 1- Substitute the values of x", y"; and
we see, by the values of x', y', that this is also the tangent of (i.) at P.
4558 A system of coaxal circles (4161), reciprocated witli
respect to one of the limiting points 1) or D\ becomes a
system of confocal conies.
Pkoof. — The origin B is one common focus of the reciprocal conies, by
(4844). The polar of P with respect to any of the circles is the same line,
by (41GG). P and its polar (both fixed) reciprocate (4858) into the line at
infinity and its polar, which is the centre of the conic. The centre and one
focus bcinc: the same for all, the conies are confocal.
ANALYTICAL CONICS
IN
TRILINEAR COORDINATES.
THE RIGHT LINE.
For a description of this system of coordinates, see (4006).
The square of the distance between two points aj3y, a'/3'y is,
with the notation of (4008),
4601
4602
= ^ [a cos^ (a-a )-hl) cos^ (i8-/8')^+r cosC(y-y)1.
Proof. — Let P, Q be the points. By drawiag the coordinates /?y, fi'y',
it is easily seen, by (702), that
pg^ = [(/3-/3T+{y-yT + 2(/3-/5')(r-y')cos^]cosec^4 (I).
Now, by (4007), a (a- a') + b(/5-/3') + 0(7-7') =0,
from which b(f3-i3'y = -a(a-a')(/3-/3')-c(/3-/3')(7-y'),
and a similar expression for 0(7 — 7'/. Substitute these values of the square
terms in (1), reducing by (702).
Coordinates of the point which divides the straight line
joining the points afty, a^^'y in the ratio I : m :
4603 '^±^\ '-^j^, '-^^. By (4032).
l-\-m l-\-7ri l-{-m
ABC being the triangle of reference, and a = 0, |3 = 0,
7 = the equations of its sides, the equation of a line passing
through the intersection of the lines = 0, (5 = is
4604 /a-myS = or a-A:.5 = 0.
4 K
618 TRILINEAR ANALYTICAL CONIC S.
Proof. — For this is the locus of a point whose coordinates a, ft are in the
constant ratio vi : I or k (4099).
When I and m have tlie same sign, the line divides the external angle C
of the triangle ABC; when of opposite sign, the internal angle C.
The general equation of a straight Hne is
4605 /a+my8+«7 = 0,
and it may be referred to as the Une (/, m, ??).
Proof: la + mft = is aritj line through the point C, and (la + mft)-^ny
= is any line through the intersection of the former line and the line 7=0
(4604), and therefore any line whatever according to the values of the arbi-
trary constants /, m, n.
The same straight hne in Cartesian coordinates is
4606 (/ cos a+ m cos ^+?i cosy) cV
+ (/ sin a-\-m siu /8+ n sin y) ?/— (/;?!+ >m;>2+ ups) = ^•
Proof. — By substituting the values of o, ft, y at (4000).
Or, if the equations of the sides of ABC are given in the
form Arr-\-B^y-\-Gi = 0, &c., the hne becomes
4607 {lA,-{-mA,-]-nA,) .v-i-{lB,+mB,-\-nB,)i/
-\rlC,-\-mC,-\-nC,= 0.
Proof. — By (4095), the denominators like \/(AI + II\) being included in
the constants /, »j, v.
4608 If '' = ^K '■ = ^i '" = ^ are the general equations of
the hnos a, /3, y, then it is obvious that lu-\-mv = is, like
(4604), a line passing through the intersection of u and Vy and
Jn-\-mn^nw = represents any straight line whatever.
To make an equation such as a = p (a constant) liomo-
geneous in a, /3, y; multiply by the equation S = aa -f^/S + Cy
(4007), thus
(a;i^-S)n-fb;v3 + Cjiy = 0,
wliicli is of the same foi'iti as ( llJOo).
THE BIGHT LINE. 619
4610 The point of intersection of tlie lines
/a + m/3 + ?iy = and ra-\-m(5-\-ny =
is determined by the ratios
/3 y
mn — mn nl' — n'l Im — I m
The values of a, (5, y are therefore
and (4017).
VI ei i ^ (mn — m'?i) t {nV — n'l) t (Ini —I'm)
4611 j^ , 2> ' D '
where D = n (mn — m'n) +1) [nl' — n'l) -f t {Im' — I'm) .
Proof.— By (4017), or by solving the three equations
aa + 1)/3 + cy = S, ^a + m/3 + ny = 0, I'a + m'ft + n'y = 0.
The equation
4612 aa+l)i8+Cy = or a sin ^+i8sin J5+y sin C =
represents a straight line at infinity.
Proof. — The coordinates of its intersection with any other line
Za+WjS + ny = are infinite by (4611).
4613 Note: ar« + B/3 + cy = 2, a quantity not zero. The equation
(^a-\-hft-{-(.y = is therefore in itself impossible, and so is a line infinitely
distant. The two conceptions are, however, together consistent; the one
involves the other. And if, in the equation Za + m/3 + ?iy = 0, the ratios
I : m : n approach the values a : fc : c, the line it represents recedes to an
unlimited distance from the trigou.
4614 The equation corresponding to (4612) in Cartesian coordinates is
Oa; + 0?/ + 6^ = 0, the intercepts on the axes being both infinite. Cartesian
coordinates may therefore be regarded as trilinear with the x and y axes for
two sides of the trigon and the other side at an infinite distance.
4615 The condition that three points
aii3i7i, a.,l^zy2, «3/3373 may lie on the same
straight line is the determinant equation,
tti /3i
ri
^2 A
72
a. A
rs
= 0.
Proof. — For it is the eliminant of the three simultaneous equations,
la^-\-m(ji + nyi = 0, Za^ + m/3j + ?iyj, la^ + mfii-\-ny;— 0. (583)
620
TRIUNE AB AXALYTICAL CONK'S.
4616 Cor. — Tlie above is also tlie equation of a straight line
passing tlirough two of the fixed points if the third point be
considered variable.
4617 Similarly, the condition that the three following
straight lines may pass through the same point, is the deter-
minant equation on the right,
= 0.
4618 The condition of parallehsm of the two straight lines
is the determinant equation
Proof. — By taking tbeliue at infiuity (4(jl2j for the third Hue in (4617).
I )n n
I' m' n
a 1) r
= 0,
4619 Otherwise the equations of two parallel Hues differ by
a constant (4076). Thus
/a+m)8+«y+A- (aa+byS+ry) = (4007)
or (/+A-a)a+(m+A'l))/8+(;i + AT)y=
represents any line i)arallel to /a + zyz/S-h^/y = by varying
the value of k.
The condition of perpendicularitv oP the two lines in
(•1618) is
4620 W -\-mm' -\- nn' — {uni' -\- m h) i'osA—(nl'-\-n'l) cos B
4621 or V (l—m vosC—ii cos B) -\-7h {m—n cos A — I cos C)
-\- n {n — I cos B — m cos ^ ) = 0.
Proof. — Transform tlie two ccjnations into Cartesians, by (4G06), and
apply the teat AA' + BJi' = (4078), remembering that
voaift-y) = -eoKJ, &c. (kill).
THE lUailT LINE.
621
When the second line is AB or 7 = 0, the condition is
4622 " = m cos .4 +/ cos B.
It also appears, by (4676), that (4620) is the condition that
tlie two lines may be conjugate with respect to the conic
whose tangential equation is
4623 /^4-m'+/i' — 2m/j cos A —2nl vosB — 2lm cos C = 0.
The length of the perpendicular from a point a(i'y' to the
line la-{-'}}ij3-\-uy = :
la + m ^' + ny __^__^^__
4624
a/ } i:--{-)n~-\-n-—'2mn cos A — 2nl cos ii— 2/m cos C]
Proof.— By (4095) the perpendicular is equal to the form in (4006), with
x', y' in the place of x, y, divided by the square root of sum of squares of
coefficients of x and y. The numerator ~ la' + miy + ny'. The denominator
reduces by cos (/3 — y) = —cos J., &c.
4625 Equation of tlie same perpendicular :
a a l—m QOsC — n cos B
^ )8' m — /icos^— / cos C =0.
y y n — I cos B — m cos A
Proof. — This is the eliminant of the three conditional equations
ia + Mj3 + A> = 0, La->rMi^'-\-Ny' = 0, and equation (4621).
4626 Equation of a line drawn through a'j3'/ parallel to the
line (/, m, n) :
a a Xm—hn
y y I)/ — am
Proof. — It is the eliminant of the three conditional equations
la-\-mii->rny = 0, la' + mjj' + tiy' = 0, and the equation at (4618).
= 0.
4627 The tangent of the angle between the lines (Z, m, n)
and {V, m\ n) is
{mn—mn) sin A-\-{nJ!—nl) sini?+ {Im' — l'm) sin (7
ll'j^r^nwi-\-nn— {mn-\-m'n) GOsA—{nl'-\-nl) cosjB — (/m'+Z m)cosC
Proof.— By (4071) applied to the transformed equations of the lines,
(4606), observing (4007).
622 TlilLINEAR ANALYTICAL CONIC S.
EQUATIONS OF PARTICULAR LINES and COORDINATE RATIOS
OF PARTICULAR POINTS IN THE TRIGON.
4628 Bisectors of the angles A, B, C :
/B-y = {), y-a = 0, a-y8 = 0.
4629 Centre of inscribed circle (or in-centre)* 1:1:1.
The coordinates are obtaiaed from their mutual ratios by the formula
(4017).
4630 Bisectors of the angles y^, ir — B, tt — C:
/3-y=0, y+a = 0, a+y8 = 0.
Centre of the escribed circle which touches the side a (or
a ex-circle) —1:1:1.
4631 Bisectors of sides drawn through opposite vertices :
P^inB = y siu C, y smC = asluA, asinA = fi siu B.
4632 Point of intersection (or mass-centre) :
cosec^ : coseci^ : cosecC.
Proof. — Assume inp — ny = 0, by (4G04), as the form of the equation of
a line through A, and determine the ratio ih : n from the value of y : /3
when a = 0.
The coordinates of the point of intersection may be found by (4G10), or
thus :
o : ft = sin B : sin A = cosec A : cosec B,
ft : y = sin C : sin B = cosec B : cosec G,
therefore a : /3 : y = cosec A : cosec B : cosec C.
4633 Perpendiculars to sides drawn through opposite
vertices :
)8cosB = y cosC, y cos C= a cosyl, acosA = ficosB.
4634 Orthocentre: secyl : secB : secC.
* This nomenclature is suggested by Professor Hudson, who proposes the following : —
" In-circle, circum-circle, a ex-circle ... mid-circle for inscribed circle, circumscribed circle,
circle escribed to the side a, and nine-point circle; also in-centre, circum- centre, a ex-
centre, ... viid-ceritre, for the centres of these circles ; and in-radiufi, circum-radiits, a ex-
radius, ... mid-radius, for their radii ; central line, for the line on which the circum-centre,
mid-centre, ortho-centre, and mass-oentre lie ; and central length for the distance between
the circum-centre and the urtho-ccntro."
THE BIGHT LINE. 623
If the Cartesian coordinates of A, B, G be x^y^, x^iji, x^y-i,
the coordinates of the centre of the inscribed circle are
4635 ^ = '±!v±*fi+^3_ m+bm+e>h
Proof. — By (4032). Find the coordinates of D where the bisector of
the angle A cuts BC in the I'atio 6 : c (VI. 3), and then the coordinates of E
where the bisector of i? cuts AD in the ratio b + c : a.
4636 For the coordinates of the centre of the a ex-circle,
change the sign of a in the above values of x and y.
4637 The coordinates of the mass-centre are
4638 The coordinates of the orthocentre are obtained from
the equations of the perpendiculars from X2y2i ^Hlzi viz.,
{x^—x.^x-\-{y^—y.^y = x.,{w^—A\^-\-y^{y^—y.^.
Perpendicular bisector of the side AB :
4639 a^mA-^^\nB-\-y^m{A—B) = 0,
4640 or acos^-y8cos5--^sin(v4-jB) = 0,
4641 or
/ , rtsiiiBsinCX . /« , ft sin C sin ^\ ^ ^
\ 2 sm A / \ 2sinB J
4642 Centre of circumscribed circle (or cir cum- centre) :
cos^ : cos^ : cosC.
Proof. — A line through the intersection of y and a sin ^1-/3 sin 5 (4G31)
is of the form a sin^ — /3 sinB + ?(y = 0, and, by (4622),
n = —?,\nB cos x4-)- sill ^1 cos 5 = sin (A — B).
Otherwise, by (4633) and (4619),
a cos J. — /? cosB + k =
is any line perpendicular to AB ; and the constant k is found by giving a : (3
the value which it has at the centre of A l>.
624 T.R I L IXl<!A h' A SA L YTK 'AL COXTrS.
4643 Centre of tlie nine-point circle (or mid-cpntre) :
cos (B-C) : cos (G-A) : cos (A-B).
Proof. — By (955) the coordinates are the arithmetic means of the corres-
ponding coordinates of the orthocentre and circum-ceutre. Therefore, by
(4(334, '42) and (4017),
a ^
sec A , cos A 7
s'mA sec.l + sin5secJ5-|-sin(7 6ecO sinJ. cos^4 + sini?cosi?-|-sin(7cosC )
which reduces to cos (B — C)x constant.
4644 Ex. 1. — In any triangle ABC (Fig. of 955), the mass-centre E, the
orthocentre 0, and the circum-centre Q lie on the same straight line ;* for the
coordinates of these points given at (4632, '34, '42), substituted in (4615),
give for the value of the determinant
cosec A (sec B cos (7— cos B sec G) + &c.,
which vanishes.
Similarly, by the coordinates in (4643), it may be shown that the mid-
centre iV lies on the same line.
Equation of the central line :
Ex. 2. — To find the line drawn through the orthocentre and mass-
centre of ^-BC. The coordinates of these points are given at (4632, '34).
Substituting in the determinant (4616) and reducing, the equation becomes
a sin 2^ sin (B-C) +13 sin 25 sin (C-yl) -|-y sin 2C sin (A-B) =0.
Ex. 3. — Similarly, from (4629, '42), the line drawn through the centres
of the inscribed and circumscribed circles is
a (cos B— cos 6')+/3 (cos C—co{iA) + y (cos J. — cos B) = 0.
Ex. 4. — A parallel to ^B drawn through C:
a sin A +/3 sin B= 0.
For this is a line through a/3, by (4604), and the equation difiVrs only by a
constant from y = 0, for it may be written
(a sin.l + /^ sinL' + y sin C)—y sin ("■ = 0.
Ex. 5. — A poi })endicular to BG drawn through C is
« cos -h /3 = 0.
For a peipeiidicnlar is /3 cos 7? — y cos (7 = (4633) (1),
and a line through C is of the form /d + »i/j = 0. Hence, by (4619), the
constant Ic (a sin A + ji sin B + y sin C) must be added to (I) so as to elimi-
nate y. Thus
/J sin C cos B + a sin A cos (7 -|- /3 sin 5 cos C = 0,
(3 sin {B+C) + « sin .1 cos C = or /3 + o cos C = 0.
Tlif Luutriil \\uv. iSce uut«f tu ;4G29).
ANHAEMONIC RATIO.
625
ANHARMONIC RATIO.
For the definition, see (1052).
4648 The three ratios of that article are the vakies of the
ratio h : k' in the three following pencils of four lines respec-
tively —
a = 0, a-kfi = 0, /8 = 0, a+///3 = 0... (i.) (Fig. 34),
a=0, a-^y8==0, a^k'l3 = 0, /3 = 0... (ii.) (Fig. 35),
a = 0, /3 = 0, a+A-/3 = 0, a+k'/S = 0...(iu:} (Fig. 36).
4649 The anharmonic ratio (i.) becomes harmonic when
h = h'. Hence the lines a-\-h^, a — l'P form a harmonic pencil
with the hnes a, j3, the first dividing the external and the
second the internal angle between a and /3 (Fig. 37).
4650 Similarly, the anharmonic ratio of four lines whose
equations are
is the fraction
(a^i— M2) i^h—fh)
Proof. — Let OL be the Hue a = 0, and
OB, 13 = 0.
fx^—/d^ = difference of perpendiculars from
A and B upon OL, divided by p.
Similarly, ^3 — i^i, &c. These differences
are proportional to the segments AB, CD,
AD, BC, and jj is a common divisor.
4651 Homographic pencils of hnes are those which have the
same anharmonic ratio. Thus the two pencils
a — /iiii3, a —^^2/3, a — ^tg/S, a — ^ii/3,
and a'— ^ii/3', o' — ^i./B', a—f.L.^(i', o'— ^ii/3',
are homographic pencils.
4 L
626
TBILINEAE ANALYTICAL CONIGS.
THE COMPLETE QUADRILATERAL.
4652 Def. — Any foiii' riglit lines together with the three,
called diagonals, which join the points of intersection, make a
figure called a complete quadrilateral.
4653 Let be any point in the plane of the trigon ABC.
Draw AOa, BOh, GOc, and complete the figure. The equa-
tions of the different lines may be written as under, with the
aid of ijroposition (4604), the ratios I : m : n being arbitrary
and dependent upon the position of 0.
In;
ca,
nb.
Aa, ))(/B-~iiy = 0,
J3b, ny -la = 0,
Cc, la —)n^ = 0,
)n^-\-iiy — la = {),
ny -\-la —ni^ = 0,
la -\-nifi — ny = 0,
AP, mP-\-ny = 0,
BQ, ny +/a = i),
CR, la -\-m^= i);
OP, m/i-\-ny -2la = i),
OQ, ny +/a -2/;/^ = 0,
OR, la -^m^-'Iny = i),
PQR, la-\-nil3-\-ny = 0.
Proof. — Aa, Bb, Cc are concurrent by addition, he is concurrent with
lib and ft, and with Cc and y, by (4G04). AP and OP are each concurrent
with be and a. PQB is concurrent with each pair of lines be and a, ca and
ft, ah and y. Similarly for the rest.
4654 Every pencil of four lines in the above figure (supply-
ing AP, BQ, Git) is a liarmonic pencil.
Proof. — By the test in (4G4'J), the alternate pairs of equations being tlie
sum and difl'erence of the other two in every case.
Otherwise by projection. Let PQRS be the quadrilateral, with diagonals
UP, QS meeting in C. (Supply the lines AC, hC in the figure.) Taking the
plane of projection parallel to OA 11, the figure projects into the parallelogram
2jqrs; the points yl,7j pass to infinity, and therefore the lines J L-, ]>C become
THE GENERAL EQUATION.
e>27
lines harmonically divided by the sides of the parallelogram, the centre,
and the points at infinity.
4655 Theorem (974) may bo proved by taking o, /', y for the lines BC,
CA, AB, and Va^-mj^ + ny, la^-m'fl ^ny, Ja + mft + n'y for he, ca, ah, the last
form being deduced from the preceding by the concurrence of Aa, Bh, and Cc.
THE GENERAL EQUATION OF A CONIC.
The general equation of tlie second degree is
4656 aa:'+b/3' + cf-\-2fl3y-{-2fiya-\-2hal3 = 0.
This equation will be denoted by <^ (o, /3, 7) = or n = 0.
Equation of the tangent or polar :
4657 i(a'Ci-\-u^l3-\-Uyy = or u^^a+u^P' -\-u^y = 0,
the two forms being equivalent and the notation being that of
(1405). The first equation written in full is
4659
628 TBILINEAB ANALYTICAL CONICS.
Proof. — By the methods in (4120). Otherwise by (4678); let a/3y
be on the curve ; then <p (a, /3, y) = 0. Next let the point where the line cuts
the curve move up to afty. Then the line becomes a tangent and the
ratio n : n' vanishes; the condition for this gives equation (4G58).
Cor. — The polars of the vertices of the triangle of refer-
ence are
4660 aa-]-hft+oy = 0, /,a+^,;8+/y=:0, o-a4-/;8+fy = 0.
4661 The condition that u may break up into two linear
factors representing two right lines is, by (4469), A = 0,
where
4662 A = abc-\-2fgh-af'-bg—cJr. (4454)
a h g
X
h b f
/i
^'f c
V
\ ^ v
= 0.
4663 The general tangential equation
of the conic (4656) expresses the condi-
tion that the line XaH-/n|3 + vy may touch
the curve and is the determinant equation
annexed. The same written in full is
4664 {bc-n \^ + {ca-g') fi'-{-{ab-1r) v'
+2 {gh -af) /xv+2 {hf-bg) vX+2 ifg-ch) X^ = 0,
4665 or AX'+BfM'-^Cv'^2Fiiv-{-2Gv\+2H\fx = 0;
writing, as in (1642),
A = bc—f\ B = ca—g\ C = ab—h\
F = gh-(if, G = hf-bg, H =fg-ch.
The tangential equation will be denoted by (1> (X, ^t, v) = or
Z7= 0, to correspond with (4656).
Proof. — The determinant is the eliminant of the equation of the line
Xa + fi/3 + j^7 = 0, and the three equations obtained by equating X, ^u, v to the
coefficients of o, /3, y in (4059).
Otherwise. — Assume Xa + ^/3 + >'y = for the tangent. Substitute the
value of the ratio /3 : y obtained from it in the equation of the curve, and
express the condition for equal roots (4119).
4666 Conversely, if the line Xa -h^t/3
-\-vy has the coefficients X, ^t, v con-
nected by the equation of the second
degree Z7 = (4664), then the enve-
lope of the line is the conic in the
A H G a
H B F IB
G F C y
a /B y
= 0.
THE GENERAL EQUATION. 629
determinant form annexed corresponding to (4GG3), or in full
4667 {BC-F') a'^(CA-G') /B'-]-{AB-H')f
+ 2 (GH^AF) /8y+2 (HF-BG) ya+2 (FG-CH) ayS = 0.
4668 or A {aa'-^bfi'+cf+2fl3y-^2oya-\-2ha/B) = 0.
Proof. — Eliminate )' from U =z and the given line. The result is of
the form I/\^ + 2EAjLt + JIju'^ = 0, and therefore the envelope is LM — B^, by
(4792). This produces equation (46G7). The coefficients are the first
minors of the reciprocal determinant of A (1G43), and therefore, by (585),
are equal to aA, 6A, &c.
4669 The condition that U may consist of two Hnear factors
is, as in (4661), D = 0, where
4670 D = ABC+2FGH-AF'-BG'-CH' =^\ (1643)
In this case ?7 becomes the equation of two points, since the
line Aa+^ij3-|-vy must pass through one or other of two fixed
points. See (4913).
4671 The coordinates of the pole of Xa+/i/3 + vy are as
A\+H,.i-\-Gv : H\-\-Bfi+Fv : GX-^F^i+Cv,
4672 or U^:U,: U,.
Proof. — By (4659) we have the equations in the '^^^.io,T!^'Z I'
margin, the solution of which gives the ratios of a : (3 : ■/. "' _^ j ^ _|_*^ _ ^.^^^
AiVJQ « — /^ V — ii
^"''^ A\ + H/ii + Gy II\ + Bfi + Fp G\ + Ffi + Gy A'
Hence the tangential equation of the pole of X'a + //3 + v'y,
i.e., the condition that Xa + ^ijS + vy may pass through the
pole ; or, in other words, that the two lines may be mutually
conjugate, is
4674 \U^'+^iU,'-\-vU.' = or XX^;,+itt'C/^ + v'f7.= 0,
the two forms being equivalent, and each
4676 = ^xv+i^^i^/+Cvv'
+ F(^v'+/i'v) + G(vV+v'X) + i/(X^'+XV).
The coordinates of the centre oq, /3o, yo are in the ratios
4677 An+m + Gt : m+Bh + Ft : Gn-\-Fh + Ct,
where a, tj, C are the sides of the trigon.
630 TRIUNE AB ANALYTICAL CONIGS.
Proof. — By (4-G71), since the centre is the pole of the line at infinity
aa + bfl + cy = (4612).
Tlie quadratic for the ratio n : n' of the segments into
which the Une joining two given points a/3-y, a'/3'y' is divided
by the conic is, with the notation of (4656-7),
4678
<!> {a', p', y) n'-\-'2 ((^„a +(/»,/8'+(^y) nn'-{-ct> {a, 0, y) n" = 0.
Proof.— By the method of (4131).
The equation of the pair of tangents at the points where 7
meets the general conic u (4656), is
4679 aitl-{-2hujip-\-bui= 0.
Pjjoof. — The point a'/3', where y meets the curve, is found from
au"- + 2ha'iy + hiy'- = [y = in (4G56)]. The tangent at such a point is
ny + u^jy' = (4658). Eliminate a', /3'.
The equation of a pair of tangents from al3'y is
4680 ^ (aW) ^ i^Py) = {<t>.ci'+<t>,li'-\-cl>yy7.
Proof. — By the condition for equal roots of (4678).
By actual expansion the equation becomes
(^]>y2+Glo'-2Flh) cP + iCa' -{-Ay'-2Gya) /3'H(^l/3' + iV -2/fa/5) y'-
+ 2(-Al3y + nya + Gui3-Fa')tyy'
4681 +2(Hl3y-Bya + Fal3-Giy)y'a'
+ 2 (Gi3y + Fya-Cafi-Ey') u'lY = 0.
In which either a', /i', y' or o, /?, y may be the variables, for the forms are
convertible.
Otlierwise the equation of the two tangents is
4682 4>(/3y-)8'7, ya-ya, a/3'-a/3) ^ 0. (4665)
Proof. — By substituting py' — /)'y) ^^- f^*'' ^^ P^ '' "^ (4664), the condition
that the line joining "'/j'y' to any point a/Sy on either tangent (see 4616)
should touch the conic is fulfilled. The expansion produces the preceding
equation (4681).
The equation of tlie asymptotes is
4683 (^(a,A7) = <^(ao,A.7o)-A-S (1),
wluM'c (/„, /^,„ y,, :ii't' tiie coordinates of the centre.
THE GENERAL EQUATION. 631
Otherwise tlie equation, in a form homogeneous in
a, /3, y, is
4684 (aa,+lj;8o+rro) ^(^,^,7) = k{na-\-hl3+CyY (2),
where n, I), t are the sides of the trigon.
And, finally, if the tangential equation (4664) be denoted
by O (A,^i, v) = 0, the equation of the asymptotes may be
presented in the form
4685 a>(a,Ij,r)(^(a,A7) = (aa+l)y8+r7rA (3).
Proof. — (i.) The asymptotes are identical with a pair of tangeuts from
the centre ; therefore, put a^, jj^, y^ for o', /3', y' in (4680) ; thus
<t> ("> fi, y) f ("o' l\, ra) = ^•' ('^« + fV' + cry = k'T (4),
since the polar becomes the line at infinity.
N'ow, multiplying the three equations in (4672) by a, /3, y respectively,
and adding, we obtain (j) (a, ft, y) = Jc (Xa + nft + yy), and therefore
0K,/3o,yo) = A-(ar. + [v3 + cy) = /.-2 (5),
since the line at infinity (4612) is the pole of the centre.
From (4) and (5), by eliminating A-, equation (1) is produced ; and by
dividing (4) by (5), we get equation (2).
Again, taking the values of a, ft, y from (4673), we have
^- + ^^^ + ^y = HKj^^ ^,a therefore ^^o+J^/'o + ^/o = *1^^M).
k ^ k A
By the last equation, (2) is converted into (3). See also (40GG).
CoK. — Since the centre (oo, 1%, y^) is on the asymptotes,
we have
4686 <^ K, A, 7o) = ^'^ - * i^> ^^ 0-
4687 The semi-axes of the general conic (4656) are tlie
values of /■ obtained from the quadratic
(«+*^>
h,
A''»
a
hs CO?
■1).
f,
I)
('
. W cos C\
+ ,,. >
r
b,
r,
a,
where a, 1), t are the sides of the trigon, and
s = al)CA-f4>(aI)t).
= 0,
632 TRILINEAB ANALYTICAL CONICS.
Proof. — The centre being a„/3„y„, put a — a,^ = x, /3—l% = y, y — yo = s.
a/3y being a point on the conic, and /• the radius to it from the centre, we
have, by (4G02),
r- = ^(x\\ COS A + 7fb cos B + zh cos C) (1).
Also (4656), f (a, 13, y) = <p (a, + x, /3„ + y, y, + z) = 0.
Expand and write /, m, n for aa^ + hl3^ + (jY,„ lia^,+ hi%+j\, f/"o +//^o + <'7u-
The terms in x, y, z become
U-\-r,:y^nz= Z (n -a^) +&c. = 2-2 = (4007) (2),
and we obtain (.<■, y,z)=-<p (a„, /3„, y,) = :s=A-f-<I» (a, i\ c) (4686) (3).
The maximum and minimum values of r" and therefore of
3racos^ + 2/"^cosI?-|-s^ccos6' (4)
required, subject to the equations (2) and (3). By the method of ^ unde-
ained multipliers (1862), the quadratic above is found.— i^erre/'^'s Tril.
are
tei'mined
Coord., Ch. 4, Art. 18
4688 The area of the conic = TrSalJCA
Proof.— If the roots of the quadratic (4687) are ±rr\ rfcr.;'-, the area
will be 7rri?-2. The coefficient of r"'' reduces by trigonometry to — SV, and
the absolute term is —4' (a, t, c). Hence the product of the roots is found.
4689 The conic will be an ellipse, hyperbola, or parabola,
according as ^^ (n,i),C) (4664) is positive, negative, or zero.
Proof. — The squares of the semi-axes have opposite signs in the hypei*-
bola. Therefore the product of the roots of the quadratic (4687) must for
an hypei'bola be negative, and therefore <1» negative in (4688).
4> (a, h, c) = makes the curve touch the line at infinity (4664), a pro-
perty which distinguishes the parabola.
The condition that the general conic (4656) may be a
rectangular hyperbola is
4690 r« + ^*+c= 2/cos^4-2i,'- cosi^+2/i cosC.
Proof. — Let the asymptotes be
/a + mft + ny = 0, I'a + m'(3 + n'y = 0.
Forming the product, equating coefficients with (4685), and denoting
(j) (a, b, c) by <?>, we get the proportions
W _ ))im' _ u.ii' _ mn' + m'n
a^j — a-A ~ h(p — b-:l ~ c^ — rA ~~ 2(/^ — bcA)
id' + 11' I hii'+l'vt
2 (g<p — ca A) 2 (^hf — ab A)
TEE GENERAL EQUATION. 633
We may therefore substitute these denominators in (4620) for the condition
of perpendicularity of the asymptotes. The result reduces to the equation
above, by (837).
For another method, see (5002).
4691 The general conic (4656) will become a circle when
the following relation exists between the coefficients :
b sin^C+c sm'B-2fsmB sinC
= c sinM+« sin^C— 2^smC siii^
= a shr B+b siu^^ — 2/i sin A shiB.
PuooF. — Equate coefficients of the equation of the conic (465G) with
those of the circle in (4751).
4692 The equation of the pair of lines drawn from a point
a(5'y to the points of intersection of the conic (/> and the line
L = \a-\- 1^(3 ^vy = is, writing L' for Aa' + |ii/3' + vy , with the
notation of (4656-7),
V'cl> (a, fi, y) - 2LL'((^y + <^,/3' + <^,y ) + ^'* (« ' ^'^ V) = ^
Proof.— By the method of (4133).
4693 The Director-Gircle of the conic, that is, the locus
of intersection of tangents at right angles, is, in Cartesians,
C {.v'^-if)-2G.v-2Fij^A^B = 0.
Proof. — Let the equation of a tangent through xy be
mi,—n + (y — inx) =0.
Therefore in the tangential equation (4665) put \ = vi, yu = — 1, v=:y — mx,
and apply the condition, Product of roots of quadratic in m = —I (4078).
The equation of the same circle in trilinears is
4694 iB+G + 2FcosA)a'-\-{C-{-A + 2GcosB)fi' + iA-i-B + 2HcoiiC)y"'
+ 2 (A cos A- H cosB- G cos G-F)(3y
+ 2(-HcosA + B cos B-F cos G-G)7a
+ 2 (-G cos A-F cos B+CcosG-H) aft = 0;
or, in the form of (4751),
4695
(aa + fe/3 + cy)(^+^+|^^^^a + &c.) = ^l^Ail (a^y + 6y« + c«/3).
4 M
634 TBILINEAB ANALYTICAL CONIGS.
Proof. — The equation of a pair of tangents (4681) through a point o^y
in trilinears, when the tangents are at right angles, represents the limiting
case of a rectangular hyperljola. Therefore the ecination referred to must
have the coefficients of a', /3'", &c. connected by the relation in (4690),
which thus becomes the equation of the locus of the point a^y ; i.e., the
director-circle.
4696 When the general conic is a parabola, C = in (4G93)
and ^ {a, h, c) = m (4G95), by (4430) and (4689), and these
equations then represent the directrix.
PARTICULAR CONICS.
4697 A conic circumscribing the quadrilateral, the equa-
tions of whose sides are a = 0, i3r=0, 7 = 0, S = 0, (Fig. 38)
ay = A-/38.
Proof. — This is a curve of the second degree, and it passes through the
points whei'e a meets /3 and h, and also where y meets /3 and d.
4698 The circumscribing circle is a-y = d=y8S; + or — , as
the origin of coordinates lies without or within the quadri-
lateral.
Proof. — Transform (46'>7) into Cartesians (4009) ; equate coefficients of
X and y and put the coefficients of xy equal to zero.
4699 A conic having a and y for tangents and j3 for the
chord of contact : (Fig. 39)
ay = k^K
Proof. — Make ^ coincide with ft in (4698).
4700 A conic having two common chords o and /3 with a
given conic ^S^ : (Fig. 40)
S = ka^.
4701 A conic having a common chord of contact a with a
given conic 8 : (Fig. 41)
S = Aa'.
4702 Coil. — If UPQ be drawn always parallel to a given
line, rN'ozRP.FQ, by (4317).
FABTIGULAB CONICS. 635
4703 A conic Laving a common tangent T at a point x'y'
and a common chord witli the conic S : (Fig. 42)
4704 A conic osculating ^S' at the point xij where T touches
at one extremity of the common chord / (x — iG)-\-m (y — ?/) :
■ (Fig. 43)
S=T {Lv + 7mj - Iv - my') .
4705 A conic having common tangents T, T' at common
points Avith the conic 8 : (Fig. 44)
S = kTT.
4706 A conic having four coincident points with the conic
S at the point where T touches : (Fig. 45)
S = kTK
4707 The conies S+L^ = 0, S+M' = 0, S-\-N' = 0,
(Fig 40) having respectively L, M, N for common chords of
contact with the conic S, will have the six chords of inter-
section
L±M=:0, M±N=0, N±L=0,
passing three and three through the same points.
Proof.— From (S + iP) - (S + N') = (M+N){M-N), &c.
By supposing one or more of the conies to become right lines, various theorems
may be obtained.
4709 The diagonals of the inscribed and circumscribed
quadrilaterals of a conic all pass through the same point and
form a harmonic pencil.
Proof.— (Fig. 47.) By (4707), or by taking LM=E? and L'W = E''
for the equations of the conic by (4784).
4710 If three conies have a chord common to all, the other
three chords common to pairs pass through the same point.
Proof.— (Fig. 48.) Take 8, S + L3I, S + LN for the conies, L being the
chord common to all ; then M, N, M—N are the other common chords.
636 TEILINEAB ANALYTICAL CONIGS.
4711 The hyperbola .vy = (Ocr+0//+j>)-
is of the form (4699), and has for a chord of contact at infinity
Oxi-Oy+2^ = 0, (i; y being the tangents from the centre.
4712 The parabola y^ = (0^-\-Oy+p) .v
has the tangent at infinity Ox-{-Oy-{-p = 0.
4713 So the general equation of a parabola may be put in
the form of (4699). Thus
(a.r+/3^)-^+(2^-.v+2/i/+c)(0^+0^/+l) = 0.
Here ax + fty is the chord of contact, that is, a diameter; 2gx + 2fy + c is
the finite tangent at its extremity, and Ox + Oy + 1 the tangent at the other
extremity, supposed at infinity.
4714 The general conic may be written
{cLv'+2kvy-{-bt/)-\-(2g.v-\-2fy-hc){0x-j-0y-^l) = 0.
For this is of the form ay + kl3o, S being at infinity.
4715 The conies 8 and S-k{0.v-\-Oij+iy
have double contact at infinity, and are similar.
4716 The parabolas 8 and S-k''
have a contact of the third order at infinity.
Proof. — For S and S—(0x + 0y-\-Jiy have the line at infinity for a chord
of contact ; and, by (4712), this chord of contact is also a tangent to both
4717 All circles are said to pass through the same two
imaginary points at infinity (see 4918) and through two real
or imaginary finite points.
Proof. — The general equation of the circle (4144) may be written
(x + iy)(x-iy) + (2gx + 2fy + c)(0x + 0y + l) = 0;
and this is of the form (4G97). Here the lines x±iy intersect Ox + Oy + 1
in two imaginary points which have been called the circular poi)ds at {njinity,
and 2(jx + 2fy + c in two finite points F, Q ; and these points are all situated
on the locus x^ + y'^ + 2gx + 2fy + c — 0.
4718 Concentric circles touch in four imaginary points at
infinity.
P ARTICULAR CONIC S. 637
Proof. — The centre being the origin, equation (4136) maybe written
{x + iij)(x—{y) = {Ox + Oi/ + ry\ which, by (4699), shows that the lines a; ± iy
have each double contact with the {supplementanj) curve at infinity, and the
variation of r does not affect this result. Compare (4711).
4719 The equation of any conic may be put in the form
^v'+y' = ey.
Here x = 0, y = are two sides of the trigon intersecting at
right angles in the focus ; y = 0, the third side, is the directrix,
and e is the eccentricity.
The conic becomes a circle when e = and y = oo , so that
ey = r, the radius, (4718).
4720 Two imaginary tangents drawn through the focus
are, by (4699),
{^v+ii/){a:—ii/) = 0.
These tangents are identical with the lines drawn through
the two circular points at infinity (see 4717). Hence, if two
tangents be drawn to the conic from each of the circular
points at infinity, they will intersect in two imaginary points,
and also in two real points which are the foci of the conic.
All confocal conies, therefore, have four imaginary com-
mon tangents, and two opposite vertices of the quadrilateral
formed by the tangents are the foci of the conies.
4721 If the axes are oblique, this universal form of the
equation of the conic becomes
The two imaginary tangents through the focus must now
be written
{cV-\-y{cosQ}-\-i siuco)} {^v+y (cosw— i siuw)} = 0.
4722 Any two lines including an angle 9 form, with the
lines drawn from the two circular points at infinity to their
point of intersection, a pencil of which the anharmonic ratio
is e^('^-2«).
Proof. — Take the two lines for sides /3, y of the trigon. The equation
of the other pair of lines to the circular points will be obtained by elimin-
638 TBILINEAB ANALYTICAL CONICS.
ating- a between the equations of the line at infinity and the cii'cum-circle,
viz.,
a« + tv3 + cy = and ^ + -^ + ^ = 0. (4738)
« n y
The result is /3^ + 2/>y cos ^ + y- = ;
or, in factors, (/3 + (''"y ) (/3 + e-'V) = 0.
The anharmonic ratio of the pencil formed by the four lines ft, /S + e'^y,
y,ft-\-c-''y is, by (4G48, i.),
— e'« : p-'« = — e"« = e'" '"--"'.
4723 Cob. — If 9 = ^Tr, the lines are at right angles, and the
four lines form a harmonic pencil. [Ferrers' TrU. Coords., Ch. VIII.
THE CIRCUMSCRIBING CONIC OF THE TRIGON.
4724 The equation of this conic (Fig. 49) is
Wy-^mya+nap = or 1 + !± + ^ = 0.
Proof. — The equation is of the second degree, and it is satisfied by
a = 0, ft = simultaneously. It therefore passes through the point aft.
Similarly through fty and ya.
The tangents at A, B, and G are
4726 ^^ + -;=0, ^+^ = 0. 1 + ^ = 0.
Proof, — By writing (4724) in the form
mya + ft(Iy + na) = 0,
ly-\-na = is seen, by (4G97), to be the tangent at ay ; for the intersection's
of a and y, with the curve, now coincide, and ^ (now ly + na) passes through
the two coincident points.
4729 The tangent, or polar, of the point a'(5'y is, b}^ (4G59),
(my' + *j/3') a+ {na' + ly') ^8+ (Ifi'-^ma') y = 0.
4730 The tangents at A, D, G (Fig. 49) meet the opposite
sides respectively in P, Q, R on the right line
a , Aj_>:==0. By(4G04).
I ^ m ^ n
4731 The line "- — ^- passes through (D), the intersection
of the tangents at A and B.
PARTIGULAB GONICS. 639
4732 The diameter tlirougli the intersection of the tan-
gents at A and B is
nna—nhl^+{ltl-mh) y = 0.
Proof. — The coordinates of the point of intersection are I : vi : —n, by
(4726-7), and the coordinates of the centre of AB are b : a : 0. The
diameter passes through these points, and its equation is given by (4616).
4733 The coordinates of the centre of the conic are as
/(-/a + >>*t) + /ir) : m(/a — mlj + wr) : n{l^ + mh-nt).
Proof. — By (4610), the point being the intersection of two diameters
like (4732). Otherwise, by (4677).
4734 The secant through {a^ftai), {c^-Ay-^, ^^ny two points
on the conic, and the tangent at the first point are respectively,
Proof. — The first is a right line, and it is satisfied by a = Oi, &c., and
also by a = cu, &c., by (4725). The second equation is what the first be-
comes when n'2 = "i' &c. For the tangential equation, see (4893).
4735 The conic is a parabola when
f^'-^m'h^-{-nH'-2mnht-2nUil-2lmnh = 0,
4736 or y(/a) + v/0Hl))+y(»a) = O.
Proof. — Substitute the coordinates of the centre (4733) in aa + 6/3 + Cy = 0,
the equation of the line at infinity (4612).
Otherwise, the conic must touch the line at infinity ; therefore put a, b, C
for A, ^i, r in (4893).
4737 The conic is a rectangular hyperbola when
/ cos J + m cos B-\-ii cos C = 0,
and in this case it passes through the orthocentre of the
triangle.
Proof.— By (4690), and the coordinates of the orthocentre (4634).
THE CIRCUMSCRIBING CIRCLE OF THE TRICON.
4738 Py siu4 + 7a sinB+afi siuC = 0,
siu^ , siiiB . sin (7 ..
640 TBILINEAR ANALYTICAL CONICS.
Proof. — The values of the ratios I : vi : n, in (4724), may be found geo-
metrically from the equations of the tangents (4726-8).
For the coordinates of the centre, see (4642).
THE INSCRIBED CONIC OF THE TRIGON.
4739 ra'-]-m'P'-\-ny-2m7i/3y-2nlya-2lmalB = 0.
4740 or ^{la) + y (»*« + ^(ny) = 0.
Proof. — (Fig. 50.) The first equation may be written
ny (ny-2la-2mi3) + (la-r>i(3y = 0.
By (4699) this represents a conic of which the lines y and ny — 2la — mft are
the tangents at F and /, and la — m^ the chord of contact. Similarly, it may
be written so as to shew that a and /5 touch the conic.
4741 The three pairs of tangents at i^,/, &C.5 are
2my8+2«y-/a ) 2ny-\-2la-mfi ) 2la-\-2m^-ny ?
and a J ' and 13 -> and y J
and they have their three points of intersection P, Q, B on
the right line /a + mjS + ^iy. By (4604).
4742 The coordinates of the centre of the conic are as
nh-\-mC : ZC+wcl : ?/ia+/I).
Proof. — By putting a and ft = zero alternately in (4739), we find, for
the coordinates of the points of contact,
at D, /3 = -i^^', and at E, a = ^"^'^
nh + mi ' ' m\ + h '
therefore the equation of the diameter through C bisecting DE is, by (4603),
a ^ ft
Similarly the diameter bisecting DF is
la + lb 7ib + mc
Therefore the point of intersection, or centre, is defined by the ratios given
above.
Otherwise, by (4677), and the values in (4665), writing for a, h, c,f, g, h
the coefficients in (4739).
4743 The secant through ai/3iyi, nj^.^y, any two points on
the curve.
■^y^/n (\/^o+v/^x) = 0.
PARTICULAR CONICS. 641
Proof. — Put a^fi^yi for afiy, and shew that the expression vanishes by
(4740).
4744 The tangent at tlie point ai/3iyi :
Proof.— Put a^= a^, &c., in (4743), and divide by ^^(a^fi^yi).
4745 The equation of the polar must be obtained from
(4739) by means of (4659).
4746 The conic is a parabola when
a * b c
Proof.— Similar to that of (4736).
THE INSCRIBED CIRCLE OF THE TRIGON.
4747 a' cos* A +^2 cos*|-f f cos* ^
-2l3y cos^:|cos2-| -2ya cos^^cos^i _2a/8 cos^^-cos^^.
4748 or cos^ ya+cos^ V^/8+cos-^ ^y = 0.
4749 The rt-escribed circle : (4629)
cos— v'^^+siu— y^+siu— \/y.
Proof. — At the point of contact where y = 0, we have, in (4740), geo-
metrically, r being the radius of the circle,
Z : m = i(5 : a = rcot-^sin^ : root— sin5 = ±cos^|J. : cos^-B;
+ for the inscribed; — for the escribed circle and tt — B instead of B.
4750 The tangent at a(5'y\ by (4744), is
eosA -^ +COS 4 ^ +COS I -f , = 0.
The polar is obtained as in (4745).
GENERAL EQUATION OF THE CIRCLE.
4751 (/a+?«)8-f «y) (a siu ^+^8 sin 7?+y siu C)
■\-}i [j^y siu ^+ya siu /^ + a/3 siu C) — 0.
4 N
642
TinLINEAB ANALYTICAL CONIC S.
Proof. — The second term is the circumscribing circle (4738), and the
first is linear by (4G()9) ; therefore the whole represents a circle. By varying
k, a system of circles is obtained whose radical axis (4161) is the line
la + mft + ny, the circumscribing circle being one of the system.
4752 If l'a-^'}n'(5-\-n'y be tlie radical axis of a second system
of circles represented by a similar equation, the radical axis
of any two circles of tlie two systems defined by Ic and Jc
will be
^•'(/a^-my8 + ?iy)-A•(ra+m'^+»^V) = ^
Proof. — By eliminating the term
/3y sin A + ya sin i? + a/3 sin C.
4753 To find the coefficient of X' + if in the circle when
only the trihnear equation is given.
Rule. — Mahe a, /3, y the coordwates of a j^oint from wliich
the length of the tangent Is hioivn, and divide by the square of
that length; or, if the point he within the circle, substitute
^' half the shortest chord through the poinV^ for ^Hhe tangent."
Proof. — If S = be the equation of the circle, and m the required co-
efficient ; then, for a point not on the curve, S -^ m ■= square of tangent or
serai-chord, by (4160).
THE NINE-POINT CIRCLE.
4754 a' sin 2A +)8-' sill '2n-\-y sin 2^
— 2 (/8y sin/1-fya sill /> + a/3siiir) = il
PARTICULAR CONICS. 643
Proof. — The equation represents a circle because it may be expressed iu
the form
(a cos J. 4-/5 cosi) + y cos C)(a sin J. + /3 sin i? + y sin G)]
— 2 (/3y sin A + ya sin B + afi sin C) = 0.
See Proof of (4751). Now, when a = 0, the equation becomes
(/3 sinS — y sin(7)(/3 cos5-y cos G) = 0,
whicli shews, by (4631, '3), that the circle bisects BC and passes through D,
the foot of the perpendicular from A.
4754« The equation of tlie nine-point circle in Cartesian
coordinates, with the side BC and perpendicular on it from A
for X and y axes respectively, is
x^ + y--B, sin {B- G) x-E cos (B-G) y = 0,
where B is the radius of the circum-circle.
THE TRIPLICATE-RATIO CIRCLE.
47546 *Let the point S (Fig. 165) be chosen, so that its
trilinear coordinates are proportional to the sides of the trigon.
Draw lines through S parallel to the sides, then the circle in
question passes through the six points of intersection, and the
intercepted chords are in the triplicate-ratio of the sides.
[The following abbreviations are used, a, h, c, and not a, i\ c, being in
this article written for the sides of the trigon ABG-I
K=a- + b' + c-; \= ^Qrc" + e'er + d-J)-) ; A = ABC ;
f^ = ~; u> = A BFD = DE'F', &c. ; 6 = DFD' = DE'D', &c.
A
By hypothesis, i?- = A = X = __1A__ (4007)=^ (1),
BF c y ^ ^ BD'
therefore BF . BF' = BD.BD', therefore F, F', D, D' are concyclic.
If AS, BS, CS produced meet the opposite sides in I, m, n,
B)i _ g sin BC n _ aa __ a? , ^n ,^.
An 6sin^C'« Z>/3 6"
* The theorems of (1 to 36) are for the most part due to Mr. R. Tucker, M.A.
The original articles will be found in The Quarterly Journal of Pure and Applied Mathematics,
Vol. XIX., No. 76, and Vol. xx., Nos. 77 and 78.
Other and similar investigations have been made by MM. Lemoine and Taylor and
Prof. Neuberg, Mathesis, 1881, 1882, 1884.
644 TBILINEAB ANALYTICAL CONICS.
SF' = BD = -;^ = v^ (1) = ^. Similarly BF' = ^, &c. ... (3).
smB KsmB A J^
7)D' = 7)P^=^.-^ = $,&c (4).
sin (7 Ac A
^D' = ^D + DI>' = ^^^^^^, &c ..(5).
FD = y(BD' + BF' -2BD.BF cos B) = ^, by (2) and (5) (G).
Hence DBF and B'B'F' are triangles similar to ABC, and tliey are equal
to each other because FSF = B'SF = E'SF', &c. (Euc. I. 37.)
BF' = y(BB'- + BF"-2BB.BF' cosB) =^ (7).
Hence BF' = FB' = EB'.
B'F = ^BB'=^-^\+^ &c (8).
a A
^^^^^BF^±FB^-mi^cr + <f+]^ (5 & 6) = -^ (9).
^^^"^ 2BF.FB 2X ^ ^ 2\ ^ ^
sin
-"=V^-£)=ir('««) (^°)-
A , , , J, a- cos ^ + 6c ^-1 -1 s
COS0 = cos (/I — w), &c. = (.iij.
A
AFE' + BBF' + CEB' = ^^"^^'^"^^'^ + &c. = /x^A = BEF, by (G) ... (12).
Or, geometrically, by Euclid I. 37.
Radius of T. R. circle, p = ^B, by (G) (B = circum-radius) (13).
The trilinear equation of the T. R. circle is
a6c(cr + /3Hr)=^(«" + ^/^ + ^y)' + «'PV + ^V + c'«/3 (1-i),
or (h' + c') a' + (r + tr) /r + (a^ + Ir) y' = {{a- + h') (cr + r) + hV' ] ^
+ ((L= + r)(Z.= -ffr) + cV} ^^ + {(c' + cr)(,y + K-) + a'h'} ^... (15).
Obtained by substituting the trilinear coordinates of B, E, F, through
which points the circle passes, in (4751), to determine the ratios I : 7n : n
and k. The coordinates of B are
^ g (a^ + IJ^) sin C ac' sin J?
^' K ' ' K •
Similiuly those of B and F.
PARTICULAR CONIC S. G45
THE SEVEN-POINT CIRCLE *
4754c Let lines be drawn tlirougli A, B, G (Fig. 1G5)
parallel to the sides of the triangles DEF, B'E'F', as in the
figure, intersecting each other in P, P\ L, M, N. Let Q be
the circam-centre ; then the seven points P, P', L, M, N, Q, S
all lie on the circumference of a circle concentric with the
T. R. circle. (IG)
The proof depends on Euclid III. 21, and the similar triangles DEF,
B'E'F'.
The radius p' of the seven-point circle is
' P /(V^~ Qx^N 2PP^ sin 2a; C(17),
obtained from p'- = p' + 8D"^—2pSD cos (B- TDD') .
Expand and substitute cos TDD' = — — = -r-j^, by (3) and (5),
sm TDD = COS (11), cosi? = — !— , s]n_B = — , cosJ. = — -~ .
^ ^ 2ca etc 2bo
3p + p'= R\ by (17) and (13) ; ^ = V'cS^' ^^ *^^^^ ^"^ "^^^ - 1 (20).'
The triliuear equation of the seven-point circle is
a6c(aHiff' + y') = a'/8y + iV« + c'a/3 (21),
or o/3y + Jya-t-ca/3 = ^ (5ca-fca/3 + a5y) (aa + Z)/3 -fey) (22).
If the coordinates of P are a^, l\, y^, and those of P' a{, ft'i, y[ ; then
"i"! = /3i/5i = yiyi (23).
The equation of STQ is, by (4615),
a sin (B-C)+f3 sin (C-yl) + y sin (A-B) (24).
And the equation of PP' is
JL(a'-hV-) + 4-(h'-chi') + ^(c'-a'b') = (25).
a c
The point S has been called the Symraedian point of the
triangle. It has also this property. The line joining the viid-
* This circle was discovered by M. H. Brocard, and has hoen called " The Brocard Circle,"
the points r, I" being called the Brocard points.
646 TFILINEAB ANALYTICAL CONICS.
point of any side to the mid-point of the perpendicular on that
through S.
Proof.— Let X, Y, Z (Fig. 166) be the feet of the perpendiculars ; x, y, z
the mid-points of the same, and X', Y', Z' the mid-points of the sides. Now
the trilinear coordinates of X', /S', and x in order are proportional to
0, c, h This determinant vanishes ;
a, h^ c . therefore the three points are on
1, cos (7, cos I? the same right line, by (4015).
That the three lines X'x, Y'y, Z'z are concurrent appears at once by (970),
since CX- 2Y'x, &c.
The Symmeclian point may also be defined as the intersec-
tion of the three lines drawn from ^, i?, C to the corresponding
vertices of the triangle formed by tangents to the circum-
circle at A, B, G.
Let Ba, (7/3, Ay be taken = CX, AY, BZ respectively.
Then Aa, B^, Gy meet in a point 2, by (976), and this point
by similarity of figure is the Symmedian point of the triangle
formed by lines through A, B, G parallel to the sides BG,
GA, AB.
If the sides of X'Y'Z' be bisected, similar reasoning shews
that a, the Symmedian point of the triangle X'Y'Z', lies
on S^.
It can also be shewn that, if A'B'G' be any triangle having
its sides parallel to those of xiBG and its vertices on SA, SB,
8G, the sides of the two triangles intersect in six points on a
circle whose centre lies midway between the circum-centres
of the same triangles. When A'B'G' shrinks to the point Sy
the circle becomes the T. R. circle.
A more general theorem respecting the triangle and circle
is the following —
Take ABG any triangle, and let DD'EE'FF' be the points in order, in
which any circle cuts the sides.
Let BD = pc, CE = qa, AF - rb | ^^O)
CD' = p'b, AE'=qc, BF'=r'a)
From BD.BD' = BF. BF', &c., Euclid III. 35, we can write three equations
which are satisfied by the values
p = r' = tac, q=p'=iah, r = q' = the (27),
and from these equations it appears that
DF=ffc; B'F' = aa, &c., where ff --= /(/-.V-/A'-M) (28),
BO that BEF and B'E'F' are both similar to ABC.
SELF-CONJUGATE TRIANGLE. 64 <
Also DF'=tabc, therefore DF' =^ FE' = ED' (29).
^ . -^^-^ tac sin B i, .
From &mBFD = we can obtain
a
cot BED = cot (l> = =F^-^^=^ (30).
The radius of the circle = (tR (31),
and the coordinates of its centre are
a = E j cos ^ + t(Kcr-a'-h'-c*) 1 _ gij^ii^^j.]y ;3 ^nd y (32).
(. zbc )
The equation of the circle is
al3y + hya + cai3 = t (aa + b[3 + cy) {ahc(l — ta-)+&c.} (33),
or tabc[a\l-ta')+l3'(l-th')+y'{l-tc')}
= a(3y{(l-tb'){l-tc') + ebx~\ +&c (34).
When t = 0, <t = 1 and the circle is the circum-circle (35).
When tK=l, (T = tX=— and the circle is the T. R. circle (36).
CONIC AND SELF-CONJUGATE TRIANGLE.
When the sides of the trigon are the polars of the opposite
vertices, the general equation of the conic takes the form
4755 /V+m2)82-?iy = 0.
Proof. — (Fig. 51.) The equation may be written in any one of the
three ways,
/V = {ny-\-m(3){ny—m(3), m^^^ = (ny + la)(ny—la),
n^y^ = {la + im^){la—imf3).
Hence, by (4699), a or BG is the chord of contact of the tangents
ny:hml3(AQ, AS) drawn from A, and ft is the chord of contact of the
tangents ny ± la (BR, BP) drawn from B. Hence a, ft are the polars of A, B
respectively ; and therefore y or AB is the polar of G (4130). Also y may
be considered to be the chord of contact of the imaginary tangents Za±im/3
drawn from G.
4756 If the points of intersection of a and j3 with the conic
be joined, the equations of the sides of the quadrilateral so
formed are
QR, Ia-\-mfi+ny = 0, SF, la-\-mfi-ny = 0,
PQ, -^la-\-tnl3-{-ny = 0, RS, la—m3-{-ny = 0.
Hence QB, SP and PQ, BS intersect on the line y in A'
and B'.
648 TEILINEAB ANALYTICAL CONICS.
4757 Eacli pencil of four lines in the diagi^am is a liarmonic
pencil, bjtlie test in (4649).
4758 The triangle A'B'C is also self- conjugate with regard
to the conic.
Proof. — The equations of its sides GB\ CA', A'B' are
lu-mft = 0, Za + m/3 = 0, y = 0.
Denote these by a, /3\ y, and put a, /8 in (4755) in terms of a', /S'. The
equation referred to A'B G thus becomes a'^-|-)8'^— 2ft'y^ = 0, which is of the
same form as (4755).
4759 It is clear that the triangles AQS and BPB, formed
by a pair of tangents and the chord of contact in each case,
are also self-conjugate.
4760 Taking A'B'G for the trigon, and denoting the sides
by a, /3, y, the equations of the sides R8, PQ, QB, SB of the
quadrilateral become respectively
ny :^la= 0, mfi i ny = 0.
Ex. — As an example of (4611), we may find the coordinates of P from
the equations
aa + b/3 + Cy = S -s C " ~ ^lm^(amn + hil + dm)
+ m(3 — ny = ^ from which < /S = '^mn-i-(amn+bnl + clm)
— lu+ +ny = 0) ^7= ^)il^(amn + hnl + clm).
To obtain the coordinates of Q, B, and S, change the signs of m, n, and I
respectively.
ON LINES PASSING THROUGH IMAGINARY POINTS.
4761 Lemma I. — The right line passing through two con-
jugate imaginary points is real, and is identical with the line
passing through the points obtained by substituting unity for
^/ — \ in the given coordinates.
Proof. — Let {a + ia\ b + ib') be one of tlie imaginary points, and therefore
(a — ia, b — ib') the conjugate point. The equation of the line passing through
them is, by (4083) and reducing, b'x — a'y + a'b — ab' = 0, which is real.
But this is also the line obtained by taking for the coordinates of the
points (a + a'j b + b') and (a— a', b — b').
Lemma II. — If P, S and Q, B are two pairs of conjugate
imaginary points, the lines BS and QR are real, as has just
been sliown, and, tliorefore, also their point of intersection is
SELF-CONJUGATE TRIANGLE. 649
real. The otlier pairs of lines PQ, BS and PR, QS are
imaginary. But the points of intersection of each pair are
real, and are identical with the points which are obtained by
substituting unity for \/ — 1 in the given coordinates, and
drawing the six lines accordingly.
Proof. — Let the coordinates of the four points be as under —
P a + ia, h + ih', Q a + ia, jj + ifl':
S a — ia, h — ih', R u—ia, (o—ift'.
The equations of PB and QS, by (4083), are L + iM and L—iM, where
L = (b—p) ,i' — (a — a) y + a jj — ah +a'iy — a'b',
M= (h' + i3')x—(a—a)^j + afi-ab — aiy +cib'.
Now the Unas L :iz iM = intersect in the same real point as the lines
i ± ill = 0, because the values L = 0, M = satisfy both equations
simultaneously. Hence, to determine this point, we have only to take i as
unity in the given coordinates.
Lemma III. — If P, S are real points, and Q, B sl pair of
conjugate imaginary points, the lines PS and QB are both
real, by Lemma I., and consequently their point of intersec-
tion is real. The remaining pairs of hues PQ, BS and PB, QS
and their points of intersection are all imaginary. But the
line joining these two imaginary points of intersection is real,
and is jdentical with the line obtained by substituting unity
for \/ — l in the given coordinates and drawing the six lines
accordingly.
Proof. — Let the coordinates of the four points be as under —
P x,ij„ Q a + ia, i8+//3',
S x.2y,, B a — ia, ft—ifi'.
Since the coordinates of B are obtained from those of Q by merely changing
the sign of i, the equations of the four imaginary lines will take the forms
PQ A-iB, SQ C-iD,
PB A + iB, SB C+iD.
Now let the coordinates of the point of intersection of PQ and SB be
L+iM, L' + iM', then will L — iM, L' — iM' be the coordinates of the intersec-
tion of PB and SQ, for the equations of this pair of lines are got from those
of PQ and SB by merely changing the sign of i. The points of intersection
are therefore conjugate imaginary points, and the line joining them is real,
by Lemma I. Also, since that line is obtained by writing 1 for i in the co-
ordinates of those points, it will also be obtained by writing 1 for i in the
original coordinates of Q and B and constructing the figure as before.
4
650 TBILINEAE ANALYTICAL CONICS.
4762 To find a common pole and polar of two given conies :
(i.) If the conies intersect in four real points P, Q, R, S,
construct the complete quadrilateral (4652). Then A'B'G
(Fig. 51) is a self-conjugate triangle for each conic, by
(4758), and therefore each vertex and the opposite side form
a common pole and polar to the conies.
(ii.) If the conies do not intersect at all in real points, the
triangle A'B'G is still real, by Lemma II. (4761), and can be
constructed in the manner shown.
(iii.) If two of the points (P, S) are real, and two {Q, B)
imaginary, then, by Lemma III., the vertex A' and the side
B'G are real, and may be constructed, and they form a common
pole and polar of the given conies.
Returning to the triangle of reference ABG,
4763 Let la = ny cos (j>, m(i= ny sm<p; then the chord
joining two points <^i, <^2 is
la cosi (<^i+(^2)+»^/3 sini (<^i+<^2) = ny cosj (<^i — (^o)^
and therefore the tangent at the point <(>' is
4764 loi cos ^'-\-mfi sin <f> = ny.
4765 Putting V = L, m' = M, n''=-N, the conic (4755)
becomes
La:'JrMP'-\-Ny' = (1).
4766 The tangent or polar of a(5'y' is
Laa+Ml3fi'-\-Nyy' = (2).
4767 Hence the pole of \a-\-iJ,p-\-vy — i)
'' H'-M'i)' (^^-
4768 The tangential equation is
i+i+i=' (^)'
and this is the condition that the conic (1) may be touched
by the four lines
\a±fil3±vy = 0.
SELF-CONJUGATE TRIANGLE. 651
4769 In like manner,
La'^+M)8'^+iVy- = (5)
is tlie condition tliat (1) may pass through the four points
(a, ±^\ ±y').
4770 The locus of the pole of the line Xa+^^+vy with
respect to such conies is
a ^ /8 "^ r
Proof. — By (3), if (a, /3, y) be the pole, " = y &c., .-. L = —, in (5),
the equation of condition.
4771 The locus of the pole of the line la-\-mB-\-ny, with
respect to the conies which touch the four hues Xa ± /ti^ ± vy
I tn n
Proof.— By (3), if (a, ft, y) be the pole, « = y &c., .*. i = -, &c., in
(4), the equation of condition.
4772 The locus of the centre of the conic is given in each
case (4770, '1) by taking the line at infinity
asin J.+j3 sin 5 + 7 sin (7
for the fixed line, since its pole is the centre.
4773 Thus the locus of the centre of the conic passing
through the four points (a±/8'±y') is
a ^ sin A ^"■^ sin B y'~ sin C _ q
~^ +^i8~~+ y
4774 The coordinates of the centre of the conic (1) are
, La M^ Ny
given by __ = _^ := —Z-
Proof. — Let the conic cut the side a in the points (0^i7i), (O/S.^y.^. The
right line from A bisecting the chord will pass through the centre of the
conic, and its equation will he ft : y = (3^ + ^.2 : yi + y.2- Now A + A is the
sum of the roots of the quadratic in y8 obtained by eliminating y and a from
the equations La' + Mft^ + Ny^ = 0, a = 0, and aa + hft + Cy = 2. Similarly
for yi + y2 eliminate a and /3. The equation of the diameter thi'ough A being
found, those through B and G are symmetrical with it.
652 TEILINEAB ANALYTICAL CONICS.
4775 The condition that the conic (1) may be a parabola is
Proof. — This is, by (4), the condition of touching the line at infinity
i\a + hft-\-Cy = 0.
4776 The condition that (1) may be a rectangular hyperbola
is L-\-M-\-N = i), and in this case the curve passes through
the centres of the inscribed and escribed circles of the trigon.
Proof. — By (4G90), (a, b, c are now L, M, N). (1) is now satisfied by
a := ±/3 = ±y, the four centres in question.
4777 Circle referred to a self-conjugate triangle :
a' sin 2A-\-fi' sin 2B+y' sin 2C = 0.
Proof. — The line joining A to the centre is ~- = — ~ (4774). Therefore
— -, the condition of perpendicularity to a by (4622). Similarly
iicosB ccosO
N ^ L
c cos G a cos A^
therefore (1) takes the form above.
IMPORTANT THEOREMS.
CARNOT'S THEOREM.
4778 If J, B, G (Fig. 52) are the angles of a triangle, and
if the opposite sides intersect a conic in the pairs of points
a, a ; h, h' -, c, c ; then
Ac.Ac'.B(i.Ba'.Cb.Ch'= Ah.Ah'.Bc.Bc'.Ca.Ca'.
Proof. — Let a, /3, y be the semi-diameters parallel to BC, GA, AB ; then,
by (4317), Ah . Ah' : Ac : Ac = /3^ : y^. Compound this with two simihir
ratios.
4779 Coil — If the conic touches the sides in n, h, r, then
Ac\Ba\ Cb' = Ah\ Bc\ Ca\
THEOREMS. 653
4780 The reciprocal of Carnot's theorem is : If A, B, G
(Fig. 52) are the sides of a triangle, and if pairs of tangents
from the opposite angles are a, a' ; h, h' ; c, c' ; then
sin (Ac) sin (Ac) sin (Ba) sin (Ba) sin (Cb) sin {Cb')
= sin (Ab) sin (Ab') sin (Be) sin (Be) sin (Cm) sin (Ca),
where (Ac) signifies the angle between the lines A and c.
Proof. — Reciprocating the former figure with respect to any origin 0,
let A, B, G (i.e., BQ, QP, PB) be the polars of the vertices A, B, C. Then,
by (4130), Q, B will be the poles of AB, AG; and b, h', the polars of the
points h, V, will intersect in B and touch the reciprocal conic. Similarly, c, c'
will intersect in Q. A, h' are perpendicular to OA, Oh', and therefore
/.Ah'= Z AOh', and so of the rest.
PASCAL'S THEOREM.
4781 The opposite sides of a hexagon inscribed to a conic
meet in three points on the same right line.
Proof. — (Fig. 53.) Let a, ft, y, y', ft', a be the consecutive sides of the
hexagon, and let u be the diagonal joining the points au and yy'. The
equation of the conic is either ay — Jcftu = or a'y'—k'ft'u = 0, and, since
these expressions vanish for all points on the curve, we must have ay — l-ftu
= ay' — k'ft'to for miy values of the coordinates. Therefore oy — a'y'
= li (hft — k'ft'). Therefore the lines a, a' and also y, y' meet on tlie line
kft — k'ft'; and ft, ft' evidently meet on that line.
Otherwise, by projecting a hexagon inscribed in a circle with its opposite
sides parallel upon any plane not parallel to that of the circle. The line at
infinity, in which the pairs of parallel sides meet, becomes a line in which
the corresponding sides of a hexagon inscribed in a conic meet at a finite
distance (1075 et seq.).
4782 With the same vertices there are sixty different
hexagons inscribable in any conic, and therefore sixty dif-
ferent Pascal lines corresponding to any six points on a conic.
Proof. — Half the number of ways of taking in order five vertices B, G,
D, E, F after A is the number of different hexagons that can be drawn, and
the demonstration in (4781) applies equally to all.
BRIANCHON'S THEOREM.
4783 The three diagonals of a hexagon circumscribed to a
conic pass through the same point (Fig. 54).
664 TBILINEAR ANALYTICAL GONIOS.
Proof.— Let the three conies /S + L\ S + M\ S + N\ in (4707), become
three pairs of right lines, then the three lines L—M, M—N, N—L become
the diagonals of a circumscribing hexagon.
Pascal's and Brianchon's theorems may be obtained, the one from the
other, by reciprocation (48-40).
THE CONIC REFERRED TO TWO TANGENTS AND
THE CHORD OF CONTACT.
Let L = 0, M=0, B = (Fig. 55) be the sides of the
trigon ; L, M being tangents and B the chord of contact.
4784 The equation of the conic is LM = R\ (4G99)
4785 The lines AP, BP, and GP are respectively
IlL = R, iiR= M, irL = M. [By (4604).
Since the point P on the curve is determined by the value
of ^5 it is convenient to call it the point /n.
4788 The points /i and — ju (P and Q) are both on the line
ju^Jv = M drawn through G.
4789 The secant through the points ^, /x' (P, P'} is
/m/x L- (ft+fi) R + M = 0.
Proof. — Write it fi(fi'L — ]i) — (fi'Ii—M), and, by (4604), it passes
through the point ^'. Similarly through /j. Otherwise, determine the co-
ordinates of the intei-section of ^L — B and f.ili~M, and of {.I'L — U and
fx'B—M by (4610), and the equation of the secant by (4616).
4790 Cor. — The tangents at the points /n and — ^i (P, Q)
are therefore
4791 These tangents intersect on B. [Proof by subtraction.
4792 Theorem. — If the equation of a right line contains an
indeterminate ^ in the second degree, it may be written as
above, and the line must therefore touch the conic LM= R^.
4793 The ])olar of the point (//, M\ B') is
LM'-2RR-\-L'M = {).
TEE come LM = Rl 655
Proof. — For /i + ju' and ju^', in (4789), put the values of the sum and
product of the roots of /li'^L' -2fiB' + 31' = (4790).
4794 Similarly the polar of the point of intersection of
aL — B and bB — M is
abL-2aR-^M=0.
4795 The line GE joining the vertex G to the intersection
of two tangents at ^t and /.i, or at —/n and —jn', is
iHLL—M= 0.
Otherwise, if two tangents meet on any line ciL — M, drawn
through G, the product of their ^u's is equal to a.
Proof. — Eliminate R from the equations of the two tangents (4790).
4796 The chords PQ, FQ and the line GE all intersect in
the same point on B.
Proof.— The equations of FQ', P'Q are, by (4789),
l^fi'L± d-i-i-i') E-M = 0,
and, by addition and subtraction, we obtain fxfx'L — M = (4795), or i?- = 0.
4797 The lines i^fx'L + M (GD) and B intersect on the chord
PP' which joins the points jn, ^t/; or — The extremities of any
chord passing through the intersection of aL-\-M and B have
the product of their ^'s equal to a.
4798 The chord joining the points ^i tan ^, /i cot ^ touches a
conic having the same tangents L, M and chord of contact B.
Proof. — The equation of the chord is, by (4789),
fi^L— fiE (tan (l) + coi<p) + M= 0,
and this touches the conic JyMsin^2^ = jB^ at the point yu, by (4792).
4799 The tangents at the points ;ii tan <}), /i cot (p intersect
on the conic LM = B^ sin^ 2(j>.
Proof. — Write the equations of the two tangents, by (4790), and then
eliminate )u.
4800 Ex. 1. — To find the locus of the vertex of a triangle circumscinbing
a fixed conic and having its other vertices on two fixed right lines.
Take LM = B"- for the conic (Fig. 56), aL + M, hL + M for the lines CD,
GE. Let one tangent, DE, touch at the point /x ; then, by (4796), the others,
656 TEILINEAU ANALYTICAL CONICS.
PD, PE, will toiicli at the points — , — , and therefore, by (4790), their
equations will be r- r-
l!^L-~li + M, ^L--B + M.
Eliminate /.«, and the locus of P is found to be (^a-\-h)-LM =■ A^abp-.
[Salmon, Art. 272.
4801 Ex. 2. — To find the envelope of the base of a triangle inscribed in
a conic, and whose sides pass through fixed points P, Q.
(Fig. 57.) Take the line through P, Q for E ; LM- P- for the conic ; ciL—M,
hL — M for the lines joining P and Q to the vertex G. Let the sides through P
and Q meet in the point fx on the conic ; then, by (4797), the other extremi-
ties will be at the points — — and , and therefore, by (4789), the
equation of the base will be a&L + (a + ?0 A'-R + /-3'^ = 0. By (4792), this
line always touches the conic ^ah LM = (a + by B^. \_Ibid.
4802 Ex. 3. — To inscribe a ti-iangle in a conic so that its sides may pass
through three fixed points. (See also 4823.)
We have to make the base ahL+ (a + h) ixB + in^M (4801) pass through
a third fixed point. Let this point be given by cL = P, dB = M. Elimi-
nating L, M, B, we get ab + (a + b) ixc + fx'^cd = 0, and since, at the point /u,
nL = B, fj}L = M, that point must be on the line abL + (a-\-b) cB + cdM.
The intersections of this line with the conic give two solutions by two posi-
tions of the vertex. [^Ibid.
RELATED CONICS.
4803 A conic having double contact with the conies S and
S' (Fig. 58) is
where E, F are common chords of S and 8', so that
S-S' = EF.
PiioOF. — The equation may be written in either of the ways
(fiE +Fy = 4^N or (fxE - P)- = 4^. S',
showing that fxE =h F are the chords of contact ^IP, CL. There arc three
such systems, since there are three pairs of common chords.
4804 C'oR. 1.^ — A conic touching four given hues A,B, C, D,
the diagonals being J^, F (Fig. 59) :
ii:'E'-2ii {AC-\-IJD)-\-F = i\.
Here S = AG and S' = BB, two pairs of right lines.
ANEABMONtO PENCILS. 657
Otherwise, if L, M, Nhe tlie diagonals and L±M±N the
sides, the conic becomes
4805 ii'L'-iM {U+3P-N')+M' = 0.
For this always touches
(L' + M'-Ny-WM' or (L + 3f+N)(M+N-L){N + L-M)iL-\-M-N).
[Sahnon, Art. 287.]
4806 Cor. 2. — A conic having double contact with two
circles G, C is
4807 The chords of contact become
^+C-C' = and ^-C+(7' = 0.
4808 The equation may also be written
which signifies that the sum or difference of the tangents
drawn from any point on the conic to the circles is constant.
ANHARMONIC PENCILS OF CONICS.
4809 The anharmonic ratio of the pencil drawn from any
point on a conic through four fixed points upon it is constant.
Proof. — Let the vertices of the quadrilateral in Fig. (38) be denoted by
A, B, C, I), and let P be the fifth point. Multiplying the equation of the
conic (4697) by the constants AB, CD, BC, DA, we have
, AB.CD _ ABa . CDy ^ PA . PB sin APB . PC . PD sin CPD
BC.DA BCfi . DAS PB.PC sin BPG. PD . PA sin DP A
sin APB. sin CPD
sin BPC. sin DP A
Compare (1056).
4810 If the fifth point be taken for origin in the system
(4784, Fig. 55), and if the four lines through it be
L—iJLj^R, L — ix,R, L—fJiJi, L—jM^R,
4 p
658 TBILINEAU ANALYTICAL CONICS.
the anliarmonic ratio of tlie pencil is, by (4G50),
4811 CoE. 1. — If four lines through any point, taken for
the vertex LM, meet the conic in the points ^tj, /n.,, fx-s, /t^, the
anharmonic ratio of these points, with any fifth point on the
conic, is equal to that of the points —/Hi, — a'o, — /"s, —l^a i^
which the same lines again meet the conic.
4812 CoE. 2. — The reciprocal theorem is — If from four
points upon any right line four tangents be drawn to a conic,
the anharmonic ratio of the points of section with any fifth
tangent is equal to the corresponding ratio for the other four
tangents from the same points.
4813 The anharmonic ratio of the segments of any tangent
to a conic made by four fixed tangents is constant.
Proof. — Let fx, n^, fj^, yug, fx^ (Fig. 60) be the points of contact. The
anliarmonic ratio of the segments is the same as that of the pencil of four
lines from LM to the points of section; that is, of nn^L—M, iiii.Jj — M,
HH-Jj—M, fifi^L- M, a pencil homographic (4651) with that in (481U).
4814 If P, P' are the polars of a point with respect to the
conies 8, S\ then P-\-hP' will be the polar of the same point
with respect to the conic S-\-h8'.
4815 Hence the polar of a given point with regard to a
conic passing through four given points (the intersections of
8 and 8') always passes through a fixed point, by (4101).
If Q, Q' are the polars of another point with respect to
the same conies, Q-\-hQ is the polar with respect to 8-\-h8'.
4816 Hence the polars of two points with regard to a
system of conies through four points form two homographic
pencils (4651).
4817 The locus of intersections of corresponding lines of
two homographic pencils ha^ang fixed vertices (Fig. 61) is a
conic passing through the vertices ; and, conversely, if the
conic be given, the pencils will be homographic.
Proof.— For climinatiiio- A- fnmi P + AT'= 0, Q + IQV, we get !'(/= r'Q-
CONSTRUCTION OF CONICS. 659
4818 Cor. — The locus of the pole of the line joining the two
points in (4816) is a conic.
PROOr. — For the pole is the intersection of P + hP' and Q + hQ'.
4819 The right lines joining corresponding points AA', &c.
(Fig. 62) of two homographic systems of points lying on two
right lines, envelope a conic.
Proof. — This is the reciprocal theorem to (4817) ; or it follows from
(4813).
4820 If two conies have double contact (Fig. 63), the an-
harmonic ratio of the points of contact A, B, G, D of any
four tangents to the inner conic is the same as that of each
set of four points (a, b, c, d) or {a, h\ c', <V) in which the
tangents meet the other conic.
Proof. — By (4798). The ^'s for the points on the latter conic will be
equal to the /.t's of the points of contact multiplied by tan <p for one set, and
by cot^) for the other, and therefore the ratio (4810) will be unaltered.
4821 Conversely, if three chords of a conic aa , hb', cc be
fixed, and a fourth rZc?' moves so that \ahcd\ = [ctb'c'd'l, then
dd' envelopes a conic having double contact with the given
one.
For theorems on a right line cut in involution by a conic, see (4824-8).
CONSTRUCTION OF CONICS.
THEOREMS AND PROBLEMS.
4822 If a polygon inscribed to a conic (Fig. 64) has all its
sides but one passing through fixed points A, B, ... Y, the
remaining side az will envelope a conic having double contact
with the given one.
Proof. — Let a,h, ...z be the vertices of the polygon, and a, a', a", a'"
four successive positions of a. Then, by (4811),
I a, a, a", a" } = { ^j ^' ■> ^" ■> ^"' | = &c. = | 2;,
Therefore, by (4821), the side az envelopes a conic, &c.
660 TBILINEAB ANALYTICAL CONIGS.
4823 Poncelet's construction for inscribing in a conic a
polygon having its n sides passing through n given points.
Inscribe three polygons, each 0/ n + 1 sides, so that n of
each may pass through the fixed points, and let the remaining
sides he a'z', af'z", ?\1"t:!" , denoted in figure (65) hy AD, CF, EB.
Jjet MLN, tlie line joining the intersections of opposite sides of
the hexagon ABCDEF (4781), meet the conic in K; then K
will be a vertex of the required polygon.
Proof. — iD.KAGIj] = iA.KDFB\, each pencil passing through
K, P, N, L; therefore the anharmonic ratio | KAGE ] = \ KDFB \ for any
vertex on the conic, by (4809) ; i.e., I Kaa'a" \ = | Kz'z"z"' \ . But, if az
be the remaining side of a fourth polygon inscribed like the others, we have
by (4811), as in (4822), | aa'a'a" \ = i zz'z'z" | . Hence K is the point
where a and z coincide.
4824 Lemma. — A system of conies passing through four
fixed points meets any transversal in a system of points in
involution (1066).
Proof. — Let u, u be two conies passing through the four points ; then
u-\-'kitj' will be any other. Take the transversal for x axis, and put ?/ = in
each conic, and let their equations thus become ax- + 2qx + c = and
aV + 2(7'a)+c' = 0. These determine the points where the transversal
meets u and u . It will then meet u + Jcu' in two points given by
ax^ + 2gx + c + k (a V + 2g'x + c') = 0, and these points arc in involution with
the former, by (1065).
Geometrically (Fig. QQ),
[a.AdhA'] = [c.AdbA'] (4809),
therefore { AGBA' ] = { AB'G'A' } = { A'G'B'A } , therefore by (1069).
4825 Cor. 1. — One of the conies of the system resolves
itself into the two diagonals ac, bd. Hence the points B, B\
G, G' are in involution with D, D' , where the transversal cuts
the diagonals.
4826 CoR. 2. — A transversal meets a conic and two tan-
gents in four points in involution, so as to meet the chord of
contact in one of the foci of the system.
For, in (Fig. 66), if h coincides with c, and a with d, the
transversal meets the tangents in G, G', while B, B\ D, B', all
coincide in F (Fig. Ql), one of the foci on the chord of
contact.
CONSTRUCTION OF CONIC S.
661
4827 The reciprocal theorem to (4824) is — Pairs of tangents
from any point to a system of conies touching four fixed hnes,
form a system in involution (4850).
4828 The condition that \x-\-iiiij-\-v:: may be cut in involu-
tion by three conies is the vanishing of the determinant
As
^1
B,
B.
a.
h
^1
2/i
2^x
2\
a.2
h
c-z
2/;
2.^/2
2h,
a.
h
^3
2/3
2i/3
2Ji,
X
V
Ai
^'■
V
X
„
/t
A
where A^, H^, i?i belong to the first conic and have the values
in (4988).
Peoof. — The quadratic A.^x- + 2IIx7j + B^y- = 0, obtained in (4987), deter-
mines the pair of points of intersection with the first conic. The similar
equation for the third conic will have A^ = A-^ + XAi, &c., if the points are
all in involution (1065). The third equation is therefore derived from the
other two; therefore the determinant vanishes, by (583).
By expanding and dividing by p^, the second determinant above of the
sixth order is obtained.
Newton's Method of Generating a Conic.
4829 Two constant angles aPb, aQh (Fig. 68) move about
fixed vertices P, Q. If a moves on a fixed right line, h de-
scribes a conic which passes through P and Q.
Proof. — Taking four positions of a, we have (see 1054),
{P.hbV'h'"} = {P.aa'a'a'"] = {Q.aaa'a"} = {Q.hl'h"h"'] .
Therefore, by (4817), the locus of 6 is a conic.
Maclaurin's Method of Generating a Conic.
4830 The vertex F of a triangle (Fig. 69), whose sides pass
through fixed points A, B, C, and whose base angles move on
fixed lines Oa, Oh, describes a conic passing through A and B.
Proof. — The pencils of lines through ^4 and B in tbe figure are both
homogi'aphic with the pencil through C, and are therefore homographic with
each other. Therefore the locus of F is a conic, by (4817).
tlTNI^Zirli
irNI^ZiirlblTY
662 TEILINEAB ANALYTICAL CONICS.
Otherwise, let a, fl, y be the sides of ABC; la + mft + ny, I'u + ra'ft + n'y
the fixed Hues Oa, Oh ; and a = /x/3 the moving base ah.
Then the equations of the sides will be
(Ifi + m) l3 + ny — 0, (I'fi + m) a + n'fiy = 0.
Eliminate /i; then Imajj— (ml3 + ny)(l'a+n'y), the conic in question, by
(4697).
4831 Given five points, to find geometrically any number of
points on tlie circumscribing conic, and to find the centre.
Let A, B, 0, D, E {Fig. 70) be the five points. Draw any
line through A meeting CD in P. Draw PQ through the inter-
section of AB and DE meeting BO in Q; then QE ivill meet
PA in F, a sixth point on the curve, as is evident from PascaVs
theorem (4781).
To find the centre, choose AP in the above construction
parallel to CD, and find two diameters, as in (1252).
4832 To find tbe points of contact of a conic with five right
lines.
Let ABCDE {Fig. 71) be the pentagon. Join D to the
intersection of AG and BE. This line unll pass through the
point of contact of AB, and so on.
Proof. — By (4783), supposing two sides of the hexagon to become one
straight line.
4833 To describe a conic, given four points upon it and a
tangent.
Let a, a', b, h' {exterior letters in Fig. 52) be the four
points. Then, ifAB is a tangent, c, c' coincide, and GarnoVs
theorem (4778) gives the ratio A.& : Bc'l Then by (4831).
Since there are tivo values of this ratio, + (Ac : Be), two
conies may be drawn as required,
4834 To describe a conic, given four tangents and a point.
Let a, a , b, V {interior letters in Fig. 52) be the four
tangents. Then, ifQ be the given point on the curve, the lines
c, c' must coincide in direction, and (4780) gives the ratio
-sin2(Ac) : sin^ (Be), by ivhlch the direction of a fifth tangent
through Q is- determined. Then by (4832). Tlie two values
+ (sin Ac : sin Be) furnish ttvo solutions.
Otherwise by (4804), d.elcrmiiiivg ^i by tlie coordinates of
the given point.
CONSTBUCTION OF CONIGS. 663
4835 To describe a conic, given three points and two
tangents.
Let A, A', A'' he the points {Fig. 67, supplying obvious
letters). Let the two tangents meet AA' in the points C, C.
Find F, F', the foci of the system AA\ CC in involution (1066)
determining the centre by (985). Similarly, find Gr, G', the
foci of a system on the line KK". Then, by (4826), the chord
of contact of the tangents may be any of the lines FG, FG',
F'G, F'G'. There are accordingly four solutions, and the
construction of (4831) determines the conic.
4836 To describe a conic, given two points and three
tangents.
Let AB, BO, CA {Fig. 167) be the tangents, and P, P' the
points. Draw a transversal through PP' meeting the three
tangents in Q, Q', Q". Find F, a focus of the system PP', QQ'
in involution (1066, 985) ; G a focus for PF, qq',and H/or
PP', Q'Q''. Cunstruct a triangle luith its sides passing through
F, G, H, ami ivith its vertices L, M, N on BC, CA, AB,
by the method of (4823), lohich is equally applicable to a recti-
lineal figure as to a conic. L, M, N will be the points of
contact. The reason for the construction is contained in
(4826). There will, in general, be four solutions.
If the conic be a parabola, the foregoing constructions
can be adapted by considering one tangent at infinity always
to be given.
4837 To draw a parabola through four given points a, a, b, b' .
This is problem 4833 with the tangent at infinity.
In figure (52), suppose cc to coincide and AB to remove to infinity so as
to become the tangent at c, the opposite vertex at infinity of a parabola, and
therefore to be perpendicular to the axis. Cc then becomes a diameter of
the parabola, and Caruot's theorem (4778) shows that
Ca.Ga ^4^Ba^^ sin' AGc
Gb.Cb' AV'' Bo'' sin' BGc
since the points C, a, a, h, V are all on the axis of the parabola relatively to
the infinite distance oi AcB. This result, however, is at once obtained from
equation (4221), Ga.Ca : Cb . Gb' being the ratio of the products of the roots
of two similar quadratics. Thus a diameter of the parabola can be drawn
through C by the known ratio of the sines of AGc and BCc.
Next, describe a circle round three of the given points a, a', b. By the
property (1263) and the known direction of the axis, the other point in
which the circle cuts the parabola can be found.
Five points being known, we can, by Pascal's theorem, as in (4831),
664 TEILIKEAR ANALYTICAL CONICS.
obtain two parallel cliovds, and tben find P, the extremity of tlieir diameter,
by the proportion, square of ordinate cc abscissa (1239).
Lastly, draw the diameter and tangent at P, and then, by equality of
angles (1224), draw a line from P which passes through the focus. By
obtaining in the same way another pair of parallel chords, a second line
through the focus is found, thus determining its position.
4838 To draw a parabola when four tangents are given.
This is effected by the construction of (4832, Fig. 71). Let AB, BC, AE,
ED be the four tangents, and CD the tangent at infinity. Then any line
drawn to C will be parallel to BC, and any line to D will be parallel to ED.
4839 To draw a parabola, given tliree points and one
tangent.
This is effected by the construction of (4835, Fig. 67). Let hC be the
tangent at oo ; then the centre of involution must be at C, so that
CC.CC = 0. CO = CA.CA' = CF-, determining F. F', another point on
the chord of contact, being found by joining AA" or A' A", FF' will be the
diameter through a, since the other point of contact h is at infinity.
4840 To draw a parabola, given one point and tliree
tangents.
This is the case of (4834), in which one of the given tangents h' is at
infinity. B must therefore be at infinity, and QB, FB and the tangent h,
since they all join it to finite points, must be parallel. The ratio found
determines another tangent, and the case is reduced to that of (4838).
4841 To draw a parabola, given two points and two
tangents.
This is problem (4836). Suppose AC in that construction to be the
tangent at infinity. F, G, H will be determined as in (4830) by mean
proportionals. The chords LM, NM will become parallel, since M is at
infinity; and we have to draw LA'' and the parallel lines from L and N to
pass through F, G, H in their new positions, so that the vertices L, N may
lie on BC and AB.
Otherwise by (4509), the intercepts s and i can readily be found from the
two equations furnished by the given points.
4842 To describe a conic touching three right lines and
touching a given conic twice.
Let AD, CF, EB {Fig. 65) he the three lines as they cut
the given conic. Join AB, AF, BC, BE, and determine K hi/
the Pascal line MLN. K will he one point of contact of the
two conies, hy (481^2) aiid the proof in (4823), since AD, CF,
EB, and the tangent at K are four iJositions of the " remaining
side " in that proposition. The prohlem is thus reduced to
BECIPEOGAL P0LAE8. 665
(4834), since four tangents and K the i?oint of contact of one of
them are noio Jcnoivn.
4843 To describe a conic toucliing eacli of two given conies
twice, and passing through a given point or touching a given
line.
Proceed by (4803), determining ^ hij the last condition.
To describe a conic touching the conies S-^-L^, S-\-M'\
S-\-N'^ (4707) and touching S twice. ISahmn, Art. 387.
THE METHOD OF RECIPROCAL POLARS.
Def. — The polar reciprocal of a curve is the envelope of
the polars of all the points on the curve, or it is the locus of
the poles of all tangents to the curve, taken in each case
with respect to an arbitrary fixed origin and circle of recipro-
cation.
4844 Thus, in figure (72), to the points P, Q, B on one
curve correspond the tangents qr, rp, and chord of contact j^q
on the reciprocal curve ; and to the points p, q, r correspond
the tangents QB, BP, and chord PQ.
The angle between the tangents at P and Q is evidently
equal to the angle j^Oq, since Oj), Oq, Or are respectively per-
pendicular to QB, BP, PQ.
4845 Theorem. — The distance of a point from a line is to its
distance from the origin as the distance of the pole of the line
from the polar of the point is to its distance from the origin.
Proof.— (Fig. 73.) Take for origin and centre of auxiliary circle, FT
the polar of c, pt the polar of C, CP perpendicular on polar of c, cp perpen-
dicular on polar of G. Then
r- - 00. Ot = Oc.OT ■) Therefore, by subtraction, OO.mt=Oc.MT,
and 0G.0iH=0c.0M) or 00 .cp =Oc.GP;
that is, OP : 00 :: cp : cO. Q. E. D.
CoE. — By making CP constant, Ave see that the reciprocal
of a circle is a conic having its focus at the origin and its
directrix the polar of the circle's centre.
4 Q
TBILINEAB ANALYTICAL CONICS.
GENERAL RULES FOR RECIPROCATING.
4846 ^ point hecoiues the polar of the point, and a rigid line
hecoines the pole of the line*
4847 ^ ^^'^e through a fixed point becomes a point on a fixed
line.
4848 The intersection of two lines becomes the line ivhich
joins their poles.
4849 Lines passing through a fixed point become the same
number of points on a fixed line, the polar of the point.
4850 ^ right line intersecting a curve in n points becomes n
tangents to the reciprocal curve passing through a fixed point.
4851 Tivo lines intersecting on a curve become tivo points
wliose joining line touches the reciprocal curve.
4852 Tivo tangents and the chord of contact become tioo
points on the reciprocal curve and the intersection of the tan-
gents at those points.
4853 -4 pole and polar of any curve become respectively a
polar and pole of the reciprocal curve; and a point of contact and
tangent become respectively a tangent and its point of contact.
4854 The locus of a point becomes the envelope of a line.
4855 -4n inscribed figure becomes a circumscribed figure.
4856 Four points connected by six lines or a quadrangle
become four lines intersecting in six points or a quadrilateral.
4857 The angle between tivo lines is equal to the angle sub-
tended at the origin by the corresponding points. (4844)
4858 The origin becomes a line at infinity, the polar of the
origin.
4859 Tivo lines through the origin become two points at
injiniiy on the polar of the origin.
4860 Tivo tangents through the origin to a curve become two
points at infinity on the reciprocal curve.
4861 The points of contact of such tangents become asymptotes
of the reciprocal curve.
4862 The angle between the same tangents is equal to the
angle betiveen the asymptotes. (4857)
* Tliat is, with respect to the circle of reciprociitiou, and so throughout with the excep-
tion of (4853).
BEGIPBOGAL POLABS. 667
4863 According as the tangents from the origin to a conic are
reat or imaginary, the reciprocal curve is an hyperbola or
ellipse.
4864 if l^i'& origin be tahen on the conic, the reciprocal curve
is a parabola.
For, by (4860, '1), the asymptotes ai'e parallel and at infinity.
4865 A trilinear equation is converted by Teciprocation into
a tangential equation.
Thas ay = hfth is a conic passing through four of the intersections of
the lines o, /3, y, <). Reciprocating, we get a tangential equation of the same
form AC = hBB, and this is a conic touching four of the lines which join
the points whose tangential equations are A=- Q, JB = 0, (7 = 0, D = 0.
See (4907).
4866 The equation of tlie reciprocal of tlie conic ahf-\-l>^ii?
= aW witli the same origin and axes is
where h is the radius of the auxiliary circle whose centre is
the centre of the conic.
Proof. — Let p be the perpendicular on the tangent, its inclination ;
then fcV-2=/ = a2cos^0 + 6-sin2 6l (4732).
4867 The same when the origin of reciprocation is the
point xy\
{aw' -\-yii' -\-W)'^ = a\v'^-{-b\i/\
Proof: A;V~'=p= -/a^ cos^d + b'^ sm-d—{x' cosd + y' sinO).
4868 The reciprocal curve of the general conic (4656), the
auxiliary circle being x^ + y^ = ¥ or iG^-\-y^ + z^ = in tri-
linears, will be symmetrically
replacing ^ by — A'^
Proof. — Let i/j be a point on the reciprocal curve, then the polar of i>/,
namely, a'^ + i/'/ — A''' = 0, must touch the conic, by (4853). Therefore, by
(4(365), we must substitute 4, ?/, — Jc' for A, /,/, u in tlie tangential equation
^\- + &c. = 0.
4869 From the reciprocal of a curve with respect to the
origin of coordinates, to deduce the reciprocal with respect to
an origin xy', substitute in the given reciprocal equation
'*' for .V and — , '^'\i.i ^^^ ^•
a?.z--\-iji/'-{-k'' d\v-\-yij-\-k'
668 ANALYTICAL GONIGS.
Proof. — Let, 7' he the perpuiidicular from the origin on the tangent and
PB = ^■. The perpendicular iVom x'y' is F — x'cosd — y'sind,
h' Jr , a ' • a . ^^' ccx +yy' + Jc^ ^
.•. — = —-—3! cos — y smU, .. - = --^ ;
p It Ih p
Icp cos
•. Rcos
aV + 2/2/' + ^''
TANGENTIAL COORDINATES.
4870 By employing these coordinates, theorems which are
merely the reciprocals of those already deduced in trilinears
may he proved independently. See (4019) for a description
of this system.
The following proposition serves to transform by recipro-
cation the whole system of trilinear coordinates of points and
equations of right lines and curves, into tangential coordinates
of right lines and equations of points and curves.
THEOREM OF TRANSFORMATION.
4871 Griven the trilinear equation of a conic (4656), the
tangential equation of the reciprocal conic in terms of X, fx, v,
the perpendiculars from three fixed points A\ B' , C upon the
tangent (Fig. 74) will be as follows, being the origin of
reciprocation and 0A\ OB', OC =p, ^/, r: —
4872 ^^X^ ■ hiL' cv" 2ffiv 2^-v\ 2h\fi _ ^^
jr ff r^ qr rp pq
Prook. — Let a = 0, /3 = 0, y = be the sides of the original trigon ABC.
The poles of these lines will be A', B', C, the vertices of the trigon for the
reciprocal curve. Let BS be the polar of a point P on the given conic ;
a, /3, y the perpendiculars from P upon BC, GA, AB ; i.e., the trilinear co-
ordinates of P. Let X, n, V be the perpendiculars from A', B', C upon BS;
i.e., the tangential coordinates of the p6lar of P referred to A', B', C. Then,
by (484.5), ^ = ^^„ A=^,, -r = ^. Substitute those values
of a, ft, y in (4656) and divide by 0P\
4873 The angular relation between the trigons ABG and
A'B'C is
JiVCr = TT-A, evil' = iT-B, A'OB' = TT-C,
TANGENTIAL G00BDINATE8. 669
4874 If ^BG be self-conjugate with regard to tlie circle of
reciprocation, it will coincide with A'B'C.
4875 Now let be the circum-centre (4629) of A'B'G'
(Fig. 74), then it will be the in-centre of ABC, and, by (4873),
2A'='n-A, 2B'=n-B, 2C'=7r-C.
Also p = q = r in (4872), which becomes ^ (X, fi, v) = 0, so
that the conic and its reciprocal are represented by the satiie
equation. Consequently any relation in trilinear coordinates
has its interpretation in tangential coordinates. We have
then the following rule : —
4876 Rule. — To convert any expression in trllinears into
tangentials, consider the origin of the former as the in-centre of
the trigon, change a, |3, y into X, ^, v, and interpret the result
by the rules for reciprocating (4846-65). If the angles of the
original trigon are involved, change these by (4875) into the
angles of the reciprocal trigon, of luhich the origin ivill now be
the circum-centre.
4877 Referring trihnears and tangentials to the same trigon
ABC, the equation of a point, as shown in (4021), becomes
AX+l/x+X.^0;
Ih Ih Ih
4878 01^5 by multiplying by -|-2,
BOC\^COAiL-^AOBv = {). (Fig. 3)
The equation of a point can generally be obtained directly
from the figure by means of this formula.
EQUATIONS IN TANGENTIAL COORDINATES.
For direct demonstrations of the following theorems, the reader may
consult Ferrers' Trilinear Coordinates, Chap. vii.
4879 The point dividing AB in the ratio a : i\ that is, the
intersection with the internal or external bisector of C, is
aXdbV = 0- Centre of AB X+^ = 0.
The point in (4878) is now on the side AB.
4881 Mass-centre, X+/li+i/ = 0. [For BOC=:COA^AOB.
670 ANALYTICAL CONICS.
4882 In-centre, nX+V+Cv^O. r By (4878), for
4883 «- ex-centre, — aX+b/x+Cv = 0. L a t^
0. L— 7
4884 Circum-centre X sin 2 A +/x siu 2B-\-v sin 2C = 0.
Proof.— For JJOC = ^B' sin 2.4, &c. in (4878). 0//ie?7t!ise.— By recipro-
cation (4876), a sin J. + ^ sin Z? + y sin (1 = is the line at infinity referred
to the trigon ABC ; therefore
X sin ^ + /Li sin J5 + »' sin (7 =
is the equation of the pole of that line referred to A'B'C ; that is,
\ sin 2^' + ^ sin 21/' + J' sin 26'', by (4875).
4885 Foot of perpendicular from C npon AB,
Xtan J[+/u,tan^ = 0.
4886 Orthocentre X tan A +fi tan B-^v tan C = 0.
4887 Inscribed conic of ABG, [Proof below.
4888 Point of contact with AB,
MX+Lfi = 0.
4889 In-circle (4629),
(^-a) iiv-\-{^-h) vX+(s;~r) x^ = o.
4890 Point of contact witli AB, (d-l)) X + (5-a) /m = 0.
4891 ii ex-circle, (6—1)) X/A+(d — r) vX—^fxv = 0.
Proof. — Since the coordinates of AB of the trigon are 0, 0, v, the equa-
tion of the inscribed conic must be satisfied when any two of the coordinates
X, /i, V vanish, therefore it must be of the form (4887). Otherwise by
reciprocating (4724).
If the circle touches AB in D (Fig. 'S), X : - n = AB : BD = S-ci : t^-h
(Fig. of 700), which proves (4890).
(4888) is the equation of the point of contact, because the line (0, 0, r)
passes through it and also touches the conic (4887).
(4889) is the in-circle by (4887) and (4890) and what precedes.
4892 Circumscribed conic, [By (487G) applied to (4739, '40).
L-X-+M>'+iVV-2MA>v-2iV2:vX-2LJ7X/i, = 0, (4740)
TANGENTIAL GOOEPlNATES. 671
4893 or ^{L\)-\-y/Mii-^^Ny = 0.
4894 Tangent at A, Mfx, = Nv.
4895 Cir cum- circle
4896 or a^/x+IJv//*+ryv=0.
Proof.— By (4876) applied to (4747, '8), and by cos— = sin^' (4875)
4897 Eolation between tlie coordinates of any right line :
4898 Coordinates of tlie line at infinity :
X = /x = v.
Proof. — The trilinear cooi'dinates of the origin and centre of the re-
ciprocal conic are o = /3 = y, (4876). It is also self-evident.
4899 The point IX -{- m/LL -{- uv = will be at infinity when
l-^m-i-n = 0.
Proof. — By (4876), for the line la + mft + ny = will pass through the
origin a = /3=:y when l + m + n:=0.
4900 A curve will be touched by the line at infinity when
the sum of the coefficients vanishes.
Proof.— By (4876), for this is the condition that the origin in trilinears,
a = /3 = y shall be on. the curve.
4901 The equation of the centre of the conic (p (X, ^t, v) is
4902 or (a+h-\-g')\+{h+b+f)iM-\-{g^f+c)v = 0.
Proof.— The coordinates of the in-centre of ABC (4876) are a=fi'=y\
therefore the polar of this point with regard to the conic f (o, ft, y) is
0^ + ^ _1_,^ =0 (4658). This point and polar reciprocate into a polar and
point, of which the former, being the reciprocal of the in-centre, or origin, is
the line at infinity, and therefore the latter is the centre of <p (X, u, y), while
its equation is as stated.
672 ANALYTICAL CONICS.
4903 The equation of the two points in which the line
(X', /Li, v) cuts the conic is
<l> (V, im\ y') <f> (X, II, v) = ((^^V+<^,,x'+</,,v7-. (4680)
4904 The coordinates of the asymptotes are found from the
equations
(l> {\, fjL, v) = and <^a + <^m+<^''= 0.
Proof. — These are the conditions that the line (A, /x, j-) should touch the
curve and also pass through the centre (4901).
4905 The equation of the two circular points at infinity is
a^(X-^)(x-v)+lj^(^-v)(^~x)+cHv-x)(v-ft) = 0.
Proof. — Put X'= ii'= v' in (4903) to make the line at infinity, and for
the conic take the in-circle (4889).
4906 The general equation of a circle is
a^ (x-^)(x-v)+I)'^ {i,-v){i.-\)^e {v-\){v-ix)
= {lX-\-mix-^nvy (1),
where l\-\-m^i-^nv = is the equation of the centre.
Proof. — The general equation of a conic in trilinears may, by (4601), be
put in the form
a(/3-/3o)(y-yo) + K7-yo)(«-«o)+c(«-"o)(/5-/5o) = {la + mr^ + ny)\
where la + m/3 + Ky = is the directrix, and ao/3oyo the focus. Now let the
focus be the in-centre of the trigon, and therefore a„ = /3o=yo= |SS~^ (709).
By this relation and aa + 6^ + cy = 2, the equation is expressed as
a(S-a)(a-/S)(«-y) + &c. = (Z'a+w'/3 + 7t'y)-,
or (a— /3)(a— y) cos" |^'1 + &C. = (ra + m'/i +«'y)''
Reciprocating by (4876), this becomes
(\—fx)(X — v) ain- A' -\- &.C. = (Z\+ "'A' +«»')',
the constant factor introduced on the right being involved in I, m, n; and
sin7i' = cos-j^, by (4875). And we know that this is a cii'cle by (4845
Cor.), and that the directrix of the conic reciprocates into the centre of the
circle.
Otherwise. — The left side of (1) represents the two circular points at
infinity (4905), and, if for the right we take the equation of a point, the
whole represents a conic, as in (49(»9), of the form AC = B'-. In this case,
A, C, the points of contact of tangents from B, being the circular points, the
conic must be a circle with i' = for its centre.
TANGENTIAL COORDINATES. 673
Abridged Notation.
4907 Let ^ = 0, B = 0, G = 0, D = (Fig. 75) be the
tangential equations of the four points of a quadrangle, where
A = a^X-\-hiiii-\-Ci^v, B = a.2\-\-biiii-\-C2v, and so on. Then
the equation of the inscribed conic will be AG = hBD.
Proof. — The equation is of the second degree in \, ju, v; therefore the
line (\, /J., J') touches a conic. The coordinates of one line that touches this
conic are determined by the equations A = 0, B = 0. That is, the line
joining the two points A, B touches the conic, and so of the rest.
4908 If the points B, D coincide (Fig. 76), the equation
becomes AG = JiB- ; and A = 0, G = are the points of
contact of tangents from the point B = 0.
4909 Referring the conic to the trigon ABG (Fig. 78), and
taking AG = k^B^ for its equation, let a tangent ef be drawn,
and let Ae : eB = h : m. The equations of the points f and
/ will be
mA + hB = 0, mhB + 6' = 0.
Proof. — The first equation corresponds to (4879). For the equation of/,
eliminate A from mA + kB = and AG — Jc^B^-.
4910 Let e, h (Fig. 77) be two points on AB whose equa-
tions are mA-\-hB = 0, 7iiA-\-kB = 0. The equation of the
point jj, in which tangents from e and h intersect, is
mmA-lr{m-\-m) kB-^C = 0.
Proof. — The equation may be put in the form
(mA + JcB)(mA + l-B) = 0,
because k'^B'^=AG if the line touches the conic. The equation being of the first
degree in A, B, G, must represent some point. That is, the relation between
A, /.I, V involved in it makes the stivaight line \a-\-i.ip-\-vy pass through a
certain point. But the equation is satisfied when mA + lcB = 0, a relation
which makes the straight line pass through e. Hence a tangent through e
passes through a certain fixed point. Similarly, by '^n'A + A-Jj = 0, another
tangent passes through li and the same fixed point.
4911 CoE. — Let m = m, then the equation of the point of
contact of the tangent joining the points ma + liB and ml'B-\-G
(4909) (e and/, Fig. 78) will be
m'A-\-2mkB-\rC = ().
4912 If ill Fig- (78) the trilinear coordinates of the points
4 R
674 ANALYTICAL OONtCS.
A, B, G are ii\, y^, z„ x.„ 7/2, ^2, ^h. y.s, ^3, the coordinates of tlie
point of contact p of the tangent defined by m will be
'in\ + 2mJcx.2 + ^3, m^T/i + 2?ri/i;?/2 + Va, mh^ + 2 » Jj^a + z-^,
and the tangent at j^ divides the two fixed tangents in the
ratios h : m and mh : 1, by (4909).
4913 Note.— The equation f7 or $ (A, ^, v) = (4G65) expresses the
conditiou that Xa+fift + i^y shall touch a certain conic. When U is about
to break up into two factors, the minor axis of the conic diminishes (Fig. 79).
Every tangent that can now be drawn to the conic passes very nearly
through one end or other of the major axis. Ultimately, when the minor
axis vanishes, the condition of the line touching the conic becomes the con-
dition of its passing through one or other of two fixed points A, B. In this
case, Z7 consists of two factors, which, put equal to zero, are the equations
of those points. The conic has become a straight line, and this line is
touched at every point by a single tangent.
4914 If TJ and V (Fig. 80) be two conies in tangential
coordinates, hJJ-[-TJ' is then a conic having for a tangent
every tangent common to Z7 and TJ' ; and kU-\-AB is a conic
having in common with U the two pairs of tangents drawn
from the points A, B.
The conic U' in this case merges into the line AB, or,
more strictly, the two points A, B, as explained in (4913).
4915 If either hJJ^JJ' or hU+AB breaks np into two
factors, it represents two points which are the opposite ver-
tices of the quadrilateral formed by the four tangents.
ON THE INTERSECTION OF TWO CONICS.
INTRODUCTORY THEOREM.
Geometrical meaning' of ^{ — 1).*
4916 In a system of rectangular or oblique i)lanc coordinates, let the
operator \/— I prefixed to an ordinate ij denote the turning of the ordinate
about its foot as a centre through a right angle in a plane perpendicular to
the plane of xy. The repetition of this operation will turn the ordinate
* [The fiction of imaginary lines and points is not ineradicable from Geometry. The
theory of Quaternions removes all imaginarincss from the symbol V -\, and, as it appears
that a partial application of that theory presents the subject of Projection in a much clearer
light, 1 have here introduced the notion of the multiplication of vectors at right angles to
each other.]
ON THE INTERSECTION OF TWO GONICS. 675
through another right angle in the same plane so as to bring it again into
the plane o£ xy. The double opei'ation has c onv ert ed y into —y. Bu t the two
operations are indicated algebraically by v — 1 . -s/ — 1 .y or (V —lYy = —y,
which justifies the definition.
It may be remarked, in passing, that any operation which, being per-
formed twice in succession upon a quan tity, changes its sign, offers a con-
sistent interpretation of the multiplier v/ — 1.
4917 With this additional operator, borrowed from the Theory of
Quaternions, equations of plane curves may be made to represent more
extended loci than formerly . Con sider the equation a^ + y^= dr. For values
of » < a, we have y = zk Va'—x^, and a circle is traced out . For values of
x>a, we may write y = ±i \/x^ — a^, where i = v/ — 1. The ordinate
\/x' — c^ is turned through a right angle by the vector i, and this part of the
locus is consequently an equilateral hyperbola having a common axis with
the circle and a common parameter, but having its plane at right angles to
that of the circle. Since the foot of each ordinate remains unaltered in posi-
tion, we may, for convenience, leave the operation indicated by i unperformed
and draw the hyperbola in the oi-iginal plane. In such a case, the circle may
be called the principal, and the hyperbola the supplementary, curve, after
Poncelet. When the coordinate axes are rectangular, the supplementary
curve is not altered in any other respect than in that of position by the
transformation of all its ordinates through a right angle ; but, if the coordi-
nate axes are obliqiie, there is likewise a change of figure precisely the same
as that which would be produced by setting each ordinate at right angles to
its abscissa in the xy plane.
In the diagrams, the supplementary curve will be shown by a dotted line,
and the unperfoi'med operation indicated by i must always be borne in mind.
For, on account of it, there can be no geometrical relations between the
principal and supplementary curves excepting those which arise from the
possession of one common axis of coordinates. This law is in agreement
with the algebraic one which applies to the real and imaginary parts of the
equation x^ — (iyy = al When y vanishes, x = a in both curves.
If either the ellipse b^x^ + a-y'^ = d-b- or the hyperbola b-x'' — a-y' = d-b-
be taken for the principal curve, the other will be the supplementaiy curve.
It is evident that, by taking diflTerent conjugate diameters for coordinate
axes, the same conic will have corresponding diflFerent supplementary curves.
The phrase, "supplementary conic on the diameter DD," for example, will
refer to that diameter which forms the common axis of the principal and
supplementary conic in question.
4918 Let us now take the circle x~ + y- = d^ and the right line x = h.
When & is > a, the line intersects the su pplementary right hyperbola in two
points whose ordinates are ±^' v &' — o.^. By increasing b without limit, we
get a pair of, so-called, imagiyiary points at infinity. These lie on the
asymptotes of the hyperbola, and the equation of the asymptotes is
(^x + iy)(x-iy) = 0. _ _
We can now give a geometrical interpretation to the statements m (4/'20).
The two lines drawn from the focus of the conic b'-x' + a^y- = d'b'' to the
" circular points at infinity " make angles of 45° with the, major axis, and
they touch the conic in its supplementary hyperbola b'^x^—d' {iy)- = d'lt\
An independent proof of this is as follows.
Q1& ANALYTICAL GONICS.
Draw a tangent from S (Fig. 81) to the supplementary liypei'bola, and
let a;, y be the coordinates of the point of contact P. Tiicii
^ = ^, (115^0) = _-f^; and >j = A s/(^^-a^) - ^'
by the value of x. Also
SN = x-CS= ,^'f ,.. - y^F^'
Therefore y = SN, therefore SP makes an angle of 45'^ with CN.
The following results are required in the theory of projec-
tion, and are illustrated in figures (82) to (86). Two ellipses
are taken in each case for principal curves, and the supple-
mentary hyperbolas are shown by dotted lines. As the planes
of the principal and supplementarj^ curves are really at right
angles, the intersections of the solid lines with the dotted are
only apparent. The intersections of the solid lines are real
points, while the intersections of the dotted lines represent
the imaginary points.
4919 Two conies may intersect —
(i.) in four real imints (Fig. 82);
(ii.) in ttvo real and two imaginary ])oints (Fig. 83) ;
(iii.) in four imaginary i^oints (Fig. 84).
[When the two hyperbolas in figures (83) and (84) are similar and
similarly situated, two of their points of intersection recede to infinity (Figs.
85 and 8G). Hence, and by taking the dotted lines for principal, and the
solid for supplementary, curves, we also have the cases]
(iv.) in ttvo real finite 2^0 ints and two imaginary points at
infinity ;
(v.) in tivo imaginary finite points and two imaginary
points at infinity ;
(vi.) in two imaginary finite points and two real points at
infinity ;
(vii.) in ttvo real finite points and two real points at
infinity.
4920 Given two conies not intersecting, or intersecting in
but two points, to draw the two supplementary curves which
have a common chord of intersection conjugate to the
TEE METHOD OF PBOJEGTION. 677
diameters upon whicli tliey are described, or in other words,
to find tlie imaginary common chord of the conies.
Poncelet has shewn by geometrical reasoning (Proprietes des Projectlves,
p. 31) that such a chord must exist. The following is a method of deter-
mining its position —
Let (abcfgh'^xi/iy = and (a'b'c'fg'li''^xyiy = (i.)
be the equations of the conies G, C (Fig. 89), the coordinate axes being
rectangular. Suppose PQ to be the common chord sought. Then the
diameters AB, A'B' conjugate to PQ bisect it in B, and the supplementary
curves on those diameters intersect in the points P, Q. Now, let the coor-
dinate axes be turned through an angle B, so that the y axis may become
parallel to PQ, and therefore also to the tangents at A,B, A', B' . This is
accomplished by substituting for x and y, in equations (i.), the values
XG0s9 — y &\nd and 7/ cos + a; sin 0.
Let the transformed equations be denoted by {ABGFGH'^xyiy = and
{A'B'G'F'G'H''^xyiy = 0, in which the coefficients are all functions of 0,
excepting c, which is unaltered. Solving each of these equations as a quad-
ratic in y, the solutions take the forms
y = a,-(; + /3±\//* {x^ — 2]px + q), y = a'x + fV ± \^fJ.' (x- — 2p'x + q) ...(n.),
with the values of a, 13, /j., p, q given in (4449-53), if for small letters we
substitute capitals. Thus, a, /3, fi, p, q are obtained in terms of and the
original coefficients a, h, h,f, g, Ji.
Now, the coordinates of D being 4 = ON, n = DN, we have // = aE + l^
and jy = a'^ + /3', therefore a^ + ft = a^ + fi' (iii-)-
The surd in equations (ii.) represents the ordinate of the conic conjugate
to the diameter AB or A'B'. For values of x in the diagram > OM and
<0B, the factor \/—l appears in this surd, indicating an ordinate of the
supplementary curve on AB or A'B'. Hence, equating the values of the
common ordinate PD, we have
I, (e-2p-Uq) = 1^' (e-2p'Uq) (iv.).
Eliminating t, between equations (iii.) and (iv.), we obtain an equation for
determining 0; which angle being found, we can at once draw the diameters
AB, A'B'.
THE METHOD OF PROJECTION.
4921 Problem. — Given any conic and a right hne in^ its
plane and any plane of projection, to find a vertex of projec-
tion such that the line may pass to infinity while the conic is
projected into a hyperbola or ellipse according as the right
line does or does not intersect the given conic ; and at the
same time to give any assigned proportion and direction to
the axes of the projected conic.
678 ANALYTICAL CONIGS.
Analysis. — Let HCKD be the given conic, and BB the right line, in
Fig. (87) not intersecting, and in Fig. (88) intersecting the conic. Draw
UK the diameter of the conic conjugate to BB. Suppose to be the
required vertex of projection. Draw any plane EGGD parallel to OBB,
intersecting the given conic in CB and the line UK in F, and draw the
plane OEK cutting the former plane in E, F, G and the line BB in A ; and
let the curve EGGD be the conical projection of HCKD on the plane parallel
to OBB.
By similar triangles,
FF^ _ OA ^FG_^aA . EF.FG ^ OA^ .^.
HF~HA^^ FK AK' "EF.FK HA.AK ^ ^'
Let a, ft be the semi-diameters of the given conic pai'allel to UK and GB ;
+1, ^^'' - (^ ■ ^^'' - P^'-SA.A K .^.
^"^ EF.FK ~ a'' •- EF.FG~ a'.OA' ^^"
Now, since parallel sections of the cone are similar, if the plane of ECKD
moves parallel to itself, the ratio on the right remains constant ; therefore, by
(1193), the section EGGK is an ellipse in Fig. (87) and an hyperbola in
Fig. (88). Let a, h be the semi-diameters of this ellipse or hyperbola
parallel to EG and GD, that is, to OA and BB ; then, by (2),
^ = £ Ej^, ... OA' = $^HA..AK (3).
a^ ct OA- b-a-
But ^EA.AK= AB-, where AB in Fig. (88) is the ordinate at A of the
a
given conic, but in Fig. (87) the ordinate of the conic supplementary to the
given one on the diameter conjugate to BB. Therefore
AO'=^AB' (4).
Hence AO, AB are parallel and propoi'tional to a and b. And, since AB
is given in magnitude and direction, we have two constants at our disposal,
namely, the ratio of the semi-conjugate diameters a and b and the angle
between them, or, which is the same thing, the eccentricity and the direction
of the axes of the ellipse or hyperbola on the plane of projection.
4922 The construction will be as follows : —
Determine the point A as the intersection of BB ivith the
diameter HK conjugate to it. Choose any j^Iane of inojection,
and in a plane through BB, parallel to it, measure AO of the
length given by equation (3) or (4), maJiing the angle BAO
equal to the required angle hetioeen a and b. will he the
vertex of projection^ and any plane IJM.'^ parallel to OBB will
serve for the plane of pirojection.
4923 OoR. T.— 'If AO = AB, the projected curve in Fig. (88)
will in every case be a right hyperbola.
TBI] METHOD OF PROJECTION. 679
4924 CoE. 2. — If BAO is a right angle, tlie axes of tlie pro-
jected ellipse or hyperbola are parallel and proportional to
AO and AB. Hence, in this case, the eccentricity of the
hyperbola will be e = OB : OA.
4925 Cor. 3. — If AO = AB and BAO = a right angle,
the ellipse becomes a circle and the right hyperbola in Cor. 1
has its axes parallel to AO and AB.
4926 To project a conic so that a given point in its plane
may become the centre of the projected curve.
Tahe for the line BB the polar of the given jjoiiit, and con-
struct as in (4922). For, ifV be the given point, and BB its
polar {Fig. 87 or 88), p the projection ofP ivill have its polar
at infinity, and ivill therefore be the centre of the projected
ellipse or hyperbola, according as P is within or luithout the
original conic.
4927 To project two intersecting conies into two similar
and similarly situated hyperbolas of given eccentricity.
Take the common chord of the conies for the line BB {Fig.
88), and project each conic as in (4922), employing the same
vertex and plane of projection. Then, since the point A and
the lines AB and AO are the same for each projection, corres-
ponding conjugate diameters of the hyperbolas are parallel and
proportional to AO and AB ; therefore, Sfc.
4928 To project two non-intersecting conics into similar
and similarly situated ellipses of given eccentricity.
Tahe the common chord of a certain two of the supple-
mentary curves of the conics (4920), in other words, the
imaginary common chord of the conics, for tlie line BB, and
proceed as in (4927).
4929 To project two conics having a common chord of
contact into two concentric, similar and similarly situated
hyperbolas.
Tahe the common chord for the line BB, and construct as in
(4922). The common pole of the conics projects into a common
centre and the common tangents into common asymptotes.
680 TBILINEAB ANALYTICAL CONICS.
4930 To project any two conies into concentric conies.
Find the common pole and polar of the given conies by
(4762), and take the common polar for the line BB in tlte
construction of (4922). The common pole projects into a
common centre.
*xuOX. Ex. 1. — Given two conies having double contact with each other,
any chord of one which touches the other is cut harmonically at the point of
contact and where it meets the common choi'd of contact of the conies.
\_Salmons Conic Sections, Art. 354.
Let AB be the common chord of contact, PQ the other chord touching
the inner conic at G and meeting AB produced in D. By (4929), project
AB, and therefore the point D, to infinity. The conies become similar and
similarly situated hyperbolas, and C becomes the middle point of PQ (1189).
The theorem is therefore true in this case. Hence, by a convei'se projection,
the more general theorem is inferred.
49oA Ex. 2. — Given four points on a conic, the locus of the pole of any
fixed line is a conic passing tlirough the fourth harmonic to the point in
which this line meets each side of the given quadrilateral. [Ibid., Art. 354.
Let the fixed line meet a side AB of the quadrilateral in D, and let
AGBD be in harmonic ratio. Project the fixed line, and therefore the point
D, to infinity. C becomes the middle point of ^ J5 (1055), and the pole of
the fixed line becomes the centre of the projected conic. Now, it is known
that the locus of the centre is a conic passing through the middle points of
the sides of the quadrilateral. Hence, projecting back again, the more
genei'al theorem is inferred.
4i70o Ex. 3. — If a variable ellipse be described touching two given
ellipses, while the supplementary hyperbolas of all three have a common
chord AB conjugate to the diameters upon which they are described ; the
locus of the pole of AB with respect to the variable ellipse is an hyperbola
whos^e sup})lementary ellipse touches the four lines CA, CB, C'A, C'B, where
C, C are the poles of AB with respect to the fixed ellipses.
(Salmon, Art. 355.)
Pkoof. — Project AB to infinity and the three ellipses into circles. The
poles P, C, C become the centres 2', c, c of the circles. The locus of p is a
hyperbola whose foci are c, c . But the lines Ac, Be now touch the supple-
mentary ellipse of this hyperbola (4918). Therefore, projecting back again,
we get AC, BC touching the supplementary ellipse of the conic which is the
locus of P. Similarly, AC, BC touch the same ellipse.
4934 Any two lines at riglit angles project into lines wliicli
cut liannonically tlie line joining the two fixed points which
are the projections of the circnlar points at infinity.
Proof.— This follows from (4723).
INVARIANTS AND COVAEIANTS. 081
4935 The couverse of tlie above proposition (4931), wbicli is the theorem
in Art. 356 of Salmon, is not nniversallj true in any real sense. If the lines
drawn through a given point to the two circular points at infinity form a
harmonic pencil with two other lines through that point, the latter two are
not necessarily at right angles, as the theorem assumes.
The following example from the same article is an illustration of this —
Ex.— Any chord BB (Fig. 88) of a conic HCKD is cut harmonically by
any line PKAH through P, the pole of the chord, and the tangent at K.
The ellipse BKB here projects into a right hyperbola ; B, B project to
infinity. The harmonic pencil formed by PK and the tangent at K, KB and
KB projects into a harmonic pencil formed by 2^^ fii^d the tangent at h, kh
and l-b, where 6, h are the circular points at infinity : but j>fc is notat right
angles to the tangent at J: of the right hyperbola. The harmonic ratio of the
latter pencil can, however, be independently demonstrated, and that of the
former can then be inferred. (Note that h is G in figure 88.) _
If we may suppose the ellipse to project into an imaginary circle havmg
points at infinity, the imaginary radius of that circle may be supposed to bo
at right angles to the imaginary tangent. The right hyperbola, however, is
the real projection which takes the place of the circle in this and all similar
instances ; and it is only in the case of principal axes that the radius is at
I'ight angles to the tangent.
INVARIANTS AND COVARIANTS.
4936 Let u ={al>cfr/]tjxyzy, u' = {a'h'c'fg'h'Xxyzf
be two conies as in (4401) witli the notation of (1620).
The three values of /.-, for which ku + n' = represents two
right lines, are the roots of the cubic equation
4937 AA:-^+0A--+0'A'+A' = 0,
4938 where A = uhc-\-'lfgh—af- — hg--ch\
4939 = Aa'^Bh'-^Cc'^2Ff'-^2Gg-^'lHh',
and A = bc-f\ F=gh-af, &c. (46(;5)
For the values of A' and 9' interchange a with a, h with h\
&c.
ka+a', kh + h', kg + g'
kh + h', kh+h\ kf+f
Proof. — The discriminant of ku + u, which
must vanish (4661), is evidently the determi-
nant here written, and it is equivalent to the
cubic in question.
• + g', ¥+f'^ ^"'+^'
4940 A, e, e', and a' are invariants of the conic ku-\-i('.
4 s
682 ANALYTICAL CONICS.
That is, if the axes of coordinates be transformed in any
manner, the ratios of the four coefficients in (4937) are
unaltered.
Proof. — The transfoi-mation is effected by a linear substitution, as in
(1704). Let «, «' thus become v, v'. Then hu + u' becomes Icv + v', and
k is unaltered. If the equation ku + u' = represents two right lines, it will
continue to do so after transformation ; but the condition for this is the
vanishing of the cubic in h ; aud k being constant, the ratios of the coeffi-
cients must be unalterable.
4941 The equation of the six lines which join the four
points of intersection of the conies u and u is
Proof.— Eliminate k from (4937) by A-h + «' = 0.
4942 The condition that the conies ir and vf may touch is
(&&-9SAy = 4 (©•^-3A0')(0'--3A'0),
4943 or 4A0 3+4A'0^+27A2A'^-18AA'00-0-0^
PuoOF. — Two of the four points in (3941) must coincide. Hence two out
of the three pairs of lines must coincide. The cubic (4937) must therefore
have two equal roots. Let a, a, ft be the roots ; then the condition is the
result of eliminating a and ft from the equations
A ('2a +/-5) = -e, A (o- + 2a/3) = G', Aa'/S = -A' (400).
4944 The expression (4943) is the last term of the equation
whose roots are the squares of the differences of the roots of
the cubic in /.-, and when it is positive, the cubic in h has two
imaginary roots ; when it is negative, three real roots ; aud
when it vanishes, two equal roots.
Proof.— By (543) or (579). The last term of /(a;) in (543) is now
= 27F(a) F(ft), a, ft being the roots of 3Ax' + 2ex + e' = 0. When this
term is positive, f(x) has a real negative root (409), and therefore F (x) has
then two imaginary roots ; for, if (a — &)'•' = —c, a — & = ic, and a and b are
both imaginai'y. When the last term of /(;c) is negative, all the roots of
/ (x) are positive, aud therefore the roots of F (.^) are all real.
INVARIANTS OF PARTICULAR CONICS.
4945 ^y\wn u = aiv^-\- by- + cz^ and n = a^ -{- 1/- -\- :r ,
A = afjc, = bc-^-ca + ab, 0' = a-{-b-{-c, A '= 1.
INVARIANTS AND GOVABIANTS. 683
4946 When u= {abr.fghXxijzy and u'=x'+i/^ + z\
e = A + B-\-C, 0' = rt+6+e, A'=l.
4947 When u = x^ + if - r^ and n = {x - a)- + (// -(^f- s\
A = -r\ A' = -^s\
4948 The cubic for k reduces to
(k-^l) {s'k'+{r'+s'-a'-fi') k-{-r'} = 0.
4949 When
u = h\e' + aY-a%'' and v' = (->'-«)' + (^-/3)2-r%
A = - a'b\ = (rb' { a' +/3- - a' - W - r^ } ,
0' = u'^'-\-h'a^-a'¥-r' («'+6^), A' = -rl
4950 When u = if-4^mx and u' = {x-a)-^{y-^Y-r\
A = —^m\ = —4m (a + m), 0' = P^—4ima—7'\ A'= — r'.
4951 When ?(,= {(^^(^fg^K^^^y^y and ?f' = ^^ + 2rt37/ cos w + ?/,
A, A'=0, 0=c(..+6)-f-g-+2(/g— c/0cos6>,
0' = (• sin^co.
Hence the following are invariants of the general conic,
the inclination of the coordinate axes being w.
4952 abc-\-2f{j:h-af'-bcr^-ch' ^ A p.
c shr 0) 0'
4953 c (a-\-b)-f--^'-\-2 ifg-ch) cos 6) ^ .2),
c siu" 0) 0'
4954 "^^ (3). and "+b--^l"^-o^^ (4).
snroi snvQ)
For these are what (1) and (2) become when the axes are
transformed so as to remove /and g.
684 ANALYTICAL CONIC S.
If the origin be unaltered, c is invariable, and transforma-
tion of the axes will then leave invariable
4956 ^^ffA-qf-y- .^j^j . f+ff--^fffcos6) ^
siu" CO siu"- (t)
as appears by subtracting (3) from (1) and (2) from (4).
4958 Ex. (i.)— To find the evolute of the conic h'X'-{-ah/
= crlr. See also (4547).
Pkoof. — Denote tlie conic by n, and by u' the hyperbola c"xy -\-lh/o: — ci?x'y
(4335), which intersects « in the feet of the normals drawn from x'y'. Two of
these normals mnst always coincide if x'y' is to be on the evolute. ti, and n'
must therefore touch. We have
A = -a'h\ e = 0, e' = -ft-i" (aV + ty-c*), A'= -2a^h-c\vy.
Substitute in (4942), and the equation of the evolute is found to be
(a-x'' + bY-cy-h27a'h'o'xhf = 0.
4959 Ex. (ii.) — Similarly the evolute of the parabola is
obtained from
u = y- — 4<mx, u' = 2a;?/ + 2 {2m— x") y — imy',
A = —4<m\ e = 0, e' = -4 (2m— x), A' z= 4my,
producing the equation 27ruy- = 4 (x — 2my. See also (4540).
4960 Ex. (iii.) — The locus of the centre of a circle of radius
E, touching the conic h\'-\-(rif—a-h\ is called a par all d to
the conic. Its equation is
Ii;V-27^V {r (a'Hi'O + id'-2K-) x' + {2a'-V-) f]
+ R' [ c' (a" + 4a-6- + h') - 2c- (a" - n^lr + 3&*) x- + 2c- {^a'-a-V + V) i/
+ (a* - C)^'^ + G60 ;o' + {Cm' - Ga'b' + h') y' + {(Sa' - lOcr't^ + (j?>0 xhf }
+ E- { - 2ar}rc' (rr + ^0 + 2c- (3rt'- a-h' + Z/') x- - 2r {a'-aV + 3h') i/'
-{(ja'-10aV- + Gb'){b\c' + dY) + {W^-6a*b--6a;'b' + 4b'') xSf
+ 2 {a--2b') 6V + 2 (&--2a') aV-2 {a'-a'b-^Zb') xSf
-2(3a*-a-/r + 6^).^y|
+ (/rV + aV--a'Z>-j-{(-«-c)' + r} {(:i! + c)- + r} = 0.
PROor.— Tf the curves in (4940) be made to touch, o/5 will be a point on
the curve pai-allel to u at a distance r. Therefore put the values of A, O, 9',
and A' in equation (4042). Itiahnon, p. 325.
INVARIANTS AND COVABIANTS. 685
4961 When u of (4936) represents two right lines. A'
vanishes, and
4962 0' = is the condition that the two lines should
intersect on u ;
4963 9 = is the condition that the two lines should be
conjugate with regard to u.
Proof. — Transform n' = into 2xy = 0, so that the axes x, y are the
right lines. This will not affect the invariants (4940). We now have, by
(4937), A' = 0, e = 2(/^-c70, e' = -c.
c = makes « pass through the origin xy ; fg = ch makes x and y conjugate.
For in (4671), if Xx + fxy + v becomes y = 0, then \ = v = 0, and the pole
is given by H : B : F. But a; = a = at the pole, therefore II =.
fg-ch = 0.
4964 The condition that either of the lines in u' should
touch u is, by (4943),
0^ = 4A0' or AB = 0,
with the above values of O and 9'.
4965 The equation of the two tangents to u, when Xa3 +^t^ + v
is the chord of contact, is, with the notation of (4665),
u^ (X, fjL, v) = {\a;-{-iii/-{-vzf A.
Proof. — The conic of double contact with u, ku + (\x + fiy + i'y- (4G99),
must now become two right lines. In (4937) A' = and Q' = 0, therefore
A-A + 6 = 0. But = <& (\, i-i, v). Hence eliminate Jc.
4966 Cor.— Taking the Hne at infinity <ix + bjj + C^, we obtain
the equation of the asymptotes (4685).
The invariant 9 of the conic l-u + u' vanishes —
4967 (i-) Whenever an inscribed triangle of u is self-con-
jugate to u.
4968 (ii-) Whenever a circumscribed triangle of u is self-
conjugate to II .
4969 0' vanishes under similar conditions, transposing u
and li in (i.) and (ii.)
Proof.— (i.) u becomes ax- + hy- + cz"- (47G5), and/= g = li — 0. There-
fore e in (4937) vanishes if a'=l'=c = (); i.e., if u' is of the form
f'yz+g'zx+h'xy (4724).
686 ANALYTICAL CONIGS.
(ii.) In this case, /'= 9'= h'= and vanishes if hr = /", &c., i.e., if
the line x = touches u, &c.
4970 If ?') u' be two conies, and if 6'^ = 4A9', any triangle
inscribed in u' will circumscribe w, and conversely.
Proof, — Let u = x^ + 1/ + z^—2yz — 2zx—2icy and u = 2fyz + 2gzx + 2hzi/,
both referred to the same triangle, (4739) and (4724). Then
A=-4, Q = 4.(f+g + ]i), Q'^-(f+g + hy, A'=2fg1i',
therefore 6^ = 4A6', a relation independent of the axes of reference (4940).
4971 Ex. (i.) — The locus of the centre of a circle of I'adius r, circum-
scribing a triangle which circumscribes the conic h-x-+a-i/ = a'b-, is
(x"" + 2/- - a- - Z>- + r-)2 + 4 [ h'x- + ahf - a-lr - r (ft- + b')} = 0,
from 9- = 4A9' and the values in (4949).
4972 Ex. (ii.) — The distance between the centres of the inscribed and
circumscribed circles of a triangle is thus found, by employing the values of
e, e', and A in (4947), to be D= ^/(/-±2jt'), as in (936).
4973 The tangential equation of the four points of intersec-
tion of the two conies u = 0, u = is
with the meanings
4974 U={ABCFGHJ\iivy; (4GG4)
U'= {A'B'C'F'G'H'JXiivf.
4976 V = {A"B"C"F"G"H"J\iivY.
4977 A = bc-f\ &c.; A' = h'c'-f\ &G.,
as in (4665), and
4978 A" = bc'+b'c-2fr, F" = gh'-\-<r'h-af-a%
4979 B"^ca'+c'a-2si^\ G"^hf^h'f-hi>:' -h'^,
4980 C" = ab'-\-a'b-2hh', H"=f<r'-\-fo'-cf/-c'h.
Proof. — The tangential equation is the condition that Xa + fip + yy may
pass through one of the four points of intersection of ?t and «'. The tan-
gential ecjuation of the conic u + Jiit' is obtained by putting a + ha' for a, &c.
in U (4GG5), and is U+kV + JrU' =: 0. The tangential equation of the
envelope of the system is V" = 4UU' (4911). This is the condition that the
line (\, /i, v) may pass through the consecutive intersections of the conies
obtained by varying A-. Put these conies always intersect in the same four
points. The above is thcrefoi'e the tangential equation of the four points.
INVARIANTS AND COVABIANTS. 681
4981 The equation of the four commou tangents of two
conies a, u is
where F = (a"h"c"f"^"h" X ^^y)'->
and a'' = BC'-\-B'C-2FF\ &c.,
f" = GH-\-G'H-AF'-A'F, &c.,
as in (4978-81).
Proof. — This is the reciprocal of the last theorem. ZJ+^ZJ' is a conic
touching the four common tangents of the conies U and TJ'. The trilinear
equation formed from this will, by (4007), be u^-^-hT + k'u'iX' =■ 0. The
envelope of this system of conies is the equation above, which must therefore
represent the four common tangents.
The curve F passes through the points of contact of w and lo with the
locus represented by (4981).
4982 Hence the eight points of contact of the two conies
with their common tangents lie on the curve F.
4983 The reciprocal theorem from equation (4973) is —
The eight tangents at the intersections of the conies envelope
the conic V.
4984 F = is the locus of a point from which the tangents
to the two given conies u, u' form a harmonic pencil.
Proof. — Putting 7 = in (4081), we get a quadratic of the form
ad' + 2]iu^+bl3^' =■ 0, which determines the two points in which the line y is
cut by tangents from o', /3', y'. Let the similar quadratic for the second conic
be a'cr + 2h'al3 + h'(3- = 0. Then, by (1064), ab' + a'b = 2hh' is the condition
that the four points may be in harmonic relation. This equation will be
found to pi'oduce F = 0.
4985 The actual values of a, h, h, suppressing the accents
on a', )3', y\ are
CP'-\-Bf-2Ffiy, G^y-^Fya-Ca^-Hy\
Af-\-Cd'-2Gya',
and similarly for a, h', h', with A' w^ritten for A, &c.
4986 If the anJictrmonic ratio of the pencil of four tangents
be given, the locus of the vertex will be F" = Jcuu'. If the
given ratio be infinity or zero, the locus becomes the four
common tangents in (4981).
ANALYTICAL CONICS.
4987 V == is tlic envelope of a conic evciy tangent of
which is cut harmonically by the two conies u, u' ; i.e., the
equation is the condition that Xa+^/S + vy should be cut har-
monically by the two conies.
Proof. — Eliminate y between the line (\, /u, v), and the conies n and m'
separately, and let AiC + 2Hal^j + Bjy' = and A'a' + 2ircily + ]l'ft- = stand
for the resulting equations. Then, by (10G4), AB' -\-A'1j = 21111' produces
the equation V = 0, which, by (4GGG), is the envelope of a conic.
4988 The actual values of A, H, B are respectively
and similarly for A\ H', B', with a' for a, &c.
4989 F'^ = 4AA'?n/ is a covariant (1629) of the conies u, u'.
For the four common tangents are independent of the axes of reference.
4990 C7=0 and V = (4973) are both contravariants
(1814) of u and n\
Proof. — For ?7= is the condition that Xo + /j/3 + ry = shall touch the
conic u ; and V = is the condition that the same lino shall be cut har-
monically by u and u'; and if all the equations be transformed by a recipro-
cal substitution (1813, '14), the right line and the conditions I'emain
unaltered.
4991 Any conic covariant with u and ii' can be expressed in
terms of u, u, and F ; and the tangential equation can be
expressed in terms of U, U' and V.
4992 Ex. (1). — The polar reciprocal of u with respect to u'
is Qn = F.
Proof. — Referring u, n' to their common self -conjugate triangle,
n = ax- + bij- + cz\ It' = .c'^ + y- + z-,
F = a(h + c) x' + h (c + a) if + c (» + h) z-.
The polar of ^, r], ^ with respect to «' is ^x + riy + i^z, and the condition that
this may touch u is hc£i'^ + caTi^ + abi,'^ = (4664), or, which is the same
thing, (bc + ca + ab)(x^ + y^+z^) = P or Gw' = P (4945).
4993 Ex. (2). —The enveloping conic V in (4987) may also
be written
0//' + 0'// = F.
INVARIANTS AND COVAlilANTS. 689
Proof. — With the same assumptions as in Ex. (1), V in (-i'.To) becomes
(b + c) X'+(c + a) fM-+ (a + h) r- = 0. The triliuear equation is, therefore,
by (4667),
(c + a) (a + b) X- + {a+b){b + c) if + (i + c) (c + a) £' = U,
or (be + ca + ab) {x- + if + z') -\- {a -{- b + c) (ax' + bif + cz") = P.
4994 Ex. (3). — The condition that F may become two right
hnes is A A' (00' -A A') = 0.
Proof.— Referring to Ex. (1), A = be, B = ca, C = ab, F= G = H=0,
A'= B'= C'= 1 ; therefore, in (4981), a" = B + G = a(b+c), &c. Heuce
the discriminant A of F = ahc (b -{- c)(c + a)(a-\-h),
or abc | (a + & + c)(&c + ca + a?;) — ate | = the above, by (4945).
4995 To reduce the two conies u, u to the forms
By (4945), o, /3, y will be the roots of the cubic
AF-eF-l-e7.--A' = (1),
and cif, //, ?:^ will be found in terms of x, v' and F, by solving
the three equations ,1'- + //- + ::'^= n, a,r-f-/3y^ + yr == u and (by
4994), a(i3 + y).r+/3(7 + a)/ + y(«+|3)^:-^=:F (2).
4996 Ex. (1) : Given ;t;- + r + -2(/ + 2i^ + 3 = 0; a;- + 2^H4;/ + 2a; + 6 = ;
to be reduced as above. To compute the invariants, wef take
and
therefore
and
Therefore (4938, '9) A = 1, 6 = 6, 6'= 11, A'= 6. The roots of equa-
tion (1) are now 1, 2, 3. Therefore (2) becomes 5X- + 8r^ + 9Z- = T.
Computing P also by (4981) with the above values of A, B, &c., we get the
three equations as under, introducing z for the sake of symmetry,
X-+ Y-+ Z-= a;-+ 7/+ 32-+ 2//z+ 22a;,
X- + 2r- + 3;;-= xr + ^f+ 6.-'-+ 4(/2+ 2zx,
5X- + 8YH9i^- = 5a!2 + 82/- + 22r + 16!/2 + 10za;,
The solution gives X=x + \, Y = \j -\-l, Z = I, and the equations in the
forms required are (x + l)- + (y + iy + l = 0, (a; + l)'' + 2 (^ + l)- + 3 = 0.
4997 Ex. (2). — To find the envelope of the base of a triangle inscribed
in a conic u so that two of its sides touch a.
4 T
a
= 1
= 1
b
1
2
c
3
6
/
1
2
y
I
1
h
(J in the first equation,
in the second.
A
= 2
= 8
B
2
5
G
1 ■
2 •
F
-1 -
-2 -
G
-1
-2
H
1 in the first equation.
2 in the second.
690 ANALYTICAL CONICS.
Let « = X- + If -\- z' — 2yz — 2zx — 2xy — 2hkxy,
and «' = 2fy3 + 2gzx + 2hxy,
X and y being the sides touched by u. Then u + hi' will be a conic touched
by the third side z. By finding the invariants, it appears that 9^^ — 4Ae'
= 4AA'A;, whence k is determined, and the envelope becomes
Compare (4970).
4998 The tangential equation of the two circular points at
infinity (4717) is
Proof. — This is the condition that Xx+fuy + v should pass through either
of those points, since x:^iy =■ c is the general form of such a line.
4999 U = being the tangential equation of a conic, the
discriminant of h U-{- TJ' is
Proof. — The discriminant of hJI-^- TJ' is identical in form with (4937),
but the capitals and small letters must be interchanged. Let then the dis-
criminant be AA;HeA;H©'A; + A' = 0. We have
A = A^ (4G70), e = (BG-F') A' + &c. = A'ai\ + &c. (4668) = A0'.
Similarly 6' = A'G, A' = A'\
5000 If ©, 0' be the invariants of any conic U and the pair
of circular points X^ + ^t'^ (4998) ; then G = makes the conic
a parabola, and 9' = makes it an equilateral hyperbola.
Proof.— The discriminant of kU+X^ + in^ is k'^A' + k (a + h) A + ab-h\
For, as above, A = A^ ; Q = A'aA + B'bA = (a + b) A since A' = B' = I,
C &c. = ; e' = {A'B'-m) G=G = ab-Jv'; and A' = 0. The rest fol-
lows from the conditions (4471) and (4474).
5001 The tangential equation of the circular points is, in
trilinear notation (see the note at 5030),
X-+/u,-4-»'"— 2/w,v cos A—2v\ cosB—2\fjL cos C.
Proof : A.^' + yir = 0, in Cartesians, shows that the perpendicular lot fall
from any point whatever upon any line passing through one of the points is
infinite. Therefore, by (4624).
5002 The conditions in (4689) and (4090), which make the
gcmoi-al conic a parabola or equilateral hyperbola, may be
obtained by forming 9 and 9' for the conic and equation
(5001) and applying (5000).
INVARIANTS AND GOVABIANTS. 691
5003 If 0'"' = 49, the conic passes througli one of the
circular points.
5004 When It in (4984) reduces to \^-\-ix\ that is, to the
circular points at infinity, F becomes the locus of intersec-
tion of tangents to n at right angles, and produces the equa-
tions of the director-circle (4693) and (4694).
5005 The tangential equation of a conic confocal with TJ is
5006 And if the left side, by varying k, be resolved into
two factors, it becomes the equation of the foci of the system.
Proof. — Since \^ + /x^ represents the two circular points at infinity (4998),
;^f/'_^\2-(.^2 _ Q^ jjy (4914), is the tangential equation of a conic touched by
the four imaginary tangents of Z7 from those points. But these tangents
intersect in two pairs in the foci of U (4720) ; and, for the same reason, in
the foci of ]cU+y-+iu-, which must therefore have the same foci.
If W + X^+fi^ consists of two factors, it represents two points which, by
(4913), are the intersections of the pairs of tangents just named, and are
therefore the foci.
5007 The general Cartesian equation of a conic confocal
with u = (4656) is
k'Au+k {C{a^'-^y')-2G.v-2Ft/-{-A+B}+l = 0.
Proof.— (5005) must be transformed. Written in full, by (4664), it
becomes (kA + 1) X^ + (kB + l) fir + kCi''. Hence, by (4667), the trilinear
equation will be
{(l-B + l) W-l-F'} a- + &c. = F- (BG-F') a- + kCcr + &c.
= k-a^cr + ]cCa' + &c., (4668)
and so on, finally writing x, y, 1 for ct, /3, y.
TO FIND THE FOCI OF THE GENERAL CONIC (4656).
(First Method.)
5008 Substitute in kU + XHA^'^ eithrr root of its discriminant
k-A' + k(a + b) A + ab— h^ = (5000), and it becomes re-
solvable into tioo factors (XxiH-/iyi-fv)(Xx.2+/iy2H-v). _ The
foci are Xiyi and x.^ya, real for one value of k and imaginary
for the other.
Proof.— By (5006) the two factors represent the two foci, consequently
the coordinates of the foci are the coefficients of X, /x, y in those factors.
692 ANALYTIGAL CONTGS.
(^Second MetJiod.')
5009 ^^cf' xy 1)6 afocuii; then, by (4720), the equation of an
iinaginari) tangent through that point is (^ — x)+i (») — y) =
or ^-hirj — (x-f-iy) = 0. Therefore substitute, in the tangential
equation (4665), the coefficients A = 1, /ti = i, v = — (x + iy),
and equate real and imaginarij parts to zero. The resultirig
equations for finding x and y are, with the notation of (4665),
5010 2(0.r-G)^ = A[..-6+v/{4/r+(«-6)'^}].
5011 2{Cn-Ff = ^[h~aJrV \^^^'^{^i-WW
5012 If the conic is a parabola, = 0, and the coordinates
of the focus are given by
(F2_,_ Q') cv = FH-\-l (A-B) G,
(F^+G^) y = GH-i (A-B) F.
5013 Ex. — To find the foci of 2x' + 2xy + 2>/ + 2x = 0. By the first
method, we have
a, h, c, f, g, h '\ from which A = — 2. The quad-
= 2, 2, 0, 0, 1, 11 I'atic for k is
and A, B, (\ F, G, Hi /rA'H4^-A + 3 = (2^— 3)(2A— 1) = 0,
= 0, -1, 3, 1, —2, J therefore ^- = f or |.
Taking |, hU+X' + /Li' = :] (-/ + 3.'- + 2At>'-4A) +X' + m' = 0,
or 2X'-12v\-^i- + 9y' + 6fiy = 0.
Solving for X, this is thrown into the factors
{ 2\ + /xy2-3 (2+ v/2) V } { 2A-/iv/2-3 (2- x/2) y } .
Therefore the coordinates of the foci, after I'ationalizing the fractions, are
2- 72 v/2-1 , 2+72 72 + 1
8 ' 8- '^"'^ 3-' -T-
5014 Otlierwise, by the second method, equations (5010, '1) become, in
this instance, (3;i; + 2)^ = ± 2, (3/y — 1)- = ±2, the sohUion of which pro-
duces the same values of a; and y.
5015 When the axes are oblique, the coordinates x, y of a
focus are found from the equations
{C (.v+f/ c,oH(o)-F cosco-G]' = iA(\/i^-4./+2«-/)
[Cy-Ff Hiii^o) = \A (y/2_4./^2«+/),
where I ami J are the invariants (4955) and (4954) respec-
INVARIANTS AND GOVAEIANTS. 693
tively. The equations may he solved for x' = x -\-y cos a> and
y = J sin w, mhiclh are the rectangular coordinates of the focus
with the same origin and x axis.
Proof. — Following' the method of (5009), the imaginary tangent thi'ough
the focus is, by (4721), £— .r-f- (»? — ;(/)(cos w + i sin w). The two equations
obtained from the tangential equation are, writing Aa for BG — F", &c.
(4668),
X^—Y'-= —A (a + h — 2h cos w — 2a sin^ w), XY = A (h sinw — a sin w cos w) ;
where X = G {x + y cosc») — F cosu — G and Y = (Gy — F) sin u).
5016 If the equation of the conic to oblique axes be
a.v^ -f 2hoci/ + hif -}- c = ,
the equations for determining the foci reduce to
y {cV-\-y cos a>) _ cv(i/-\-cr cosw) c
acos(o—h bcosQ)—h ~ ab — h^
5017 The condition that the line Xa?+^?/ + v^ may touch the
conic u-\-(X'x-{-fiy-\-vzY is
U-\-(l) {fxv—ii'v, v\' — v\, Xji'—X'/j) = 0. (4656, 4936, 74)
5018 or {A+u')U=n:\ (4938)
where 20 = \' l\-\- fj,' [J^^v U,. (4674)
Proof. — Put a + \'^ for a, &c. in U of (4664). The second form follows
from the first through the identity
A^ (^/ — /uV, &c.) = UU' — lP.
5019 Otherwise, let F = u,.,x-{-Uy,y + u^,z, the polar of
X, y, z (4659), then the condition that P' may touch u-\-V"'^
becomes, in terms of the coordinates of the poles,
5020 (1 + "'') "' = ^^,x"^^,.y"^i>,,Z' (See 4657).
Pkoof. — If we put ;«^.,, »,,., u,,, from (4659), for \, ^, v in ?7 to obtain the
condition of touching, the result is A?^' ; and similar substitutions made in
n give A (0^,»" + &c.), therefore (50l8) becomes (l4-^'")«' = {(\>^.x" -\-kc).
5021 The condition that the conies
u + {}!x + /t/ + v',<;)^ u + {}!'x + /V + v'zf
may touch each other is
(A+ r)(A+ V") = (A ± n)^ (49B8-74)
694
ANALYTICAL CONIC 8.
Proof. — Make one of the common cbords
(X'x + ix'y + v'z) ± (\"x + /u' V + v"z)
touch either conic by substituting X' ± X" for X, &c. in (5018). The result
is (A+ t7')(Z7'± 2Ii-\-TJ") = (U'± U)-, which reduces to the form above.
5022 The condition, in terms of the coordinates of the poles
of the two lines, is found from the last, as in (5019), and is
(l+«')(l + <'") = ll±{<l>...v"+<t>,!/"+<l>..z")V-
5023 The Jacobian, J, of three conies it, v, iv, is the locus
of a point whose polars with respect to the conies all meet in
a point. Its equation is
(ha.^-\-h,i/-\-g\z, a2^+hi/+g2^, fh^+fhy-^g^z
Jh^v+b,ij-\-f\ z, Jh.v-\-b,i/+f, z, fh.v+b,2/-\-f, z
gi^'^-^fi!/+fi -» g2^v+fz!/+c.,z, g;^'-{-fij/+c, z
Proof. — The equation is the eliminant of the equations of the three
polai's passing thi'ough a point ^r)i, viz., ?t j.^ + ?t^7j + ««4 = 0, ViX + v^ri -{-i\i^ :=0,
wJ+w^f]-\-^tKi; = 0. See (4657) and (1600).
a^ B^ y^ fi y y a a /3
«! ^1 yl Ayi ricii ctiA
«2 ^2 yi ^lyz yi^i ^i^i
as 0:^ r' Ayij y-iO^i o.^^^
a! ^4 y\ ^174 74 a^ ^4^
a' ^5 75 A75 7n% ttoA
= 0.
5024 The equation of a
conic passing through five
points ai/3iyi, 00/3.270, &c. is
the determinant equation
annexed ; and the equation
of a conic touching five
right lines X,/tiVi, X^^iaVo, &c.
is the same in form, X, ^, v
taking the place of a, /3, -y.
Proof. — The determinant is the eliminant of six equations of the type
(4656) in the one case and (4665) in the otlier. By (583).
5025 If three conies have a common self -con jugate triangle,
their Jacobian is three right lines.
Proof. — The Jacobian oi aiX^-\-'b^if-'rc^z-, a^x' + h^^^f + Cf-, a.^x'-^-h.^^i' + c^z^
is, by (5023), x,jz = 0.
For the condition tliat three conies may have a common point, see
Salmons Conic<, 6th edit., Art. 389a, and rroc. Land. ISlalJi. Sec., Vol. iv.,
p. 404, /. /. Walker, M.A.
mVAUIANTS AND COVARIANTS. 695
5026 A system of two conies has four covariant forms
i(, n, F, •/, conneeted by the equation
-{-Fun {&&'-3AA')-AA''u'-A'Ahi''
-^A'uhi {2AQ'-&')-\-Au''h (2A'0-0'2).
Proof. — Form the Jacobian of u, u\ aud P. This will be the equation of
the sides of the common self-conjugate triangle (4992, 5025). Compare the
result with that obtained by the method of (4995).
5027 By parity of reasoning, there are four contravariant
forms t/, U' V, r where F is the tangential equivalent of */,
and represents the vertices of the self-conjugate triangle. Its
square is expressed in terms of U, f/', V and the invariants
precisely as J^ is expressed in (5026).
5028 The locus of the centre of a conic which always
touches four given lines is a right hue.
Proof. — Let IT = 0, 77' = be the tangential equations of two fixed
conies, each touching the four lines ; then, by (4914), TJ+kTJ' = is another
conic also touching the four lines. The coordinates of its centre will be
G±kG;_ ^^^ ^1±M^, by (4402). The point is thus seen, by (4032), to lie
G-{-kG G-\-kC
on the line joining the centres of the two fixed conies and to divide that line
in the ratio hC' '. G.
5029 To find the locus of the focus of a conic touching four
given lines.
In the equations (5010, '1) for determining the coordinates of the focus,
write A-\-kA' for A, &c., and eliminate k. The i-esult in general is a cubic
curve. If 2, 2' be parabolas, S + /v2' is a parabola having three tangents in
common with 2 and 2'. If (7 = 0'= the locus becomes a circle. If the
conies be concentric, they touch four sides of a parallelogram, and the locus
is a rectangular hyperbola.
Note on Tangential Coordinates.
5030 It must be borne in mind that a tangential equation in trilinear
notation (that is, when the variables ai-e the coefficients of o, /3, y in the
tangent line la + mlS + nY) will not agree with the equation of the same locus
expressed in the tangential coordinates \ fj., v of (4019). Thus, to convert
equation (5001), which, for distinctness, will now be written
l^-\-m^-Yn' — 'imn cos A—'2nl cos B — 2lm cos C* =
into tangential coordinates, we must substitute, by (4023), a\, &/u, c»' for
I, m, n. The equation then becomes
a-\'"+tV + cV-26ccos^/xv-2cacos5^X-2aOcos(7A^ = 0.
Put 2tc COS J. = b^ + c^ — a^, &c., and the result is the equation as presented
in (4905).
Corrigenda.— In (4678) and (4692) erase the coefficient 2 ; and in (4680) and (4903) supply
the factor 4 on the left of the equation.
THEORY OF PLANE CURVES.
TANGENT AND NORMAL.
5100 Let P (Fig. 90) be a point on tlie curve AP ; FT, PN,
PG, the tangent, ordinate, and normal intercepted by the <k
axis of coordinates. See definitions in (1160). Let /.PTX
5101 tant/; = g,, by(1403); siuV' = J; cosr^ = g.
5104 Suh-taiigent NT = i/d\j, Sub-normal NG = i/y^.
5106 PT=i/V'OT^), pr=ciV(l +?/;).
5108 PG = i/x/(i^), PG' = uVii-\-4)-
Let OP = r (Fig. 91), n = r-\ AOP = e, OPT = ^ ;
Arc AP — s. Then, by infinitesimals,
lie . dr , _ J (je
dr
5110 sin (l) = r—, cos <^ = j:* ^^^^ <^ = >'
5113 {d.vy-\-{duy = (ds)\ s., = x/(n-//a.
5114 tanV>^ ^'-""^ + ^'""^ . (1768)
5115 lidercei)t6 of Normal OG = r~, OG = Vj-. (Fig.«JO)
TANGENT AND NORMAL. 69'
^^''''' ^^=^i^^PGN=^h:NPf-'l^-'^-
5116 se = ^/{r'+rl), s„ = y/{l+r%y
Proof.— By rOg = sin^ and tan^ = r8,. (5110).
EQUATIONS OF THE TANGENT AND NORMAL.
The equation of the curve beiug y =f{a') or u = (}> {.r, y)
= 0, the equation of the tangent at oij is
5118 ^-^ = ^(^-.0, (4120)
5119 or ft/.,-^ = ^m^-i/^
5120 or ^w,+>;«, = .r?/,+//«,. (1708)
5121 If (p {a;, y) = i'» + f'«-i+ ••• +^o> where v„ is a liomogciicous function
of .« and y of bhe n^^ degree, the constant part forming the right member of
equation (5120) takes the value
— v„_i — 2y„_2— ... — (« — 1)^1 — ^%
By Euler's theorem (1621) and (p (x, y) = 0.
The equation of the normal at xij is
5122
''-•"=-i(^-^')'
(4122)
5123 or
Iv.^rj = aw.+y,
5124 or
iUy — V^(x = ^ri(y—!/U:c-
(1708)
POLAR EQUATIONS OP THE TANGENT AND NORMAL.
Let r, be the coordinates of P (Fig. 91), and E, those
of S, any point on the tangent at P ; and let n = r~^, U=B~^i
T = e — O; the polar equation of the tangent at P will be
5125 R= , , ^ . ■^J or t/= t« COS T+«, slur.
dg [r sin T)
The polar equation of the normal is
5127 R = —r^ — r, or U= ucosT- u-0„ sin r .
f/0(?*COST)
P.OOF.-From X = OP ^ sinO|P_ sml^^-O ^ ,,a from iau = rO,
jB OS sm OPS sm fp
(6112). Similarly for the normal.
4 U
698 THEORY OF PLANE CURVES.
Let OY ~p be tlie perpendicular from the pole upon the
tangent, then
5129 ;> = r siu(^ = {nr-\-u'^-\ (5112)
5131 /n'7^, - ^!^^W^ - (40G4&5119,'20)
OS, drawn at right angles to r to meet the tangent, is
called the 2^olar sub-tangent.
5133 Polar sub-tangent = r-6,. (5112)
RADIUS OF CURYATURE AND E VOLUTE.
Let ^, J/ be the centre of curvature for a point xy on the
curve, and p the radius of curvature ; then
5134 i.v-iy^{^j-yjy = p^ (1).
5135 {.v-$)-^(^-r})ii, = (2),
l+2/l+(//->?)z/2.= (3).
Proof. — (2) and (3) are obtained from (1) by differentiating for x, con-
sidering ^, T) constants.
The following are different values of p :
5137 ^ = (1+^^ ^Ft^^: ^ ^.
5139
5141 = J- ^ :!± ^_i^
5144 ^ (>''+/1)^ _ (^+t<D'^
5146 = .sv = y>+/),^ = >•>',,.
PuooFS. — For (5137), eliminate x — l and y~n between equations d),
(2), and (3). 1 W'
(5138) is obtained from the i)rcccding value by substituting for y^. and
EABIUS OF GUEVATUEE AND EVOLUTE. 699
y.,^ the values (1708, '9). The equation of the curve is here supposed to be
of the form (x, y) = 0.
For (5139) ; change the variable to t. For (5140) ; make t = s.
For (5141-3) ; let PQ=QE = ds (Fig. 92) be equal consecutive elements
of the curve. Draw the normals at P, Q, E, and the tangents at P and Q to
meet the normals at Q and E in T and S. Then, if FN be drawn parallel
and equal to QS, the point N will ultimately fall on the normal QO. Now
the difference of the projections of PT and PN upon OX is equal to the pro-
jection of TN. Projection of PT = dsx, ; that of PN or QS = ds (x, + x.,,ds)
(1500); therefore the difference = dsx.,,ds = TN cos a. But TN : ds =
ds : p, therefore px.,g = cos a. Similarly pi/2s = sin a.
For (5144) ; change (5137) to r and 6, by (1768, '9).
(5145) is obtained from p = rr„ = - ''-^ and (5129) ; or change (5144)
from r to u by r = n'^.
(5146.) In Fig. (93), PQ = p, PP' = ds, and PQP' = dx^.
(5147.) In Fig. (93), let PQ, P'Q be consecutive normals ; PT, P'T'
consecutive tangents; OT, OT', ON, ON' perpendiculars from the origin
upon the tangents and normals. Then, putting p for OT = PN, q for
PT= ON, and d^ for Z TPT' = PQP', &c., we have
q = ^, QN=% and p = PQ = p + QN = p+p,.,.
dtf/ dip
(5148.) dp = r cos (p d\p and cos (p = r, . Eliminate cos (p.
5149 Def.— The rvolnte of a curve is the locus of its centre
of curvature. Eegarding the evolute as the principal curve,
the original curve is called its involute.
5150 The normal of any curve is a tangent to its evolute.
Proof.— By differentiating equation (5135) on the hypothesis that ^ and
V are variables dependent upon x, and combining the result with (3), we
obtain i/^j/j = — 1. ^ , l r>
In (Fig. 94), the normal at P of the curve AP touches the evolute at Q.
Otherwise the evolute is the envelope of the normals of the given curve.
If xij and ^n are the points P, Q, we have the relations
5151 i=f.f+!^.,= .p, jS, = ^ = f-
Proof.— Take Qn = dk and ns = dr,, then Qs = dp. The projection of
Qn, ns gives dp in (5151) and proportion gives (5152).
5153 The evolute and involute are connected by the for-
mulae below, in which r\ /, s in the evolute correspond to
r, 2^i s in the involute.
5154 /)±*' = constant; p'' = r'-p'; r' = r'+p'-2pp.
700 THEORY OF PLANE CURVES.
Proof. — From Fig. (94), Jp = ±(?s', &c., s being tlie arc RQ measured
from a fixed point R. Hence, if a string is wrapped upon a given curve, the
free end describes an involute of the curve. (3155, 'G) from Fig. (93).
5157 To obtain tlie equation of the evolute; eliminate x and
ij from equations (5135, '6) and the equation of tlie curve.
5158 To obtain the polar equation of the evolute ; eliminate
r andp from (5156) and (5157) and the given equation of the
curve r = ^(lO-
5159 Ex.— To find the evolute of the catenary y = -^ (e' + c' '). Here
2/^=1 (e^-e''^) = ■^^y'-"'') ; y.,^ = 1 {e^- + e"^) = ^ ; so that equations
c 2c c
(5135, '0) become
(a;-0 + (y-v) ^^-^'~''^ =0 and 1 + t^ + (2/-v) J = 0.
From these we find y=^, x = ^— j- •/{yf—4c-). Substituting in the
equation of the curve, we obtain the required equation in i, and rj.
INVEESE PROBLEM AND INTRINSIC EQUATION.
An inverse question occurs when the arc is a given func-
tion of the abscissa, say s = (p (x) ; the equation of the curve
in rectangular coordinates will then be
5160 «/ = J v/(4- 1) d.V. [From (5113).
5161 The intrinsic equation of a curve is an equation inde-
pendent of coordinate axes. Let y = <l> {x) be the ordinary
equation, taking for origin a point on the curve (Fig. 95),
and the tangent at for x axis. Let s = arc OP, and xp the
inclination of the tangent at P ; then the intrinsic equation of
the curve is
5162 s = J sec t/».r^ (Ixfj ;
where x^, is found from tan x^ = <l>'{x).
ASYMPTOTES. 701
To obtain the Cartesian equation from the intrinsic equa-
tion :
5163 Let s = F(-^) be the intrinsic equation. Eliminate t//
between this and the equations
,v = f cos xff (Is, y = J sin xft ds.
5165 The intrinsic equation of the evolute obtained from
the intrinsic equation of the curve, s = F{\p), is
4^+*'=/, a constant. (5154)
5166 The intrinsic equation of the involute obtained from
s' = F(\p), the equation of the curve, is
For cl\l, is the same for both curves (Fig. 94), ^ only differing
by ^TT, and s = ^ pd^.
ASYMPTOTES.
5167 Def. — An asymptote of a curve is a straight line or
curve which the former continually approaches but never
reaches. {Vide 1185).
GENERAL RULES FOR RECTILINEAR ASYMPTOTES.
5168 Rule I. — Ascertain if y^ has a limiting value when
X = 00 . If it has, find the intercept on the x or y axis, that
is, X— yxy or y— xy^ (5104).
There toill be a.n asijmptote parallel to the y axis ivhen y^ is
infinite, and the x intercept finite, or one parallel to the x axis
ivhen y^ is zero and the y intercept finite.
5169 Rule II. — When the equation of the curve consists of
homogeneous functions of x and y, of the m*^, n^^, Si'c. degrees,
so that it may be written
■Hf ) Wi)+*'=- = » (^)
702 THEORY OF PLANE CURVES.
put ^ix 4-/3/0?' y and expand <pUi+ - ), &c., hij (1500). Divide
(1) hy X™, and make x infinite; then ({> (^) = determines fi.
Next, put this value of /n in (1), divide hy x™~\ and malce x
infinite; thus |3f/)' (/i) + -j^ (^) =: determines /3. Should the
last equation he indeterminate, then
gives two values for /3, and so on.
When n is <in — 1, /3 = 0, and luhen n is >m — 1, /3 = oo .
5170 Rule III. — If </> (x, y) = he a rational integral equa-
tion, to discover asymptotes parallel to the axes, equate to zero
the coefficients of the highest powers of x and j, if those co-
efficients contain y or x respectively.
To find other asymptotes — Suhstitute |itx-|-|3 for y in the
original equation, and arrange according to powers of :k.. To
find fx, equate to zero the coefficient of the highest power ofx.
To find (3, equate to zero the coefficient of the next power of x,
or, if that equation be indeterminate, take the next coefficient in
order, and so on.
5171 Rule IV. — If the polar equation of the curve he v = i{0)
and if r^ CO makes the polar suhtangent v^Q^^q, a finite quan-
tity, there is an asymptote whose equation is r cos {0 — a) = c ;
where a + ^tt = f~^ (00 ) = the value of B of the curve when r is
infinite.
5172 Asymptotic curves. — In these tlie difference of corres-
ponding ordinates continually diminishes as x increases.
As an example, the curves y = (^ {'•i') and y = <}> (^') H — are asymptotic.
5173 Ex. 1. — To find the asymptotes of the curve
(a + S.c)(x^ + f) = 4.c'^ (1).
The coefficient of 7/^, a + Sx = 0, gives an asymptote parallel to the 1/ axis.
Putting y = fix + ft, (1) becomes
(a + 3x)(x- + i.'x- + 2fiilt + rr)-4x' = (2).
The coefficient of x\ 3 (1 +a^-)-4 = gives /i = ± -^. Substituting this
V "J
value of /J in (2), the coefficient of »' becomes - - ± — ~ ; and this, equated
to zero, gives ft = ^ . Hence the equations of two more asymptotes
are a^z/S = ± (3.c-2a).
SINGVLAUITIES OF CURVES. 703
Ex. 2. — To find an asymptote of the curve r cos 6 =: a cos 2d. Here
^.2 de _ a^ cos 2d
dr a cos 20 sin — 2a sin 20 cos
When ?• = CO , d = lir, and rfl,. = —a. Hence the equation of the asymp-
tote is r cos = —a.
SINGULARITIES OF CURVES.
5174 Concavity and Convexity. — A curve is reckoned convex
or concave towards the axis of x according as yv/gx is positive
or negative.
POINTS OF INFLEXION.
5175 ^4, 2)oint of Injiexion (Fig. 96) exists where the tangent
has a Umiting position, and therefore where v/.^. takes a maxi-
mum or minimum vahie.
5176 Hence 7/2,, must vanish and change sign, as in (1832).
5177 Or, more generally, an even number of consecutive
derivatives of 7/ = ^ (,t) must vanish, and the curve will pass
from positive to negative, or from negative to positive, with
respect to the axis of re, according as the next derivative is
negative or positive. [See (1833).
MULTIPLE POINTS.
5178 ^ multiple jpoint, known also as a node or crunode,
exists when y^. has more than one value, as at B (Fig. 98), If
(}> {x, y) = be the curve, f ,. and (j>y must both vanish, by
(1713). Then, by (1704), two values of v/,. determining a
double point J will be given by the quadratic
<^2.^!. + 2(^..^/.. + <^2,. = (1).
5179 If i>ij;, ^2</3 'Pxy also vanish; then, by (1705), three
values of y,., determining a triple jw I nt, will be obtained from
the cubic
<l>Syyl + ^2y.ryl-^H,^.I/.r-^<l>S. = (2).
5180 Generally, when all the derivatives of (p of an order
704' TBIIORY OF PLANE CURVES.
less than n vanish, the equation for determining /^. (put = z)
may be written
(=;rf,+rf.,)«.^G,.,y) = 0.
Proof, — Let ah be the multiple point. Then, by (1512),
<p (a + /^, h + h) = -, Qid^ + hdyYcp {x, y)
n \
+ terms of higher order which vanish when h and h ai'e small.
h (p„ dx
And -^ = — ®-^ = -^ in the limit,
CUSPS.
5181 When two branches of a curve have a common tan-
gent at a point, but do not pass through the point, they form
a cusj)i termed also a spinode or stationary jJoint.
5182 In the first sjiecies, or ceratoid cusp (Fig. 100), the
two values of ^j.^x have opposite signs.
5183 In the second species, or raniphoid cusp (Fig. 101),
they have the same sign.
CONJUGATE POINTS.
5184 A conjugate point, or acnode, is an isolated point whose
coordinates satisfy the equation of the curve. A necessary
condition for the existence of a conjugate point is that ^^. and
(j)y must both vanish.
Pkoof. — For the tangent at such a point may have any direction, there-
fore — is indeterminate (1713).
5185 There are four species of the trij^le point according as
it is formed by the union of
(i.) three crunodes, as in (Fig. 102) ;
(ii.) two crunodes and a cusp, as in (Fig. 103) ;
(iii.) a crunode and two cusps, as in (Fig. 104) ;
(iv.) when only one real tangent exists at the point.
5186 Ex. — The equation if = {x-a){,v-h){x-c)^ when
a -Ch <ic represents a curve, such as that drawn in (Fig. 07).
* Salmon's JUigher I'lauc Curves, Arts. 39, 40.
(..
UN:
SINGULARITIES OF GURV^S. 705
When b = c the curve takes the form in (Fig. 98). But
if, instead, h = a, the oval shrinks into a point A (Fig. 99).
li a = b = c the point A becomes a cusp, as in (Fig. 100).
A geometrical method of investigating; singular points.
5187 Describe an elementary circle of radius r round the
point X, y on the curve (p {x, y) = 0, intersecting the curve in
the point x-\-h, y-\-h. Let h = r cos B, k = r sin 0. Expand
^ {x-\-h, y-\-h) = by (1512), and put «/>,, = ii siny, ^y =
K cos y. We thus obtain
K sin (y + 0)-\-^ (<^2,, cos^ d + 2./,.,^ sin cos + (p,^ sin- e)-\-— =0
Bj being put for the rest of the expansion (.'■)'
According as the quadratic in tan 0,
(^,.r + 2(/),^^ tan + (^2i/ tan- = 0,
has real, equal, or imaginary roots ; i.e., according as
i>ly — hxi^iy is positive, zero, or negative, xy will be a crunode,
a cusp, or an acnode. By examining the sign of B, the
species of cusp and character of the curvature may be deter-
mined.
Figures (105) and (106), according as B and ^2^ liave
opposite or like signs, show the nature of a crunode ; and
figures (107) and (108) show a cusp.
Proof. — At an ordinary point the circle cuts the curve at the two points
given by 6 = —y, 9 = 7r — y. But, if f^^ and ^^ both vanish, there is a
singular point. Writing A, B, G for <po^, f-,^, (poy, equation (1) now becomes
C'cos^^[tan-^0+^tane+||+^ = O (2).
(i.) If B^>AGy, this may be put in the form
G cos^ d (tan d - tan a) (tan 6 - tan /3) + ^ = 0,
and the points of intersection with the circle are given by 6 = a, ft, tt + o,
and7r + /3. (Figs. 105 and 106.)
(ii.) When B^ = AG, we may write equation (1)
27?
(7cos'0(tan0-tana)-+^ = 0.
If B and G have opposite signs, there is a cusp with a for the inclination of
the tangent (Fig. 107). So also, if B and G have the .same sign, the inclina-
tion and direction being v + a (Fig. 108). The cusps exist in this ca.se
because B changes its sign when tt is added to 6, B being a homogeneous
function of the third degree in sin 6 and cos 0.
(iii.) If B^< AG, there are no real points of intersection, and therefore xij
is an acnode.
4 X
706 THEORY OF PLANE CURVES.
CONTACT OF CURVES.
5188 A contact of the n^^^ order exists between two curves
when n successive derivatives, y^., ... y^^ or Tg, ... r^g, corres-
pond. The curves cross at the point if n be even. No curve
can pass between them which has a contact of a lower order
with either.
Ex. — The curve y =■ <p («) has a contact of the n^^ order, at the point
where x = a, with the curve y = (f> (a) + (x — a) ^'(a) -f ... + ^ — ~- ch" (a).
5189 Cor. — If the curve y =f(x) has n parameters, they
may be determined so that the curve shall have a contact of
the {n — iy^' order with y = (j> (x).
A contact of the first order between two curves implies a
common tangent, and a contact of the second order a common
radius of curvature.
Conic of closest contact with a given curve.
5190 Lemma. — In a central conic (Fig. of 1195),
tan CPG = 4 -^•
S ds
Proof.— Putting PCT=d, CrT=(p, GP = r, CD = R, wc have, by
(1211), r'' + R:' = a- + b\ .•.rr, = -RR, (i.).
Also ii-;- sin </) = a&, by (1194), .-. i2r0, = a&, by (-5110) (ii.).
Now tan CPG = -cot^) = - ^ (5112) = — *^ = ^, by (i.) and (ii.).
r rdg ab .
But p = ^ (4538), .-. -V ^ = ^ = tan CPG.
ah S ds ab
5191 To find the conic having a contact of the fourth order
with a given curve at a given point P.
If be the conic' s centre, the radius r = OP, and the
angle v between r and the normal arc found from the equations
, 1 (/p vosp 1 (h
tail v= -f, =
o as r p as
and these determine the conic.
ENVELOPES. 707
Proof. — In Fig. 93, let be the centre of the conic and P the point of
contact. The five disposable constants of the general equation of a conic
will be determined by the following five data : two coordinates of 0, a com-
mon point P, a common tangent at P, and the same radius of curvature PQ.
Since V = POT, cW = POP', dx^ = TOT', and ds = PP', we have, in
passing from P to P', di' = P'OT—POT = d^P-dO. Now rdd = ds cos v,
therefore = -~- = — — ; and tan v has been found in the
lemma. ' ^'^ ^°' ^ P '^'
The squares of the semi-axes of the same conic are the
roots of the equation
{Q+b'Sacy.v'-Oa' {lS+2b'-3ac){9-\-b'-3ac) .v+7'29a'
= 0,
a, h, c being written for p, pg, p^g. The eccentricity is found
from 9(e^-2y _ (18+26^-3ae)^
1-e' ~ 9-\-b'-3ac '
Also the equation of the conic referred to the tangent and
normal at the point is
Aa'+2Bay-{-Cif = 2i/,
where A = ^, B = -^, c = ^ + ^-Pf.
p op p vp 6
Ed. Times, Math. Reprint, Vol. xxr., p. 87, where the demonstrations by
Prof. Wolstenholme will be found.
ENVELOPES.
5192 An envelope of a curve is the locus of the ultimate
intersections of the different curves of the same species, got
by varying continuously a parameter of the curve ; and the
envelope touches all the intersecting curves so obtained.
5193 Rule. — If F (x, y, a) = be a curve having the para-
meter a, the envelope is the curve obtained by eliminating a
betiveen the equations
F (a?, ?/, a) = and d^F (.i', y, a) = 0.
708 THEORY OF PLANE CURVES.
Proof. — Let a change to a + h. The coordinates of the point of intersec-
tion of F (x, y, a) = and F (:c, y, a + h) =0 satisfy the equation
F(x,y, a + h)-F(x, y, a) _ ^^ ^^^^ (IF (x, y, a) _ ^^ (1404)
h ' ' da
5194 If F{Xi y, a,h,c, ...) = be the equation of a curve
liavhig n parameters a, b, c, ... connected by n — 1 equations,
then, by varying the parameters, a series of intersecting
curves may be obtained. The envelope of these curves will
be found by differentiating all the equations with respect to
a, h, c, &c., and eliminating da, dh, ... and a^b, ...
Ol95 Ex. — In (2) of (5135), we have the equation of the normal of a
curve at a given point xy ; ^, >/ being the variable coordinates, and x, y the
parameters connected by the equation of the curve F (x, y) = 0. By differ-
entiating for X and y, (5136) is found, and the elimination as directed in
(5157) produces the equation of the evolute which, by (5194), is the envelope
of the curve.
INTEGRALS OF OUHVES AND AREAS.
FORMULA FOR THE LENGTH OF AN ARC S.
5196 s=^ds= [v/(l+Z/i-) d^^' = J \/l+iI% (5113)
5200 = j v/K+2/?) dt = Jy(r+r^) cW (5116)
5201 = fv/(>'^^;+l) dr = r J'!'' .,. . (5111)
5203 Legendre's formula, 5 = j>^H- \ pd"*^.
/»2Tr
5204 The whole contour of a closed curve = \ pdxf/.
Jo
Pkoof. — In figure (93), let P, P' be an element Js of the curve ; PT^ P'T'
tangents, and 07', 07'' the perpendiculars upon them from the origin ; OT=p
PT = q. Then ch + P'T'-PT = TL, i.e., ds + dg = pd^ ; therefore s + q
= lpd\p. But qdij/ = —dp; therefore s = p^ + jjfdi}/. Also, in integrating
all round the curve, P'T'—PT taken for every point vanishes in the summa-
tion, or dq = 0. Therefore I (?*• = I pd\p.
INVEBSE CURVES. 709
FORMULA FOR PLANE AREAS.
5205 If y = 1^ (,^) t)e tlie equation of a curve, the area
bounded by the curve, two ordinates (x = a, x = h), and the x
axis, is, as in (1902).
A = rV {^^) dx-
5206 With polar coordinates the area included between two
radii (0 = a, B = ^) and the curve is
Proof. — From figure (91) and the elemental area OPP'.
5209 The area bounded by two circles of radii «, h, and the
two curves = <^ (r), B = ^ (r) (Fig. 109).
rdrdd = \ r {^j^ (r) - <!> (r)} dr.
4> (r) J a
Here r{Tp{T) — ^{r)}dr is the elemental area between the
dotted circumferences.
5211 The area bounded by two radii of curvature, the curve,
and its e volute (Fig. 110).
^ = \\p^dy\i = l\pds.
Proof. — From figure (93) and the elemental area QPP' .
INVERSE CURVES.
The following results may be added to those given in Arts. (1000-15).
5212 Let r, r be corresponding radii of a curve and its
inverse, so that rr' = F ; s, s' corresponding arcs, and ^, (p'
the angles between the radius and tangents, then
-—7 = — r and <f) = <f>'.
ds r
Proof. — Let PQ be the element of ai-c ds, P'Q' the element els', and
the origin.
Then OP. OP' = OQ.OQ', therefore OPQ, OQ'P' are similar triangles;
therefore PQ : P'Q' :: OP : OQ' = r : r' ; also z OPQ = OQ'P'.
710 THEORY OF PLANE CURVES.
5214 K p, p be the radii of curvature,
r+4=2siiif
9 9
Proof. — From p = r sin 0, p' = r sin 0, we have
V' = h^ ■^, therefore ^f ' = F '^:!2^^i=2rp _ _ _
r^ dr r*
(i.)-
Also »• = —, therefore -— = 77-
r (ir r -
("•)•
Now p' = r' fl (5148), therefore ^,=^f-'^ = ^P^ _ JL^
ap p r dr dr rr pr
by (i.) and (ii.).
Therefore 4 + - = ^ = 2sin0.
p p r
5215 To find the equation of the inverse of a curve in
rectangular coordinates, substitute
, , ., and , / ,
for X and y in the equation of the given curve.
5216 The inverse of the algebraic curve
where u^^ is a homogeneous function of the 7i"' degree, will be
5217 The inverse of the conic zt^ + 1^1 + 1*0 = is
k'u,+k'u, {.v'+if)Jruo W+ff = 0.
5218 If the origin be on the curve, this equation becomes
k'u,-^u, {x'^if) = 0.
5219 The angle will also be unaltered in any curve,
r =^f(0), if the inversion be effected by putting
r = A:r " and 6 = nd' .
Proof. —
tan 0' = r'fl;.; (5112) = rO'^r^. = r'^''/n;/"-' = W'd^ — r8^ = tan0.
PEDAL CURVES. 711
PEDAL CURVES.
5220 The locus of tlie foot of the perpendicular from the
origin upon the tangent is called ^ jpeclal curve. The pedal of
the pedal curve is called the second pedal, and so on. Re-
versing the order, the envelope of the right lines drawn from
each point of a curve at right angles to the radius vector is
called t\ie first negative jpedal, and so on.
5221 The pedal and the reciprocal polar are inverse curves
(1000, 4844.)
AREA OF A PEDAL CURVE.
5222 Let C, P, Q be the respective areas of a closed curve,
the pedal of the curve, and the pedal of the evolute ; then
P-Q=C, P+Q = i jVv/t/f, 2P = C+iJ r'dxff.
Proof. — With figure (93) and the notation of (5204), we have, by (5206),
P = i f p-dij;, Q = il q'dxP ; therefore P + Q = U Ci'' + 5') ^'Z' = 5 I '•'#•
Also, taking two consecutive positions of the triangle OPT = A, we get
OPT- OP'T' =hA = SG + SQ-^P. Therefore, integrating all round,
[^clA = = G+Q-P.
5225 Steiner^s Theorem. — If P be the area of the pedal of a
closed curve when the pole is the origin, and P' the area of
the pedal when the pole is the point xy,
P'-P = ^{j,^J^^f)^ax-hy,
where a=\ ]3 cos Odd and h=\ 2^&m9d6;
Jo Jo
9 being the inclination of j9.
Proof. — (Fig. 111.) Let LM be a tangent, »S' the point xy, perpendiculars
OM = p and SB = 2/. Draw SN perpendicular to OM, and let ON = p^ -^
then p' = i [/-# = ^{(p-piy- # = i J /f?^+ f J p;^^/'- J m^^^
= P+ ^ OS^- \p (x cos + 7/ sin 0) clB, by (4094), and dd = #.
And g Pld^ = twice the area of the circle whose diameter is OS.
712 THEORY OF PLANE CURVES.
5226 Cor. 1. — If P' be given, the locus of .ri/ is a circle
whose equation is (5225), and the centre of this circle is the
same for all values of P', the coordinates of the centre being
a J b
— and — .
TT TT
5227 Corv. 2. — Let Q be the fixed centre referred to, and
let Q8 = c. Let P" be the area of the pedal whose origin is
Q; then P'-P" = ^c\
For a and h must vanish in (5225) when the origin is at the
centre Qj and ,7]^ + ?/^ then = c^
5228 CfoE. 3. — Hence P" is the minimum value of P'.*
ROULETTES.
5229 Def. — A Roulette is the locus of a point rigidly con-
nected with a curve which rolls upon a fixed right line or
curve.
AREA OF A ROULETTE.
5230 When a closed curve rolls upon a right line, the area
generated in one revolution by the normal to the roulette at
the generating point is twice the area of the pedal of the
rolling curve with respect to the generating point.
Proof. — (Fig- 112.) Let P be the point of contact of the rolling curve
and fixed straight line, Q the point which generates the roulette. Let B bo
a consecutive point, and when B comes into contact with the straight line,
let P'Q' be the position of itQ. Then PQ is a normal to the roulette at Q,
and P is the instantaneous centre of rotation. Draw (^N, QS perpendiculars
on the tangents at P and R. The elemental area PQQ'P', included between
the two normals QP, Q'P', is ultimately equal to PQB+QRQ'. But PQB
= dC, an element of the area of the curve swept over by the radius vector
QP or r round the pole Q ; and Q22Q' = ^r^d^^ ; therefore, whole area of
roulette = C+\ \''r'd^ = 2P, by (5224).
5231 Hence, by (5228), tliere is one point in any closed
curve for which the area of the corresponding roulette is a
* For a discuseion of the pedal curves of an ellipse by the Editor of the JSduc. Times and
others, see Ecpritit, Vol. i., p. 23 ; Vol. xvi., p. 77 ; Vol. xvii., p. 92 ; and Vol. xx., p. 106.
BOULETTES. 713
minimum. Also the area of the roulette described by any
other point, distant c from the origin of the minimum roulette,
exceeds the area of the latter by wc^.
5232 When the line rolled upon is a curve, the whole area
generated in one revolution of the rolling curve becomes
Jo \ p '
where p, p are the radii of curvature of the rolling and fixed
curves, and G is the area of the former.
Proof. — (Fig. 113.) Instead of the angle d\j/, we now have^ie sum of
the angles of contingence at P of the rolling curve and fixed curve, viz.,
since pdvj/ =: ds = p'dijj', by (5146),
LENGTH OF THE ARC OF A ROULETTE.
5233 If (^ and I be corresponding arcs of the roulette and
the pedal whose origin is the generating point ; then, when
the fixed line is straight, a = I; and when it is a curve,
5234 ^da = ^{l+Pr)dC
Proof. — (Fig. 112.) Let B be the point which has just left the straight
line, Q the generating point, N, 8 consecutive points on the pedal curve.
Draw the circle circumscribing BQN8, of which BQ -— r is a diameter, and
let the diameter which bisects NS meet the circle in K. Then, when the
points P, B,, P' coincide, KN and BQ are diameters, and SKN = SPN = f/i/'
= QBQ' ; tlierefore SN or d^ = rdij/ = QQ' or da. When the fixed line is a
drr = rdif; (l+ ^), as in (5232).
RADIUS OF CURVATURE OF A ROULETTE.
5235 Let a (Fig. 113) be the angle between the generating
line r and the normal at the point of contact; p, p the radii
of curvature of the fixed and rolling curves, and E the radius
of curvature of the roulette ; then.
^^ cosa— r
4 Y
714 THEORY OF PLANE OUBVES.
Proof. — Let consecutive normals of the roulette meet in ; then
OQ = li, PQ = r, MPT=u.
11 — r PM ds cos a , , /7, , 7/'\ /ds,ds\
B QQ d<T VTT/ \ p p /
from which B is obtained. If the curvature of the roulette is convex to-
wards P (Fig. 114), we must write B + r instead of B — r above.
5236 The curvature is convex towards P when B is posi-
tive, that isj wlien the carried point Q falls within the circle
whose diameter measured on the normal of the rolling curve
= ^^ , . When Q falls without this circle, the curvature is
concave ; and when Q falls upon the circumference, the point
is one of inflexion. The circle has for this reason been called
the circle of inflexions.
5237 In figure (163) let PA = p, PB = p, PQ = r, OQ = R,
as in (5235). Draw PGD, the circle of inflexions, with its
diameter PG = -^^—„ and therefore PD = ^^ , cos a. From
P-^P P-^P
these values and proportion it follows that BG : BP : BA and
QD : QP : QO. Also, if the circle on diameter PE = PG be
drawn, AE : AP : AB and OF : OP : OQ.
5238 A simple construction for the centre of curvature of
the roulette is the following. (Fig. 164, with letters as in
5237.) At P draw a perpendicular to PQ to meet QB in N.
Join NA, which will meet Q,P produced in 0, the required
point.
Pkoof. — From equation (5235), assuming to be the centre of curvature,
we can deduce the relation {BA : AP){PO : OQ)(QN : NB) = 1, therefore,
by (9G8), A, 0, N are collinear points.
THE ENVELOPE OF A CARRIED CURVE.
5239 When a curve is rigidly connected with a rolling
curve, it will have an envelope. The path of its point of
contact with the envelope is a tangent to both curves, and
therefore the normal, common to the carried curve and its
envelope, passes through the point of contact P of the rolling
and fixed curve.
5240 The centre of curvature of the envelope is obtained as
follows.
TBAJECT0BIE8, ^c. 715
In Fig. (163), from P draw a normal to the carried curve meeting it in
Q, and let 8 on PQ he the centre of curvature of the envelope for the point
Q ; and that of the cai'ried curve. Then PS is found from
P 9
DS a f \- -—-
\PS PO J
5241 When the envelope is a right hne, the centre of curva-
ture lies on the circle of inflexions (5236). When the carried
curve is a right line, the same point lies on the circle PEF
(Fig. 163), and if the right line always passes through a fixed
point, that point lies on the circle PEF.
5242 If 2^ ^® ^^® perpendicular from a fixed point upon a
carried right line whose inclination to a fixed line is ^ ; the
radius of curvature of the envelope is p = jj+j;,^, by (5147).
INSTANTANEOUS CENTRE.
5243 When a plane figure moves in any manner in its own
plane, the instantaneous centre of rotation is the intersection
of the perpendiculars at two points to the directions in which
the points are moving; and a line from the instantaneous
centre to any point of the figure is the normal to the path of
that point.
Ex. — Let a triangle ABG slide with its vertices A, B always upon
the right lines OA, OB. The perpendiculars at J., B to OJ., OB meet in Q,
the instantaneous centre, and QC is the normal at G to the locus of G.
Since AB and the angle AOB are of constant magnitude, OQ, the
diameter of the circle circumscribing OAQB, is of constant magnitude.
Hence the locus of the instantaneous centre (^ is a circle of centre and
radius OQ.
5244 Holditch's Theorem. — If a chord of a given length LM
moves completely round a closed curve, the area enclosed
between the curve and the locus of a point P on the chord is
equal to ircc where c = LK, d = MK.
5245 If the ends of LM move on different closed curves
whose areas are X, /u, while the area described by K is /c, then
K = '-^T TTCC .
c-\-c
Proof. — (5244). Let the innermost oval in figure (134) be the envelope
of LM, € its area, and E the point of contact. Let EL = I, EM ^ m,
716 THEORY OF PLANE CURVES.
EK=.k, l + m = a = c + c' ; d, the inclination of LM. Then, integrating
in every case from to 2ir,
i^mHd = fJi-€^ Also J J(Z + »0 dd = 7ra\
.-. ahdd = 7rft- + \— /z (i.)- Similarly c^ldd= ttc' + X — k (ii.),
the last being obtained from ^ ^ (P—Jc-) dd = \ — i^. k is then found by
eliminating the integral between (i.) and (ii.).
(5245.) If the curves X, fi coincide, \ = // and therefore \—k = ttcc'.
TRAJECTORIES.
5246 Def. — A trajectory is a curve which cuts according to
a given law a system of curves obtained by varying a single
parameter.
The differential equation of the trajectory which cuts at a
constant angle |3 the system of curves represented by
^ {x, y, c) = is obtained by ehminating c between the
equations
c/, Cv, y, c) = and tau jB = ^f^^\
the derivatives of <^ being partial, and y^ referring to the
trajectory.*
Proof. — At a point of intersection we have for the given curve
m = —0^-1-^^^, and for the trajectory m' = yx- Employ (4070).
If the trajectory is to be orthogonal, tan /3 = oo , and the
second equation becomes
Ex. — To find the curve which cuts at a constant angle all right linos
passing through the origin.
Let y = ex represent these lines by varying c ; then, writing n for tan ft,
the two equations become y — cx = and n (1+cy:,) = y^ — c. Eliminating
c, ^Vx—y = '>i(yyx + x)- Divide by x' + y^' and integrate; thus
tan-'l=7ilogy(.VHr) + 0,
X
which is equivalent to r = ae~>, the equation of the logarithmic spiral (5289).
* For a very full investigation of this problem, see Eulor, Novi Com. Fetrop., Vol. xiv.,
p. 46, XVII., p. '10f> ; and Nova Acta Petrop., Vol. i., p. 3.
CAUSTICS, Sfc. 717
CURVES OP PURSUIT.
5247 Def. — A curve of pursuit is the locus of a point wliich
moves with uniform velocity towards another point while the
latter describes a known curve also with uniform velocity.
Let f(x, 7/) = be the known curve, xy the moving point
upon it, ^rf the pursuing point, and n : 1 the ratio of their
velocities. The differential equation of the path of ^rj is
obtained by eHminating x and y between the equations
f{x,lj) = (i.), ij-^y} = 7}^{.v-i) (ii.),
^/{^l-\-yl) = n^/{l+v!) C^^-)-
Proof. — (ii.) expresses the fact that xy is always in the tangent of the
path of ^r].
(iii.) follows from 1 : ?i = •/(d^'^ + cW) : ^/{iW-\-cly') ; the elements of
arc described being proportional to the velocities.
Ex. — The simplest case, being the problem usually presented, is that in
which the point xy moves in a right line. Let x =■ a be this line, and let
the point ^i/ start from the origin when the point xy is on the x axis. The
equations (i.), (ii.), (iii.) now become, since x^ = 0,
x = a, y = rj + r}^(a—^), ?/, =«y(l + ?jp.
From the second y^= r/o^^ (a— ^), therefore (a—^) r}^ = n \/(l + 7;p.
Putting ,.=^, _^&__ = _ii^^-.
Integrating by (1928), we find
log {p-\- \/l+p^) = — ?ilog (a — |) + n log a,
so that p and i, vanish together at the origin ;
therefore \/\ +p'''-{-p = ( — — ) , and therefore ^/l+p^^—p = f- — ^j ;
the equation of the required locus, the constant being taken so that t = >/ =
together. If, however, w=l, the integral is
4<a
718 THEORY OP PLANE CURVES.
CAUSTICS.
5248 Dep- — If right lines radiating from a point be reflected
from a given plane curve, the envelope of the reflected rays is
called the caustic by reflexion of the curve.
Let (p {x, y) = 0, xp(x, y) = be the equations of the
tangent and normal of the curve, and let 1th be the radiant
point; then the equation of the reflected ray will be
(^(/^,A•)VG^^;/)4-V'(/^,A:)(^G^^^/) = 0,
and the envelope obtained by varying the coordinates of the
point of incidence, as explained in (5194), will be the caustic
of the curve.
Ex.— To find the caustic by reflexion of the circle x" + i/ = r, the radiant
l^oint being hJc.
Taking for the tangent and normal, as in (4140), a; cos a + ?/ sin a = r,
and X sill a — y cos a = 0, the reflected ray is
(h cos a + Jc sin a — r)(:e sin a — y cos a)
+ (h sin a — A: cosa)(.i; cos a + y sin a — r) = 0.
Reducing this to the form
A cos2a + I> sin 2a + (7 sin a — D cos a = 0,
and differentiating for a,
-2^ sin 2a + 25 cos 2a+ 6' cos ct + D sin a = 0.
The result of eliminating a is
{4^(h' + k'){x' + y')-r\x + hy-r\y + hyy = 27 0cx-hi/y(x' + y'-h'-k'y,
the envelope and caustic required.
5249 QuetelcVs Theorem. — The caustic of a curve is the
evolute of the locus of the image of the radiant point with
respect to the tangent of the curve.
Thus, in the Fig. of (1178), if S bn the radiant point, W is the image in
the tangent at P. The locus of W is, hi this case, a circle, and the evolute
and caustic reduce to the single point S'.
Since the distance of the image from the radiant point is
twice the perpendicular on the tangent, it follows that the
locus of the image will always be got by substituting 2r for r
in the polar equation of the pedal, or ^ for r in the polar
equation of the reciprocal of the given curve with respect to
the radiant point and a circle of radius k.
TRANSCENDENTAL AND OTHER CURVES. 719
TRANSCENDENTAL AND OTHER CURVES.
THE CYCLOID.* (Fig. 115)
5250 Def. — A cycloid is the roulette generated by a circle
rolling upon a right line, the carried point being on the cir-
cumference. When the carried point is without the circum-
ference, the roulette is called ^ iwolate cycloid; and, when it is
within, a curtate cycloid.
5251 The equations of the cycloid are
,1-= a (^-f-sin ^), ?/ = a (1 — cos ^),
where B is the angle rolled through, and a the radius of the
generating circle.
Proof. — (Fig. 115.) Let the circle KPT roll upon the line BE, the
point P meeting the line at D and again at E. Arc KP = KB ; therefore
arc FT = AK = OT. Also 6 = PGT, the angle rolled through from A, the
centre of the base EB. Then
x= OT+TN= ad + a sin d; y = FN = a-a cosd.
5253 If s be the arc OP and p the radius of curvature at P,
s = IPT = v/(8«//), p = 2PK.
Proof.— (i.) The element Pp = Bh = 2 (OB- Oh) ultimately; therefore,
by summation, s = 20B. Also OB = FT = ^{TK. TR) = ^/{2ay).
(ii.) Let two consecutive normals at P and p intersect in L. Then FL,
pi are parallel to BA, bA ; therefore PLp is similar to BAi. But Pp = 2Bi ;
therefore p or FL = 2BA = 2PK.
5255 CoE. — The locus of L, that is the evolute of the
cycloid, consists of two half-cycloids as shown in the diagram.
5256 The area of a cycloid is equal to three times the area
of the generating circle, and the curve length is four times
the diameter of the same circle.
Proof. — (i.) Area FpvN = FprR = BhqQ ultimately. Therefore, by
summation, DiJ.^0 — cycloid = Tral But BE .AO = 2wa.2a = 47ra^; there-
fore cycloid = Stto,^.
(ii.) Total curve length = 8a, by (5253).
* The earliest notice of this curve la to bo found in a MSS. by Cardinal do Cusa, 1454
See Leibnitz, Opera, Vol. m., p. 96.
720 THEORY OF PLANE CURVES.
5257 The intrinsic equation of the cycloid is
* = 4a sin t/>.
Proof : s = 2VT = 4a sin PET, and PKT = PTN = if/.*
THE COMPANION TO THE CYCLOID.
5258 This curve is the locus of the point 11 in Fig. (115).
Its equation is
^=a(l-cos^).
Pkoof. — From x = ad and y = a (1 — cosS).
5259 The locus of ;S^, the intersection of the tangents at P
and B, is the involute of the circle ABO.
Proof : B8 = BP = arc OB.
PROLATE AND CURTATE CYCLOIDS. (5250)
5260 The equations in every case are
a: = a {6-\-7n siu 6), y = a (1 — m cos 6).
The cycloid is prolate when m is > 1 (Fig. 116), and curtate
when m is < 1 (Fig. 117), m being the ratio of GP to the
radius a.
EPITROCHOIDS AND HYPOTROCHOIDS. (Fig. 118)
5262 These curves are the roulettes formed by a circle
rolling upon the convex or concave circumference respectively
of a fixed circle, and carrying a generating point either within
or without the rolling circle.
The equations of the epitrochoid are
5263 cr = {a-\-b) cos e-mO cos'-^ 6,
5264 // = (a-\-b) siu 6-nib siii^ 6,
* For other properties, sec Pascal, Uistoirc dc la Roulette ; Carlo Dati, History of the
Cycloid; Wallis, Trail e de Cyclnidc ; Groningius, Ilistoria Cycloidis, Bibliuthcca Univ.; and
Lalouere, Gcomctria promota in sup/on dc Cycluidc liliris ; Bernoulli, Op., Vol. IV., p. 98 ;
Eulor, Comm. I'et., 17G6 ; and Logondre, ]£xerc%ce du Calcul. Int., Tom. ii, p. 491.
TRANSCENDENTAL AND OTHER CURVES. 721
where a, h are the radii of the fixed and rolling circle
(Fig. 118), B is the angle OGX, Q is the generating point
initially in contact with the x axis, and m is the ratio OQ : b.
The dotted line shows the curve described. For the hypo-
trochoid change the sign of b.
Proof: x= CN+MQ; CN = (a + h) con B;
MQ = OQ cos OQM =-0Q cos (^ + 0), where (p = FOR, cand b(t> = aO.
5265 The length of the arc of an epitrochoid is
= («+^') J
l-{-m'-2mQ0s'^['(W,
which is expressed as an elliptic integral E (L; <p) by substi-
tuting a9 = 2b<p.
For the arc of a hypotrochoid, change the sign of b.
Proof : s = ^s,dd = J y(xl + yl) dd (5113). Find x, and y, from (5263-4).
EPICYCLOIDS AND HYPOCYCLOIDS. (Fig. 118)
5266 For the equations of these curves make 111 = 1, in
(5263, '4). P is then the generating point, and the curve is
shown by a solid line in Figure (118).*
5267 If 4" be the inclination of the tangent at a point P on
any of these curves,
/J a-\-b a
cos u— 7)1 cos — -! — a
tau xjj = —-- = taii-^^^ — 0, if m = 1.
• n • « + «/) 'lb
sm u—7n siu — ~ u
5268 Hence, in the epicycloid, xjf = ^-^ — 0,
Jib
and the equation of the tangent is
X sm ' 6—1/ COS ; 6= {a-\-2b) sm — t'.
5269 The equation of the normal will be
x cos — 1^ — G-\-y sm — In — u = a cos — - V.
2b ^ 2b 2b
* Prof. Wolstenholme has investigated these curves considered as the envelopes of u
chord whose extremities move on a fixed circle with uniform velocities in the ratio 111 : n or
m : {-n).—Proc. Lond. Math. 80c., Vol. iv., p. 321.
4 z
722 THEOIiY OF PLANE CURVES.
5270 The length of the arc of an epicycloid or hypocycloid
included between two successive cusps is
— (u ih f>), ^nd the included area is — (3a ± 26) .
a ^ a ^
Proof. — Putting m = 1 into (5265) and aO =■ h(p, the length becomes
— (a ±6) sm-L-d(b = —(a±b).
a Jo 2 a
Otlierwise by (5234) ; the pedal being the cardioid whose perimeter = 8a
(5333).
(ii.) The area, by (5232), is 7r6- + i f "^Z^' sin^l" (l+ ^^) # 5 since, in
Jo 2 \ ci '
Fig. (118), dif/ of (5232) = dPOB = d<p and r = PB = 2h sin|-.
5271 The evolute of an epicycloid is a similar epicycloid.
Proof. — The equation of the tangent referred to an x axis drawn through
the summit of the curve will be (by turning axes through an angle hir — a),
X cos ' d-\-y sin ^ 0= (a + 2&) cos —0.
Comparing this with (5270), which is the equation of the tangent of the
evolute, we see that the epicycloid and its evolute are similar curves having
their parameters in the ratio a + 'lh : a; and that the radius drawn through
a cusp of either of the curves passes through a summit of the other.
5272 When h = —\a, the hypocycloid becomes a straight
line, namely, a diameter of the fixed circle.
THE CATENARY. (Fig. 119)
5273 Gharacteristic. — The perpendicular TP from the foot
of the ordinate upon the tangent is of a constant length c, and
therefore equal to OA, the perpendicular from the origin on
the tangent at the vertex, r is the parameter of the curve.
The equation is
5274 y = ^{e^ + e--c).
id Q
PuoOF: tanPCT = f^ = — -f-— , .-. x = c ', log (y + ^/ ,/ -cr) -log c}
(1928), since x = when ij = c. Therefore
e?=l (y+y(y/2_c^)} therefore e'^^ ~ {y- ^^Of- c')].
TRANSCENDENTAL AND OTHER CURVES. 723
5275 If s = arc AG, s = ^ {e^ -e"^) = CP.
Peoof: s = ^^(l + 7jl)dx (5197)
= \ij{l+-^)dx = ^'jdx = ^(e^-e-^)=y(f-c') = CP.
5276 The area OAGT = cs. (5205)
5277 The radius of curvature at G = — , and is therefore
c
equal to the tangent intercepted by the axis of ic.
Proof: 0051^ = — , .'. —s'm\l/ilyg = -Vg, .'. p = s^ = — (5146).
y r c
5278 The catenary derives its name from a chain, which,
when suspended from its extremities, takes the form of this
curve.
For the equation of the evolute of the catenary, see (5159).
THB TRACTRIX. (Fig. 119)
5279 Gharacteristlc. — The length of the tangent intercepted
by the x axis is constant. This curve is the involute of the
catenary, being the locus of P in Figure (119).
The equation of the tractrix is
5280 .r = c log {c-\-s/{c'-u')} -c \ogi/-^{c'-i/).
Proof. — Let the tangent FT = c, then the differe ntial e quation of the
curve is therefore yx^ = —\/G- — 'if. Substitute z — s/c' — y\ and integrate
by (1937).
5281 The area included by the four branches = 7^c^
Proof. — Area = 4 ?/(Z.c = — 4 ■/c-—y"-dy = 7^c^ by (1933).
THE SYNTRACTRIX.
5282 This curve is the locus of a point Q on the tangent of
the tractrix in Fig. (119). Let Q^ be equal to a given con-
stant length d ; then the equation of the syntractrix will be
5283 07 = clog {d+ s/id'-y')} -c logy-y/{d'-f).
724 THEORY OF PLANE CURVES.
THE LOGARITHMIC CURVE.* (Fig. 120)
5284 Gharacteristic. — The subtangenfc is constant.
The equation of the curve is either
5285 y = aen, or x = n\og^,
where n = NT^ the constant subtangent, and a is the intercept
on the y axis.
5287 If ^i ^6 an even integer, y may take negative values.
The most general form of the equation may perhaps be
assumed to be
y
= e^l cos \-i sm — ).t
\ n n J
THE EQUIANGULAR SPIRAL. (Fig. 121)
5288 Gharacteristic. — The angle OFS between the tangent
and radius is constant. The equation of the curve is either
- r
5289 r = ae^^ or = n log — .
5291 tan (/) = n, s = r sec <l>,
measuring s from the pole.
Proof.— By (5112) and (5200).
5293 Hence the length of the spiral measured from the pole
to a point P (Fig. 121) is equal to PS, the intercept on the
tangent made by the polar subtangent OS.
5294 The locus of S is a similar spiral, and is also an invo-
lute of the original curve.
5295 The pedal curve, which is the locus of Y, is also a
similar equiangular spiral.
Proof. — The constancy of the angle ^ makes the figure OPTS always
similar to itself. Therefore P, Y, and S describe similar curves. Hence, if
ST is the tangent to the locus of >S, OST = <p = OPS ; therefore PST is a
right angle ; therefore the locus of *S' is an involute of the original spiral. J
* Originated l)y James Gregory, Geometricv Pars Unit^crsalis, 1668.
t Sec Elder, Anal. Injin., Vol. ii., p. '290 ; Vincent, Aim. de Gergoiuie, Vol. xv., p. 1 ;
Gregory, Camh. Math. Journal, Vol. i., i)p. 231, 264 ; Salmon, Uigher Plane Curvet, p. 274.
X For additional propertica, see Bernoulli, Opera, p. 497.
TRANSCENDENTAL AND OTHER CURVES. 725
THE SPIRAL OF ARCHIMEDES.* (Pig. 122)
5296 Gharacteristic. — The distance from the pole is propor-
tional to the angle described. Hence the equation is
5297 r = aO. Also tan (^ = ^. By (5112).
5299 The intercept, FQ, on any radius between two succes-
sive convolutions of the spiral, is constant and = 2<X7r.
5300 The area swept over by any radius is one third of the
corresponding circular sector of that radius.
5301 This curve is one of the class the general equation of
which is
Q
r = a0'\ with tan (j) = — .
THE HYPERBOLIC OR RECIPROCAL SPIRAL. (Fig. 123)
5302 The equation is r=—-.
u
5303 An asymptote is the line ?/ = «. (51^1)
5304 The spiral is also an asymptote to itself.
For when the radius is of the first order of smallness, the distance
between two successive convolutions is of the second order. Hence the
distance to the pole measu.red along the curve is infinite.
The area between the radiants i\^ r.^ is = \a {vi—r.^.
5305 The equation of the Lituus is r =
^0
THE INVOLUTE OF THE CIRCLE. (Fig. 124)
5306 The equation is
Proof: ^ = OPY = cos-^- and ^(r'-a-) —BP = nxcAD = a (0 + 0).
5307 The pedal of the involute is the spiral of Archimedes.
*^Inveiited by Conon, b.c. 250.
726 THEORY OF PLANE CURVES.
Proof. — Let ?•', 0' be the coordinates of Y on the pedal curve. Then
r'=J]P = arc .42?= a(d'+U). (See 5297).
5308 The reciprocal of the involute is the hyperbolic spiral.
Proof. — (Fig. 124.) Let P' on OY correspond to P, and let r', d' be the
polar coordinates of P'. Then ?•' = OP' = -— -.
But OY=BP = ^vcAB = a (6' + i,7r), .■.r' = —^—. See (5302).
THE CISSOID.* (Fig. 125)
5309 Characteristic. — A line drawn from the end, 0, of
a fixed diameter of a circle to the end, Q, of any perpendicular
ordinate intersects the parallel ordinate equidistant from the
centre in a point, P, whose locus is the cissoid. The equation
of the curve is
fiSlO tr(2a-v)-a.^' and ^(V _ {6a-2.v) ^w
b6W y {^a .1) - a. ana -^ - ^ ^{^a-.vf '
Proof. — By similar triangles, y : x =■ \/{2ax—x-) : 2a— x. Two mean
proportionals between the radius a and OS are given by the curve, for it
appears that a^ : GT '.: GT : GS, and therefore a : GT : s/CS.GT : GS.
5311 The tangent of the circle at B, the other end of the
diameter, is an asymptote to both branches of the cissoid.
5312 The area between the curve and its asymptote is equal
to three times the area of the circle.
Proof: In ydx substitute x^2ashrd.
I>
THE CASSmiAlNr OR OVAL OF CASSINL (Fig. 126)
5313 Gharactcristic. — The product PA.PB of the distances
of any point on the curve from two fixed points A, B is con-
stant ; the equation is consequently
{i/^ia+xf} {i/+{a-af} = m^
or (.r-+7/-+«-)-— 4rt-.i'- = m\
where 2a = AB. The equation in polar coordinates is
r*— 2aV cos 2^+a'— m* = 0.
* Pioclos, A.D. 600.
TBANSGENDENTAL AND OTHER CURVES. 727
5314 K a be > m, there are two ovals, as sliown in the
figure. In that case, the last equation shows that if OPP'
nfeets the curve in P and P', we have OP. OP' =^{a^—m'^) ;
and therefore the curve is its own inverse with respect to a
circle of radius = \/(a*— m^).
5315 being the centre, the normal PG makes the same
angle with PB that OP does with P.l.
Proof.— From {r + dr)(r' — dr') = vi' and r?-' = vi^ ; therefore rdr' = r'dr
or r : r' = dr : dr' = sin : sin 6', if 6, d' be the angles between the normal
and r, r . But OP divides APB in a similar way in reverse order.
5316 Let OP = B, then the normal PG, and the radius of
curvature at P, are respectively equal to
THE LEMNISCATE.t (Fig. 126)
5317 Characteristic— This curve is what a Cassinian be-
comes when m = a. The above equations then reduce to
{.v^^i/y = 2rt' i^v^-f) and r'^ = 2a^ cos 26.
5318 The lemniscate is the pedal of the rectangular hyper-
bola, the centre being the pole.
5319 The area of each loop = a\ (5206)
THE CONCHOID. J (Fig. 127)
5320 Characteristic— If a radiant from a fixed point in-
tersects a fixed right line, the directrix, in P, and a constant
length, BB = J), be measured in either direction along the
radiant, the locus of P is a conchoid. If OB = a, be the per-
pendicular from upon the directrix, the equation of the
curve with B for the origin or for the pole is
5321 ^vY = {a+i/Y (b'-if) or r = a sec e±b.
* B. WiUiamson, M.A., Educ. Times Math., Vol. xxv., p. 81.
t Bernoulli, Opera, p. 609.
X Nicomedes, about a.d. 100.
728 TUEOBY OF PLANE CURVES.
5323 When rt < &, there is a loop; when a = h, a cusp;
and when a > ^5 there are two points of inflexion.
5324 To draw the normal at any point of the curve, erect
perpendiculars, at B to the directrix, and at to OP. They
will meet in S the instantaneous centre, and SP will be the
normal at P (5242).
5325 To trisect a given angle BON by means of this curve,
make AB = 20N, and draw the conchoid, thus determining
Q; then AON = SAOQ.
Pkoof.— Bisect QT in S ; QT = AB = 20N, therefore /S'^= SQ=ON;
therefore NOS = N80 = 2NQ0 = 2A0Q.
5326 The total area of the conchoid betwceu two radiants each making
an angle 6 with OA is
aHan0 + 26^9 + 3ay(&- — a^) or aHsxud + 2h'd,
according as h is or is not >a.
The area above the directrix 7 _ o 7 1 o- fm ( ^4- ~\ A-l-6
between the same radiants ) ~ *= \ 4 2 /
The area of the loop which exists when 5 is >a is
b a— y(h^ — a^)
THE LIMAgON.* (Fig. 128)
5327 Characteristic. — As in the conchoid, if, instead of the
fixed line for directrix, we take a fixed circle upon OB as
diameter. This curve is also the. inverse of a conic with
respect to the focus. The equation, with OB for the initial
line and axis of x is
5328 r = a cos 6:^b or (ci''-^+?/^ — rt.r)' = b- {d^-\-/f),
where a = OB, h = PQ.
5330 With h > (/, is a conjugate point.
With /> < «, is a node. [For m = a, see (5332).
5331 The area = it {la'-\-l)').
AVliun a = 2h, the limaf;on has been called the Irlscctrix.
* Blaise Pascal, 1643.
TRANSCENDENTAL AND OTHER CURVES. 729
THE VERSIERA.* (Fig. 130)
{Or Witch of Agnesi.)
5335 Gharacterlstic. — If upon a diameter OA of a circle as
base a rectangle of variable altitude be drawn whose diagonal
cuts the circle in i?, the locus of P, the point in which the
perpendicular from B meets the side parallel to OA, is the
curve in question. Its equation is
5336 .17/ = 2a v/(2«.i'-cr2),
where a = 00 the radius.
5337 There are points of inflexion where x = f «.
The total area is four times the area of the circle.
THE QUADRATRIX.t (Fig. 131)
5338 Characteristic. — The curve is the locus of the inter-
section, P, of the radius OD and the ordinate QN, when these
move uniformly, so that x : a :: 9 : -J-tt, where x = ON,
a = OA, and = BOD. The equation is
= a^ tau ( . -J- ).
\ a 2/
5339 The curve effects the quadrature of the circle, for
OG : OB :: OB : arc ADB.
Proof: 00 : OB y. CP : BD. But CP = x in the limit when it is small,
therefore CP : BD :: a : ADB.
5340 The area enclosed above the x axis = 4rt'7r~^ log 2.
Proof. — In the integral x tan ( — j dx substitute tt (a — x) = 2ay,
and integrate yjj inn yihj by parts, using (1940). The integrated terms
produce log cos ^t — log cos ^^tt at the limit |7r, which vanishes though of
the form go — 00 . The remaining integral is j log cosydy, and will be found
at (2635).
THE CARTESIAN OVAL. (Fig. 131)
5341 Gharacteristic. — The sum or difference of certain fixed
multiples of the distances of a point F on the curve from two
* Donna Maria Agnosi, InstUiiziani Analitichc, 1748, Art. 238. t Dinostratus, 370 b.c.
5 A
730 THEORY OF PLANE CURVES.
fixed points A, B, called tlie foci, is constant. The equations
of the inner and outer ovals are respectively
5342 im\-\-lr2 = /JCg, mr^ — lr. = nc^,
where Vi = AP, rg = BP, c^ = AB, and n > m > I.
534:3 ^° draw the curve, put — = ^ and — - =-a ; therefore r^^ /jir^ = a,
m m
where a is > AB and )u < 1 (1). Describe the circle centre A, and radius
AR = a. Draw any radiant AQ, and let P, Q be the points in which it cuts
the ovals, then, by (1),
5344 PR=fxPB and QR = fiQB (2).
Hence, by (932), we can draw the circle which will cut AR in the required
points F, Q. Thus any number of points on the oval may be found.
5345 By (2) and Euc. vi. 3, it follows that the chord BBr bisects the
angle PBQ.
Draw Ap thi^ough r, and lot PB, QB produced meet Ar in p and q. The
triangles PBR, qBr are similar, therefore qr = ju-qB ; therefore q is ou the
inner oval. Similarly p is on the outer oval. By Euc. vi. B., PB.QB
= PR. QR + BRr ■ therefore, by (2), (l-yu^) PB.QB = BR\ Combining
this with PB : Bq = BR : Br, from similar triangles, we get
5346 BQ.Bq = ^f^ = p4 (3).
5347 Draw QG to make Z BQG = BAq; therefore, A, Q, C, q
being concyclic, we have, by (3),
BQ.Bq = AB.BC = p^ (4).
1 —jM
Hence C can be found if a, /t, and the points yl, B are
given. G is the third focus of the ovals, and the equation of
either oval may be referred to any two of the three foci.
Putting BC = Ci, AG = c^, AB = c^, the equation between I, m, n is
obtained from (4) thus: c^Ci{l—fi^) = a^—cl; therefore Ci(cs + Ci) = a' + frciCy
But C3 + C1 = a„ a = — -, u = — , and the result is
m m
5348 l'c,-Jrn% = m% or l'BC-\-m'CA-\-7rAB = 0... (5),
where GA =—AG.
Putting 7',, r^, r.j for PA, PB, PG, the equations of the
curves arc as follows —
TRANSCENDENTAL AND OTHER CURVES. 731
Inner Oval. Outer Oval.
5349 mri + Zra =710^ ... (6), mr^—lr-^ = nc^ ... (7),
5351 nri+li's = mc.2 ... (8), nry-lr^ =mc2... (^0'
5353 mr^—nr.2=lci .-.(10), nri^mr^=lc^ ...(11).
That (6) and (7) are equations of the curve has been shown. To deduce
the other four, we have Z APB = AqB = ACQ (5347) ; therefore ACQ,
APB are similar triangles. But, by (6), mAP + lBP = nAB, therefore
mAG+lGQ = 7iAQ or nAQ—WQ = mAG, wliich is equation (9). Again,
ABQ, APG are similar. But, by (7), mAQ-lBQ = nAB ; therefore
mAG-lGP = nAP or nAP+lGP = mAG, which is eqiiation (8).
Equations (10) and (11) are obtained by taking (G) from (8) and (7)
from (9), and employing (5).
5355 AP.AQ= AB.AC= coustaiit.
Proof. — Since A, Q, G, q are concyclic, Z QGA = QqA = ABB ; there-
fore P, Q, G, B are concyclic; therefore AP.AQ = AB.AG = constant (12).
5356 CP. CP' =CA.CB = constant.
Proof: Z PGB = PQB = Bpq = BGq. Hence, if GP meets the inner
oval again in P', GBq, GBP' are similar triangles. Again, because Z BPG
= BQG = BAq = BAP', the points A, B, P', P are concyclic ; therefore
CP.CP' =GA.GB = constant. Q. E. D.
Hence, by making P, P' coincide, we have the theorem : —
5357 The tangent from the external focus to a series of tri-
confocal Cartesians is of constant length, and = ^{GB.GA).
5358 To draw the tangents to the ovals at P and Q. De-
scribe the circle round PQGB, and produce BB to meet the
circumference in T; then TP, TQ are the normals at P
and Q,.
The proof is obtained from the similar triangles TQB, TBQ, which show
that sinTQA : sin TQB = Z : w, by (2), and from differentiating equation (7),
which produces — ^ : — -? = Z : m*
as as
5359 The Semi-cubical parabola y^ = aoc^ is the evolute of a
parabola (4549). The length of its arc measured from the
origmis "=2>R^+4"'V"^r
* For the length of an arc of a Cartesian oval expressed by Elliptic Functions, see a paper
by S. Roberts, M.A., in Froc. Lond. Math. Soc, Vol. v., p. 6.
732 THEORY OF PLANE CURVES.
5360 The FoJiinn of Descartes, .i^^— 3«,i7/+?/ = 0, has two
infinite brandies, and the asymptote ^v-\-i/+(i = 0.
For the lengths of arcs and for areas of conies, see (6015), et seq.
LINKAGES AND LINKWORK.
5400 A jjk?ir^ linkage, in its extended sense, consists of a
series of triangles in the same plane connected by hinges, so
as to have but one degree of freedom of motion ; that is, if
any two points of the figure be fixed, and a third point be
made to move in some path, every other point of the figure
will, in general, also describe a definite path. With two points
actually fixed, the hnkage is commonly called a piece-ivorlc,
and if straight bars take the place of the triangles, it is called
a linh-iDorlc.
THE FIVE-BAR LINKAGE.
5401 Mr. Kempe's fundamental five-bar linkage is shown
in Figure (135). A, B, D' are fixed pivots indicated by small
circles. G, I), B', C , in the same plane, are moveable pivots
indicated by dots. The lengths of the bars AB, BG, GD, DA
are denoted by a, b, c, d. The lengths of AB\ B'G\ G'D', D'A
are proportional to the former, and are equal to ha, kh, kc, Jed,
respectively. Hence ABGD, AB'G'D' are similar quadri-
laterals, and A AUG' = ADG. P being any assigned point
on BG and />'P = X, F' must be taken on D'G' so that
D'F' = A — . Draw FN, F'N' perpendiculars to AB. Then,
ah
throughout the motion of the linkage in one plane, NN' is a
constant length.
Proof: NN' = ]}D'-(BN+N'D'). But BB' = a-U, and
BN+N'B' = \ cos B-\ '^ cos B-^(2ah cos B-2ccl cosD)
ah 'lab
= _A_ („'^ + h^ - c^ - d') (702). Hence
2ab
LINKAGES AND LINEWOEK. 733
5402 NN' = u-kd- JL^{a'~J\-b'-c'-cP).
2(10
5403 Case I. -(Fig. 136.) If A = ^(^^;M)_^, then
NN' = -^ ; consequently, if tlie bars PO = BB and F'O =
F'B' be added, the point will move in the line AB.
If, in this case, d = ha and h = c, then X = h and F coin-
cides with G, P' with C, and B' with D, as before moving
in the line AB.
5404 Case IL — (Fig. 137.) If, in Case I., M = a and
a^-\-J)^ ^ c^^(p^ X is indeterminate; that is, P may then be
taken anywhere on BG. D' coincides with B, and NN' = 0.
PP' is now always perpendicular to AB. If the bars PO,
P'O be added, of lengths such that PO^-P'0'~ = PB--P'B\
will move in the line AB. If, on the other side of PP' , bars
P0'= P'B and P'O' = PB be attached, then 0' will move in a
perpendicular to AB through B.
5405 Case III.— (Fig. 138.) If, in Case I., U = «, h = d,
and G = — a, the figure A BCD is termed a contra-parallelogram.
BP = X is indeterminate, BC'=hc = -'^^^ and BP' = -X.
Hence BG' and BP' are measured in a reversed direction ;
PP' is always perpendicular to AB, and if any two equal bars
PO, P'O are added, Avill move in the line AB.
5406 If three or more similar contra-parallelograms be
added to the linkage in this way, as in Figure (139), having
the common pivot B and the bars BA, BG, BE, BG in geo-
metrical progression ; then, if the bars BA, BG are set to any
angle, the other bars will divide that angle into three or more
equal parts.
5407 If, in Figure (138), AD be fixed and DG describe an
angle ADG, then B'G' describes an equal angle in the opposite
direction. Mr, Kempe terms such an arrangement a recersor,
and the linkage in Figure (139) a multiplicator. With the aid
734 THEORY OF PLANE CURVES.
of these, and with a translator (Fig. 140), for moving a bar
AB anywhere parallel to itself, he shows that any plane curve
of the 7^*'' degree may, theoretically, be constructed by link-
work.*
5408 Case IV. — (Fig. 141.) If, in the original linkage
(Fig. 135) M = a, D' coincides with B. Then, if the bars
BPO, RP'O' be added by pivots at P, F, and B ; and if
OP = PB = BP' and O'P' = P'B = BP ; the points 0, 0'
will move in perpendiculars to AB. For by projecting the
equal lines upon AB, we get jVL = BN' and BN = N'L\
therefore BL = BL' = NN' = a constant, by (5402).
5409 Case V.— (Fig. 142.) Make ha = d and X = h. Then
B' coincides witli D, P with G, and P' with G'. Replace B'G\
G'D by tlie bars DK, KD' equal and parallel to the former.
Also add the bars GO = DK and OK = CD. Draw the per-
pendiculars from 0, G and G' to AB. Then by projection,
NL = N'D' ; therefore BL = BN+NL = BN+N'U = BD'
—3/7V"'= constant. Hence the point will move perpendi-
cularly to AB.
5410 Case VI.— (Fig. 143.) In the last case take k = l.
Therefore d = a, U coincides with ,Z?, BK = BG, and GDKO
is a rhombus. This is Peaucellier's linkage.
5411 Casi^ VII. — (Fig. 144.) In the fundamental linkage
(Fig. 135), transfer the fixed pivots from A, B to P, 8, adding
the bar ^S^, so that PBHA shall be a parallelogram. Then,
since NN' is Constant (5 102), the point P' will move perpen-
dicularly to the fixed line PS.
5412 Join AG cutting PS in Z7, and draw UV parallel to AD.
Then UV : AD = PU : AB = GP : GB = constant ; there-
fore P?7 and f/Fare constant lengths. Hence it follows that
the parallelism of AB to itself may be secured by a fixed pivot
at U and a bar UV instead of the pivot S and bar SA.
5413 In Case VII. (Fig. 144), with fixed pivots P and 8
* Froc. of the Loud. Math. Soc, Vol. vii., p. 213.
LINKAGES AND LINKWOBK. 735
and bar SA, make h = a, d = c, ha = d, \ = h. Then B' coin-
cides with D, N' with N, F with G and L, and F' with G' ;
and we have Figure 145, DG, DG' are equal, and they are
equally inclined to AB or G8; because, in similar quadri-
laterals, it is obvious that AB and GD and the homologous
sides DG' and AD' include equal angles. Therefore GG' is
perpendicular to GS, and G' moves in that perpendicular only.
5414 If two equal hnkages like that in (5413), Figure (145),
but with the bars AS, GS removed, be joined at D (Fig. 146)
and constructed so that GDy, jDG' form two rigid bars, then
AB, a|3 will always be in one straight hue. Let A, B be made
fixed pivots, then, while G describes a circle, the motion of
the bar oj3 will be that of a carpenter's plane.
5415 On the other hand, if the linkage of Figure (145), with
AS and GS removed as before, be united to a similar inverted
Hnkage (Fig. 147), with DG, DG' common, then, with fixed
pivots A, B, D', the motion of the bar a/3 will be that of a Hft,
directly to and from AB.
5416 The crossing of the Hnks may be obviated by the
arrangement in Figure (148). Here the bars (7'/3, G'D, G'D'
are removed, and the bars FD, FE, FG added in parallel ruler
fashion.
5417 Case YIII.— (Fig. 149.) In Case YIL, substitute the
pivot TJ and the bar UV for S and SA. Make d = a, and
therefore Jv = l. Then h' = b and c = c, making BGDG' a
contra-parallelogram ; D' coincides with B, and B' with D.
The bars AB, AD are now superfluous. Take BF = X; then
BF' = X f ; therefore FF' is parallel to GG', therefore to BD,
h
therefore to FV (5412) ; therefore V, F, F' are_ always in one
right line. F' , as in Case VII., moves perpendicularly \>oFV
and AB. This arrangement is Hart's ^i;e-Z>ar Unhage.
5418 When a point F (Fig. 152) moves in a right line FS,
it is easy to connect to P a linkage which will make another
point move in any other given line we please in the same
736 THEORY OF PLANE CURVE 8.
plane. Let QR he sucla a line cutting PS in Q. Make Q a
fixed pivot, and let 0(2, OF, OB be equal bars on a free pivot
0. Then, if the angle FOR be kept constant by the tie-bar
Fll, FQB, being one half of FOB (Euc. iii. 21), will also be
constant, and therefore, while F describes one line, B describes
the other.
If the bar FO carries a plane along with it, every point in
that plane on the circumference of the circle FQB will move
in a right line passing through Q.
THE SIX-BAR INVERTOR *
5419 If in the linkwork (5410, Fig. 143) the bar AD be
removed, and D be made to describe any curve, will describe
the inverse curve, just as, when D described a circle, moved
in a right line which is the inverse of a circle.
Proof.— Let BOD and CK intersect in E. Then BO.OD = BE- - OE'
= BG- — OC^- = a constant called the mochilus of the cell.
THE EIGHT-BAR DOUBLE INVERTOR.
5420 Two jointed rhombi (Fig. 150) having a common
diameter AB form a double Peaucellier cell termed positive
or negative according as P or Q is made the fulcrum. We
have P(2.PB = PQ.Q8 = AP^-AQ\ the constant modulus
of the cell.
THE FOUR-BAR DOUBLE INVERTOR.
5421 If, on the bars of a contra -parallelogram ABCD
(Fig. 151) four points j), q, r, s be taken in a line parallel to
Ac or BD, then in every deformation of the linkage, the
points 2h <h '■' ^ ^^ill ^ic in a right line parallel to AG ; and
pq .pr = pq . q,^ = a constant modulus. Thus, if _p be a ful-
crum and r describes a curve, q will describe the inverse
curve. If q be the fulcrum, j? will describe the inverse curve.
Proof. — Let Ap = mAB, therefore pq = mBD, &nd pr = (I — m) AG,
thereiore pq.pr = m (1— w) AG.BD = m (l—m)(AD'—AB-) = constant.
* Since the curve described is the inverse and not the i^olar reciprocal of the guiding curve,
it seems bettor to call this linkage an invtrtor rather than a reciprocator.
LINKAGES AND LtNKWOEK. 737
THE QUADRUPLANE, OR VERSOR INVERTOR.
5422 Let the bars of the contra-parallelogram invertor
(5421, Fig. 151) carry planes, and let P, Q, B, S be points in
the planes similarly situated with respect to the bars which
contain p, q, r, s respectively, so that /.PAp = QA<i and
AP : Ap = AQ : Aq; and similarly at G. Then, if P be the
fulcrum and B traces a curve, Q will trace the inverse curve
and the angle QPB will be constant.
PjjOof. — Let PA = nAB and PB = n'AB, therefore, by similar triangles,
PAQ, BAD, PQ = nBD. Also, by the triangles PBU, ABG, PB = n'AG ;
PA PTi
therefore PQ.PB - nn'AG.BD = ^^j~- (AD' - AB') , a constant.
Again, the inclination of PQ to BD = that of AP to AB, which is con-
stant. Similarly, by the triangles PBB, ABG, the inclination of PB to AG
= that of BB to BG, which is also constant ; therefore QPB, the sum of
these two inclinations, is a constant angle.
THE PENTOGRAPH, OR PROPORTIONATOR.
5423 Let ABGD (Fig. 153) be a jointed parallelogram, J, B
fixed pivots, q a tracer placed at any assigned point in BG
produced ; then a pencil at p will evidently reproduce any
figure traced by q diminished in linear proportions in the ratio
of Bq to BG.
THE PLAGIOGRAPH, OR VERSOR PROPORTIONATOR.
5424 In the same figure, make an angle qBQ = pBP,
BQ = Bq, and DP = Dp, and let a tracer Q and pencil P be
rigidly connected to the arms BG and DG. Then P will pro-
duce a similar reduced figure as before, but no longer similarly
situated. It will be turned round through an angle QBq.
This is Prof. Sylvester's Plagiograph.
Proof.— Let BG = h.Bq', therefore AD = kBQ, DP = kAB, and Z ABQ
= PDA ; therefore (Euc. vi. 6) AP = hAQ. Also PAQ is a constant angle,
for PAQ = BAD-BAQ-PAD = BAD-BAQ-BQA = BAD- (n- ABQ)
= BAD-TT + ABG+QBq = QBq.
THE ISOKLINOSTAT,* OR ANGLE-DIVIDER.
5425 This linkage (Fig. 154) accomplishes the division of
an angle into any desired number of equal parts. The dia-
* Invented and so named by Prof. Sylvester.
5 B
738 TEEOBY OF PLANE CURVES.
gram shows the trisection of an angle by it. A number of
equal bars are hinged together end to end, and also pivoted
on their centres to the same number of equal bars which
radiate, fan-like, from a common pivot. The alternate radial
bars make equal angles with each other.
The same thing is accomplished in a different way by
Kempe's Multiplicator (5406, Fig. 139).
A LINKAGE FOR DRAWING AN ELLIPSE.
5426 In the arrangement of (5413, Fig. 145) the locus of
any point P, on DC, excepting D and G', is an ellipse.
Proof. — Take 08, 00' for x and y axes ; P the point xy ; SOB — 0,
and therefore ODC' = 29; PD = h. Then we have x = (c—h)cosO,
y = (c + h) sinO, therefore ^ + -^ .^ = 1 is the equation of the locus.
(^C — ll) i^C-Tll)"
Any point on a plane carried by DC also describes an ellipse round ; but
if the point lies on a circle whose centre is D and radius DO, the ellipse be-
comes a right line passing through 0, as appears from (5418).
A LINKAGE FOR DRAWING A LIMAQON, AND ALSO A
BICIRCULAR QUARTIC.*
5427 (Fig. 155.) Let four bars AF, AQ\ BG, GD be
pivoted at A, B, G, D, and let AB = BG = BQ' = a ; AD
= DG = DP' = 6. Take a fulcrum F on BG, a tracer at P,
and a follower at Q, so that PQ is parallel to BD. Let FP = p,
FQ = r; then, if P traces out a circle passing through F, Q
will describe a lima9on.
Proof.— Let BQ = ma, therefore PD = mh ; r = 2?u . BN, p = {m. + 1) . DN
+ (1 - m) BN. Also BN^ - DN^ =o?- h\ Eliminate BN and DN, and the
equation between r and p is
r+(l — m) rp — mp^ = m (ju + l)'- (ci' — b-) = A;^
If P describes the circle p = c cos 0, Q describes the locus
r"+ (1 — m) cr cos B — mc- cos'- Q = Zr,
which is the inverse of a conic, that is, a limayon (5327).
If be made the fulcrum, the equation reduces to r—p^ = 4 (a- — h-).
5428 With the same fulcrum F, drawing FH parallel to
AG, if a tracer at II describes the circle, then a follower at K
on GD will trace out a bicircular quartic.
* W. Woolsey Johnson, Man. of Matli,, Vol. v., p. 15'.'.
LINKAGES AND LINKWORK. 739
Proof. — Draw FL, LK parallel to BA, AD. Let FH = p, FK = r,
CK = /3, CF = a = nFB, and therefore CL = np. Now
2 (cc + rr) = r + ny+ ("'-fy\
r'
Therefore, if H moves on the circle p = c cos 0, K will describe the curve
7-* + vrcV- cos'0-2 (a-+/3') r+ (a-— /3'-)- = 0,
or (a;Hr)'+ Kc'-2a'--2/3^) x^-2 {a^-\-(i^) y'+ (a'-/?)- = 0.
A LINKAGE FOR SOLVING A CUBIC EQUATION.*
5429 Let the tliree-bar linkwork (Fig. 156) have the bars
AB, DC produced to cross each other. Let AB = AD = a,
BC ^h, CD = c ; and let b and c be adjustable lengths.
Suppose x^ — qx-\-r = a given cubic equation.
Make <?=2"\/( '/ + — )? ^ — i\/(^~~)j ^^^n deform the
quadrilateral until EG = CD ; DE will then be equal to a real
root of the cubic.
Proof : conE = — = ^ — --^-^ — ,
2cie 4c {x + a)
from which .t3«-2 (r + &') ai + 2a {c'-lr) = 0.
Equate coefficients with the given cubic.f
ON THREE-BAR MOTION IN A PLANE.
5430 If a triangle ABG (Fig. 157) be connected by the
bars AO, BO' to the fulcra 0, &, the locus of G is called a
three-bar curve.
OA, O'B meet in Q, the instantaneous centre of rotation of
the triangle, since QA., QB are perpendicular to the movements
of J and B respectively. Therefore GQ is the normal to the
locus of G.
5431 If a triangle similar to ABG be placed upon 00'
(homologous to AB), the circum-circle of the triangle will
pass through the node, and the vertices of the triangle are
called the foci of the curve.
* M. Saint Loup, Comptcs Rendus, 1874.
t The foregoing account of linkages is taken chiefly from a paper by A. B. Kempe,
F.R.S., in the Proc. of the Eoyal Soc. for 1875, Vol. xxiii. Other results by the same author
will be found in the Froe. of the Lond. Math. Soc, Vol. ix., p. 133 ; and by H. Hart, M.A.,
ihid., Vol. VI., p. 137, and Vol. viii., p. 286. See also The Messenger of Mathematics, Vol. v.
740 THEORY OF PLANE CURVES.
Figures (158) and (169) exhibit different varieties of the
curve according to the rehxtive proportions between the lengths
of the bars.*
MECHANICAL CALCULATORS.
The Mechanical Integrator. i
5450 This instrument computes not only the area of any
closed plane curve, but the moment and also the moment of
inertia of the area about a fixed line. The principle of its
action is shown in Figure (160). OP is a bar carrying a
tracer at P, and a roller J at some point of its length. The
end is constrained to move in the fixed line ON. When
the tracer P moves round a closed curve, the length OP mul-
tiplied by the entire advance recorded by the roller is equal to
the area of the curve.
Proof. — Let the motion of the tracer from P to a consecutive point Q be
decomposed into PP' and P'(2 parallel and perpendicular to ON. Let
OP = a and PON = 6. When the pointer moves from P to P', the roll
accomplished is PP' sin 6. The roll due to the motion from P' to Q will be
neutralized by the exactly equal and opposite roll in the motion of the pointer
from q to ^j', since the bar will there have again the same inclination. Con-
sequently the product of the entire roll and the length a is equal to the sum
of such terms as aPP' sin d. But this is the area OPP'O' = NPP'N'. The
algebraic addition of such rectangles gives the entire area, and the instru-
ment effects this, for the area SN is subtracted, by the motion of the roller,
from the area QN which is added.
5451 The instrument itself is shown in Figure (161). A
frame moving parallel to OX by means of the guide BB
carries two equal horizontal wheels geared to a central wheel
which has two circumferences, such that its rate of angular
motion is half that of the lower wheel and one third of that of
the upper. The latter wheels carry two rollers, M and I, on
horizontal axles ; and the middle wheel carries an arm OP, a
pointer at P, and a roller A. In the initial position, the
* Tho curve is a tricircular trinodal sextic, and is completely discussed by S. Roberts,
F.R.S., and Prof. Cayley, in the rruc. of the Lond. Math. Soc, Vol. vii., pp. 14, 136.
t Invented and manufactured by Mr. J. Amsler-Laflbn, of Schaflfhauson. The demon-
strations (which in clearness and elegance cannot bo surpassed) of the action of this instru-
ment, and of the rianimetcr which follows, were communicated to the author by Mr. J.
Macfarlane Gray, of the Board of Trade.
3IEGHANICAL CALCULATORS. 741
rollers J and /are parallel, while M is at right angles to A.
The frame is thus supported above the paper on the three
rollers ; and if the arm OP be moved through an angle AOA',
the axles of the rollers M and I will describe twice and three
times that angle respectively. Putting OP = a as above, and
A, My and I for the linear circumferential advances recorded
by the three equal rollers respectively, we have the following
results —
I. — The area traced out by the pointer P = Aa.
II. — The moment of the area about OX = M
a-
III. — The moment of inertia about OX = (3^4 + /)--.
Proof. — I. Since moves in the line OX, while the pointer P moves round
a curve, the roller A will, as shown above, make the rolling 2A. sin 6, where
Ji = PP' in Figure (160), and the area of the curve = a'^h sin or a X roll.
II. The moment of the area about OX
= 2 (a/i sin X ^^^) = -^ (2/i-2/i cos 2^).
Now 2A vanishes when P returns to the starting point, and — 2/; cos 2d is the
roll recorded by M. For, when OP makes an angle with OX, the axis of
31 will make an angle — (90^ + 26) with OX. In this position, while P makes a
parallel movement h, the roll produced thereby in M will be —h sin (90° + 2^)
= —Ji COS 20. Therefore — X roll of M = moment of area.
4
III. Lastly, the moment of inertia of the area about OX
= :^( ah sin ex^-~^\ = ~%{Shsind-hsm^d).
Now, when OP makes an angle 6 with OX, the axis of I makes —30 ; there-
fore — 2/t sin 30 is the entire roll of I. Hence the moment of inertia
= -^ X roll oiA+^x roll of I.
4 12
The Planimeter. (Fig. 162)
5452 This instrument * is a simpler form of area computer.
is a fixed pivot ; OA, AP are two rods having a free pivot
at ^ ; (7 is the roller, and P the pointer. The area of a closed
curve traced by the pointer is equal to the total roll multiplied
by the length AP.
* Like The Integrator, the invention of Mr. Amsler.
742 TREOBY OF PLANE CURVES.
Proof. — Decompose the elementary motion PQ of the pointer into PP',
effected with a constant radius OP, and P'Q ah^ig the radius OP', and so all
round the curve. The roll of G accomplished while P moves from P' to Q
will be neutralized by the equal contrary roll when P moves from q to j:)' on
the radius Op' = OP. Thus the total roll recorded will be the sum of the
rolls due to the movements PP', QQ', &c.
Draw OB perpendicular to AP, and, when P comes to P, let B' be the
altered position of B. The area PQSP = \{0P-- OR-) w, where u, = POQ.
But 0P^= A' + PA- -2PA.AG-2PA.bg (Euc. ii. 13) ; therefore, since
BG is the only varying length on the right, we have PQSR = PA{BG^BG')u).
But BGio is the roll of G due to the angular motion w of the rigid frame OAP,
and the subtraction of the area OSR from 0P(^ is effected by the instrument,
since when the pointer moves from 8 to R the direction of the roll must be
reversed. Hence the total area = PA x the total recorded roll.
APPENDIX ON BIANGULAR COORDINATES.*
5453 In the figure of (1178), the biangular coordinates of a
point P are defined to be ^ = F88' and <p = PS'S, or
a = cot B and /3 = cot ^.
5454 The equation of a right Hne YY' is
aa-\-b^ = 1,
where a = cot SYS' and h = cot SY'S'.
Proof. — Supplying the oi'dinate PN in the tigure and denoting the angle
S'SY by i/', the equation is obtained from GN cos \p + PN sinif/ = p the per-
pendicular on the tangent, SS' &\n\j/ =. YY' and SS' cos '4' = SY—S'Y'.
5455 cot xjj = a — b.
5456 Equation of a hne through G : a — /8 = const.
5457 Equation of the fine at infinity : a-\-fi = 0.
5458 Let >S^/S" = c, then the distance between two points
aji„ a,/3, is
* Quarterly Journal of Mathematics, Vole, 9 and 13 ; W. Walton, M.A,
APPENDIX ON B I ANGULAR COORDINATES. 743
5459 The equation of a line tlirougli tlie two points is
;8-A ~ A-A"
5460 The length of the perpendicular from a'/3' upon the
line aa-^h[i = I is
a'+A Vl(«-6y^+i}'
5461 Cor. — The perpendiculars from the poles S, S' are
therefore
^/{{a-by+ly ^{{a-by+l}'
5463 When the point a (5' is on SS' at a distance h from 8,
_ (a — b) h-\-bc
^'~ v^{{a-by-\-iy
With two lines aa-\-h^ = 1, aa-{-b'(5 = 1, the condition
5464 of parallelism is a — b=^a—b',
5465 of perpendicularity {a — b){a—b')-]-l = 0.
5466 The equation of the line bisecting the angle between
the same lines is
aa-\-bfi-l a'a-\-b'^-l
s/{{a-by-^l] V{{a'-b'y-\-l\
5467 The equation of the tangent at a point a^' on the
curve F{a, jS) = is
5468 And the equation of the normal is
a- a _ ^—^'
{a^'-l) F,. + (l + a^) F, = (g A-1) F, + (1+^^) F,:
5469 The equation of a circle through /S', 8' is
a^-l = m(a+/8),
where m = cot 8P8' the angle of the segment.
744 THEOUY OF PLANE CURVES.
5470 If G be the centre, tlie equation becomes
a/3= 1.
5471 And, in this case, the equations of the tangent and
normal at a (5' are respectively
4 + 4 = 2 and a-y8 = a'-/8'.
a p
5472 The equation of the radical axis of two circles whose
centres are S, S\ and radii a, b, is
uating the tangents from a/3 to the
actively
Proof. — By equating the tangents from a/3 to the two circles, their
lengths being respectively
5473 To find the equation of the asymptotes of a curve
when they exist, —
Eliminate a and j3 between the equations of the line at
infinity a+)8 = 0,
the curve F (a, y8) = 0,
and the tangent (a — a ) F,,-\-{^—^') F^> = 0.
Ex. — The hyperbola a^ + ft' = vi" lias, for the equation of its asj-mptotes,
a — /> = ± m \/2.
SOLID COORDINATE GEOMETRY.
SYSTEMS OF COORDINATES.
CARTESIAN OR THREE -PLANE COORDINATES.
5501 The position of a point P in this system (Fig. 168) is
determined by its distances, x = PA, y = PB, z = PC, from
three fixed planes YOZ, ZOX, XOY, the distances being
measured parallel to the mutual intersections OX, OY, OZ of
the planes, which intersections constitute the axes of coordi-
nates. The point P is referred to as the point xyz, and in
the drawing x, y, % are all reckoned positive, ZOX being the
plane of the paper and P being situated in front of it, to the
right of YOZ and above XOY. If P be taken on the other
side of any of these planes, its coordinate distance from that
plane is reckoned negative.
FOUR.PLANE COORDINATES.
5502 In this system the position of a point is determined
by four coordinates a, j3, y, I, which are its perpendicular
distances from four fixed planes constituting a tetrahedron of
reference. The system is in Solid Geometry precisely what
trilinear coordinates are in Plane. The relation between the
coordinates of a point corresponding to (4007) in trilinears is
5503 ^a+i^yg+Cy+DS = 3F,
where A,B,G,D are the areas of the faces of the tetrahedron
of reference, and V is its volume.
TETRAHEDRAL COORDINATES.
5504 In this system the coordinates of a point are the
volumes of the pyramids of which the point is the vertex and
5 c
746 SOLID GEOMETRY.
tliG faces of tliG tetrahedron of reference the bases : viz., ^Aa,
\B^, \Cyi -^D^. The relation between them is
5505 a'+i8'+y+8'= V.
POLAR COORDINATES.
5506 Let be the origin (Fig. 168), XOZ the plane of
reference in rectangular coordinates, then the polar coordi-
nates of a point P are r, B, (/>, such that r = OP, = LPOZ^
and = Z XOO between the planes of X^OZ and POZ.
THE RIGHT LINE.
5507 The coordinates of the point dividing in a given ratio
the distance between two given points are as in (4032), with a
similar value for the third coordinate t-
5508 The distance P, Q between the two points xijz^ x'y'z is
PQ =^{(^aj-w'f^-{y-yj^-{z-^j}. (Euc.i.47).
5509 The same with oblique axes, the angles between the
axes being X, /t, v.
PQ = y {(.^^_.^.7+(^,_,y7+(^-^7+2 {y-2j){z-z') cosX
+ 2 {z-z%v-.v) cos/x+2 {x-a'){y-ij) cos v\ . (By 702).
5510 The same in polar coordinates, the given points being
PQ = y[r2+/^-2r/ [cos 6 cos ^'+sin 6 sin & cos (<^-f ) } ].
Proof. — Let P, Q be the points, the origin. Describe a sphere cutting
OF, OQ in 1J,C and the z axis in A; then, by (702), PQ- = OF'+OQ^
— 20P. OQ cos FOQ and cos POQ, or cos a in the spherical triangle ABC, is
given by formula (882), since b = 6, c = d', and A = <p — <p'.
DIRECTION RATIOS.
5511 Through any point Q on a right line QP (Fig. 169),
draw lines QL, QM, QN parallel to the axes, and through any-
other point P on the line draw planes parallel to the coordi-
THE BIGHT LINE. 747
nate planes cutting the lines just drawn in L, M, N; then the
direction ratios of the line OP are
5512 /-^ m-^ n-^
The angles PQL, PQM, PQN are denoted by a, j3, y ; and the
angles YOZ, ZOX, XOY between the axes by X, in, v.
5513 When X, ^, v are right angles, the axes are called
rectangular, and the direction -ratios are called direction-
cosines, being in that case severally equal to cos a, cos/3,
cosy.
5514 When L, M, N are the direction-ratios (or numbers
proportional to them) of a line which passes through a point
abc, the line may be referred to as the line {LMN, ale), or, if
direction only is concerned, merely the line LMN.
EQUATIONS BETWEEN THE CONSTANTS OF A LINE.
5515 The relation between the constants of a line with
rectangular axes is
Z'^+m^+n- = 1 ;
and with oblique axes, it is
5516 / cos a+m cos /8+71 cosy = 1.
Proof.— The first by (Euc. i. 47). The second by projecting the bent
Hue QLGP (Fig. 169) upon PQ, thus PQ = QL cosa + LOcos/^+OPcosy,
and QL = PQ.l, &c., by (5512).
5517 Also, when the axes are oblique,
cos a = Z+m cos v-\- n cos fi,
cos^ = m+ w cos X+ Zcosv,
cosy= n-\- Z cos ju,-|-m cosX.
Peoof.— By projecting QP in figure (169) and the bent line Q,LGP upon
each axis in turn, and equating results; thus PQcosa= QL + LG cos(3
+ GP cos y, applying (5512).
5518 The relation between I, m, n and X, ^t, v is
P-{-m^-\-n^+2mn cosX+2/i/ cos/ut-f 2/m cos v = 1.
Proof. — By eliminating cos a, cob/3, cosy between (5516) and (5517).
748 SOLID GEOMETRY.
5519 The relation between cos a, cos/3, cosy and X, ^, v is
cos^ a siii^ X+cos^ /8 siu" ft+cos^ 7 siir v
+2 cos fi cos y (cos /a cos v— cos X)
+2 cosy cos a (cos V cos X— cos /a)
+2 cos a cos y8 (cos X cos /x— cos i/)
= 1 — cos"X— cos'/x — COS-1/+2 cosX cos ft cos V.
Pkoof. — By eliminating I, vi, n between the four equations in (5516) and
(5517).
5520 The angle Q between two right lines Imn, I'm'n, the
axes being rectangular :
cos 6 =: ll'-\-7nm-{-nti.
Proof. — In Figure (160), let QP be a segment of the line Imn. The
projection of QP upon the line I'm'n will be QP cos 6. And this will also
be equal to the projection of the bent line QLCP, upon I'm'n, for, if
planes be drawn through Q, L, C, and P, at right angles to the second line
I'm'n, the segment on that line intercepted between the first and last plane
will be = QP cosQ, and the three segments which compose this will be
severally equal to QL .1' , LG.m, CP.n, the projections of QL, LG, GP.
Then, by (5512), QL = QP.l, &c.
5521 sin^ e = (mn'-m'ny-\-{nl'- n'iy-Y{lm' -I'mf.
Proof. — From
l-cos-0 = {l' + m'' + n^){r'^m'-^n'-)-{ll' + mm' + nu')- (5515, '20).
5522 With oblique axes,
cos 6 = ll'-\-7n7u-^nn-\-{7n)i-\-inn) cosX
-\-{7iV-\-nl) cos/x+(/m'+rm) cos v.
Proof.— As in (5520), substituting from (5517) the values of cos a, &c.
EQUATIONS OF THE RIGHT LINE.
5523 ^^ = U^ = ^^ or .iz:i^^.y^^£^c
L M N I m ?i
Here abc is a datum point on the line, and if r be put for the
value of each of the fractions, r is the distance to a variable
point xyz. L, M, N are proportional to the direction ratios of
THE BIGHT LINE. 749
the line, which ratios must therefore have the values
5524 /= ..... ,\r., ^.^ ^ri= ^^
- ^
5525 Note. — Instead of <x, b, c in the equation we may use
IcL + a^ ]cM-\-b, Jd + c, where h is an arbitrary constant.
5526 The equations of a line may also be written in the
forms .1' = \z-\-a, y = iiz-\-^.
5527 These are the equations of the traces on the planes of
xz and yz, and are equivalent to
A -a _ ;/-/8 _ £-0
X ~ ^ ~ 1 ■
5528 If the line is determined as the intersection of the two
planes Ax-{-Bij-{-Cz = D and A'xi-B'y + C'z = V, we may
write equations (5523) by taking
L = BC'-B'C, M= CA'-C'A, N= AB-A'B,
DB'-D'B ^ DA-DA
N ' N
= 0.
Proof. — Eliminate z between the equations of the planes, then the
reciprocals of the coefficients of x and y will be L and M.
5529 The projection of the line joining the points xyz and
ahc upon the line Imn is
I {.v-a)-{-7n (y—b)+n (z—c).
5530 Hence, when the line passes through ahc, the square
of the perpendicular from xyz upon it is equal to
5531 Condition of parallelism of two lines LMN, L'M'N' :
750 SOLID GEOMETRY.
5532 Condition of perpendicularity :
LL'-^MM'-{-NN' = 0. (5520)
5533 Condition of the intersection of the lines (LMN, ahc)
and (L'M'N\ a'h'cf) (5514) :
{a-a'){MN'-M'N)^{h-h'){NL'-N'L)
J^{c-c')(LM'-L'M) = 0.
Proof. — Eliminate x, y, z between the equations
X — a _ y — h _ z — c_ ■, x — a'_ y — h' _z — c /
by subtracting in pairs, and then eliminate r and r' .
5534 The shortest distance between the same lines is
{a-a'){M}^'-M'N)^{h-h'){NL'-N'L)-\-{c-c'){LM'-L'M)
^{{MN'-M'NY-^{NL'-N'LY-\-{LM'-L'My\
Pkoof. — Assume A, n, v for the div-cos. of the shortest distance. Then,
by projecting the line joining ahc, a'h'c upon the shortest distance, we get
pz=z {^a-a)\ + ih-l') ix + {G — c')y. Also, by (5520), iX + iU/x + iV.' = and
L'X + M'n + N'v = 0, giving the ratios X : /n : y = MN'—M'N : NL'—N'L
: LM'—L'M; and (5524) then gives the values of \, n, v.
5535 The equation of the line of shortest distance between
the lines {Imn, ahc) and {I'lii'ii , a'h'c) is given by the inter-
section of the two planes
l{^^-a) + m(y-b) + n(s,-y) = 'l±4^ (i.),
I {x-a)-\-m {y-h)-\rn {z-y) = -_L___ ... (n.),
where it = I {a' — a)-\-m {b' — h)-\-n {c' — c),
u'=I'(a — a')-{- m' {b — h') + n {c — c) ,
and cos 6 = 11' -{-nini' -\- nn' .
Proof. — (Fig. 170.) Let be the point xyz on the line of shortest dis-
tance AB ; P, Q the points abc, a'h'c on the given lines AP, BQ. Draw BR
and PR parallel to AP and AB ; RT perpendicular to BQ ; and QN, TM per-
pendicular to BR. Then Z RBQ = d, RN = u, QT = «', therefore NM
= u'cosi6 and RM = RN+NM = 7t + u' cos 6, and in the right-angled tri-
angle RTB, RM cosGc^d = RB, the projection of OP upon AP, that is, the
THE BIGHT LINE. 751
left member of equation (i.). Similarly for equation (ii.). It should be
observed that (i.) and (ii.) represent planes through AB respectively per-
pendicular to the given lines AP and BQ.
5536 Otherwise, the line of shortest distance is the inter-
section of the two planes whose equations are
I {w—a)-\-m{y — b)-\-n (^ — c)
^ I (.r- (Q + m {ii-h')-\-n (z-c)
i {.V — a)-\- m {y — h') -\-n {z — c) '
For these equations state that cos is the ratio of the projections of OP
or of OQ upon the given lines, and this fact is apparent from the figure.
5537 Equations of the line passing through the two points
ahc, a'h'c :
x — a __ y — h _ !z — c
a — a h~b' c — c
5538 A line passing through the point ahc and intersecting
at right angles the line Imn :
w — a _ y—h
L ~~ M ~ N '
where L = Im {b — b')-\-nl(c — c) — {m^-\-n^){a — a),
and symmetrical values exist for M and N.
Proof. — The condition of perpendicularity to Inm is
Ll + Mvi + Nn = 0; (5520)
and the condition of intersecting the line is
(a-a)(iMn-mN) + {b-b')(Nl-iiL) + (c-c')(Lm-m) = 0.
These equations determine the ratios L : M : N.
5539 Equations of the line passing through the point abc,
parallel to the plane Lx-\-My-\-Nz = D, and intersecting the
line (I'm'n, a'h'c) :
00— a _ y—b s—c
I m w '
where l, m, n are found, as in the last, from
LI + Mm + iV/i = 0, and
{a-a'){7nn' -m'n)-\-{b-b'){nl' -n'l)-\-{c-c'){lm' -I'm) = 0.
752 SOLID GEOMETRY.
5540 Equations of the bisector of the angle between the
two lines htn-^ii-^^ li>ihn2 :
cC
_ y
Ix + h >Wl + ?% **1+W2
Proof. — Taking the intersection of tlie lines for origin, let x-^y-^Zi, 3522/2^2 ^^
points on the given lines equidistant from the origin ; then, if xijz be a point
on the bisector midway between the former points, x = ^(x-i + x^), &c.
(403o) ; and the direction-cosines of a line through the origin are propor-
tional to the coordinates.
5541 The equations of a right line in four plane coordinates
are = Ci — lLz=z^ — ^= (i.),
L M N R ^ ^'
where a/SyS is a variable point, and a'fi'^'^' a fixed point on
the line. The relation between L, M, JSf, B is
5542 AL-^BM^CN+DR=^Q (ii.).
Proof. — For, since equation (5503) holds for a/3y^ and also for afi'y'(i\
we have A (a-a') +-B (/3-/3') + G (y-y') +-D (^-^') = 0.
Substitute from (i.) a — a =■ rL, /? — /3' = rM, &c.
5543 In tetrahedral coordinates the same equation (i.) sub-
sists, but the relation between L, M, N^ R becomes, by-
changing Aa into a, &c.,
5544 L-{-M+N-\-R = 0.
THE PLANE.
5545 General equation of a plane :
A.v-}-Bi/+Cz-\-D = 0.
5546 Equation in terms of the intercepts on the axes :
a ' b ^ c
5547 Equation in terms of p, the perpendicular from the
origin upon the plane, and /, m, n, the direction-cosines of p :
Lv-\-mi/-\-7iz = p.
THE PLANE. 753
Proof. — If P be any point xyz upon the plane, and the origin, the pro-
jection of OP upon the normal through is j^ itself; but this projection is
Ix + my + nz, as in (5520).
5548 The values of I, m, n, j; for the general equation
(5545) are
j_ A _ -D
Proof. — Similar to that for (4060-2) : by equating coefficients in (5545)
and (5547) and employing V' -\- i)V -\- v? = 1.
5550 The equation of a plane in four-plane coordinates is
•XL / «i A yi ^1
with / : /« : w : r = — !- : -^ : -i-!- : — ,
Pi Ih Ih Ih
where a^, jSj, yi, S^ are the perpendiculars upon the plane from
A, B, G, D, the vertices of the tetrahedron of reference, and
i^i) P-Zi psi pi ^1'® the perpendiculars from the same points upon
the opposite faces of the tetrahedron.
Proof. — Put y = ri = for the point where the plane cuts an edge of the
tetrahedron, and then determine tlie ratio I : in by proportion.
See Frost and Wolstenholme, Art. 81.
5551 The equation of a plane in tetrahedral coordinates is
also of the form in (5550), but the ratios are, in that case,
/ : m : n : r = a^ : /8i : 71 : S^.
The relation between the three-plane and four-plane coor-
dinates is a = 79 — Lv— my — n& .
5552 The equation of a plane in polar coordinates is
r {cos Q cos ^+siu Q siu & cos (<^— <^')} = p.
Proof. — Here jj is the perpendicular from the origin on the plane, and
p, 0', 0' the polar coordinates of the foot of the perpendicular. Then, if ^ is
the angle between p and r, we have ^ = r cos i/' and cos i// from (882).
5553 The angle between two planes
Ix + my -\'7iz=: p and I'x + my + 11 z = p
5 D
754 SOLID O^OM^TBt.
is given by formula (5520), and the conditions of parallelism
and perpendicularity by (5531) and (5532), since the mutual
inclination of the planes is the same as that of their normals.
5554 The length of the perpendicular from the point x'l/z'
upon the plane Ax -{-Bi/-\-Gz-{-D = is
Proof.— As in (4094).
5556 The same in oblique coordinates
= ^^ ! — i^-! ! i- = p — x COS a—?/ cosp—z cosy,
P
where p is found from (5519) by putting A, Bj G for p cos a,
P cos |3, p cos y. This gives
( A'sm^X + B'- &'ur n + C'^ sin' p + 2BG(cosfJ. cos i^ — cosX)
CKKQ 2_ (. +26'J. (cos V cos \ — cos /x) + 2^-5 (cos \ cos /i— cos y)
1 — cos^X— cos^/x— cos"''i' + 2 cosX cos^cosj'
5559 The distance r of the point ici/z from the plane
Ax + By-{-Gz-\-D = 0, measured in the direction Imn, the axes
being oblique :
_Aa^;±W±C^±D
Al-\-Bm + Cn
Proof. — By determining r from the simultaneous equations of the line
and the plane, viz.,
'-i^ = y-~-L = tZlll = r and Ax + B>j-\-Cz + D = 0.
I m n
Otherwise, by dividing the perpendicular from x'lj'z' (5554) by the cosine of
its inchnation to Imn, viz., — —^ — -^.
EQUATIONS OF PLANES UNDER GIVEN CONDITIONS.
5560 A plane passing through the point aJ>c and pcrpen-
diculai- to the direction Imi} -.
/(.r-«)+m (//-/>) + // {z-v) = 0.
THE PLANE. 755
5561 A plane passing througli two points ahc, ctl/c :
^ 7 + ^r— 77+v 7 = 0,
5562 with X+/t+v = 0.
PuoOF. — By eliminating n between the equations
l{x — a)+m{y — 'b)-\-n{z — c) =0, l(a — a')^-mil) — h')-\-n{c — c)-=0,
and altering the arbitrary constant.
5563 A plane passing througli the point of intersection of
the three planes tt = 0, y = 0, -zy = :
lu-\-mv-\-nw = 0.
5564 A plane passing through the line of intersection of the
two planes u = (), ?; = :
lu-\-mv =■ 0.
5565 A plane passing through the two points given by
7t = 0, -y = 0, H' = and u = a, v = h, w = c:
lu-\-mv-{-nw = with la-i-mb-\-tic = 0.
.V y z 1
a.\ 7/1 Zy 1
•^2 yi ^i -'-
^t's Ih -3 1
= 0.
5566 The equation of a plane passing
through the three points x^ij^z^^, x.^j.f..^, 9'zy-^z^
or A, B, G, is given by the determinant
annexed, in which the coefficients of x, y, z
represent twice the projections of the
area ABG upon the coordinate planes.
Peoof. — The determinant is the eliminant of Ax + Bt/ + Gz =. l, and
three similar equations. Expanded it becomes
^ (2/2^3-2/3^2 + 1/8-1-2/123 + 2/1^2— 2/2-1 ) + ?/ (&c.) + z (&c.)+:r,?/./3 — &c. = 0.
Hence, by (4036), we see that the coefficients are twice the projections of
ABC, as stated.
5567 The sum of squares of the coefficients is equal to
four times the square of the area ABG.
Proof. — For, if I, m, n are the dir-cos. of the plane, and ABC = S, the
coefficients are = 2Sl, 2Sm, 2Sn, by projection.
5568 The determinant {xi, y^, ^3), that is, the absolute term
in equation (5566), represents six times the volume of the
tetrahedron OABG, where is the origin.
756
SOLID GEOMETRY,
Proof. — Writing the equation of the plane ABC, Ax-^-By + Gz-\-I) = 0,
we have for the perpendicular from the origin, disregarding sign,
B
therefore B = 22jS = G X the tetrahedron OABG.
5569 If 0:njz be a fourtli point, P, not in tlie plane of ABG,
the determinant in (6566) represents six times the volume of
the tetrahedron PABG.
Proof. — By the last theorem the four component determinants represent
six times (OBCP+OCAB+OABB + OABC) for an origin within the
tetrahedron.
5570 A plane passing through the points ahc, a'b'c\ and
parallel to the direction Imn :
= 0.
A— a
y-b
b-b'
m
z—c
a — a
I
c-c'
n
a — a b — b'
Proof. — Eliminate X, ^
Z\ m
between the equations (5561-2) and
= - — '-r, H ; = 0, the condition of perpendicularity between Imn
and the normal of the plane (5561).
5571 A plane passing through the point ahc and parallel
to the lines hnn. I'm'n :
x — a
I
r
,,-h
m
m
z—c
n
n
= 0.
Proof. — The equation is of the form \ (x — a) +fx (y — h) +v (z — c) = 0,
and the conditions of perpendicularity between the normal of the plane and the
given lines are l\ + mfx + 7iy = 0, VX + m'/u+n'y = 0. Form the eliminant
of the three equations.
5572 A plane equidistant from the two right lines (ahCf Imn)
and {a'h'c\ I'm'n) :
cV—^(a-\-a') I I'
y—\{b-\-b') m m
z—i{c-{-c') n n By (5571).
TBANSFOBMATION OF COOBBINATES. 757
5573 A plane passing tlirougli the line {abc, Imn) and per-
pendicular to the plane l'x-\-my-\-nz — i^ :
The equation is that in (5571).
For proof, assume X, fi, v for dir-cos. of the normal of the required plane,
and write the conditions that the plane may pass thi'ough abc and that the
normal may be perpendicular to the given line and to the normal of the
given plane.
TRANSFORMATION OF COORDINATES.
5574 To change any axes of reference to new axes parallel
to the old ones :
Let the coordinates of the new origin referred to the old
axes be a,l,c', xyz and xy'z, the same point referred to the
old and new axes respectively ; then
5575 To change rectangular axes of reference to new
rectangular axes with the same origin :
Let OX, or, OZ be the original axes, and OX, OF, OZ'
the new ones,
k mi ni the dir-cos. of OX' referred to OX, OY, OZ,
km.^n.2 do. OY' do. do.
liin^ih do. OZ' do. do.
xyz, int the same point referred to the old and new axes
respectively. Then the equations of transformation are
5576 A^=k^+ky)^hi (i-)>
y = m^^-\-m.{q-\-m.X (ii-)^
% = ih^^-n,r)-\-n,t, (iii.).
And the nine constants are connected by the six equations
5577 /: + m? + w!= l...(iv.), l,h^-m,m,-Vn,ih = ^ ... (vii.),
Z^+m^+n^ = 1 ... (v.), yiH-m3r?ii+«3«i = ... (viii.),
tl-^-irh^-k-thi = 1 ...(vi.), /i/,+mim2+ni«2 = ... (ix.),
so that three constants are independent.
758 SOLID GEOMETRY.
Proof. — By (5515) and (5532), since OX', 0Y\ 0^' are mutually at
right angles.
5578 The relations (iv. to ix.) may also be expressed thus —
'■ '"^ 5i— =±1 (x.),
= ±1 (xi.)
1 1 m.i n.i
= ±1 (xii.).
Obtained by eliminating the third term from any two of equations
(vii. — IX.). Also, by squaring each fraction in (x.) and adding numerators
and denominators, we get
(z;+m;+7.;)(z;+m^+rip-(/,/3+^v«.+'¥'3)~ ' ^^''''^■
5579 If the transformation above is rotational, that is, if it
can be effected by a rotation about a fixed axis, the position
of that axis and the angle of rotation are found from the
equations 2 cos^ = l^-^-m.i-^-n^,—!,
Kcop. cos^a __ cos'yS 00 s^ y
where a, /3, y are the angles which the axis makes with the
original coordinate axes.
Proof. — (Fig. 171.) Let the original rectangular axes and the axis of
rotation cut tlie surface of a sphere, whose centre is the origin 0, in the
points X, y, z, and I respectively. Then, if the altered axes cut the sphere in
i, n, I, we shall have 6 = Z xR in the spherical triangle ; Ix ■= I^ =. u ; ly =■
Ji; = /3 ; Jz = 7; = y, and by (882) applied to the isosceles spherical triangles
xlk, &c., 7j = cos xl = cos" a + sin' a cos d, vi., = cos yrj = cos"/3 + sin'"' /5 cos 6,
Wg = cos ciC = COS" y + sin'- y cos 0. From these cos y, coso, cos/3, and cosy
are found.
5581 Transformation of rectangular coordinates to oblique :
Equations (i. to vi.) apply as before, but (vii. to ix.) no
longer hold, so that there are now six independent constants.
th:b sphere. 759
THE SPHERE.
5582 The equation of a sphere when the point ahc is the
centre and r is the radius,
5583 The general equation is
a;'+i/-^z'+A.v+Bi/+Cz + D = 0.
The coordinates of the centre are then — ^5 — lyi ~ ~o'*
and the radius =^^{A'+ B' + G'- 4D) .
Proof. — By equating coeflBcients with (5582).
5584 If <''//2! be a point not on the sphere, the value of
{x — ay'-^{y — hy^ + {z — cy^ — r is the product of the segments
of any right line drawn through xijz to cut the sphere.
Proof. — From Euc. iii., 35, .36.
THE RADICAL PLANE.
5585 The radical planes of the two spheres whose equations
are u = 0, u = 0, is
U — l(' =z 0.
5586 The radical planes of three spheres have a common
section, and the radical planes of four spheres intersect in the
same point.
Proof. — By adding their equations, and by the principle of (4608)
extended to the equations of planes.
POLES OF SIMILITUDE.
5587 T)ef. — A 2)ole of shwUitiLcle is a point such that the
tangents from it to two spheres are proportional to the radii.
5588 The external and internal ]}oles of similitude are the
vertices of the common enveloping cones.
760 -SOLID GEOMETRY.
5589 Tlio locus of the pole of similitude of two spheres is a
sphere whose diameter contains tlie centres and is divided
harmonically by them.
CYLINDRICAL AND CONICAL SURFACES.
5590 Def. — A conical surface is generated by a right line
which passes through a fixed point called the vertex and
moves in any manner.
5591 If the point be at infinity, the line moves always
parallel to itself and generates a cylindrical surface.
5592 Any section of the surface by a plane may be taken
for the guiding curve.
5593 To find the equation of a cylindrical or conical surface.
Rule. — EJiininojte xyz from tlie equations of the guiding
curve and the equations ^^ = llzE = — of any generating
line ; and in the result put for the variahle parameters of
the line their values in terms of x, y, and z.
5594: Ex. 1. — To find the equation of the cylindrical surface whose
generating lines have the direction Inm, and whose guiding' curve is sriven
hy bV + ay = a'b' ?ind z = 0.
At the point where the line ^Zlf = y~h> _ ^ meets the ellipse, 2 = 0,
I m n 1 » >
a; = a, y=ft. Therefore tV + a^/3^ = a'i". Substitute in this, for the
variable parameters, a, /3, a = x -, /3 = i/— — ; and we get, for the
n n
cylindrical surface h"^ (nx — lzy- + d-()iy — mzy- = d'h-n-.
5595 A conical surface whose vertex is the origin and
guiding curve the ellipse lrx--{-dhf = a~b", z = c, is
«- o' c~
Proof. — Here the generating lino is — = -^ = — . At the point of inter-
I m n
Gomcows. 761
^. . ,, ,. J Ic VIC . fc-/V , a m c _ ■.,■,
section of the line and curve z = c, x = —, y = — ; .. — 7, — r t- — a u .
n n 'It' n
Substitute for the variable parameters I : m : n the values x : y : z, and the
result is obtained.
CIRCULAR SECTIONS.
5596 Rule. — To find the circular sections of a qaadrlc curve,
express the equation in the form A (x'^ + / + z'^ + c^)-t-&c. = 0.
If the remaining terms can be resolved into tivo factors, the
circular sections are defined by the intersection of a sphere and
two planes.
5597 Generally the two quadrics
ai<r + bif + G:^-\-2fyz + 2gzx-{-2Jixy = 1
and {a -\-X)x^ + {b-\-X) f-{-ic-\-X) z' + 2fyz + 2yzx + 2hxy = 1
have the same circular sections.
Pkoof.— Let r, p be coincident radii of the two surfaces having hm for a
common direction. Then — = aP + h»r + or + 2fmn + 2gnli-2Mm and —
r P
= the same +\. Therefore, if r has a constant value throughout any
section, p is also constant throughout that section.
5598 Ex.— An oblique circular cone whose vertex is the point a, 0, h,
and guiding curve the circle x'^ + if ■= c"' ; z = ; is
{az-hx)- + bh/' = c- {::-h)-.
The equation may be written
h^(x'' + y^ + z'-c") = z {2ahx + (h'' + c''-n') z-2hc''},
and therefore the cone has two series of parallel circular sections, z = h and
2abx + {b^ + c''-a^) z-2bc^ = f (5583). {Frost and WolstenlxolmeS)
CONICOIDS.
5599 Defs.— A conicold is a surface every plane section of
which is a conic.
The varieties are the ellipsoid, the one-fold and two-fold
hyperbololds, the eUlptic and hyperbolic _ paraboloids, the
spheroid of revolution, the cone, and the cylinder.
5 E
762 SOLID GEOMETRY.
In any of the following equations of a conicoid, by making one of the
variables constant, the equation of a section parallel to a coordinate plane is
obtained, and the equation of the surface is by that means verified. Thus,
in the equations of (5600) or (5617), Figs. (172) and (173), if z be put
= ON, we get the equation of the elliptic section BPQ, the semi-axes of
which are NQ = - V(c'-ON') and NR = ^y{c'-ON% a, b, c being the
c c
principal semi-axes of the conicoid ; that is, OA, OB, OC in the figure.
THE ELLIPSOID.
5600 The equation referred to tlie principal axes of the
figure is
^ + T^ + ^= 1- (Fig. 172)
5601 There are two planes of circular section whose
equations are
with ayhyc.
is a cone having a common section with the conicoid and a sphere of radius
r. If the common section be plane, one of the three terms must vanish
in order that the rest may be resolved into two factors.
Since a >b > c, the only possible solution for real factors is got by
making r = b.
5602 Sections by planes parallel to the above are also
circles, and any other sections are ellipses.
5603 The umbilici of the ellipsoid (see -■)777) are the points
whose coordinates are
V u—r ^ tr—e"
Proof. — The points of intersection of the planes (56' >1) and the el lipsoid
(5600) on the xz plane are given by x = =b a * / ''., .„ 2' = ± c \ — r, •
Since, by (5602) the vanishing circular sections are at the points in the xs
plane conjugate to x and 2', we have, by (4352), x =■ -■/, z = x .
C0NIG0ID8. 763
5604 If a = h, in (5600), the figure becomes a spheroid, and
every plane parallel to tij makes a circular section. Hence
the spheroid is a surface of revolution. It is called prolate
or oblate according as the ellipse is made to revolve about its
major or minor axis.
THE HYPERBOLOID.
5605 The equation of a one-fold hyperboloid referred to its
principal axes is
£1_|-J^_4 = 1. (Fig. 173)
(t fr cr
5606 The planes of circular section, when ayhyr, are all
parallel to one or other of the planes whose equations are
Proof. — As in (5(501), putting r = a.
5607 The generating lines of this surface belong to two
parallel systems (i.) and (ii.) beloAV, with all values of 9.
5608
^= cos^+ — sin ^/ — = cos^— — sin ^ /
4- = sill ^- - cos ^ \ 4-= sill 6-^ — cos e
b ^ ) ^ f'
For the coordinates which satisfy either pair of equations,
(i.) or (ii.), satisfy also the equation of the surface. The
equations may also be put in the forms
5610
.1 — rt cos Q _ ij — h sin Q
a sin ~~
5612 If ?-' = 0, 33 = (/. cos 9 and // = h sin 9. Hence 9 is
the eccentric angle of the point in wliich the lines (i.) and (ii.)
intersect in the ;/'^ plane.
5613 Any two generating lines of opposite systems intersect,
but no two of the same system do.
764 SOLID GEOMETRY.
5614 If two genei-ating lines of opposite systems be drawn
through the two points in the principal elliptic section whose
eccentric angles are 0-\-a, d—a, a being constant, the coordi-
nates of the point of intersection will be
ii = a cos 6 sec a, ?/ = b sin 6 sec a, z = ^c tan a,
and the locus of the point, as 9 varies, will be the ellipse
5615 -,-^+j-^=l; z = ±ciaua.
(r sec^ a o- sec" a
Pkoof. — From (i.) and (ii.), patting 6±o for 6*
5616 The asymptotic cone is the surface given in (5595).
Proof. — Any plane through the z axis whose equation is ?/ = mx cuts the
hyperboloid and this cone in an hyperbola and its asymptotes respectively.
5617 The equation of a two-fold hyperboloid is
il_|l_4 = l. (Fig. 174)
a^ ¥ c^
and the equation of its asymptotic cone is
5618 4-€--4 = 0.
a^ ¥ c^
Pkoof. — Any plane thi'ough the x axis, whose equation is y = mz, cuts
the hyperboloid and this cone in an hyperbola and its asymptotes respec-
tively.
There are two surfaces, one the image of the other with regard to the
plane of yz. One only of these is shown in the diagram.
5619 The planes of circular section when & is > c are all
parallel to one or other of the planes whose joint equation is
Pkoof. — As in (5G01), putting r- = —h'-.
5620 If ?> = '", the figure becomes an hyperboloid of revo-
lution.
THP] PARABOLOID.
5621 Tliis surface is generated by a parabola which moves
with its vertex always on another parabola ; the axes of the
two curves being parallel and their planes at right angles.
* The surface of a oiic-fold hyporboloid, as generated by right lines, may frequently be
seen in the foot-stool or work-basket constructed entirely of straight rods of cane or wicker.
CENTRAL QUADBIC SUBFAGE. 765
The paraboloid is cUipfic or hyperholw according as the
axes of tlie two parabolas extend in the same or opposite
directions.
5622 The equation of the elliptic paraboloid is
h c
h and c being the Jatera recta of the two parabolas.
Proof : QM' = h.OM; FN' = c.QN; .-. ^^ + ^ = 0M-\- QN=x.
h c
li h = c, the figure becomes the parahol old of revolution.
K-^^ = .v, (Fig. 175)
5623 Similarly the equation of the hyperbolic paraboloid i
IS
^_-^ = .l-. (Fig. 176)*
5624 The equations of the generating Unes of this surface
are JL±-l- = m and -1^^-^ = — ,
the upper signs giving one system of generators and the lower
signs another system.
5625 The equations of the asymptotic planes are
CENTRAL QUADEIC SURFACE.
TANGENT AND DIAMETRAL PLANES.
5626 Taking the equation of a central quadric %,-\-j7,-\- -,,
ar Ir c.
= 1 to include both the ellipsoid and the two hyperboloids
* The curvature of this sui-face is antklastlc, a sort of curvature, which may be seen in
the saddle of a mountain ; for instance, on the smooth sward of some parts of the
Malvern HUls, Worcestershire,
766 SOLID GEOMETRY.
according to the signs of li' and (r, the equation of tlie tangent
l^lane at xy:: is
By (r.G79).
5627 If 2^ be the length of the perpendicular from the origin
upon the tangent plane at xijz,
p' ■" a' "^ b' "^ c' '
Proof.— From (5549) applied to (5G2G).
5628 The length of the perpendicular let fall from any
point ^1)1 upon the tangent plane at xi/?: is
(5554 & 5027)
5629 Direction cosines of the normal of the tangent plane
, 7 P'^' PV P^
at xyz, I = ^—, m = -^, n = ^.
a^ b^ c-
Proof. — By (5548) applied to (562G) and the value in (5G27).
5630 If ^ '''ij *i are the direction cosines of j),
J) =z Iv + i)K/ 4- n^ and jr = irP -f b^))r-^c-}i^.
Proof. — (5GoO) By projecting- the three coordinates x, y, z upon p.
(5631) By substituting the values of ^, y, z, obtained from (5G20), in
(5G30).
5632 The equation of tlie normal at xi/z is
(f-^)-2l=(,-,y)-^=(^-^)f,
lL If <v
since the dir-cos. are tlie same as those of tlie tangent plane
at (5626).
5633 Each term of the above equations
or J) niuU i[)lied into the length of the normal.
CENTRAL Q-UADBIG 8UBFA0E. 767
Proof.— Each term squared = (LJEI! = ilLJll = iL^.
£1 yl ii
a' b* c*
Add numerators and denominators, and employ (5627).
5634 Equation (5681) is the condition that the plane
h-\-my-\-nz = 2) may touch the conicoid ; and if j; = 0, we
have for the condition of the plane lx-\-iiii/-\-nz = touching
the cone ^^^ + t^ + ^ = 0,
cr Ir c^
5635 «-/-+6-m'+r/r = 0.
5636 The section of the quadric made by a diametral plane
conjugate to the diameter through the point ,rp has for its
equation ^ 4.M_^ ^ = (). By (5688).
^ (r Ir c
5637 Hence the relation betAveen the direction cosines of
two conjugate diameters is
It mm nn __ ^
cr b^ c'^
ECCENTRIC VALUES OF THE COORDINATES.
5638 These are defined to be
,r = «X, ?/ = bjj., z = cv, with \--}-ijl--\-v~ = 1.
5640 ^, i^h V are the dir-cos. of a line called the eccentric
line; and ^ = r\, rj = tjh, l=rv are the coordinates of the
corresponding point upon an auxiliary sphere of radius r.
5641 The eccentric lines of two conjugate semi-diameters
are at right angles. By (563 7).
5642 The sum of the squares of three conjugate semi-
diameters is constant and = a'^ -{- b'' -\- c'\
Proof. — Let a', h', o be tbe semi-diameters, and .('i.'/,^i, ^hVi^i^ ^■H-.i^z their
extremities. Put the eccentric values in the equations ^'^\'^y\'^^\ = ^'i <^^->
and add. By (5641), Aj + X'^ + X', == 1, &c.
768 SOLID GEOMETRY.
5643 Tlie sum of the squares of the reciprocals of the same
is also constant.
Proof. — Put 9\ cos a^, i\ cos ft^, r, cos yj for x^, y^, z^ in the equation of the
quadric. So for x^, y^, z^ and ajj, //g, z.^. Divide by r,, r.,, r^, and add the results.
5644 The sum of squares of reciprocals of perpendiculars
on three conjugate tangent planes is constant.
Proof. — For each perpendicular take (5G27), and substitute the eccentric
values as in (5642).
5645 The sum of the squares of the areas of three con-
jugate parallelograms is constant.
Proof. — By the constant volume of the parallelepiped v^A, = p.,A., = r\A.,
(5648) and by (56M). l 1 i'l i 7 . . ia s,
5646 The sum of the squares of the projections of three
conjugate semi-diameters upon a fixed line or plane is
constant.
Proof. — With the same notation as in (5642), let (Inin) be the given line.
Substitute the eccentric values (5638) in {lx^-\-mii^-\-nzJ-+{lx.-\-mD.^-\-Hz.,)'-
+ (?«'3 + "'//3 + "^a)'- In the case of the plane we shall have
a"'—{h\ + mij-^ + ?(^i)' + &c.
5647 CoK. — The extremities of three conjugate semi-
diameters being x^y-^z^, '^•iV-^-i^ ^'zil-ihy it follows that, hy pro-
jecting upon each axis in turn,
•^•i+'* 2+^*3 = «' ' y\+iil^Hl = ^' ' -i+-2+-3 = ^''•
5648 The parallelopiped contained by three conjugate semi-
diameters is of constant volume = abc.
Pkoof. — By (5508), the volume = .\\ y^ -j I = abc \ \ yu, v^
\ A y. ^. ' \ K F-: ".
•*'» 'Ji ~3 I \ A'a ''3
by the eccentric values (5638). But the last determinant =1 by
(584, I.).
5649 Cor. — If a, //, c' are the semi-conjugate diameters,
w the angle between a and h\ and ^^ the perpendicular from
the origin upon the tangent plane parallel to (f'h', the volume
of the parallelopiped is ^>a'/^' sin w = dhr.
CENTRAL QUADRIG SURFACE.
769
5650 Hence tlie area of a central section in the plane of ab'
/,/ . abc
= tra b sin co = tt .
5651 Quadratic for the semi-axis of a central section of the
quadric ^ + "/-i + "i = 1 made by the plane lx-\-my-{-nz = :
crP
b'nr
a^—r^ 6-— r- c' — r
.4-
cw
Proof. — The equation is the condition, by (5G35), that the plane
Ix + rny -tnz = may tonch the cone
as in the Proof of (5600). For another method, see (1863).
a —
b-1-
f
= 0.
c ~ n
5652 When the equation of the quadric is presented in the
form
(Lv'-}-bi/'-]-c^'-{-2fi/::^+2g-z.v+2Juvi/ = 1,
the quadratic for r takes 1
the form of the determi-
nant equation annexed.
Or, by expanding, and
writing A' for the same
determinant, with the ^ J
fraction — erased, the / m
equation becomes
^'r^j^ [{j)j^c)l'-\-{c-^u)m--^(a-^b)ir-2fmn-2^nl-2ldm] 1-
— f — m'^ — w'^ = 0.
Pi^OOF.— The equation of the cone of intersection of the sphere and
quadric now becomes
and the condition of touching (5700) produces the determinant equation.
5 F
770 SOLID GEOMETRY.
5654 To find the axes of a non-central section of the
quadric ~ -[- ^- + ^ = 1.
a- b- c^
Let PNQ (Fig. 177) be the cutting plane. Take a parallel
central section BOG, axes OB, OG, and draw NP, NQ parallel
to them. These will be the axes of the section PNQ, and NQ
will be found from the equation v^^,, + ^^^~ = 1.
^ OA- OC-
5655 The area of the same section
ibc
Trabc ( i _p''\
where j^' and p are the perpendiculars from upon the cutting-
plane and the parallel tangent plane.
Proof.— The area = ttNP.NQ = tt^.OB.OG
= - (l- ^ OB. 00 = "^ (l- ^,^)' by (5650).
SPHERO-CONICS.
Def. — A splicro-conic is the curve of intersection of the
surface of a sphere with any conical surface of the second
degree whose vertex is the centre of the sphere.
Properties of cones of the second degree may be investi-
gated by sphero-conics, and are analogous to the properties
of conies.
A collection of fonnulaj will be found at page 562 of Roath's Rigid
Dynamics, 3rd edition.
CONFOCAL QUADRICS.
5656 Deitnition. — The two quadrics wliose equations are
^ + |; + |: = l and -^+JL^ + ^^1,
are confocal. We shall assume a>by r.
5657 As X decreases from being large and positive, the
third axis of the confocal ellipsoid diminishes relatively to the
CENTRAL QUADBIG SURFACE. 771
others until X = —c^, wlien the surface merges into the
focal elHpse on the xy plane,
b'-c^
X still diminishing, a series of one-fold hyperboloids appear
until X = —li\ when the surface coincides with the focal
hyperbola on the zx plane,
r^. = 1.
.2 A2
b' Ir-(r
The surface afterwards developes into a series of two-fold
hyperboloids until \ = —a\ when it becomes an imaginary
focal ellipse on tlie yz plane.
5658 Through any point xyz three confocal quadrics can be
drawn according to the three values of X furnished by the
second equation in (5656). That equation, cleared of frac-
tions, becomes
5659 \^^{a'^b'+c'-.v''-jf-^') \'
-\-a-b-c---b-r.i''—c-a^f/ — a'b-z^ = 0.
These three confocals are respectively an ellipsoid, a one-
fold hyperboloid, and a two-fold hyperboloid. See Figure
(178); P is the point xyz; the lines of intersection of the
ellipsoid with the two hyperboloids are DFE and FPG, and
the two hyperboloids themselves intersect in HPK.
Proof.— Substitute for X successively in (5659) a', b', c\ — cc ; and the
left member of the equation will be found to take the signs + , — , + , —
accordingly. Consequently there are real roots between cr and l'-, Ir and r,
G' and —00 .
5660 Two confocal quadrics of different species cut each
other everywhere at right angles.
Proof. — Let a,h,G; a', l/, c be the semi-axes of the two quadrics ; then,
at the line of intersection of the surfaces, we shall have
772 SOLID GEOMETRY.
"which, since a"' — d- =■ I/' — h' := c' — cr = X, becomes the condition of per-
pendicu^larity of the normals by the values in (5629). Thus, in (Fig. 178),
the tangents at P to the three lines of intersection of the surfaces are
mutually at right angles.
5661 If P be the point of intersection of three quadrics
aJ)^Ci, ajjx.,, a.Jj.^c^ confocal with the quaclric abc ; the squares
of the semi-axes, rfg, d^, of the diametral section conjugate to
P in the first quadric are (considering a^ > a.2 > a.^, and writing
the suffixes in circular order)
rf.s = ^^1 ~ (i
dl = a% al,
In the second,
il\ = iii- al
dl = rt! ~ «!,
in the third,
(IZ = (4~ al
72 2 2
rtj = «3 ^ Uo.
Or, if for a^, dl, dl we put (v^ + \, rt' + A^, d' + \^, the above
values may be read with A in the place of a and the same
suffixes.
Proof. — Put a'- — fr=:^t; then
- .. /•'-
a- h^ c' a^—fj. b-—fi r — /i
are confocal quadrics. Take the difference of the two equations, and we
obtain, at a common point x'li'z, — — '— + &c. = 0. Comparing this with
(5651), the quadratic for the axes of the section of the quadric by the plane
lx-\-mij + nz =■ 0, we see that, if Z, m, n have the values — , &c., /x is identical
with ?-^; the plane is the diametral plane of P; and the two values of ^ are
the squares of its axes. Let dl, dl be these values ; then, since there are but
three confocals, the two values of /.< must give the remaining confocals, i.e.,
a^ — d^ = al and "j — t?^ — ^r
The six axes of the sections are situated as shown in the
diagram (Fig. 179). Either axis of any of the three sections
is equal to one of the axes in one of the other sections, but
the equal axes are not those which coincide. is supposed
to be the centre of the conicoids, and the three lines are drawn
from parallel to the three tangents at P to the lines of in-
tersection.
5662 Coordinates of the point of intersection of three con-
focal (|uadrics in terms of the semi-axes :
CENTRAL QUADBIC SURFACE. 773
2 2 2 1.2 72 I 2
2 __ «1<<2% 2 _ O1O2O3
2 2 2
^1^203
- («;_ci)(6?-eD'
The denominators may be in terms of any of tlie confocals
since a\—hl = cC — K = cil — ^35 &c.
Proof. — The equation of a coufocal may be written ~ H — ^f — — +
= 1, producing a cubic in or, the prod
uct of whose roots a-, a'-, «- g'ives £'
5663 The perpendiculars from the origin upon the tangent
planes of the three confocal quadrics being jj^, jJo, J)^ :
2722 27 2 2
«lt>i^l 2 _ (l-lOiC^i
IK = /..2 J^ J .,2^ » P
{al-al){al-aiy ^ («J-«^)(«^-«5)'
Pi
27.2 2
Proof. — By (5649), 2hd->dz = aibiC^ ; then by the values in (5661).
RECIPROCAL AND ENVELOPING CONES.
5664 Def. — A right line drawn through a fixed point always
perpendicular to the tangent plane of a cone generates the
reciprocal cone.
The enveloping cone of a quadric is the locus of all
tangents to the surface which pass through a fixed point
called the vertex.
5665 The equations of a cone and its reciprocal are respec-
tively
.Ar=+%=+6V = (i.), and i^ + l + ^ = (ii.).
Proof. — The equations of the tangent plane of (i.) at any point xyz, and
of the perpendicular to it from the origin, are
Axl + Byn + Cz^ = (iii.), and £ = ^ = J- (iy.).
Eliminate x, y, z between (i.), (iii.)) ^^(^ (iv.)-
5667 The reciprocals of confocal cones are concyclic ; that
774 SOLID GEOMETRY.
is, liave the same circular section ; and the reciprocals of con-
cjclic cones are confocal.
Proof. — A series of concyclic cones is given by
A:e' + By- + Cz' + \{x'+ if + r') =
by varying X ; and the recipi'ocal cone is
5668 The reciprocals of the enveloping cones of the series
of confocal quadrics / ^ + ,./ ^ -\- ^^ = 1 , with /)//? for
ft^ + A 6^ + A r + A
the common vertex, P, of the cones, are given by the equation
Proof. — Let hnn be the direction of the perpendicular jj from the origin
upon the tangent plane drawn from F to the quadric. Equate the ordinary
value of j3^ at (5G31) with that found by projecting OP upon p ; thus
(a- + X) I- + (Z>- -I- X) m- + (c- + X) u' = (fl + gm + hn)-.
"Now p generates with vertex a cone similar and similarly situated to the
reciprocal cone with vertex P, and Z, m, n are proportional to .*■, y, z, the
coordinates of any point on the former cone. Therefore, by transferring the
origin to P, the equation of the reciprocal cone is as stated.
5669 Cor. — These reciprocal cones are concyclic ; and
therefore the enveloping cones are confocal (5667).
5670 The reciprocal cones in (5668) are all coaxal.
Proof. — Transform the cone given by the terms in (.5068) without X to
its principal axes ; and its equation becomes Ax- + Pij''+Cz- = 0. Now, if
the whole equation, including terms in X, be f-o transformed, .('■ + ?/'■ + , r will
not be altered. Therefoi'e we shall obtain
(A + \)x'+(n + X)if + (C + X);r = 0,
a series of coaxal cones.
5671 The axes of the enveloping cone are the three normals
to the three confocals passing through its vertex.
Proof.— The enveloping cone becomes the tangent plane at P for a con-
focal through P, and one axis in this case is the normal through P. Also
tliis axis is common to all the enveloping cones with the same vertex, by
(5670). But there are three confocals through P (5658), and therefore
three normals which must be the three axes of the enveloping cone.
5672 The equation of the enveloping cone of the quadric
th:e! general quadbig. 775
-f^ \- —^ \- — = 1 is, when transformed to its prm-
a'-\-\ h^ + X c^ + X
cipal axes,
x-Xi ' x-x, ' x-x, X' ' x'+^/:; x'+^/;
where X^, \,, Xg are the values of X for the three confocals
through P, the vertex, and cli, cl^ are the semi-axes of the
diametral section of P in the first confocal (5661).
Proof. — Transform equation (5G68) of the reciprocal of the enveloping
cone to its principal axes, as in (5G70). Let Xj, X.,, \ be the values o£ A
which make the quadric become in turn the three confocal quadrics through
P. Then the reciprocal (A+X) x'+(B + X) if + {G + X) z^ = must become
a right line in each case because the enveloping cone becomes a plane.
Therefore one coefficient of x'', y^', or ,:'•' must vanish. Hence A + X^ =■ 0,
B i-X.^ = 0, (7 + X3 = 0. Therefore the reciprocal cone becomes
(x-x,) x'+(x-x.;) ,f+(x-x,) z' = 0,
and therefore the enveloping cone is
-^ + -jI--+ ""' =0.
X_\^^ \-X., x-x.
THE GENERAL EQUATION OF A QUADRIC.
5673 This equation will be referred to as f{x, y, ^) = or
U = Of and, Avritten in full, is
ax'+by^+cz^-{-2fi/z-{-2gzaj-\-2Jhvi/-\-22hv-\-2qi/-]-2rz-\-d = 0.
By introducing a fourth quasi variable ^ = I, the equation
may be put in the homogeneous form
5674 cLv'+btf-^cz'-{-dtv'-\-2fi/z+2gz.v-\-2h.vij
-\-2]Kvt+2qi/t-\-2rzt = 0,
abbreviated into
(a, b, c, d,f, g', h,j), q, rjcf, i/, z, f)- = 0,
as in (1620).
Transforming to an origin xyz and coordinate axes
parallel to the original ones, by substituting x -^^, // + »?, ^' -{-I
for X, y, and z, the equation becomes, by (1514),
11^ SOLID GEOilETBY.
5675 fie'+br+d'+2f7iC+2^ii+mv
where U = f{x\ //', z) (omitting the accents).
5676 The quadratic for r, the intercept between the point
xyz and the quadric surface measured on a right line drawn
from xy':/ in the direction Imn, is
r^ (al'--^b})r-\- c}r-\-2fmn-\-2gnl-\-2hhn)
+ r {UJ,-Jr-mUy-\-nU,)-^U = 0.
Obtained by putting ^ = rl, n = rm^ I = rn in (5674).
5677 The tangents from any external point to a quadric are
proportional to the diameters parallel to them.
Proof.— From (5676), as in (1215) and (4317).
5678 The equation of the tangent plane at a point xijz on
the quadric is
(S-.V) t/,,.+(r,-//) u,Ml-~) V. =
5679 or ^ll+r,U, + Ca+TU, = 0,
with T and t made equal to unity after differentiating.
Proof. — From (5676). Since xyz is a point on tbe surface, one root of
the quadratic vanishes. In order that the line may now toucJi the surface,
the other root must also vanish; therefore lU^ + mU „ + 7ilL ^= 0. Put
rl = ^ — x, rm = r] — i/, rn = '(—z; ^tji^ being now a variable point on the
line, and therefore on the tangent plane.
5680 Again, xU^ + yU, + zU, + tU, = -lU, hy (l&2^^),
therefore x JJ^, + y U,, + z F. = —tUf,
which establishes the second form (5679).
5681 Equation (5679) also represents tlio polar plane of
any point xjjz not Ij^ing on the quadric surface. Written in
full it l)ecomes
i (,u-\-hj/+ii'Z-{-p) or ,r {ai+hrj + o-^-^p)
+17 (A.r+/>// + /i + ry) +// {^J^hr) + ^fZ-^q)
THE GENERAL QUADIilG. 777
5683 That is, the forms
iU,+y]U^-]-CU,+ U=0 and .vU,-\-i/U^-JrzU,-{- U= d
are convertible, U standing for / (r, ?/, z) in the first, and for
/(^, V, t) in the second.
5685 The intersection of the polar planes of two points is
called the^oZar line of the points.
5686 The polar plane of the vertex is the plane of contact
of the tangent cone.
Proof. — If ^ijii be the vertex and xyz the point of contact, equation
(5683) is satisfied. If x, y, z be the variables and l,ii'C constant, the second
form of that equation shows that the points of contact all lie on the polar
plane of the point i,r]ii.
5687 Every line through the vertex is divided harmonically
by the quadric and the polar plane.
Proof. — In equation (5684) put x = i. + Bl, y = r}-\-Bm, s = i^+Bn to
determine B, the distance from the vertex to the polar plane. This gives
B = —- — — -, emplovinff (5680).
lU^ + mU^ + nU^ t J to V ^
Now, if r, r' are the roots of the quadratic (5676), with |, tj, c, written for
X, y, z, it appears that — — - = E, which proves the theorem,
r + r
5688 Every line {hnn) drawn through a point xyz parallel
to the polar plane of that point is bisected at the point, and
the condition of bisection is
Proof. — The equation is the condition for equal roots of opposite si^ns in
the quadratic (56/6). Since I, m, n are the dir. cos. of the line and U^, TJy,
U^ those of the normal of the polar plane (5683), the equation shows that
the line and the normal are at right angles (5532).
5689 The last, when x, y, z are the variables, is also the
equation of the diametral plane conjugate to the direction Imn.
Expanded it becomes
{al-\-hm-\-gn) d'-\-{hl-\-bm-\-J)i) ij-\-{gl-\-fm-\-cn) z
•\-l)l-\-qm-\-rn = 0.
5 G
?78
SOLID GEOMETRY.
For the point xyz moves, when x, //, z are variable, so that every diameter
di'awn through it parallel to hnn is bisected by it, and the locus is, by the
form of the equation, a plane.
If the origin be at the centre of the quadric, p, q, and r of course vanish.
5690 The coordinates of tlie centre of the general quadric
U=0 (5673) are
A' A' A'
2^' " 2A
- = 2i' '^^
Proof. — Every line through xyz, the centre, is bisected by it. The condi-
tion for this, in (5688), is Uj.= 0, Uy= 0, and Z7, = 0, in order to be inde-
pendent of Inm. The three equations in full are
h g p
ax + Jiy + gz-\-p =
hx+hy+fz + q =
gx+fy + cz + r =
Solve by (582).
and A'
h h f q
g f r
p q r d
a h g
■ A =
h h f
9 f c
5691 The quadric transformed to the centre becomes
Proof. — By the last theorem, the terms involving ^, ?;, i^ in (5675) vanish.
The value of t)" or / (x, y, z), when xyz is the centre, appears as follows : —
U = ^Uti5680) =px + qy + rz + d = P^P + q^, + ^^r ^^ (5690) = ^ (1647).
ZA A
The last equation, being again transformed by turning the
axes so as to remove the terms involving products of coordi-
dinates, becomes
5692
a.i-+fy'+yz'+^ = 0,
5693 where a, |3, y are the roots of the discriminating cubic
R'-R' {u+b+c)-i-R {bc+ea + ah-f'-^'-h^-A = 0,
or {R-a){R-h){R-c)-iR-a)f'-{R-b)^'-'{R-c) h
-2fo'h = 0.
Proof. — It has been shown, in (1847-9), that the roots of the discrimi-
nating cubic (multiplied in this case by -;] are the reciprocals of the
maximum and minimum values of x^ + y^-\-z^. But such values are evidently
THE GENERAL QUADBIC, 779
the squares of the axes of the quadric surface. Let the central equation of
the surface be ^ + ^ + ^ = 1. Therefore \ = — ^, &c., producing
a- b' c- a- A ^ °
the equation above.
5694 The equations of the new axis of x referred to the old
axes of ^5 Vi t are
(i^+a/) .V = (G+ao-) ,/ = {H-\-ah)z;
and similar equations with j3 and y for the y and z axes.
Proof. — When Imn, in (5689), is a principal diameter of the quadric, the
diametral plane becomes perpendicular to it, and therefore the coefficients of
X, y, z must be proportional to I, m, n. Putting them equal to El, Em, En
respectively, we have the equations
(a — B)l + hm + gn = (1)"^ The eliminant of these equations is
hl+(b — B)m+fn = (2) /■ . the discriminating cubic in E al-
gl+fm-{- (c—B) n = (3) ) ready obtained in (5693).
From (1) and (2), I: m = hf-g (h-E) : gh-f(a-E),
and from (2) and (3), m : n =fg — h(c — E) : hf—g (b — E) ;
therefore (gh-af+Ef) I = (hf-bg + Eg) m = (fg-ch + Eh) n,
which establish the equations, since x : y : z = I : m : n and F = gh — af,
&c., as in (4665).
5695 The direction cosines of the axes of the quadric.
If the discriminating cubic be denoted by </> (E) = 0, and
its roots by a, |3, y ; the direction cosines of the first axis are
JZiM, jZAM, JZES
For the second and third axes write /3 and y in the place of a.
Proof.— Let F+af=L, G + ag = M, H+ah = N (i.),
(a — b)(a — c)—f- = \, (a — G)(a — a) — g'- = fx, (a — a)(a — b) — li' = v...(ii.).
Then the equation (a) = may be put in either of the forms
L- = fiv, M^ = i>\, N' = Xfi (iii.).
Now the dir. cos. of the first axis are, by (5694), proportional to
Their values are, therefore,
y^ ^/u y^'
^{X + ^ + rY y(\ + /i+v)' ^{X + fx + ry
But X =_^iM and X + M + v = ^^'^fr\ by actual diflFerentiation of the
da da
cubic in (5693).
780 SOLID GEOMETRY.
5696 Cauchy's proof that the roots of the discriminating cubic (5693)
ai'e all real will be found at (1850).
5697 The equation of the enveloping cone, vertex cri/z, of
the general quadric surface U =0 (5673) is
4 {abcfghXlmnfU = {lU,-{-mU,-]-nU,y\
-ttdth ^—x, v — y, ^— ^ substituted for /, m, 7i.
Proof. — The generating line through xijz moves so as to touch the
quadric. Hence the quadratic in r (5676) must have equal roots. The equa-
tion admits of some reduction.
5698 When U takes the form aa^-\-hi/-\-cz^^ = 1, equation
(5697) becomes
{aP-\-btn^-{-cn^){a.v--{-bif-^cz- — l) = (aLv-\-bmi/-\-cnzy.
5699 The condition that the general quadric equation may-
represent a cone is A'=0; that is, the discriminant of the
quaternary quadric, (5674) or (1644), must vanish.
Proof. — By (5692). Otherwise A' = is the eliminant of the four
equations ?7^ = 0, Z7j, = 0, U, = 0, U = 0, the condition that equation
(6675) may represent a cone.
5700 The condition that the plane
h-\-my-\-nz = may touch the cone
{ahcf(jli^:ciiz)'^ = is the determinant
equation on the right.
a
h
S
I
h
b
f
)n
8'
f
c
n
I
m
n
= 0.
Proof. — Equate the coeflBcients I, m, n to those of the tangent plane
(5681), p, q, r being zero, and xyz the point of contact. A fourth equation
is Ix + viy + nz = 0, which holds at the point of contact. The eliminant of
the four equations is the determinant above.
5701 The condition that the
plane lx-\-my-{-nz-\-f = may
touch the quadric
{ahcd/ghpqrXxyziy =
(5673) is the determinant equation
on the right.
Proof.— As in (6700).
n
h
S'
P
I
h
b
f
Q
m
S"
f
c
r
n
P
Q
r
d
t
I 7)1 n
= 0.
BEGIPBOCAL POLABS.
781
5702 If the origin is at tlie centre, 2) = q = r = 0. In that
case, transposing tlie last two rows and last two columns, the
determinant becomes
a
hg I ^
h
h f m
g
/ c M
I
m n t
t d
0,
a h g I
= t-
a h g
h b f 7n
h h f
g f c n
g f ^
I m 71
5703 The condition that the line of intersection of the
planes
lx-{-my-{-nz-\-i = ^ ...i^}) and l'x-\-r\%ij-\rnz-\-t' = ^ ... (ii.)
may touch the general quadric (ahcdfghjxjrXxyziy^ = 0, is the
determinant equation deduced below.
Multiply equation (i.) by ^ and (ii.) by rj to obtain the piano
(lE+Vn) x + (m^ + m'rj) y + (nk + nti) z + U + t'ii = (iii.),
passing tlirougb the intersection of (i.) and (ii.)- The line of intersection
•will touch the quadric if (iii.) coincides with the tangent plane at a point
xyz, and i{ xyz be also on (i.) and (ii.). Therefore, equating coefficients of
(iii.) and the tangent plane at xyz (5681), we get the six following equa-
tions, the eliminant of which furnishes the required condition,
ax+ hy+gz + 2nv= U+ I'l]
lix + % + fz + qiv = mt, + Di'ri
gx+ /2/+ CZ+ rw =^ ni-\- n'l)
])x-\- qy+rz+dw= tt,+ t'tj
lx+ my+ nz+ ho =
I'x + m'y + n'z + t'tv =
a
h
9
V
I
h
b
f
9
m
9
.t
c
r
n
V
'/
r
d
t
I
in
n
t
V
m
11
t'
r
RECIPROCAL POLARS.
5704 The method of reciprocal polars explained at page 665
is equally applicable to geometry of three dimensions.
Taking poles and polar planes with respect to a sphere of
reciprocation, we have the following rules analogous to those
on page &QQ.
782
SOLID GEOMETRY.
RULES FOR RECIPROCATING.
5705 ^ ijlane becomes a jjoint.
5706 ^ plane at infinity becomes the origin.
5707 Several ijoints on a straight line become as many 'planes
passing through another straight line. These lines are called
reciprocal lines.
5708 Points lying on a plane become planes passing through
a pointy the pole of the plane.
5709 Points lying on a surface become ptlanes enveloping the
reciprocal surface.
5710 Therefore, by rules (5708) and (5709), the points in
the intersection of the plane and a surface become planes
passing through the pole of the jjlane and enveloped both by
the reciprocal surface and by its tangent cone.
5711 When the intersecting plane is at infinity, the vertex
of the tangent cone is the origin.
5712 Therefore the asymptotic cone of any surface is
orthogonal to the tangent cone drawn from the origin to the
reciprocal surface. The cones are therefore reciprocal.
5713 The reciprocal surface of the guadric is a hyperboloid,
an ellipsoid, or a paraboloid, according as the origin is without,
within, or upon the quadric surface.
5714 The angle subtended at the origin by ttoo points is equal
to the angle between their corresponding planes.
5715 The reciprocal of a sphere is a surface of revolution of
the second order.
5716 The shortest distance betweeit two reciprocal lines
passes through the origin.
5717 The reciprocal surface of the general quadric
{abcdfghpqr\xy::\y = (5074), the auxiliary sphere being
x^ + y'^ + z^ = Jr, is
= 0,
a
h
A'"
p
^
h
h
/
n
V
^
f
c
r
C
P
V
r
d
-/r
^
V
c
-/r
or, if j) =
'[ = ''
= 0,
a h ix i
-k'
a h i>'
h b f r)
h b f
A" /' f i
^- f c
i V C(^
TBEOEt Of TORTUOUS CURVES. 783
Proof. — The polar plane of the point hjC with respect to the sphere is
^.T + r/^ + 4z — A;^ = 0. This must touch the given surface, and the condition is
given in (5701).
5718 The reciprocal surface of the central quadric
^- + ^-f— ■=1, when the origin of reciprocation is the
o?' Ir c^
point xy'z , is
or, with the origin at the centre,
5719 «^f+6V+cT = A^'.
Proof. — Let _p be the perpendicular from x'y'z' upon a tangent plane of
the quadric, and ^ril the point where p produced, intersects the reciprocal
sui'face at a distance p from x'y'z' . Then
lY^=p = W + my' + nz - ^(aH' + h\v + c'w^). (5030)
Multiplying by p produces the desired equation.
THEOEY OF TORTUOUS CUEVES.
5721 Definitions. — The osculating plane at any point of a
curve of double curvature, or tortuous ciirve,^ is the plane
containing either two consecutive tangents or three consecu-
tive points.
5722 The princiiml normal is the normal in the osculating
plane. The radius of circular curvature coincides with this
normal in direction.
5723 The binomial is the normal perpendicular both to the
tangent and principal normal at the point.
5724 The osculating circle is the circle of curvature in the
osculating plane, and its centre, which is the centre of circular
curvature, is the point in which the osculating plane intersects
two consecutive normal planes of the curve.
5725 The angle of contingence, d^, is the angle between two
consecutive tangents or principal normals. The angle of torsion,
cIt, is the angle between two consecutive osculating planes.
* otherwise named " space curve."
784 SOLID GEOMETRY.
5726 The rcdifiiing plane at any point on the curve is per-
pendicular to the principal normal ; and the intersection of
two consecutive rectifying planes is the rectifying line and
axis of the osculating cone.
5727 The osculating cone is a circular cone touching three
consecutive osculating planes and having its vertex at their
point of intersection.
The rectifying developable is the envelope of the rectifying
planes, and is so named because the curve, being a geodesic
on this surface, would become a straight line if the surface
were developed into a plane.
'&■
5728 The polar developable is the envelope of the normal
planes, being the locus of the line of intersection of two con-
secutive normal planes. Three consecutive normal jjlanes
intersect in a point which is the centre of spherical curvature :
for a sphere having that centre may be described passing-
through four consecutive points of the curve.
5729 The edge of regression is the locus of the centre of
spherical curvature.
5730 The rectifying surface is the surface of centres (5773)
of the polar developable.
5731 An evolute of a curve is a geodesic line on the polar
developable. It is the line in which a free string would lie if
stretched between two points, one on the curve and one any-
where on the smooth surface of the polar developable.
5732 In Figure (180) A, A', A", A'" are consecutive points on a curve.
The normal planes drawn through A and A' intersect in GE ; those thruugli
A' and A" in G'E', and those through A" and A'" in G"E". GE meets G'E'
in E, and G'E' meets G"E" iu E'. The principal normals in tlie normal
planes are AG, A'G', A"G", and these are also the radii of curvature at
A, A', A", while G, G', G" are the centres of curvature. Z. AG A' = dij/ and
CA'G' =dT.
The surface EGG'G"E' is the polar developable, GG'G" being the locus
of the centres of curvature, and EE'E" is the edge of regression.
EA is tiie radius and E the centre of spherical curvatui-e for the point
A. hll, nil', WW are elemental choi-ds of an evolute of the curve, AhS
being a normal at A, and A'lIW a normal at A' , and so on. The first
normal drawn is arbitrary, but it determines the position of all the rest.
THEORY OF TORTUOUS CURVES. 785
PROPERTIES OF A TORTUOUS CURVE.
5733 The equation of tlie osculating plane at a point xyz on
the curve is
(f_,,)X+(,-y)^i+«-~)^ = 0.
5734 ^, f-h V are the direction cosines of the binorrual, and
their complete values are
5735 The angle of contingence
Proof. — Let the direction of a tangent be Imn, and that of a consecutive
tangent l + dl, ni + dm, n + dn. Since the normal of the plane must be per-
pendicular to both these lines, we shall have, by (5532),
l\ + vifx + nv = () and {l-\-dl)\ + (m + dm) fx + (n + dn) v = 0,
therefore \ : fj. : v =■ mdn — ndm : ndl—ldn '. Idm—mdl,
and the denominator in the complete values of A, n, v is
^/{(?)uZn — ?itZHi)^ + &c.] = sinfZt/',
by (5521) ; that is, = c?i//. Also ?, m, n = a-,, y^, z^ and dl = x.>sds, &c.
Therefore X = {y^z.^-y^uZs)-—- Similarly, /tx and »/; and s^ — p, by (5146).
5736 The radius of curvature /o at a point xyz.
1 _ ^2 . ^,2 , .2 _ - i.+^2. + 4-4
P ^t
Proof : d^p = ^{(ysh-y-zs^.Y-^- &c.} ds, in (5735),
therefore i^, = ^ { (x] + yl + zl) (x^ + y^ + z^) - (x^x., + y.y.,, + z,Zo,y ]
= V'(a^L + 2/L + 4) 5 ^^^<^® ^s + y' + ^s = 1 5
and differentiating this equation makes XgX2s + &'0. = 0.
Otherwise, geometrically, precisely as in the proof of (5141), we find the
direction cosines of the principal normal to be
5737 cos a = pd\„ COS ^ = pifisy COS y = px^^-
Therefore p"' (a-'-^ + y^s + 'D — cos^ a + cos^ )S + cos" y = 1.
The change to the independent variable t is made by (1762).
5738 The angle of torsion, in terms of A, ^t, v of (5734), is
5 H
786 SOLID GEOMETBY.
Proof.— By (5745), we have (dry = {dxy + idfiy+ichy (i.),
which gives the first form. The third reduces to this by the method in
(5736). For the secoad form put u — ygZ^^ — yisZ,, &c., then
\ i_i V 1 ds ,f,y.r),\ ^^ dii udK „
-=- = — = -p = 77 (5/'34), d\ = — ----, &c.
Substitute in (i.), reducing by K' = ii/ + v"-\-io- and KdK — udu + vdv + wdio.
CURVATURE AND TORTUOSITY.
5739 Radius of curv., p = -^; CurYature = — = ■^.
^ dxjj p (Is
Radius of torsion, <r = ^; Tortuosity = — = -^.
(It cr (Is
If Tg changes sign while passing through the values
zero or infinity, there is a point of inflected torsion or
a cuspidal 2')oint, respeGtivelj. If t^, without changing sign,
passes through zero or infinity, there is a point of suspended
torsion or infinite torsion respectively.
If Tg is zero, identically, the curve is plane.
5740 The radius of spherical curvature,
R = Vip'^Pr)'
Proof. — In Fig. (180) B'^ = p- + EG'^ and EC = Pt by analogy with q=i>^
n a plane curve (see proof of 5147).
5741 T^he element of arc of the locus of centres of circular
curvature is
ds = RdT, and therefore R = s'^.
Proof.— In Fig. (180) ds = CG' = pdr sec<p = Edr.
5742 The radius of curvature of the edge of regression
= S': = RR, = p+p, ,
S'' being the arc of the edge of regression.
Proof. — An inspection of Figure (180) shows that R and p stand in the
same relation to the edge of regression that r and p occupy with regard to a
curve in the standard formula. In fact we may .substitute li for r, p for p,
THEORY OF TORTUOUS CURVES. 787
(p for 0, T for i(/, and <p remains (p. Tlie chosen line of reference AB being
always parallel to the tangent EC, then AEC = BAE = <p. Also the angle
of contingence CEC = GAC = dr, by the right angles at C and C. Ac-
cordingly, we have the formula p = s^ = r7'p=p+2h^ from (5146-8), and the
values above corresponding to them.
5743 ^ method of estimating the variation in direction of a
right line ivhose position is given as de'pending upon the form
of a tortuous curve at every point.
Let X, y, % be tlie direction cosines of tlie line referred to a
fixed principal normal, tangent, and binormal of the curve
[«;, y, % may either be constants with respect to the varying
principal normal, tangent, and binormal, or they may be func-
tions of the angle between the binormal and the spherical
radius] .
5744 The complete changes in x, y, r^, with respect to the
fixed origin and axes, will be
S/y = dif—xdxjj,
S:^ = dz-\-xdT.
Peoof. — In Figure (180) AC, AB are the fixed axes of a; and z. Let a
line AL of unit length be drawn always parallel to the line in question ; then,
if X, y, z be the coordinates of L, x, y, z will also be the direction cosines of
AL, and therefore of the given line.
Now, suppose A to move to A', and consequently AL to take the position
A'L' . Then the changes in x, y, z will be the changes dx, dy, dz relatively to
the moving axes, plus the changes due to the rotations dxj; round the
binormal and dr round the tangent. With the usual notation, we shall have
^x = dx + w^z — Wj?/, hj = dy + w^x — w^z, cz = dz + w^y — u).^x,
with Wj = 0, w., = —dr, Wj = —d\j/.
5745 If (^x ^® ^^® angular change in the direction of the
right line.
For dx = LL' since AL is a unit length.
Examples.
5746 The angle between two consecutive radii of circular
{d.y = (d^y+Ciry.
curvature being de
788 SOLID GEOMETRY.
Proof. — Here, in (5744), a' = 1, ?/ = 0, z = 0, therefore Sx = 0, cy z=—d\p,
Sz = dr. Substitute these values in (5745).
5747 The angle, dv, between two consecutive radii of
spherical curvature, ^ being the inclination to the binormal,
(d-qY = (dxlt.smct>Y-\-(d(l>-dTy:
Proof. — In (5744) the direction cosines of R (Fig. 180) are x = sin (p,
y = 0, z = cos (p,
therefore ^x = cos f (dijt — dr), Sy = —dij/ sin f, cz = — sin^ (dcp — dr).
Substitute in (5745).
5748 The angle of contingence of the locus of the centres
of circular curvature,
Proof. — The dir. cos. of the tangent at G to the locus (Fig. 177) are
X = cos 0, y = 0, 2 = sin (f) ;
therefore ^x = — sin ^ (df + dr), Sy = —d\p cos (p, ^z = cos (df + dr).
Substitute in (5745).
5749 The osculating plane of the same curve has its
direction cosines in the ratios
dxb . , 1 / d(b , dr \ dxb ., ,
— ^ sill 6 cos (p : —( — -^ _L — — : x cos-^ 6.
(h \^iX dx' dx
Proof. — As in the Proof of (5735), the dir. cos. of the normal to this
plane are proportional to ijlz — z^y., zLv—xSz, xcy — ycx. Substitute the
values in last proof.
5750 The angle of torsion of the same curve is found from
(5745) and (5744) as above, x, t/, z being in this case the dir.
cos. of the normal of the osculating plane as given in (5749).
5751 The direction cosines of the rectifying line are
^. dr dijj
' ^' lu-
Proof. — The rectifying plane at A' (Fig. 180) is perpendicular to the
noi'mal A'G'. Therefore its equation is ■.r — yd{j/ + zdr ^= 0. Th(^ ultimate
intersection of this plane with the rectifying plane at A (that is, the plane
of yz) is the rectifying line. Hence the equation of the latter is yd\p = zdr ;
and the dir. cosines reduce to the above by (574G).
THEORY OF TORTUOUS CURVES. 789
5752 Cor. — Tlie vertical angle of the osculating cone
= 2 tan-' '4*.
(IT
5753 The angle of torsion of the involute of the curve is
Proof. — This angle is also the angle between two consecutive rectifying
lines. Therefore, taking the dir. cosines from (5751), we must put in (5744)
-^ de' de '
therefore dx = ^-^ dib— ^ dr = ; Ey = r^de • h = \}j.,Ae.
de de '' ^e ' ^ -^
5754 The angle of torsion of an evolute of the curve
= d\lf siu (a— r).
Peoof.— (Fig. 180.) Let ER'H" be an evolute of the curve, AE the
tangent to it in the normal plane of the original curve at A, and let a = GAE,
the inclination of AE to the principal normal. At any other point E" of
the evolute, where its tangent is A"E'E", let the corresponding angle be
= G"A"S". Then = a — r, r being the sum of the angles of torsion
between A and A", or the total amount of twist of the osculating plane. Now
the normal of the osculating plane of the evolute at E",is perpendicular to
EE' and E'E", two consecutive tangents. Therefore its dir. cosines in
(5744) must be
a; = — sin (o — -), y = 0, z = cos (a — r) ;
therefore ^x = cos (a — r) dr + — cos (a — r) dr = 0,
hj = sin (a — r) d\p ; cz = sin (a — r) fZr — sin (a - r) dr = 0.
Hence the angle required =: cy = dij/ sin (a — r).
5755 Approximate values of the coordinates of a point on a
tortuous curve near to the origin in terms of the arc, the axes
of X, y, z being the principal normal, tangent, and binormal,
and the arc s being measured from the origin :
6/0^ 8/0^ hpa z4\/oo- pa/
p and <y being respectively the radii of circular curvature and
torsion.
790 SOLID GEOMETRY.
Proof. — By Taylor's theorem (1500), since x, y, z, s are the same as
dx, cly, dz, ds initially, we have x = x^-\-\x2aS- + \x3gS^ + &c., and similar ex-
pansions for y and z. The dir. cosines of the principal normal at the point
xyz will be, from (5737),
cos ( - vp) = p.'-o,, cos ^^ + «// j = py.„ cos (I - 7 ] = pz,,, ;
»// = cZi/' and t = dr being estimated positive as drawn in Figure (180) for
positive values of x, y, z.
Differentiate these equations for s, and iu the results put the initial values
a\ = Zg = 0, ys= '^, \p = T = 0, i(/^, = —, r, = — , &c.,
p '^
to determine the derivatives in the above expansions.
THE HELIX.
5756 Tlie lielix is a curve traced on a c^^lincler of radius a,
so that its tangent preserves a constant inclination, = Itt — a,
to the axis. Taking the axis of the cylinder for the z axis of
coordinates, the equations of the helix are
jr = a cos 6, y = a sin 6, js = ad tau a.
5757 The radius of curvature p = a sec^a.
5758 The radius of torsion a = 2a cosec 2a.
Proof. — p from (5806) ; since pj = a, p^ = x , and 6 = a at every point.
By (5739), a = s^. But dz = dssiua and adr ^ dz cos a.
5759 The helix of closest contact with a given curve may
be found as follows.
Determine the constants a and a from equations (5757-8), with the
known values of p and a for the given curve ; then place tlie lielix to have a
common tangent with the curve at the point, and make the osculating planes
coincide.
GENERAL THEORY OF SURFACES.
5770 Definitions. — A tangent plane passes through three
consecutive points on a surface which are not in the same
right line.
5771 The nonna.J at any point of a surface is perpendicular
to the tangent plane.
THEORY OF SURFACES. 791
5772 A normal plane is any plane through the normal.
5773 A line of curvature on a surface is a line along which
consecutive normals to the surface intersect. At every point
of a surface there are usually two lines of curvature at right
angles to each other ; and to these correspond two principal
radii of curvature. The two lines of curvature coincide with
the principal axes of the indicatrix at the point. See (5778).
5774 The surface of centres is the locus of the centres
of principal curvature. There are two such surfaces, for
there are two centres on each normal, and the normal is a
tangent to both surfaces. Either surface may be regarded as
generated by the evolutes of the lines of principal curvature.
5775 A geodesic is a line traced on a surface along which
the osculating plane at every point contains the normal to
the surface. See (5779).
5776 The radius of geodesic curvature * of a curve traced
on a surface is measured by the ratio of the element of arc of
the curve to the angle between consecutive normal sections
of the surface drawn through consecutive tangents of the
curve. Geodesic curvature, being the reciprocal of this, is
therefore the rate of angular deviation of the normal section
per unit length of the curve.
5777 An umhiUcus is a point on a surface where a section
parallel to and close to the tangent plane is a circle ; in other
words, the indicatrix is a circle.
For a definition of Indicatrix, see (5795).
5778 In Figure (182) OCT) is the normal at to a curved surface;
AOA', BOB' are the lines of curvature, therefore the normals to the surface
at J. and intersect in the centre of curvature radius pi (5773), and the
normals at B and 0, in the centre, radius pi. The normals to the line of
curvature BOB' at B and 0, drawn in the osculating plane BOB', intersect in
K, and those at B' and intersect in H. HOD is the angle between the
osculating plane of the line of curvature and the plane of normal section.
Similarly for the line of curvature AOA'.
5779 If POP' be a geodesic, its osculating plane POP' contains OD the
normal to the surface at 0, and therefore p = OD, the radius of curvature
of this section at ; but PE, the normal to the surface at P, does not inter-
sect OD, the consecutive normal at 0, unless the geodesic coincides with one
of the lines of curvatui-e, OA or OB. The angle DPE is the angle of torsion
which vanishes in the latter case.
* Not to be confounded with the radius of curvature of a geodesic.
792 SOLID GEOMETRY.
GENERAL EQUATION OF A SURFACE.
5780 Let the general equation of a surface be represented
by {x, y, z) = 0.
5781 The equations of any tangent at a point cTi/z are
tz£ = rLZlL = ^i:£^ with l6,-\-md>^+ncb,=:{).
Peoof. — At an adjacent point x-\-rl, y + rm, z + rn, we have
(j)(x + rl, y-\-rm, z + rn) = 0,
therefore, by (1514), {x, y, z) +r (/0^4-??z^,, + ?i^,) = 0,
the rest vanishing in the limit. But f (x, y, z) = 0, therefore
l(l)^-\-m^,, + n(p, = 0.
But I, m, n are the direction cosines of the line joining the two points, which
becomes a tangent in the limit ; and if kr}!!, be any point on this line distant p
from xyz, ^—x = pl, r}—y = pin, ^—z = pn, &c.
5782 The equation of the tangent plane at xijz is
Proof. — Eliminate I, m, n from Ifj. + 7n(j>y + 7i(j>, = by ^ — x = pi, &c., as
above.
TANGENT LINE AND CONE AT A SINGULAR POINT.
5783 If, in the expansion in (5781) by Taylor's theorem, all the deriva-
tives of (x, y, z) of an order up to n inclusive vanish, we have
(j) (x + rl, y + rm , z + rn) = (x, y,z) + ~ — - {Jd^ + m<J,, + n cL) " * ' ^ (a', y, z) = 0.
There are in this case n + 2 coincident points at xyz in the direction linn,
and since the equation (Id^ + mdy + nd,)"*^ cp (x,y, z) = is of the n + V^
degree in l,m,n; n + 1 tangents to the surface at a-y:; can, in general, be
drawn in any given plane through that point. This equation now takes the
place of the conditional equation in (5781),
5784 Equation (5782) is now replaced by
{{i-.v) cL^-iri-il) ^/.+ (^-^) ^/J-X.r, //, -) = 0,
the equation of the locus of all tangents at the point ,<7/.v, and
representing a conical surface generated by the motion of
those tangents.
5785 The equation of the normal at xyz is
^k ~~ ~Zl — — "^I — • (o/«-)
9^' % 9.
THEORY OF SURFACES. 793
5786 The equation of the tangent at a point xy^ on the
curve of intersection of the tangent plane at xijz with the
surface is
X jU, V '
with the two conditions
For these are the conditions of perpendicularity to the normals of the
tangent planes at xyz and x'y'z' respectively.
There are three exceptional cases in which the ratios
\ : 1^1 : V have more than one set of values ; namely —
giygty I, — When f ,, «^^„ <p. vanish simultaneously, there is a
tangent cone at xijz.
5788 II.— When .^.,., <l>y., (/)-/ vanish simultaneously, x'y'z is
a singular point on the surface.
5789 III.— When ^ = h = h. In this case the point
tv %' i>^'
xyz coincides with xyz, and the tangent there meets the curve
in more than two coincident points, the condition for which is
(X^/,+K/+»'^y'<^('*''//'-) = (i.).
with X(^,+/x(^,4-»'<^„^ = (ii.).
These equations furnish two sets of values of the ratios
X : /ti : V, giving thereby the directions of two inflexio7ial
tangents (tangents to the curve of intersection) at xyz, each
meeting the surface in three coincident points. If all the
derivatives of an order less than n vanish at xyz, equation (i.)
will be replaced by {'Xd, + i.ul,j^-^vLL)"(p {x, y, z) = 0, which,
together with (ii.), will determine n inflexional tangents at
the point.
5790 The polar equation of the tangent plane at the point
rdcj), r, 0', r/)' being the variables, is, writing u for r~\
II z={u{ios6—UgSm6) (ios6'-\-{u sm0-{-UgGOs6)GOs {<!>'— (t>)sm6'
-^u^ cosec siu {(!>' — (f>) siu 6'.
5 I
794 /SOLID GEOMETRY.
Proof. — Write the polar equation of the phine through p"/3, the foot of
the perpendicular on the plane from the origin ; thus
pto = cos fi cos a + sin sin a cos (0-/3).
Differentiate for B and to find pug and jm., and eliminate jh ", ^^^ ft- This
elimination is troublesome.
5791 The length of the perpendicular from the origin upon
the tangent plane at xijz,
.r^.+!/<t>„ + ~'l>. or ^-^-^ -,, (5782,5549)
the second form being the value of jj when the equation of
the surface is <j) (x, y,z) = c, a constant, and when «^ is a
homogeneous function of the n"' degree (1624).
5793 In polar coordinates,
1 9 , 9 , •> za r'+r^+r! cosec'^^
—r = ir-\-u;-{-tii cosec a = — ■ — 2_! — 2 .
Proof. — Add together the values of the squares of pu, pug, and. jm, found
in (5790).
For a geometrical proof, see Frost and Wolstenholme, Art. (314).
THE INDICATRIX CONIC.
5795 Dei\ — The indicatrix at any point of a surface is the
curve in which the surface is intersected by a plane drawn
parallel to the tangent plane at that point and infinitely near
to it.
5796 The following abbreviations will be employed —
The derivatives of (p {x, y, z), </>,.,, (j>,^, (p,,, (p^,, <j>,^,, </>.,„ </),., <^„ <p„
will be denoted by a, h, r, /, g, h, I, m, n.
5797 PiJOP. — The indicatrix at a point xyz of a surface
<l> (x, y,z) = is the conic in which the elementary quadric
surface
5798 I.
R'
is intersected by the tangent plane at ,«;//:;, whose equation is
5799 II. l^-{-mri+nC+iN=0.
THEORY OF SUJ^ FACES. 795
The origin of coordinates is the point cvijz in both equa-
tions. B is an indefinitely small radius from the centre of
the quadric (I.) to a point ivt on the indicatrix, and p is the
radius of curvature of the section of the surface </> by a normal
plane drawn through B ; the ratio B'^ : p being constant for
all such planes.
Proof.— Let 0, in Fi^. (181), be the point xyz on the surface (p; x + E,
2/ + r/, 2 + C an adjacent point P. Then
<p(x + ^,y + V, z + 0=(p(x, y, z) + ^^ + m»; + «^+i(a^H...+2//^^»^)+&c.
With xyz for origin, draw the quadric surface
ae + hrf^ce+2frj-C^2ga + 2Ur^ = N (i.)
and the plane ll+m^i + nli^^N = (ii.)-
Since I, V, <r are very small, N is likewise. Also the unwritten terms in the
above expansion may be neglected in the limit. Hence, any point ^?y4 lying
on the intersection of the quadric (i.) and the plane (ii.) will also lie on the
original surface f (x + t„ y + ii, ^ + = ^•
To determine N, we have the perpendicular from a-y,? upon the plane (ii.),
'-^ (5549). The radius of curvature of the section of the
^ 2 ^(P + m' + n')
surface made by a normal plane at drawn through P being p, we have
P = —, and therefore N = - — ^Q' + m' + n').
2p p
In the Figure, B = OP, p = OL, and the intersection of (i.) and (ii.) is
the conic PSQ. Since p is indefinitely small, we may put N=0 in equa-
tion (ii.). This amounts to taking the parallel section of the quadric by the
tangent plane at instead of the section PSQ. But these two will be equal
in all respects, since the section of the quadric is a central one.
5800 If m = 0, equation (II.) becomes l^-^nl = 0, and if
the inchnation of the indicatrix plane to the plane of xy be a,
tan a = — — . To obtain, in this case, the equation of the
n
indicatrix in its own plane, put ^ = ^' cos a, 1 = t sin a, and
7, = 77', in equation (I.).
5801 When none of the three constants /, m, n are zero,
the quadric (I.) simplifies as follows —
From (II.) we have /^+m»} = —nl and two similar equa-
tions. Square these, and by the results ehminate the terms
in nl, II, h] from (I.), which then becomes
5802 III. i/f+AV+^r = A^,
where H = a+ ~ Qf-mg — nh), K=h+^ (mg-nh-If),
mn ni
L = c-\-^ (nh-lf-mg).
796 SOLID GEOMETRY.
This is the equation of another quadric intersecting the
plane (II.) in the indicatrix.
5803 The equation of a surface for points near an origin
(Fig. 182), the normal at being taken for z axis, is
^ + ^=2^,
Pi ' P2
where pi, p., are the radii of curvature of the normal sections
through the x and y axes, and those sections will be proved to
be the lines of curvature at 0.
Proof. — Let AG=i a and BG = 1 he. the semi-axes of the indicatrix conic
at a small distance z from (5795). The equation of the conic will there-
fore be — + ^ = 1 ; but — = 2pi and — = 2p2, giving the equation
required.
Secondly, on a line of curvature, the normal to the surface at a point xyz
will intersect the z axis (5773). The condition for this, by (5533) [with
xyz for abc, the origin for a'h'c,
L, M, N = <p^, f,j, f, (5785) = — , % -2, and L', M', N'= 0,0,1'],
Pi ' Pi -^
gives xy { ) = 0, thei-efore x = or ■?/ = on a line of curvature.
^P^ ^^' Q.E.D.
5804 If the equation of the surface with the same axes be
,;' = ax^ -\-2hxij -\-l)y- -{-2pjz -\-2gzx-{- cz"^ -\-\\\g\\eY powers,
then p. = ^^, p, = |.
Proof. — Put y = and divide by c, therefore \ = a~ -\-2gx-\-cz-\-&:c.
When X and z vanish, we have 1 = 2api.
5805 For a normal section making an angle with AC,
— = 2{a cos' e^2h siu 6 cos O-^h siir 6).
P
Proof.— Turning the axes in (5804) through the angle by (4049), the
coefficient of x'- becomes a cos" + as above.
5806 Euler's Theorem. — If p be the radius of curvature of
c.
THEORY OF SURFACES.
797
any other normal section at 0, making an angle ACP = witli
AC (Fig. 182),
1 _ cos' 6 sill' 6
P ~ Pi Pi '
Pboof. — Let r = CP ; then x = r cos 0, y =r sin 6, and r^ = 2pz, which
substitute in (5793),
5807 Cor. — The sum of the curvatures of two normal
sections at right angles to each other is constant; or, if p, p
be the radii of curvature for those sections, and p„, p,j the
radii for the principal sections,
P P pa Pb
5808 The radius of curvature of a normal section varies as
the square of the radius of the indicatrix in that section.
Proof. — From r- = 2pz, in Figure (182).
5809 Meunier's Theorem. — The radius of curvature of an
oblique section of a surface is equal to the radius of curvature
of the normal section through the same tangent multiplied by
the cosine of the inclination of the planes.
T3 /t:.- 10o^ ' FN' PN^ ,T n 9 NO
PHOOP.-(F.g. 183.) p=— ,p=^, therefore ^ = -^
■when NO and NO vanish.
cos 0,
5810 Quadratic for ?/^. at a point on the surface z = (p {,v, y)
giving the direction of the principal normal sections, and,
therefore, of the lines of curvature (notation 1815).
+ {(l+ir)s-pqr}=0.
Proof. — (i.) The equations of the normals at the consecutive points xyz
and x + dx, y + dy, z + dz of the surface f (x, y, z) ^ are
l-x __7i-y _i:-z _^T ^ -(x + dx) ^ v- (y + dy) ^ ^ - (^ + dz)
and
5811 The condition of intersection is, by (5533),
dx dy
'Px ^y
d<t>^ d<j)y
dz
= 0, or
1
P
r + sy:c
Vx
■+tyx
p+qy^
-1
798 SOLID GEOMETRY.
by dividing the first vow by dx, and putting z^=. (Px + 9/l/xj (^fx= 'P^x + ^xyVxt
&c. The form of <p (x, y, z) being, in this case, (j) {x, y) — z, f, becomes —1,
and df^ becomes zero. The determinant equation produces the quadratic.
(ii.) Otherivise. — Consider ^»;4 the point of intersection of consecutive
normals. The equations of a normal being
tl^ = ^^y=^ or ^-x=2^(z-0 and r,-y = g (z-O-
p q -I
Differentiate both equations for x, considering ^, r], i^ constant and p, q func-
tions of X and y ; the results ai-e
l + (r + syx)(z-O+p(p + qyx) = and yx+(s + tyx)(z-i;) + q{p + qy;) = 0.
Eliminate z — ii to obtain the quadratic in y^..
Ool2 If the equation of the surface be in the form f (x, y, z) = 0, the
quadratic for y^ may be obtained in the same way. The requisite substitu-
tions in the first determinant are found from fx'^fi/Vx'^i'z^x = Oj giving ^'x'>
d<Px =■ ^2x + 0a-;/2/x + 0.r^2x! <^c., and with the notation of (5796) the determin-
ant equation and quadratic for y^ becomes
n nyx -G + myx)
I in n = 0.
an — gl+(hn — gm) yx hn—fl+ (hn—fm) yx gn —cl+ (fn — cm) i/x
Oolu The above determinant, or the corresponding one in (5810), is the
differential equation of the lines of curvature.
5814 The radii of curvature of the principal normal sections
of the surface (p {x, y, z) =: at a point xyz are given by the
following quadratic, in which A' is the bordered determinant
in (5700), and the notation is that of (5796) and (1620).
where P = l^-{-m^-]-n^.
Proof. — The quadratic in (5653) applied to a section of the quadric (I.)
(5798) by the plane (II.), becomes
A'B'+ {(b + c) l'+(c + a) vr + (a + h) n'-2fmn-2gnl-2hlm] NB'
-(l'-\-m' + 7i')N'= 0,
whose roots, being the two values of ii', are the squares of the principal
semi-axes of the indicatrix. Put B' = -^ , as in the Proof of (5797).
5815 Otherwise, the quadratic in (5651) might be applied to a section of
the (piadric (III.) (5802) by the plane (I.).
5816 If the equation of the surface be given iu the form
THEORY OF SURFACES. 799
z = (j) (^x, y), tlie quadratic becomes [writing, as in (1815),
J), (J, V, S, t tor ::j., Zy, Z^xi ^xyi ^2y]i
where Jc^ = f" + ^-"^ + 1 .
Otherwise, this equation may be found from the two equations obtained
in the second proof of (5810), by eliminating ij^ instead of z—C
5817 The radius of curvature at a point xijz on the surface
(^ (,(', 7/, z) =0 of the normal section whose tangent has the
direction cosines A, ^ii, v is, with the notation of (5796) and
(1620), ^ y(P+„t'+„^)
Proof. — From equation (I.) (5798), since ^, ri, i are respectively equal
to R\, Rfi, and Rv.
5818 The curvature at any point of a surface (p{x, y,z) =
is termed ellvptic or synclastic, liyperholic or anticlasfic, and
paraholic or cylindrical, according as the indicatrix is an
elhpse, hyperbola, or two parallel right lines, or according as
the principal curvatures have the same signs, opposite signs,
or one of them vanishes ; and this will be according as the
determinant A', in (5814), or s^—rt, in (5816), is negative,
positive, or zero.
Proof. — The i-ule follows at once from the consideration that the two
values of p in the quadratic of (5814) must have the same sign in the first
case, different signs in the second, and that one value must be infinite in the
third case.
5819 The condition for an umbilicus is that the indicatrix
must be a circle ; therefore, either (III.) (5802) must be a
sphere, or, if it be a quadric surface, the plane (II.) must
make a circular section of it, and therefore either /, m, or n
must vanish.
5820 Otherwise, the quadratic in (5814) or (5816) must
have equal roots.
5821 Otherwise, the conditions for an umbiUcus on the sur-
face (x, y, z) = are the two equations
hn^-\-cm"~-2fmn _ cP-\-an^ — 2gnl _ a})V-\-bP — 2hlm
800 SOLID GEOMETRY.
Proof. — The radius of the indicatrix, and therefore also p in (5817), is
constant for all values of X, ju, v. Now, by (5817),
P
.-. (a- -) \--f (Z*- -) f^'+ (c--) ^' + 2fi^y + 2gvX-\-2hXfx = 0,
and l\ + iufx + ni' = 0, since X/^v is always tangential, and Inm is normal to
the surface. As these equations are true for all values of X, n, v, the second
expression must be a factor of the first. The quotient, by division, is there-
fore („_l)4+(6_i)ii + (._A)L.
\ p I I \ p I m \ pin
Equating to zero each of the three coefficients of the remainder, and elimi-
nating p, we obtain the above conditions.
5822 If a common factor of the three fractions m (5821)
exists, that factor equated to zero is the differential equation
of a line of spherical curvature at every point of which there
is an umbihcus. If the fractions are identically equal, the
surface has an umbilicus at every point, and must therefore
be a sphere.
5823 The number of umbilici on a surface of the n^^^ degree
cannot exceed n{10n^—2hn-\-lQi). Sahnon, -p. 208.
5824 The condition that the indicatrix may be a rectangular
hyperbola is
{a-\-b+c){l^-\-7ri'-\-n') = {ahcfgh\lmnY.
Proof. — The quadratic in (5814) must have equal roots of opposite
signs.
Similarly, when z = (p (x, y) is the equation of the quadric, the condition
becomes 0-+P^) t-2pqs+(l + q^) r =0. (5816)
5825 The condition that the indicatrix may become two
coinciding lines.
Here equation I. (5798) must represent a cone, and the plane (TI.)
must touch it. Hence N = 0, and, if ^ be eliminated, the quadratic for the
ratio ^ : ?/ obtained is
(an^ + cP — 2gnl) ^' + 2 (clm—fnl — gmn + hn-) ^T] + (bn- + c)n- — 2fmii) »/" = 0,
and this must have equal roots.
CURVATURE OF A SURFACE.
5826 Defs. — Integral curvature of a closed surface is equal
to the area of that part of the surface of a sphere of unit
radius which is intercepted by radii drawn parallel to the
THEORY OF SURFACES. 801
normals at all points of the given surface. This area also
measures the solid angle of the cone generated by the radii.
The curve on the sphere is called the liorogrcqjli of the curve
on the original surface. In other words, integral curvature
of a closed surface is the area of the horograph of its
boundary.
5827 Average curvature is the integral curvature divided by
the area of the surface.
Specific curvature is the average curvature of a small
(dsY 9 1
element at the point ; i.e., - '^ -:- {ds}'^ = .
pipi Pip2
5828 The last is the usual measure of curvature at a point,
and its value in coordinates of the point is given by
1 ^' or ^'^'•^'' , (5796)
according as .^ (.v, i/," ^) = or z = <^ {e, y) is the form of the
equation to the surface.
Proof.— From the product of roots of the quadratics in (5814) and
(5816).
5829 In a plane curve integral curvature is the plane angle
contained by the terminal normals, and average curvature is
the integral curvature divided by the length of the curve.
5830 Another measure of curvature at a given point of a
surface is the ratio of the area of the indicatrix to the area of
the indicatrix cut off by the same plane on a sphere of unit
radius .which touches the surface internally at the point. This
measure is = \/piP2-
Proof.— Putting AC = R„ BC = R,, in Fig. (182), and OC = z, the area
of the indicatrix of the surface is ■kR^U,^ at an elhpsoidal point. But
B\ - 2p^s and Rl = 2p,z, therefore ttR.R, = 27rz /(piP.,). Also the indicatrix
of the sphere = 27rz s'ince Pi = p. = 1 for the sphere.
5831 The radius of curvature of any normal section at a
point P of an elhpsoid (Fig. 184) is equal to the square of
the semi-diameter parallel to the tangent of that section,
5 K
802 SOLID GEOMETRY.
divided by the perpendicular from P upon the diametral
plane conjugate to OP.
Proof. — Let AOB be the plane parallel to the tangent plane at P;
OA = d, the semi-diameter in it parallel to the given tangent PT. Draw
PB perpendicular to OA and PN^p perpendicular to the plane AOB. The
radius of curvature at P of the elliptic section PA = ~- (4536). Therefore,
by (5809), the radius of curvature of the normal section through the same
tangent PT, will ^e P = ^ X || = ^.
5832 The principal radii of curvature at P, viz. p^, p.2, are
found from their sum and product thus : putting y for OP,
and a, h, c for the semi-axes of the ellipsoid,
P,+P.= -^—^ ^-, p^p, = —r-.
Proof.— Let a, /3 be the serai-axes of the section AOB (Fig. 184), then
„2 + /32^y2_^2^^2^g2 (-5642) siud paP = ahc (5648). By these values
eliminate a, ft from Pi+Pa = — ^"^^ P1P2 = 2* (^831).
5833 The lines of curvature on a quadric surface are its
intersections with the confocal quadrics.
Proof. — Let the quadric and confocal be the ellipsoid and one-fold
hyperboloid in (Fig. 178) intersecting in the line DPE, and let their equa-
tions be, as in (5656),
^+•^ + 7=1 (^-^ -^ ?fx + E4n-4=^ ("■>•
At any point P on the line of intersection x, y, z satisfy the three following
equations : —
First, the differential of (ii.), 4^ + ,4% + "^ = ^■
Second, the difference of (i.) and (ii.),
2! + t + ?! =
d' (ai' + X) h' (b' + X) c' (cHX)
Third, the difference of their differentials,
xdx , y di/ zdz _ ^
a- (d- + X) Ii' {V -j- X) c- (c- + X)~
The eliminant of these equations in x, y, z pro-
duces the determinant equation here annexed, which,
by (5811), is the condition for the intersection of con- ^r b"^ c" = 0.
secutive normals. Hence this condition holds for
every point of the line of intersection of (i.) and (ii.).
dx
dy
dz
.r.
a-
V
7
dx
a'
'hi
b'
dz
THEORY OF SURFACES. 803
The general method of determining the lines of cnrvature
of a surface from the differential equation in (5811) is here
exemplijB.ed in the case of an ellipsoid.
5834: The determinant just written gives for the differential equation of
the lines of curvature
(b'-c')xdi/dz+{c''-a')ijchcU + (a'-b')zdxdy = (i.).
To solve this, multiply by -^ and substitute for z and dz from the equation
of tbe quadric. The result is of the form
Axyyl+(x--Ajf-B) y^-xy = 0,
in which A = t^% B = ^1^^^ ; or, multiplying by ^,
¥{a- — c-) ci' — c- X
A'^ {xyy^-y-)-B'& +{xyy^-f) = 0,
X X
which is of the form in (3236). Solving by that method, we find that the
two equations ^ = a and xyy^—y^ = /3 have the common pinmitive
X
ax^—y^ = fi, which, with the relation Aa^-Ba-^fi = 0, constitutes the
solution. The result is that the projections of the lines of curvature upon
the xy plane are a series of conies coaxal with the principal section of the
ellipsoid, and having their axes a, 6 varying according to the equation
At an umbilicus ^^ = 0, therefore, equation (i.) becomes \_{h- — c^) xdz
+ (a^ — 6') zdx'] dy = 0. Here dy = 0, being a solution, gives y = G = 0,
showing that the plane of zx contains a line of cnrvature. The other
factor, equated to zero, taken with the differential equation of the curve
cKxdx + a^zdz = 0, gives the coordinates of the umbilicus, as in (5603).
OSCULATING PLANE OF A LINE OF CURVATURE.
5835 Let (f> be the angle between the osculating plane and
the normal section through the same line of curvature, ds^ an
element of the other line of curvature, and p, p their radii of
curvature respectively : then
as p —p
Pkoof.— Fig. (185). Let OA, OB be the lines of curvature; OP, AP
consecutive normals along OA ; and OS, BS the same along OB. Also, let
BQ, CQ be consecutive normals along the line of curvature BG. Then,
ultimately, OP = p, OS = p', BQ = p + dp. Also, let QP produced meet the
osculating plane of ^0 in R. Join BO and RA, and draw QN at right angles
to PS. Since the tangent to ^0 at is perpendicular to the plane 0B(2P
and that at A to AG(2P, it follows that both tangents are perpendicular to
QP, which must therefore be perpendicular to the osculating plane ARO.
Hence (j> or ROP = PQN.
804 SOLID GEOMETRY.
Now ^ = 1^ = P'-^r'' , .-. tan ^ = ^ = t . f- ultimately.
ds SB p NQ, ds p —p
5836 At every point on a line of curvature of a central
conicoid _/jf? is constant, where d is the semi-diameter parallel
to the tangent at the point and j9 is the perpendicular from
the centre upon the tangent plane.
Pkoof. — Let the first and third confooals in (56G1) be fixed, and there-
fore «! and a.^ constant. Draw the second confocal through the point of
contact P of the tangent plane (Fig. 178). Then, by (5668), ^:>,fZ3 and pgf?!
are constant along the line of intersection of the first and third surface,
because, by (5661), d^ = a\—a^ and d\ = O3 — flg-
GEODESIC LINES.
5837 The equations of a geodesic on the surface ^ (x, y, z) =
cVo, y.,s z.^s
9.V 9y 9z
Proof. — The osculating plane of the curve contains the normal to the
surface (5775) ; therefore, by (5737) and (5785).
5838 A geodesic is a line of maximum or minimum distance
along the surface between two points.
Proof. — The curve drawn in the osculating plane from one point to a
contiguous point is shorter than any other by Meunier's theorem (5809),
for any oblique section has a shorter radius of curvature and therefore a
longer arc. A succession of minimum arcs, however, may constitute a maxi-
mum curve distance between the extreme points ; for example, two points on
a sphere can be joined by either of two arcs of a great circle, the one being
a minimum and the other a maximum geodesic.
5839 A surface of revolution such as the terrestrial globe affords a good
illustration. A meridian and a parallel of latitude drawn through a point
near the pole are the two lines of curvature at the point. The meridian is
also a geodesic, but the parallel is evidently not, for its plane does not
contain the normal to the surface.
5840 A geodesic is the line in which a string would lie if
stretched over the convex side of a smootli surface between
two fixed points.
Proof. — Any small arc of the string POP' (Fig. 182) is acted upon by
tensions along the tangents at P and P', and by the normal reaction of the
surface at 0. But these three forces act in the osculating plane (5775) ;
therefore the string will rest in equilibrium on the surface in that plane.
THEORY OF SUEFACES. 805
CoE. — Two equal geodesies drawn from a point and in-
definitely near to each other are at right angles to the line
which joins their extremities.
5841 If a geodesic has a constant inclination to a fixed line,
the normals along it will be at right angles to that line.
Proof. — Let Imn be the fixed line and a the constant angle ; then
Ixg + mj/g + nzg = cos a, and therefore lx2s + my2s + nzos =■ 0.
Therefore, by (5837), the principal normal is at right angles to Imn.
Example. — The helix, the axis being the fixed line.
5842 On any central conicoid 2^d is constant along a geo-
desic, where p is the perpendicular from the centre upon the
tangent plane and d is the semi-diameter parallel to the
tangent of the geodesic.
Proof. — (Fig. 186.) Let AT, BT be the tangents at the two extremities
of a small geodesic arc AB, and let the tangent planes at A and B be ABC
and BOB. AT and BT make equal angles with GB, by the property of
shortest distance, for if the plane BOB be turned about GB until it coincides
with the plane ABG, ATB will become a straight line, and therefore
/.ATB = BTG = i, say.
Let w be the angle between the tangent planes ; let the perpendiculars
upon those planes from A, B be AM ~ q, BN= q, and from the centre of the
quadric p, p ; and let xyz and x'y'z' be the points A, B. Then
q = ATsmiBinu), q' = BT sin i sin u), :. q : q — AT : BT (i.),
therefore q '. q = p' '• p (ii-)-
Again, let d, d' be the semi-diametei^s pai-allel to AT and BT. Then, by
(5677), AT : BT = d : d' ; therefore p' : p = d : d' or pd=p'd' ; that is,
pd is constant.
5843 If a line of curvature be plane, that plane makes a
constant angle with the tangent plane to the surface.
Proof. — Let PQ, QB, BS be equal consecutive elements of the line of
curvature. The consecutive normals to the surface bisect PQ and QB and
meet in a point. Therefore they are equally inclined to the plane PQB.
Similarly the second and third normals are equally inclined to the plane QBS,
and so on. Hence, if the curve be plane, all the normals are equally inclined
to its plane. Hence also the following theorem.
5844 Lancrefs Theorem. — The variation in the angle be-
tween the tangent plane and the osculating plane of a line of
806 SOLID GEOMETRY.
curvature is equal to the angle between consecutive osculating
planes.
5845 CoE. — -If a geodesic be either a line of curvature or a
plane curve, it is both, but a plane line of curvature, as in
(5839), is not necessarily a geodesic.
GEODESIC CURVATURE.
Theorem. — The square of the curvature at any point of a
curve traced on a surface is equal to the sum of the squares
of the normal and geodesic curvatures (5776), or
5846 7 = ?5 + P^'
where p is the radius of curvature of the normal section and
p" the radius of geodesic curvature. Also, if <^ be the angle
between the plane of normal section and the osculating plane,
5847 p — p" sin ^ = p cos <^-
Proof.— Let FQ = QU (Fig. 187) be consecutive elements of any curve
traced on a surface. Prodace FQ to ,S', making Q8 = FQ. Let QT = FQ
be the consecutive elements of the section of the surface drawn through
FQS and the normal at Q. Join ES, ST, TE. FQSB is the osculating
plane of the curve FQE. FQST is the plane of normal section, and there-
fore FQT is a geodesic. QET is the tangent plane, and STE is a right
angle.
Then, putting *SQ7i; = #, SQT=d^', EQT=d^l^", EST = cp, we have
ds / ds „ ds ,f,.._,.
Therefore 4. = ^^ = l?i = ^" *••
p ds.dxp Eb
Also -^ = ^^T = 1^ = ^°« •?'' ^« "^ (^S09).
p as . a\j/ oil
Thus both theorems are proved. Note that p' is the radius of curvature of
the geodesic FQT, while p" is the radius of geodesic curvature of FQE.
I
RADIUS OF TORSION OF A GEODESIC.
5848 If ^ he the angle between the geodesic and one of the
lines of curvature ; p^, p.. the principal radii of normal curva-
ture, and <T the radius of torsion.
J-z=f— -iVsiii(9eos6'.
THEOUY OP STJRPAGES. 807
Proof.— (Fig. 182.) Let OP = ds be the geodesic, OA, OB the lines of
curvature, and ~ AGP. The angle of torsion ch measures the rotation of
the normal to the surface round OP = ds. But this angle is equal to the
sum of the rotations of the normal round OA and OB resolved along ds.
For, in travelling along each of the lines GN and NP, which are iu the direc-
tions of the lines of curvature, the normal rotates only about the other.
Therefore, if Wj, Wj be the rotations round OA, OB, dr = w^ cos 9 + Wj sin 0.
Tj 4. ^-5 sin 6 ds cos d .1 dr _ ( 1 1 \ . „ ^
But Wi = , Wj = ; . . — = "7~ = I sin cos o.
p2 Pi tr ds \ pi P2 1
5849 The product _2^fZ lias tlie same value for all geodesies
which touch the same line of curvature.
Proof. — By theorems (5836) and (5842), since the product where they
touch it must be the same as that for the line of curvature.
5850 The product j)d has the same value for all geodesies
drawn through any umbilicus on a conicoid.
Proof. — The semi-diameter d, in this case, is the radius of a circular
section, and therefore equal to the mean semi-axis h for all the geodesies ;
andp is the same for all.
5851 The geodesies drawn through any point on a conicoid
to two umbilici make equal angles with either line of curva-
ture through the point.
Proof. — ipd is the same for each geodesic, by the last, and p is the same
for each ; thei'efore d is the same, that is, the diameters parallel to the two
geodesies at the point ai-e equal ; therefore they are equally inclined to each
axis of their section ; but these axes are parallel to the lines of curvature
(5803) ; therefore, &c.
5852 Hence the geodesies joining any point to two opposite
umbilici lying on the same diameter are continuations of each
other.
5853 The sum of the distances of any point on a line of
curvature from two interior umbilici is constant ; and the
difference of the distances from one interior and one exterior
umbilicus is constant.
Proof. — Geometrically, as in the analogous theorem for the focal distances
in a conic, if r, r' are the distances and r-\-dr, r' ■\-dr' the distances for a
consecutive point on the line of curvature, it follows from (5851) that
dr =■ —dr for interior umbilici and dr = dr' for exterior ones.
5854 A system of lines of curvature and the umbilici on a
808 SOLID GEOMETRY.
quadric surface has therefore analogous properties with a
system of cod focal conies and their foci in a plane, the geo-
desies corresponding to straight lines.
5855 111 tlie same way, every surface has a geodesic geo-
metry proper to itself ; spherical trigonometry, for instance,
being the geodesic geometry of the sphere.
INVAEIANTS.
INVARIANTS OF A SINGLE FUNCTION.
5856 The constancy of the ratio B^ : p in equation (5798)
gives rise to the following invariant forms. Since the quadric
surface I and the tangent plane II are the same for all posi-
tions of the coordinate axes, they have been called respec-
tively the invariable quadric and the invariable ]jlane. As a
consequence,
5857 <lyr-^4>l+i^
is an invariant of (j) {x, ij, z) .
Proof. — By (5791), since iho perpendicnlar from the origin upou tlie
invariable plane is constant. Also, the coefficients of the discriminating
cubic (5693) of the invariable quadric will not be altered by transformation
of axes. Therefore the following are also invariant forms : —
5858 <^..+«^.>.+<^2.,
5859 4>.A^,-\-^,A,^-\-4>.A^-4^.-^s-<l>i,.
5860 <^2.r ^-ly <l>2z + ^^z i^z.v 4*xy " <^2.r <t>l, " 4>ty <^lr " 4>lz i>ly •
5861 A similar theorem applied to a function (p (x, ?/) of
two variables gives the invariable conic and invariable line;
namely,
r<^,,+2£7;<^,,+f<^,, = l and i<l>,,-^yj<l>, = 1 ;
and from these the invariants,
5863 4r.+<t^, <^..+<3^.., i>.A^-<tc,,
5866 .v(l>y-ij<t>,,, <<<^,+//<^.,.
QUADBATUBE AND GUBATUBE. 809
Proof. — The last two invariants are obtained from the cosine of the
angle between the invariable line (5862) and the fixed line y'^ — x>] = 0,
joining the point xij with the origin, or the fixed line x^ + yt} = 0.
INVARIANTS OF TWO FUNCTIONS.
5868 An invariant of tlie two functions ^ {x, y), \p {x, y) is
Proof. — Form the cosine of the angle between the invariable lines
^^x + Vfi/= 1 and t,4'j;-\- r,\}/^ = 1, observing (5863).
Also tlie two following expressions are invariants,
5869 4,,,xj,,,,+^,,,f,,-2,l>...„xl..,„,
5870 .^,../.,„.+<^„^,,+2f,,.|.„,.
Proof. — From the invariable conies (5861) of ^ and ij/, we get
invariable for any value of X. Hence the coefficients of the several powers
of X in the invariant
(02. + Af ,.) i<p,, + H;,) - if., + H.,y
are also invariants. This gives (5869). Subtracting the latter from the
invariant i<p2x + f2!,)(4'2x + ^2i/) produces (5870).
INTEGRALS FOR VOLUMES AND SURFACES.
5871 If y be the volume included between tlie surface
z = (p{x,y), three rectangular coordinate planes, the cylindri-
cal surface y = 4' G^Oj ^^^^ *^® plane x = a, Fig. of (1906)
5872 V = J JJdi f/^rf;. = J P^/^f/^.
For the limits and demonstration, see (1906).
5874 The area of the surface i> {x, y, z) = or z =f{x, y)
will be
5 L
810 SOLID GEOMETRY.
Proof. — The area of the element whose projection is dxdy will be
dxdij secy, where y is its inclination to the plane of xy, and therefore the
angle between the normal and the z axis. Therefore
secy = ^(cpl + (i>l + <pi)-^cl>, = y(l + 4 + 4), by (1708).
5875 Let the equation of a surface APB (Fig. 188) in polar
coordinates be r=f{d,({>), and let V be the volume of the
sector contained by the planes AOB, AOP, including an angle
(f) = PEG, the given surface APB, and the portion OPB of
the surface of a right cone whose vertex is 0, axis OA, and
semi-vertical an^le 9 = AOB or AOP; then
"&'
V= 1 n V siu ^rf^f?<^.
Jo Jo
Proof. — Through P, any point on the surface, describe a spherical sur-
face PGD, with centre and radius r = OP. The volume of the elemental
pyramid, vertex 0, base Pe, = ^r.Pf.Pg = \r.rdd .r sin ddif). Here the
error of the small portions, like PE, ultimately disappears in the summation,
since the volume of P-E/, being equal to ^dr .rdd .r sin ddcp, is of the third
order of small quantities ; and so in similar instances.
5876 The area of the same surface APB (Fig. 188) is
Jo Jo
Proof. — Let the perpendicular from upon the tangent plane at P to
the given surface be ON = p. The element of
area Pi; = area Pe.^^= rdd.r sin Odcp.— = ':^^^d6dx{,.
ON p p
Substitute the value of p in (5793).
SURFACE OF REVOLUTION.
If 7/ =/('') (Fig. 90) be the generating curve, and the x
axis the axis of revolution, Fthe volume, and 8 the surface
included between the planes x = a, .<■ = h ;
5877 y = ^'vf(Lv, s = ("2n,f y (1 +;/:.) d.v.
Proof. — The volume of the elemental cylinder of radius y and height tZic
is ny'\lx. The element of the surface of revolution is
2iryds = '■lirySj.dx = 2wy \/{l+yl) dx. (5113)
QUADRATURE AND CUBATURE. 811
Guldin's Rules. — When the generating curve of a surface
of revolution is a closed curve, and does not cut the axis of
revolution, a solid annulus, or ring, is formed.
5879 I^ULE I. — The volume of the solid ring is equal to the
area of the generating curve multiplied hij the circumference of
the circle described by the centroid* of the area.
5880 Rule II. — The surface of the ring is equal to the
perimeter of the generating curve multiplied by the circum-
ference described by the centroid of the perimeter.
Proof. — Let A be the area of the closed curve, and dA any element of A
at a distance y from the axis of revolution. The volume generated
=z\2TrydA = 2TT\ydA = 2nyA,
by the definition of the centroid (5885), y being its distance from the axis.
Similarly, if P be the perimeter, writing P instead of A.
Quadrature of surfaces bounded by lines of constant
gradient,
5881 Defining the curve (7) as the locus of a point on the
given surface at which the normal has the constant inclina-
tion 7 to the z axis ; let F {y) be the projection of the area
bounded by the curve (7) upon the xy plane ; then the area
itself will be found from the formula,
I sec 7^(7) dy.
Jo
Proof. — The element of area between two consecutive curves (y) and
(y + (Zy) projected on the xy plane will be dF {y) = F' {y) dy ; and, since the
slope is the same throughout the curve (y), this projected element must be
equal to the corresponding element of the surface multiplied by cos y.
5882 Rule. — Equate coefficients of the equation of the
tangent p)lane ivith those of fe + mrj + nZ; = p, and eliminate I
and mfrom P-f m^+n^ = 1. The result will be an equation in
X, y and n = cos 7, representing the projection of the curve (7)
upon the xy plane. From this F (7) must be found.
5883 Ex.— Taking the elliptic paraboloid ^ + ^ = 2z ; the tangent
plane at rc^/z is ^ + ^ — ^ = -• Equating coefScients of the last with
l^ + mrj + nl^ =p, and substituting for I and m in V + m^ + n'^ = 1, we obtain
for the projection on the xy plane, ^ + j^ = tan^y. The area of this ellipse
Centre of mass, or gravity.
812 SOLID GEOMETRY.
is F (y) = Trah tan'-y, and therefore F' (y) — 27rah tany sec^. Consequently,
by (5881), S = lirah Ttan y sec'^ycZy = fraJ (sec»y-l).
CENTRE OF MASS.
5884 Definitions. — The moment of a body witli respect to a
plane is the sum of the products of each element of mass of
the body and the distance of the element from the plane.
5885 The distance (denoted by x) of the centre of mass *
from the same plane is equal to the moment of the body
divided by its mass.
5886 WoTE. — If the body be of nniform density, as is supposed to be the
case in all the following examples, assume unity for the density, and read
volume instead of mass in the above definitions.
The definition gives the following formulas for the position
of the centre of mass of a uniform body :
5887 For a p/a7ie curve,
- _ J ^d^ ^ J-^ v^(l +in) da; ^ jr cos 6 ^{r'^+r^ dS
ids i^(lJri/l)d.v i^(r^+rl)de
For y, change x into y and cos d into sin d ; but observe that in all cases,
if the body be symmetrical about the axis of x, [/ vanishes. The formula
gives the centre of volume of the portion of the curve included between the
limits of integration.
For a 2^f-cine area,
5890 -J[.vd.rd,,^[^^
)li(l'><ll/ J. '/'/•»■
The area is bounded by the curve, the a; axis, and the ordinates x = a,
a; = fc, if such be the limits of integration.
For a, jyiane sectorial area bounded by two radii SP = r,
SP' = t' (Fig. 28) and the curve r = P{^Q)\
* Al80 called centre of gravity or inertia, and more recently centroid.
CENTRE OF MASS. 813
5892
5894 y
- _ ff*'' cos edOdr __ Ijr^cos^rfg
_ JJr^ sin ^^/^f/r _ |Jr^sm^(/^
l^rdOdr ~ li-dB
The second forms for x and y give the centroid of an area like SPP'
(Fig. 28). The double integrals applied to that figure require the limits of
the integration for r to be from to F{d), aiicl afterwards for 0, from
01 = ASP to 02 = ASP'. But, if applied to the area in (Fig. 109), the order
of integration must be reversed, as explained in (6209).
For a surface of revolution round the x axis,
KOQfi - _ J.r// v/(l+;/I.) (Iv Ji- sill e cos6> v/(r^+0 dO
"" iyViX-^y^da^- j'rsiii^y(r^+rDrf^
Proof. — Bj (5885), for the moment = \x. 'Zmjcls and the area = ^iryds ;
the second form by (5116). If a; = «, a; = 6 are the limits of integration, the
surface is bounded by the parallel planes x = a, x = h; and in the second
form, the corresponding values of are the limits defining the same parallel
planes.
For any surface,
5898 ^ = ^r^!^??f? - ''''''
^\^{l-^z:.+z;)dajdi/
For a solid of revolution round the x axis,
_ _ J .ri/'d.v __ \\ r^ siu cos 6d9dr
^^^^ '""'Jfd^^^' ^\7-^mededr '
Proof. — By (5885), for the moment =\x.7ryhlx and the voUime
= j Try'^dx. The limits as in (5896).
5901 For any solid figure bounded as described in (5871),
the coordinates of the centroid are given by
F.r = I U .V dx dij dz = \\ xz dx dy,
Vy = \{[ ydxdydz = U yzdxdy,
Vz=Wzdxdydz = lA{^'dxdy,
814 SOLID GEOMETRY.
wliere V= M j d.vdi/dz = 1 i zdxdy,
as in (5872-3), and the limits are as defined in (190G).
5902 For the wedge ^Imiiecl solid {OAPB, Fig. 188) defined
by the polar coordinates r, 0, <^, as in (5875),
V.v = I J j\-^ sin'^ e cos <f> d0d(l>,
Wf = l((r' siii^ 6 sin 4>ded<t>,
Vz = i nV* sin 6 cos ed0d<l>,
where 7=1 ((r' sin eded<f>.
Proof. — By (5875); multiplying the elementary pyramid ^r^ sinddddf
separately by the distances of its centroid from the coordinate planes ; viz.,
|r sin d cos 0, |r sin sin <p, and |r cos 6.
MOMENTS AND PRODUCTS OF INERTIA.
5903 Definitions. — The moment of inertia of a body about
a given right line or axis is the sum of the products of each
element of mass and the square of its distance from the line.
5904 The square of the radius of (juration of the body about
the given line is equal to the moment of inertia of the body
divided by its mass.
5905 The moment of inertia of a body witli respect to a
plane or point- is the sum of the products of each element of
mass and the square of its distance from the plane or point.
5906 The prodiiet of inertia of a body \vith respect to two
rectangular coordinate planes is the sum of the products of
each element of mass and its distances from the two planes.
5907 I^et A, B, G be the moments of inertia of a body
about three rectangular axes ; A\ B\ C the moments of
inertia with respect to the three planes of yz, zx, and xy ; and
MOMENTS OF IN EUTI A. 8l5
F, G, H the products of inertia witli respect to the second
and third planes, the third and first, and the first and second
respectively. F, (i, H are more frequently called the products
of inertia about the axes of yz, zx, and xij respectively.
By the definitions we have the values
5908 A = Sm if+z'), F = tmi/z,
B = "^m {z^-\-,v-), G = %mza;,
C = Xtn {d^+tf)i H = Xnicvy.
5914 A^=%.n^=:S-A^ ,,^,,,s = ^-t^±^,
B' = Xmif = S-B
C = tmz'= S-C
2
= tm{.v'+i/-\-z%
5920 Theorem- 1. — The M. I. of a lamina about an axis per-
pendicular to its plane is equal to the sum of the two M. I.
about any two axes in its plane drawn through the foot of the
perpendicular axis and at right angles to each other.
Proof. — By the definition (5903), and Euc. i. 47.
5921 Theorem 11. — The M. I. of a body about a given axis,
plane, or point is equal to the M. I. about a parallel axis or
plane through the centroid, or about the centroid itself
respectively, plus the M. I. of the whole mass, supposed col-
lected at the centroid, about the given axis, plane, or point.
Proof. — In the figure, p. 168, let the given axis be perpendicular to the
paper at B; let J. be the centroid, and m an element of mass at G ; then, for
every thin section of the solid parallel to the paper,
M.I. = %n.BG' = %m{ACr- + AB--^AB.Al))
= %m .AG' + -$7n. AW - 2AB . 2m . AD.
But ^vi.AD = 0, by (5885), since ^1 is the centroid of the body, which
proves the proposition. Similarly for the plane or point.
Cor. I. — Hence, if the M. I. about any axis is known, that
about any parallel axis can be found without integration. For,
let Ii be the M. I. about a given axis, whose distance from
the centroid is a, and let Ig be the required M. I. about an
axis whose distance from the centroid is b ; then, by Theorem
JL, I, = I,-m{a'-b').
816 SOLID GEOMETRY.
Cor. II. — The M. I. has the same value for all parallel
axes at the same distance from the centroid.
5922 Theorem, III. — The iiroduct of inertia for two assigned
axes is equal to the product for two parallel axes through the
centroid of the body plus the product taken for the whole
mass collected at the centroid with respect to the assigned
axes.
Proof. — Let x = x + x', y = y + y' he the coordinates of an element of the
body with respect to the assigned axes ; ^, y being the coordinates of the
centroid, and x', y' the coordinates of the same element with respect to
parallel axes through the centroid, all axes being parallel to z. Then
"Lmxy = ^in (x + x')(ij + y') = d-y'S,m+'S:,mx'y' + xl.my' + yllmx'
= xyl^ni+'Simx'y'.
Since Swa;' and ^my' vanish by the definition of the centroid.
5923 The M. I. of a body with respect to a point is equal
to the M. I. for any plane through the point plus the M. I.
about the normal to the plane through the point.
Proof. — For the origin and yz plane,
•^mx' + %m (y' + z') = %mr\ (5908, '14, '19)
5924 G-iven the moments and products of inertia, A, B, G,
F, G, H, as above, about three rectangular axes, the moment
of inertia of the body about a line through the origin, whose
direction cosines are /, i)i, n, will be
/ = AP-{-B7n^+Cn^-2Fmn-^2Gnl-i-2Hhn.
Proof. — (Fig. 11.) Let xyz be a point P of the body, 021 the line Imn,
and PM the perpendicular upon it. Then the M. 1. about OM
= ^m (OP' - OM') = %m { {x' + 2/H z'){l' + m" + n^) -{Ix + viy + nzf ] (5530)
producing the above result, by (5908-13).
ELLIPSOIDS OF INERTIA.
5925 The equation of the Momental Ellipsoid is
AaP-+Bif^Cz^-2Fijz-2Gzx'-2mij = Me\
obtained by putting Ir^ = 71ft*. M being the mass of the body,
and £* a constant to make the equation homogeneous. Hence the
square of the radiuf^ of the momental cllip><oidfor anijiwint varies
inversely as the moment of inertia of the body about that radius.
MOMENTS OF INERTIA. 817
5926 If the products of inertia vanish, the axes are called
the princijKd axes of the body.
5927 At every point of a body there are always three
principal rectangular axes.
Proof. — These are evidently tbe princij^al fixes of tlie momental ellipsoid
of the point ; for if the coordinate axes be made to coincide with the former,
F, G, H will vanish.
5928 The equation of the momental ellipsoid referred to its
principal axes will be
5929 The moment of inertia about a line hmi will now be
I=Af+Bm'-^CnK
THE ELLIPSOID OF GYRATION.
5930 The equation of the Ellipsoid of Gyration referred to
principal axes is
,2 2 ^2 1
T'^^'^IT'^ IT
It is the reciprocal surface of the momental ellipsoid (5719),
and its property is —
5931 The moment of inertia about the ^perpendicular from
the origin upon the tangent plane varies as the square of the
perpendicular.
5932 For any other rectangular axes through the point, the
equation of the ellipsoid of gyration is, by (5717),
= 0, being the reciprocal surface of
the momental ellipsoid,
{A,B,G, -F, -G, -HXxyzy
= M,
with the radius of the sphere of
reciprocation = 1. The equation when expanded becomes
A -H -G X
-H B -F y
-G -F G z
1
y ' M
5933 (BC-F')a^+...+2{FG+CH).vy= H B F ^.
A H
G
H B
F
G F
C
818 SOLID GEOMETRY.
LEGENDRE'S EQUI-MOMBNTAL ELLIPSOID.
5934 The equation is
with the values in (5914).
5935 The mass of this elKpsoid is taken equal to that of
the body, and it has the same principal moments of inertia.
THE MOMENTAL ELLIPSOID FOR A PLANE.
5936 If -l'> I^\ G' be the moments of inertia for the three
coordinate planes, as in (5914), the M. I. for a plane through
the origin whose dir. cos. are /, in, n, will be
r = A'l'-\-BV+C'rr+2Fmn-i-2Gnl+2Hlm.
Proof : I' = 2//i (Iv + my + nzf — %nx- . l~ + &c. = AT- + &c.
5937 The momenta! ellipsoid for this plane will be
A\v'-\-By+C'z'+2Fi/z-\-2Gz.v-]-2Kvi/ = 3h\
and its property is —
5938 27ie M. I. for any ijJane passing tltroiujh the centre of
the ellipsoid is equal to the inoerse square of the radius per-
pendicular to the plane.
5939 If ''* be a radius of this ellipsoid, and a, b, c its semi-
axes, the M. I. about r
Proof. — (Fig. IL) M. I. about /•, plus M. I. for tlic plane OM perpen-
dicular to 7-
= 2mOP' = •2,mx' + :S,my' + :^mz' = ^ + ^ + 4, by (5938).
EQULMOMENTAL CONE.
5940 The equation of the equi-momental cone at any point
of a body, referred to principal axes of the body at the point.
MOMENTS OF INEBTIA. 819
is (A-I),r^-^{B-I)y^-^{C-I)z'=0.
its property being that
5941 The generating line passes through the given point, and
moves so that the M. I. about it is a constant = I.
Pkoof. — Let hn7i be the generating line in (me position, then
Al^ + Bvt} + C?r = I{1' + in' + ir) . Therefore, &c.
5942 Theorem. — If two systems have tlie same mass, the
same centroid, principal axes and principal moments of inertia
at the centroid, they have equal moments of inertia about any
right line whatever, and are termed equi-moniental. By (5906)
and (5929).
5943 If two bodies are equi-momental, their projections are
equi-momental.
Peoof. — If the projection be from the xy plane in the ratio 1 : n, the
coordinates x, ?/, 2 of a particle become x, y, nz, and the mass m becomes nm.
The conditions in (5942) will then be fulfilled.
MOMENT OF INERTIA OF A TRIANGLE.
5944 The M. I. of a triangle ABD (Fig. 190) about a side
BD, distant jj from the opposite vertex A, is
J mp^
(5
Proof. — Let
BD^a and EF-^y; I = if^'^in^V = '^ = -^{
5945 The M. I. of a triangle ABC (Fig. 190) about a
straight line BD passing through a vertex B, and distant p
and q from the vertices A and C, is
Proof.— By (5944), taking difference of M. I. of the triangles ABD, GBD.
5946 The M.I. of a triangle ABC about an axis through
its centroid parallel to BD, is
/ = ,„itH+i'. By (6921)
lo
820 SOLID GEOMETRY.
5947 Cor. — If the triangle be isosceles, so that j^ = q, the
last two moments of inertia become
5949 The M. I. of the triangle about axes perpendicular to
ABC through B and through the centroid, respectively, are
m ^ ^ ^ and ?/i^ ^-^ — ^. (5920)
5951 The M. I. about GF of the trapezoid ACGF (Fig.
190), is
^ 6 •
5952 The moments and products of inertia of a triangle
about any axes are the same for three equal particles, each
one-third of the mass of the triangle, placed at the mid-points
of its sides.
Proof. — (Fig. 190.) The M. I. of the three particles at the mid-points
of AB, BG, CA about BB, any line through a vertex, will be
3 [ 4 ^ 4 ^ 4 ) '
which is equal to that of the triangle, by (5945).
MOMENTAL ELLIPSE.
5953 If a, i3 be the radii of gyration of a plane area to
principal axes Ox, Oy, where is the centroid, the equation
of the momental ellipse for the point will be
5954 Also the area is equi-momental with three equal
particles, each one one-third of its mass placed anywhere on
the ellipse so that may be theii' centroid.
Proof. — Let xy, x'y', x"y" be the coordinates of three equi-momental
particles : then
^ {x' + x" -f x") = 7»/3- ; ''i (2/- + y" + y'") = ma' ■ xy + x'y' ^- x"y" = ;
o o
MOMENTS OF INERTIA. 821
and the two systems have the same centroid ; therefore
x + x'-\-x"=.0 and y' -\-y" -\-jj"' = 0.
Eliminating x', y', x", y" between the five equations, we find the equation of
(5953) for the locus of xy.
5955 The momental ellipse for the centroid of a triangle is
the inscribed ellipse touching the sides at their mid-points.
Proof. — (Fig. 189.) The inscribed ellipse, which touches two sides at their
mid-points, also touches the third side at its mid-point, by Carnot's theorem
(4779). Now Di^ is parallel to AG, the tangent at E ; therefore BE, which
bisects DF, passes through the centre of the ellipse. Similarly, AT)
passes through it; therefore is the centroid of the triangle.
Let OE = a , and let h' be the semi-diameter parallel to AC; then
9^ + Ml = \, But ON = ~, therefore FN' = U'\ The M. I. about
OE, by (5954), = |-?nf6'^sin^w = m— ;5-, where a, h are the semi-axes.
Hence the M. I. about OD, OE, OF varies inversely as the squares of those
lines, and therefore the ellipse in the diagram is a momental ellipse, since it
has six points which fulfil the requirements.
5956 The projections of a plane area and its momental
ellipse form another plane area and its momental ellipse. (5943)
5957 The M. I. of a tetrahedron ABGD about any plane
through A is
where a, /3, y are the perpendiculars on the plane from B, G, D.
5958 The tetrahedron is also equi- momental with four
particles, each one - twentieth of the mass, placed at the
vertices, and a particle equal to the remaining mass placed at
the centroid (5942).
5959 The equi-momental ellipsoid of a tetrahedron has the
same centroid, and touches each edge at its middle point.
Peoof. — By projecting a regular tetrahedron and its equi-momental
sphere (for the centroid) of radius = ,/3 X radius of inscribed sphere.
5960 To find the point, if it exists, in a given right line at
which the line is a principal axis, and to find the other prin-
cipal axes at the point.
822 SOLID GEOMETRY.
Let be a datum point in the line. Take this for origin,
the given line for axis of z, and OX, OY for the other axes.
Then, if h be the distance from to the required point 0',
and the angle between OX and the principal axis O'X',
5961 'i = ^ • = -^; and tan 2u
^tny 'ZntcV A — B
where A, B, H are the moments and product of inertia about
OX, OY.
Proof.— At the point 0, 0, //, ^m (z-h) x = Xm (z-h)y = 0, from
which h is found ; and the equation for 6 is that for determining the prin-
cipal axes of the elliptic section of the momental ellipsoid, whose equation
is Ax^ + 2Hxy + B7/ = Me*, as in (4408).
5964 The equality of the two ratios in (5961) is the condi-
tion that the ,^■ axis should be a principal axis at some point of
its length.
5965 If an axis be a principal axis at more than one point
of its length, it passes through the centroid of the system;
and, conversely, if it be a principal axis at the centroid, it is
so at every point of its length.
Proof. — For h mnst be indeterminate in (5901). Therefore 2myz = 0,
I,my = 0, "^.mzx = 0, SjHic = 0.
5966 The principal axes O'X', O'Y' arc parallel to the
principal axes of the projection of the body in the original
plane of xy. By (5962-3).
5967 Given the principal axes of a body at its centroid, to
find the principal axes and moments of inertia at any point in
the principal plane of xy.
Let G in the Figure of (1171) be the centroid, GX, GY
principal axes. A, B the M. I. about them, and P the given
point. Find two points 8, S', called /o^i of inertia, such that
the X and Y moments of inertia there are equal, and therefore
B + m.GS'' = A ; giving 08 = 08' = yJ'^~Jl... (i.).
The internal and external bisectors of the angle 8r8' will be
two of the princi[)al axes at /', and the third will be the normal
to the plane.
MOMENTS OF INERTIA. 823
Proof. — The X and Y principal moments being equal at *S', the moment
about every line through />' in this plane is the same. [For I = Al' + Bm'^
+ Gn^ and n = and A = B, therefore I = vl.] Therefore the moments
about 8F and 8'P are equal. Therefore the bisectors PT, PG of the angles
at P will be principal axes.
5968 Let 8Y, 8'Y' be the perpendiculars on PT, and 8Z,
S'Z' those upon PG ; then the M. I. about PT and PG will be
respectively,
SP-^&P
2
SP-S'PY
A + mSY.S'Y' = B-\-7ni^^^
A-mSZ.S'Z' = B-\-m(
Peoof.— Draw GR perpendicular to SY. The M. I. about GR (« = SGR^)
= Acos'e + Bshrd (5929) = ^-(^-P) sin-e
= A-m GS' sin' d (by i.) = A-mSR'.
Therefore M. I. about PT = A-mSR' + viRY' (5921)
= A + m(RY+8R)(RY-SR)=A + mSY.S'Y'
= A + mBG' (1178) = B + viAG' (by i.) = B + m [ ^^X^^ )'-
Similarly for the M. I. about PG.
5969 Hence, if an elhpse or hyperbola be described with
S, S' for foci, the tangent and normal at any point of the
curve are principal axes, and the M. I. about either is constant
for that curve.
5970 Similarly, for a point P in amj plane through the
centroid 0, it may be shown that the same construction will
give the axes Pf, PG about which the product of inertia
vanishes, OX, OY being the axes at in the given plane about
which the product of inertia vanishes.
5971 The condition for the existence of a point in a body
at which the M. I. about every axis through it shall be the
same, is —
There must be two princifal axes of equal moment at the
centroid, and the M. I. aljout each must he less than the third
principal moment.
Two such points will then exist situated on the axis of
unequal moment, and equi-distant from the centroid.
824 SOLID GEOMETRY.
5972 Given the principal axes at the centroid of a body and
the moments of inertia about them, to find the principal axes
and moments at any other point.
[See (5975) for the result.]
Let A, B, G be the given principal moments, and let the
mass of the body be unity. Then the ellipsoid of gyration at
the centroid 0, and a quadric confocal with it, will be
5973 Prop. 1. — The M. I. is constant for all tangent planes
of the confocal, and is equal to the
M. I. for the origin -{-\ = S+\. (5919)
Proof. — Let I, m, n be the dir. cos. of the tangent plane of the confocal,
p the jaeipendicular on the plane from 0. The M. I. for this plane
= M. I. for a parallel plane through 0+p^ (5921)
= A'r + B'm- + C'n'+p- (5936)
= (8-A) P+(S-B) m' + (S-C) n' + (A + X) r+(B + \) m'+(C + K) n'
(5914, 5G31) = S + X, which is independent of I, m, n.
5974 Pi^OP. II. — All these planes are principal planes at
their points of contact, and if the three confocals be drawn
through any point P, the tangent planes at P to the confocal
ellipsoid, two-fold hyperboloid, and one-fold hyperboloid, are
respectively the principal planes of greatest, least, and mean
moments of inertia. The normal to the confocal ellipsoid is
the axis of least moment, and the normal to the two-fold
hyperboloid is the axis of greatest moment.
Proof. — Draw any other plane through P. The perpendicular on it
from is less than the perpendicular on the parallel tangent plane to the
confocal ellipse, and greater in the case of the two-fold hyperbola. Then,
by (5921).
The solution of the problem at (5972) is now given by Proposition III.
5975 Prop. III. — The principal moments of inertia at P are
0P'^ — \^, 0P^ — \, OP'^ — Xs, and the normals to the three con-
focals at P are the principal axes.
PuooF. — The M. I. about the x axis at P
= M. I. for the origin P— M. I. for the yz plane
= S+0F'-iS + \) = OP'-X, (5921-73).
MOMENTS OF INEBTIA. 825
5976 The principal moments of inertia above, expressed in
terms of X^ of the confocal ellipsoid and d^, d^, its principal
semi-diameters conjugate to OP, will, by (5661), become
OP'-K, OP'-\+dl OP'-\,-\-dl.
5977 The condition that the line abc, hmi, referred to prin-
cipal axes at the centroid, may itself be a principal axis at
some point of its length, is
(lb h c f^ f^
I in m n n I __ 1
A-B ~ B-C ~ C-A "■ 2^ '
Here abc is any point on the Hne, and if a confocal quadric
of the ellipsoid of gyration at the centroid be drawn through
the stated principal point of the given line, p is the perpen-
dicular from the origin upon the tangent plane of the confocal
at that point.
_ mi • T x — a y — h z — c r\
PiiOOF. — The given hue -- — = ^ = {}•)
I m n
must be a normal to the confocal ~~- + ^r—r + tttt = 1 ("•)•
Therefore, by (5629), ^ = ^p^' '" ^ ^' '^ ^ 'gH ^''^'^'
Eliminate x, y, z from (i.) by means of (iii.), and from the resulting equa-
tions eliminate p, and the condition above is obtained.
Also, by (5631),
f = (A + X) r + (J? + X) m^ +(G + X) n' = AP + Bm^ + Gv? + \ . . . (iv.).
The principal point xyz is now found by eliminating \ and p from equa-
tions (iii.)> by means of (iv.) and (5977).
INTEGRALS FOR MOMENTS OF INERTIA.
By the definition (5903), the following indefinite integrals
for moments of inertia are obtained : —
5978 For a plane curve, y =f{x), the M. I. about the x and
y coordinate axes are
y^ds and x^ds; and therefore (x^ + y^) ds = r^ds
5 N
826 SOLID GEOMETRY.
is the M. I. about an axis perpendicular to both the former
through the origin (5920).
5980 Observe that ds may be changed into dx, dy, or dO
by the substitution formulae (5113, '16).
5981 For a plane area bounded by the coordinate axes, the
ordinate y and the curve y =f(^x), the M. I. about the x and
2/ axes are
1 1 u^dAdy = 1 1 y^d.v and \ I x"d.vdy = \ x-ydx.
5983 And the M. I. about an axis perpendicular to both
the former drawn through the origin,
= JJ(.v^+/) dxdy = ((r'drdd = J {r'dS,
but in the last two integrals the area has the boundaries
described in (5894).
5986 The M. I. of a solid bounded by three rectangular
coordinate planes and the surface z =f{x, y) about the z axis,
will be
JT(.^^2+/) zdxdy =\\\r' siii=^ ddrddd^i,
but in the last integral the solid is bounded as described in
(5875).
5988 The volume, which represents the mass in all these
cases, has already been expressed (5205, 5871) ; and by
dividing by the volume, the square of the radius of gyration
of the solid is found (5904).
Proofs. — FormulsE (5981-3) are directly obtainable by geometry from
figures 90 and 91, and formulae (6986-7) from figares 168 and 188. The
transition to polar coordinates may also be eifected by the formula of
(2774).
D9o9 In expressing moments of inertia, the factor m will stand for the
mass of the body, and the remaining factor will therefore bo the value of the
square of the radius of gyration.
PERIMETEBS, AREAS, VOLUMES, Sfc. 827
PERIMETERS, AREAS, VOLUMES,
CENTRES OF MASS, AND MOMENTS OF INERTIA
OF VARIOUS FIGURES.
RECTANGULAR LAMINA AND RIGHT SOLID.*
For a rectangle whose sides are a, h, the momenta of inertia
about the sides, and an axis perpendicular to both where they
meet, are respectively
6015 wi-^, m— , "*— 3—
Proof : ^'ax' dx = ~= m ^. The third by (5920) .
Jo "
6018 Hence, for a right solid, whose dimensions are 2a,
2b, 2c,
M. I. about the axis of fio^ure 2c = m. — i: — .
ARC OF A CIRCLE.
6019 Let AB (Fig. 191) be the arc of a circle whose centre
is and radius /*. Let the angle AOB = Q ; then
Length of arc AB = r6. (601)
6020 Huygens' Approximation. — Rule. — From 8 times the
chord of half the arc take the chord of the whole arc, and divide
the remainder by 3.
Proof. — The rule gives ~ ( 16 sin — — 2 sin — j .
° 3 \ 4 2/
Expand the sines by (764) as far as 0^ and the result is rd.
6021 Taking an axis OX through the mid-point of the arc
with origin 0, the centroid of the arc is given by (5889)
2rsiui^ TT r- • •,„!. - _ 2r
cV =
J_. Hence for a semi-circle cV =
e IT
For M. I. of a triangle, see (5944-52).
828 SOLID GEOMETRY.
6023 Also, for the centroid of BX, y = ^^' ^"^' ^^
where a = /_ XOB.
6025 The M. I. of the arc AB about OX and OY are
mr^/^ sin6\ -, mr^/^ . sin ^\
_^l-__) and -2-(H-^).
(5978)
6027 M. I. about axes perpendicular to XOY, through and
X the mid-point of the arc respectively, are
mr^ and m.2r'(l-^^^\ (5979)
6029 CoE.— The if. I. of a") _ /w^f
circular ring about a diameter j 2
SECTOR OF CIRCLE. AOB (Fig. 191)
6030 Area = -^, .v= -^^- . ForXO^, y = — ^.
Proof. — ^, y are respectively f of x, y in (6021, '3) ; since the centroid
of eacli elemental sector is distant fr from 0. Otherwise, by (5893, '5).
6033 The M. I. about OX and OY are
Proof.— By (5981-2) ; or integrate (6025-G) for r from to r.
SEGMENT OF CIRCLE. ABX (Fig. 191)
6035 ^«a=-(<»-smO), ^= ^^g_^^gy
6037 For CCZ, 7 '•(2-3cosa+cos»a)
o (a— sma cos a)
Proofs. — From the sector and triangle ; otherwise, the centroid, by
(5893, '5).
PEBIMETEBS, ABE AS, VOLUMES, 8rc. 829
6038 M. I. about OX and OY, (5981-2)
^ (3^-4 sin ^+sin 6 cos ^) and ^ (^-siu ^ cos 6).
6040 CoE. — Hence, for a semi-circle^ w = — — .
6041 Also, tlie M. I. of a circle about a diameter, and about
a central axis perpendicular to its plane, are respectively
??^' and =:'. (6920)
4 2
THE RIGHT CONE.
If li be the height, r the radius of the base, and I the slant,
6043 Curved surface = irrl. Volume = \7rrV1.
6045 Distance of centroid from vertex = |/t.
6046 M. I. about axis of figm-e = m i^rl
6047 ^' I' about cross axes through the vertex and centroid
respectively.
7n^o{r'+^h') and 7n-io{4^r'+h').
FRUSTUM OF CYLINDER.
Let 6 be the inclination of the cutting plane to the base,
and c the length of the axis intercepted.
6048 The distance of the centroid from the axis is
- a' tan 6
X •=■
4c
.2
6049 The M. I. about the axis = m — , being the same as
that of a cylinder of height c. Hence, by (5921) and the value
of X above, the M. I. about any line parallel to the axis can be
found.
830 SOLID GEOMETRY.
SEGMENT OF SPHERICAL SURFACE. (Fig. 191)
Let be tlie origin of coordinates ; OA = r the radius ;
and 00 = x tlie abscissa of AB tlie plane of section.
6050 The curved area of AB = 2trr {r—x) = the area of its
projection on the enveloping cylinder of the sphere.
Proof: Area, = \ 2-mj —dx = 27rr(r—x). (5878)
Jx y
6051 For centroid of surface, a = ^ (r-\-cv).
6052 The M. I. about the axes OX, OY are
^ {2r'-ra?-A^') and ^ (4^y^^r.v-{.a;').
o o
HEMISPHERICAL SURFACE.
6054 ^rea = 2'irr\ ^^= ^' (6050-1)
6056 -M". I. about OX or or = m |rl (6052-3)
SEGMENT OF SPHERE.
6057 Volmne = -^ (2r+.v)ir-.vy\ a} = ^^^±^.
6059 M. I. about OX = ^ (r-.vy (8r^+9r.r+ar).
6060 il/. I. about OY
Proof. — As in (0146-7) ; or put a = h = c in the results.
HEMISPHERE.
6061 Volume = |1^r^ .v = ^r. (6064)
Proof. — Vol. = surface (6054) x -, by elemental pyramids having their
common vertex at the centre of the sphere. Otherwise, make x = in
(6057).
FEBIMETEBS, AREAS, VOLUMES, .fc. 831
6063 M. I. about OX or OY = m frl (6059-GO)
SECTOR OF SPHERE.
6064 Volume = Ittt^ (r—h), ^= i {r+h).
Proof. — Vol. = surface (6050) X — . x = ^ oi x in (6051), since the
o
centi^oid of each elemental pyramid is distant'fths of r from the centre.
6066 For the M. I. add together the M. I. of the cone and
segment (6046, '59).
THE PARABOLA, y^ = 4ax. (Fig. of 1220)
6067 Rad. of citrv. p = ^^ = 2a (^1+ |)l (4542)
6069 Coordinates of centre of curvature
3ci^+2«, ^^,. (4545)
6071 ArcAP^s = ^{ax^ar)^a\og ^'''+ v^(^+-^^) .
\/ u
6072 = a [cot e cosec ^+log cot {10)'].
Proof: s=[^(l+'j^)dx. (5197,4206)
Substitute ^x, and integrate by (1931).
6073 Arc AL = a^2^-a\og{l■^^2).
Centroid of arc AP with above value of s.
6074 *.r = ?£+f y(..^+«..)+^log 2^+«+2y(-^'+«.»-) .
4 o tt
6075 ^^/ = f{^/«(^+«r-«^}.
6076 For centroid of arc AL, putting x = a,
. _ 6^/2 + log(3 + 2^/2) 4 (2v/2-l)a
^~8{v/2+log(l + y2)} ' ^ 3 • v/2 + log(l+y2)-
832 SOLID GEOMETRY.
Half-segment of parabola ANP.
6078 Area = fa;?/, <r = f ct% ^ = |?/.
6081 The ilf. I. about the .i; and ?/ axes are
m f a.r and 7n j,v^.
THE ELLIPSE.
6083 The equation being h\r + a-jf = a^Ir, the length of the
arc AP (Fig. of 1205), putting <^ for the eccentric angle
of P, is
Jo
Proof. — lu {dsf = {dxf -\- {d^if (5113), substitute dx = —a sin 0f?0,
% = & cos fdcp, by (4276), and use" (4260).
6084 The length of the elliptic quadrant AB is
2 I 4 2! 2! 2* 3! 3! 2« 4! 4! 2« ) *
Proof. — Expand the binomial surd above, and employ (2454) and (2472).
Similarly, from (5887) and (5978) the three following values are found.
6085 For the ccntroid of the same quadrant,
1^2 1 ^4
cV = — . - — Y"-^ — ~-^ approximately.
6086 The M. I. about the x and y axes are approximately,
and
m^ l-je'—i^e' , ma' 1
2 -l-ie'—Ae'
4^ 64'
6088 Fagnani's Theorem. — (Fig. 192.) Let P be any point
on the ellipse, GY the perpendicular on the tangent at P;
/_ACY=^d; (2 the point whose eccentric angle =-|7r — 0.
Then
6089 PY-\-AP = a(^{l-e' siu'd) dS = BQ ;
and in the hyperbola (Fig. 193)
6090 PY-AP = a I \/(l-6- siu'^) (W.
PEBIMETEB8, AREAS, VOLUMES, ^c. 833
6091 Cor. — The difference between the lengths of the
infinite curve and asymptote =: a\ ^(l — c^siird)dd, where
, a •'«
tan a =--.
b
Proofs.— By (5203),
AP + FY or s + q=[pde=a[ ^/(l-e' &{iv 6) iW = BQ, by (6083).
In the hyperbola we have q—s=- j
6092 Draw the tangent at Q and the perpendicular GU upon
it. Let X, X be the abscissae of P, Q. The following relations
subsist,
PY= "-^ = QU, CY.CU=ab, CP'+CU' = (r+bK
Proof. — Let tp = the eccentric angle of P, and let AGU = ti'. Then
tau(b = ^ = - tan 0. (4276-80)
ox a
Similarly for Q, tan ( -r — ) = cot = — tan 6',
therefore tan^ = cotfl' or f = ,^ ~^' ('•)•
The relation therefore between P and Q is reciprocal. Now FY = e^x sin d
(4296) and x' = a sin 6, therefore PF = — — = Q?7, by the reciprocity.
Again, Cf/"' = cr cos" ^' + fc'^ siu'^ 0' (4372) = «'■' sin" ^ + Z>'^ cos" ^ (ii.).
Put f in terms of by the above, and we find
pjp a'h' _ d'b'-
^^^"^^^0+T''sin^0 ~ CY''
Lastly, CP- + C U'^ = .r + 7/ -\- a- sin' f + b"' cos' 0, by (ii) , = a' + U' (42 76-7).
6095 When P coincides with Q, the point is called ' ' Fagnani's
point," GY = V{ah), PY = a-b, and x = ci} {a-\-b)-^.
6096 Oriffith>^' Theorciih.* — If an ellipse of eccentricity e,
and a hyperbola of eccentricity e~S be placed as in the figure
of 1205 (the circle representing the ellipse), P^p being con-
sidered corresponding points ; then, calling PQ, in (6088), a
Fagnanian arc, we have the following theorem : —
* J. Griffiths, M.A., Proc. Lond. Math. Soc, Vol. v., p. 95.
5
834 SOLID GEOMETRY.
The ratio of tlie difference of two FagnaDian arcs on the
ellipse to the difference of the two corresponding arcs on the
hyperbola is equal to the product of e'- and the four abscissae
of the points on the ellipse.
SECTOR AND SEGMENT OF ELLIPSE.
6097 The formulae for the sector and segment of a circle
may be adapted to the ellipse by writing a for r and multi-
plying linear dimensions parallel to the minor axis by b : a.
But a will then represent the eccentric angle of the semi-arc,
and B twice that angle. Thus, in the figure of (1205), if
AGP be the half sector, a = AGp, = 2AGp.
Sector of ellipse {2AGP in fig. of 1205) :
6098 Area = '^, 1' = *^^^, y = ^1^, ^0-2)
the last being for the half sector AGP. The M. I. about the
X and y axes are
(6033)
r^Mnt mh' I ^ sill ^\ T ina^ /\ , sin ^\
6101 -TV--r) ""* -T\^+-r}
Segment of ellipse {2ANP in same figure) :
6103 Area = '-!^{e-sme), ^ = ^^^^^. (6035-6)
6105 For ~y of the half segment ANP, and for the M. I.
about the x and y axes, replace r by b in (6037-8) and by a
in (6039).
6108 For the whole ellipse, the area = nab. (6103)
6109 For the half ellipse, .T = |^. (6104)
OTT
6110 The M. I. about the x and // axes, and a third central
axis perpendicular to both,
and ^ .' — ^. (6041-2)
4 4 4
PEBIMETEBS, AREAS, VOLUMES, ^c. 835
6113 The area of the ellipse whose equation is
(abcfghla-yiy=ii, is = ^(^ft_ft.)3 "r ^/c^-
Proof. — If a, /3 be the semi-axes of the conic, the area 7ra/3 takes this
value, by (4414) and (4407).
6114 Lamherfs Theorem. — The area of a focal sector of an
ellipse, as P8P' (Fig. 28), in terms of f, <^', the eccentric
angles of P, P\ is
^ {<^-f-e(sin(^-siiif)} = Y^^"^"^'"''^"''''^'^^-
In the second value, sin ^ and sin^- are =iJ =—
respectively, where r = SP, r = 8F, and c = PP\* a result
of use in Astronomy.
THE HYPERBOLA.
6115 The length of an arc of the hyperbola hy-ahf = a^
and the abscissa of its centroid may be approximated to, as in
(6084) for the arc of an ellipse, by the substitutions from
(4278),
j (h- = a j see (^ ^/(f- sec'^ <ji — l)d(l>
and I ,rds = er j sec' <^ \/(t'" sec'- <^— 1) dcji.
6117 Landen's Theoreni. — This theorem gives any arc of an
hyperbola in terms of the arcs of two ellipses, as follows :
^^/{(r-\-b'-\-2ab cos C) dC =
^^{a'-b'shvA)dA-^^y(b'-(rs\n'B)dB-{-2asmB+coiist,
«- *■
that is — Arc of elUfse tcliose semi-axes are a-^b and a — b
= Arc of ellipse whose major axis is 2a and eccentricity b : a
+ difference between a right line and the arc of an liijperhola
whose major axis is b and eccentricity a : &.t
* Williamson's hiteg. Calc, Art. 137. t Ibid., Art. 157.
836 SOLID GEOMETRY.
6118 Arm ANP (Fig-, of 1188) bounded by rr, ^, and tlie
curve
= " ) A^^{a--(r)-a- \og'^-:L^ ^^ . (1931)
(4271)
6119 =i [.r//-r/Hog(| + |)(.
6120 Area of i^ector between CA, CF and the curve
6121 Area between tivo ordinate^ //i, ij.,, when the asymptotes
are the coordinate axes
ah 1 Xo
Proof : sin 2.4(70 | .j dx = -^ j ^ ^. (4387)
6122 The centroid of ANP, A being the area (6118), is
given by
6124 The M. I. of ANP about the x and // axes are
6125 A(2,,.'_„-,.) y(.,.^-,r) - !f i.,g^±^t!!=I^),
THE ELLIPTIC PARABOLOID.
6126 T^nuation, '}l^!L = 2z.
6127 I 'ol. of ,njmril f = 77 ^ (^^ft) .-', -^ = ^-■
6129 ^1^. P about the axes of ,r, //, J^nd ;: respectively^
/(iz , ;::-\ /ft;:: , rJ\
PEBIMETEBS, AREAS, VOLUMES, .^r. 837
6132 The surface S of the same segment may be found from
6133 If the surface of the paraboloid be bounded by a
curve of constant gradient y (5881), the area becomes
S = i7rab{sec'y-1). (5883)
THE PARABOLOID OF REVOLUTION.
6134 Equation, .v^-\-y^ = 2az or r'- = 2az.
6136 Surface of segment, S = §77 ^a { (2z+(f)^-a^} . (5880)
6137 Volume = miz' = i7rr% z = ^z. (5887, '99)
6140 M. I. about axis of figure = '-^ (6131)
6141 For M. I. about OX and OY put a = h in (6129-30).
THE ELLIPSOID.
6142 Equation, :^ + |^ +^ = 1, semi-axes a, b, c (5600).
6143 The surface of the segment cut off by the plane whose
abscissa is x, will be found from
Proof.— By (5874) and (6629, 7), eliminating z by means of the equa-
tion of the surface.
6144 The volume of the solid segment and the centroid are
given by
^-■^^^^^^ ^' 4(2«+.r)-
Proofs. — Let (Fig. 177) represent one octant of the ellipsoid; OA, OB,
OC being the principal semi-axes. The elemental section
4PNQ = TrNF.NQdx = tt— v/a'-a;' — s/d'-x-dx.
Therefore Vol. ;
'^ {\a'-x') dx = ^ (2a«-3a-^a; + .-') = &c.
(I J -p ofl
838 SOLID GEOMETRY.
The moment with respect to the plane of yz
a" J 3, 4a"
and division by the volume gives £' as above.
6146 Tlie M. J. of the solid segment about the axis a
Trhc (6''+r)
ma'
Proof.— (Fig. 177.)
(rt-.r)'^(8r<^+9«.r+av-).
M. J. = [\nP.NQ:^^^±^cU (G112) = -M^ f'V,^-.r^^/. = &o.
6147 The M. I. about the axis h
^ ^ 5 4 («-.r)'(8«2+9«.*^+3cV-) + 2«'^-5«V+ai'= ? .
15a- ( (r )
Proof : il/. I. = {"ttNP.NQ (^ + OnA dx (5921)
= ■^— ^ (a- — x-y^dx+ '^ (a^ — .^-) a:-t?.« = &c.
4a.* Jx ti Jx
6148 The volnme of the whole ellipsoid = ^trahc.
Proof. — By making a; = in (6144).
Otherwise : Let kiqi:, be the point on the auxiliary sphere of radius r cor-
responding to xyz on the ellipsoid. By (5638-9), rx = a^, ry-= hr], rz =■ ci^.
Therefore [ dx dy dz = '^[ dk dr}di: = ^ iirr\ (6061)
6149 For the ccntroid of tlie semi-eUipsoid .r = '—. (6145)
8
6150 The M. I. about the axis a = » J^'+^") . (6146)
o
6151 The volume of a segment cut off by an)/ plaue PNQ
(Fig. 177), where 0A^=(1 is the semi-conjugate diameter, and
V — ttahv — \—rK — -•
Proof. — Taking the area of tlie section from (5655), tlic vdlunie of the
segment will be
T^ahc sin
-J>-^ )■'■■■• "■'"- -"" = 5-
V
being ilie inclination of <l to the cutting ])lanf'. Integrate, and put
x = d-l>.
PEBIMETEBS, AREAS, VOLUMES, ^r. 839
PROLATE SPHEROID.
Put c = b in equation (6142) of the ellipsoid ; then a will
be the semi-axis of revolution.
6152 The surface of the zone between the plane of ij'i and a
parallel plane at a distance x is
L e a a
Proof.— Bj (5878). ,S' = "^ ^ J (^ - xA dx.
Then by (1933). Otherwise, make h = c in (6143), and reduce.
6153 CoK. — The whole surface = 2ir6 ( 6+ — sin~^e j.
6154 The centrolcl of the surface of the zone in (6152) is
Proof. — From
^i>V(f-«')-
6155 The M. I. of the same zone is
6156 And for the whole surface, by making x = a and
doubling, / 9 i \ /I
if. I. = .aV (^ - i) sm-V+.ft'(l + ^,
Proof : M. I
IV^? "''") *'■" °^' \ "V(7 ""') ■'"■■
The first integral by (1933). For the second, by Rule VI. 2048, we obtain
the formula
6157 |.^V(«— ^«0 dx = ^ sin-i| + ^^^^ ^/(a'-x^),
in which — must now be written for a.
e
6158 For the volume, moment of inertia, and abscissa of
centroid of the solid prolate spheroid, make c = b in (6144-51),
a being the axis of revolution.
840 SOLID GEOMETRY.
OBLATE SPHEROID.
6159 Piit b = a in the equation (6142) of the elhpsoid ;
then c will be the semi-axis of revolution.
The surface of the zone between the plane of xij and a
parallel plane at a distance z, is
4> Tra // 4 , 9 9 o\ I ire- 1 at';;;4--v/(c*4-«W)
c" e &
Proof.— By (5878). /b' = ^^^ ^ iL^^j^A dz. Then by (1931).
6160 C.)K.— The whole surface = 2mr-\- !!^loff 3L±^.
e 1—e
6161 The centroid of the surface of the zone in (6159) is
given by
-_27ra\eU c^ At c^ I
Proof. — As in (6154). z for the surface of half the spheroid is obtained
in this case by making 2 = c, but in (6154) put x = a.
6162 The M. I. of the same zone is
6163 And for the whole surface, by making :: = c and
doubling,
nr r t/i c' \ , TTC" (4«'-— iV") , a(l-\-e)
PKOor: J/. I. = 2.|.y (l + g) <b = 2^"^ |(.=_.») ^(_^£L +.) ....
The first integral involved is given at (1931), and the second is obtained in
the same way as in the Proof of (6155), giving
6164 \^' v/;?T^</,. = ^^^ VCr + cr)- I log {x+ ,/(.oHaO}.
6165 For the ruin me, inomeiif of inertia^ and ahscis.^a of
rnilrui.d of the solid oblate spheroid, make h = a in (6144-51),
c being the axis of revolution.
JOINT INDEX
TO THE
SYNOPSIS
AND TO THE
PAPEES ON PURE MATHEMATICS,
CONTAINED IN THE UNDEKMENTIONED
BRITISH AND FOREIGN JOURNALS
AND
TRANSACTIONS OF SOCIETIES.
D P
" There is an immense amount of knowledge lying scattered at the present day,
and almost useless from the difficulty of finding it when wanted."
—Frofcasor J. D. Everett.
KEY TO THE INDEX.
Prefixed to each title will be found the symbol by which the work is
referred to ill the Index. The words ''with Vol." or '' ivith Year," signify
that any number following the symbol in the Index denotes, respectively,
the Volume or Year of the journal. The year is given in all cases in which
the work consists of more than one series of volumes. In order to connect
the volumes with the years of publication, a Chronological Table is prefixed
to the Index ; in which table successive series of numbers in any column
indicate successive series of volumes of the publication.
A. with Vol.— Avchiv dcr Mathematik; 1843 to 1884; 70 vols.
[B. M. C. : P.P. 1580.] *
Ac. 7vith Vol. — Acta Mathematica, Zeitschrift Journal, herausgegeben
von G. Mittag-Leffler; Stockholm, 1882 to 1885; 7 vols.
AJ. with Vol. — American Journal of Mathematics ; Baltimore. Editor :
J. J. Sylvester, F.R.S.; 1878 to 1885; 7 vols. [B. M. C. :
P.P. 1575.6.]
An. with Year. — Annali di Scienze Matematiche e Fisiche, compilati da
Prof. Barnaba Tortolini ; Eome, 1850-57; afterwards— Annali
di Matematiche pura et applica; Rome, 1858-65. Series II.,
Annali di Matematiche pura et applica, compilati da F. Brioschi
e L. Cremona; Milan, 1868-85 ; 23 vols, in all. [B. M. C. :
P.P. 1573 and 952.]
At. with Year.— Att'i della Reale Accademia delle Scienze di Napoli;
1819 to 1878; 15 vols. [B. M. C. : for 1819-55, 8 vols.,
Acad. 2813 ; for 1863 to 1878, 7 vols., Acad. 96.]
C. with Fo/.— Comptes rendus hebdomadaires des seances de rAcaderaie
des Sciences; Paris, 1835 to 1885; 100 vols. [B. M. C. :
Acad. 424 and B.B. 2099. c] t
CD. luith Vol. — Cambridge and Dublin Mathematical Journal. Editor,
W. Thomson, B.A. ; 1846 to 1854; 9 vols. [B. M. C. :
P.P. 1565.]
CM. ivith Vol. — Cambridge Mathematical Journal; 1839 to 1845;
4 vols. [B. M. C. : P.P. 1565.]
CP. ivith Fo/.— Cambridge Philosophical Transactions; 1822 to 1881 ;
13 vols. [B. M. C. : Acad. 3008.]
* i.e., British Museum Catalogue, Pressmark F.P. 1580.
t Ji.E. signifies Reading -Room volumes ivithin reach.
844 KEY TO THE INDEX.
E. with Vol. — Educational Times Reprint of Mathematical Questions
and Solutions, with additional p:ipevs ; London, half-yearly,
1863 to 1885; 44 vols. Editor: W. J. C. Miller, B.A.
[B. M. C. : 2242. c]
G. ivith Vol. — Giornale di Matematichead uso degli studenti delle univer-
sita Italiane, pubblicato per cura del professore G. Battaglini ;
Naples, 1863-85; 23 vols. [B. M. C. : P.P. 1572.]
I. with Vol. — Journal of the Institute of Actuaries, or. The Assurance
Magazine; London, 1850-84; 24 vols. [B. M. 0.: P.P.
U2S.g.g. and 126.]
J. ^vith Vol. — Journal fiir die reine und angewandte Mathematik,
herausgegeben von A. L. Crelle ; 1826-1856 ; and Journal
alsFortsctzung des von A. L. Crelle gregrundeten Journals von
C. W. Borchardt; Berlin, 1856-1884- 97 vols. [B. M. C. :
P.P. 1585 and B.R. 2022. g.]
JP. 'With Vol. — Journal de I'Ecole Polytechnique ; Paris, 1796 to
1884 ; 34 vols. [B. M. C. : T.G. l.h.]
L. with Year. — Journal de Mathematiques pures et appliquees, ou
Eecueil mensuel de raemoires sur les diverses parties des
Mathematiques, publie par Joseph Liouville; Paris, 1836 to
1884; 49 vols. [B. M. C. : P.P. 1575 and R.B. 2022.(7.]
LM. with Vol. — London Mathematical Society^s Proceedings; 1866 to
1885 ; 16 vols. [B. M. C : Acad. 4265, 2.]
M. ivith Vol. — Mathematische Annalen, in Verbindung mit C. Neumann
begriindet durch R. F. A. Clebsch unter Mitwirkung dcr
Herren Prof. P. Gordan, Prof. C. Neumann, Vols. 1-9 ; and
Prof. K. V. Miihl, gegenwartig herausgegeben von Prof. F.
Klein und Prof. A. Mayer, Vols. 10, &c. Leipsig, 1869-1885 ;
25 vols. [B. M. C. : P.P. 1581.6.]
Man. ivith Year. — Manchester Memoirs, or. Memoirs of the Literary
and Philosophical Society of Manchester; 1805 to 1884;
23 vols. [B. M. 0. : 255.fZ., 9-12, and Acad. 1360.]
Me. ivith Year. — The Oxford, Cambridge, and Dublin Messenger of
Mathematics; 1862 to 1871; 5 vols. Continued as — The
Messenger of Mathematics. Editors : W. A. Whitworth,
M.A., C. Taylor, D.D., R. Pendlobury, M.A., J. W. L.
Glaisher, F.R.S. ; Cambridge, 1872 to 1885; 14 vols.
[B. M. C. : P.P. 1565.6. and 463.]
Mel. ivith Vol. — Melanges mathematiques de I'Academie des Sciences
do Saint Petersburg; 1849 to 1883; 6 vols. [B. M. C. :
Arad. 1125/9.]
Mem. with Year. — Memoires do TAcadomio Imperialo des Sciences do
KEY TO THE INDEX. 845
Saint Petersburg; 1809 to 1883; 51 vols. [B. M. 0.:
1809-30, T.O. 9.a., 14-20; 1831-59, Acad. 1125/2; 1859-83,
Acad. 1125/3.]
Mo. ivitk Year. — Monatsbericht der Koniglich-Preussiclien Akademie
der Wissensobaften zu Berlin; 1856 to 1881 ; continued as —
Sitzungsbericbte der Koniglicb, &c. ; 1881 to 1885; 30 vols.
[B. M. C. : Acad. 855.]
N. 'With Year. — Nouvelles Annales de Mathematiques. Journal des
Candidats aux Ecoles Polytechnique et Normal. Redige par
MM. Terquem et Gerono ; Paris, 1842 to 1882; 41 vols.
[B. M. 0. : P.P. 1544.]
No. with Year. — Nova Acta Regise Societatis Scientiarura Upsalieusis ;
1775 to 1884; 26 vols. [B. M. C. : for 1775-1850, 14 vols.,
T.O. 6.b., 13-19; for 1851-1884, 12 vols., Acad. 1076.]
P. luith Year.— The Pliilosopbical Transactions of the Royal Society of
London; 1781 to 1884; 104 vols: i.e., vols. 71 to 174.
[B. M. C. : B.B. 2021. r/.]
Pr. ivith Vol. — Proceedings of the Royal Society of London ; 1800 to
1885; 39 vols. [B. M. C. : for vols. 1 to 23, Acad. 3025, 21 ;
for vols. 24, &c., B.B. 2101. cL]
Q. tuith Fo?.— Quarterly Journal of Mathematics; 20 vols. Cambridge,
1857-78; Editors: J.J. Sylvester, F.R.S., N. M. Ferrers,
F.R.S., G. G. Stokes, F.R.S., A. Cayley, F.R.S., M. Hermite,
F.R.S. ; 1878-85, N. M. Ferrers, F.R.S., A. Cayley, F.R.S.,
J. W. L. Glaishcr, F.R.S. [B. M. C. : P.P. 1566 and 25L]
TA. with FoL— Transactions of the American Philosophical Society ;
Philadelphia, 1818-71; 14 vols. [B. M. C. : Acad. 1830/3
and T.G. I8.h. 11.]
TB. ivith Fo?.— Transactions of the Royal Society of Edinburgh ; 1788
to 1880 ; 29 vols. [B. M. C. : T.O. 15.6. 1 and B.B. 2099.^.]
Tl. ivith Fo/..— Transactions of the Royal Irish Academy; 1786 to
1879; 26 vols. [B. M. C. : Acad. 1540 and BB. 2099. g.]
TN. with Fo/.— Transactions of the Royal Society of New South Wales ;
Sydney, 1867-83; 17 vols. [B. M. C. : Acad. 1971.]
Z. with Fo/.— Zeitschrift fiir Mathematik und Physik, herausgegeben
von Dr. 0. Schlomilch und Dr. B. Witzschel; Dr. 0. Schlo-
milch. Dr. E. Kahl, und Dr. M. Cantor; Leipzig, 1856-1883.
28 vols. [B. M. C. : P.P. 1581.]
Year.
A
Ac
AJ
An
At
C
CD
CM
CP
^
G
I
J
JP
Year
1801
1801
1
3
2
'.'.'.
2
3
3
4
"4
4
5
5
6
5, 6
()
7
7
7
8
s
9
'" 8
9
10
10
11
U
12
12
13
"■9
I A
14
14
15
"10
15
16
16
17
17
IS
IS
19
i.'i
19
20
"11
20
21
21
22
1
22
23
"12
2:5
24
24
25
2
2.T
26
1
26
27
"2
2
27
28
3
2S
29
4
29
30
3
5,6
30
31
7
13
.-51
32
3
8, 9
3-2
33
4
10
'14
3 .5
34
11, 12
34
35
1
"5
13, 14
15
3-)
36
2, 3
15
36
37
4, 5
16, 17
37
38
6, 7
"6
18
16
3^
39
4
8, 9
1
19
■■'i9
40
10
20, 21
4il
41
11-13
2
22
"17
41
42
14, 15
7
23, 24
42
43
1
16, 17
3
25, 26
43
44
2, 3
'"5
IS, 19
27, 28
44
45
4, 5
20, 21
4
29, 30
-1.")
46
6, 7
22, 23
1
31, 33
4ii
47
8, 9
24, 25
2
.34, 35
18
47
48
10, 11
26,27
3
36, 37
19
4S
49
12, 13
28, 29
4
9
38
49
50
14, 15
I. 1
30, 31
5
1
39, 40
50
51
16, 17
2
■'6
32, 33
6
41, 42
51
52
18, 19
3
II. 1
34, 35
7
2
43,45
52
53
20, 21
4
36, 37
8
3
46,47
'20
53
54
22, 23
5
38, 39
9
4
48
54
55
24. 25
6
"2
40, 41
5
49. 50
55
56
26, 27
7
42, 43
9
51, 52
"21
56
57
28, 29
8
44, 45
"e
53, 54
57
58
30, 31
II. 1
46, 47
7
55
58
59
32, 33
2
48, 49
56, 57
59
60
34, 35
3
50, 51
58
60
61
36, 37
4
52, 53
"s
59, 60
"22
61
62
38, 39
54, 55
9
61
62
63
40
"5
III. 1
56, 57
"1
10
62
'23
63
64
41, 42
6
58, 59
10
i, 2
2
11
63
64
65
43, 44
7
"2
60, 61
3,4
3
64
"24
65
66
45,46
62, 63
5, 6
4
12
65, 66
66
67
47
64, 65
7,8
5
13
67
"25
67
68
48
III! 1
3
66, 67
9, 10
6
68, 69
68
69
49, 50
2
4
68, 69
11, 12
7
14
70
69
70
51
3
70, 71
13, 14
8
15
71, 72
"26
70
71
52
...
4
72, 73
11
15, 16
9
73
71
72
53
74, 75
17, 18
10
16
74,75
72
73
54, 55
5
"5
76,77
19, 20
11
17
76
73
74
56, 57
78, 79
21, 22
12
77, 78
27
74
75
58
6
"6
80, 81
23, 24
13
i's
79, 80
75
76
59, 60
7
82, 83
25, 26
14
19
81
76
77
61
8
84, 85
27, 28
15
82-84
77
78
62
1
'7
86, 87
29, 30
16
20
85, 86
78
79
65, 64
2
88, 89
12
31, 32
17
21
87, 88
"28
79
80
65, 66
3
90, 91
33, 34
18
89, 90
29
80
81
67
4
92, 93
13
35, 36
19
22
91,92
30, 31
81
82
68
1,2
5
94, 95
...
37, 38
20
23
93, 94
32
82
83
69
3,4
96, 97
39, 40
21
95
33
83
84
70 ! 5-7
6
98. 99
41, 42
22
24
96, 97
34
84
1885
... 1 ...
7
100
43, 44
23
...
1885
A
Ac
AJ
An
At
c
CD
CM
CP
E
G
I
J
JP
Year.
L
LM
M
Me
Man.
Mem.
Mel.
Mo
N
No
1'
Pr
Q
TA
TE
TI
TN
Z
Year.
1831
91
1801
2
92
"8
2
3
93
9
3
4
94
4
5
i.'i
95
5
5
6
96
10
6
7
97
7
8
98
8
9
I.'i
99
9
10
2
100
11
10
U
3
101
11
12
102
"6
12
13
2
4
103
13
14
104
1
14
15
"5
1.7
105
"7
12
15
16
106
16
17
107
17
18
'6
108
1
8
13
IS
19
'3
...
109
19
20
" 7
110
20
21
8
111
21
22
'" 8
112
22
23
113
9
23
24
4
9
114
24
25
115
2
14
25
26
10
116
10
26
27
9
117
27
28
118
15
28
29
119
29
30
"11
120
2
3
16
30
31
5
II. 1
121
31
32
10
122
32
33
...
"' 2
123
33
34
124
4
34
35
125
'.'.'.
35
36
i.'i
126
36
37
2
...
127
3
5
17
37
38
3
3
128
38
39
4
11
129
6
18
39
40
5
130
40
41
6
4
131
"7
41
42
7
"6
i'. 1
132
42
43
8
2
133
4
"s
19
43
44
9
5
3
12
134
44
45
10
4
135
20
45
46
11
7
5
136
9
46
47
12
6
13
137
47
48
13
8
7
138
21
48
49
14
8
139
49
50
15
'"e
9
14
140
'5
50
51
16
9
10
141
51
52
17
10
n
142
52
53
18
7
"1
12
143
10
22
53
54
19
11
13
144
6
23
54
55
20
12
14
I. 1
145
7
55
56
II. 1
13
1
15
146
"i
56
57
2
14
' 8
2
16
147
8
1
2
57
58
3
1.5
9
3
17
"2
148
2
3
58
59
4
III. 1
"2
4
18
149
9
23
4
59
60
5
2
5
19
150
10
3
11
5
60
61
6
3
6
20
3
151
4
6
61
62
7
i.i
IL 1
4
7
II. 1
152
11
5
7
62
63
8
5, 6
8
2
4
153
12
12
8
63
64
9
"2
7
9
3
154
13
6
9
64
65
10
'"2
8
10
4
"5
155
14
10
65
66
11
"l
3
9
3
11
5
1.56
11
66
67
12
10
12
6
157
15
8
24
1
12
67
68
13
4
3
11
13
7
"e
158
16
9
2
13
68
69
14
2
1
12, 13
14
8
159
17
13
25
3
14
69
70
15
2
14, 15
15
9
7
160
18
4
15
70
71
16
"3
3, 4
'"5
"4
16
16
10
161
19
10
14
5
16
71
72
17
5
II. 1
17, 18
4
17
11
162
20
26
6
17
72
73
18
"4
6
2
19, 20
IS
12
8
163
21
7
IS
73
74
19
5
7
3
21
19
13
164
22
24
8
19
74
75
III. 1
6
8
4
20
14
9
165
23
13
25
9
20
75
76
2
7
9, 10
5
"5
"22
21
15
166
24
14
27
10
21
76
77
3
8
11,12
6
23, 24
22
16
167
25, 26
11
22
77
78
4
9
13
7
25
23
17
168
27
15
12
23
78
79
5
10
14, 15
8
"6
26
24
18
10
169
28, 29
16
28
26
13
24
79
80
6
11
16,17
9
27
25
19
170
30
29
14
25
80
81
7
12
18
10
28. 29
5
26
20
11
171
31,32
17
15
26
81
82
8
13
19
11
7
30
27
III. 1
172
33
18
16
27
82
83
9
14
20,21
12
31
6
28
173
34, 35
19
17
28
S3
84
10
15
22,23
13
8
29
12
174
36, 37
84
885
16
24,25
14
30
38, 39
20
1885
r.
LM
M
Me
Man.
Mem.
Mel.
Mo
N
No
P
Pr
Q
TA
TE
TI
I'N
Z
EXPLANATION OF ABBREVIATIONS, &o.
a. c.
areal coordinates.
i. cq.
= i)ideterminate equation.
alg.
=
algebraic.
imag.
= imaginary.
ap.
=
application.
1. c. m
= loivest coinmon multiple.
anal.
=
analytical.
num.
= mimerical.
ar. p.
=
arith. progression.
0. c.
= oblique coordinates.
c. c.
=
Cartesian coordinates.
p. c.
= polar coordinates.
en.
=
construction.
p. d. e
= partial difference equations.
cond.
=
condition.
p. e.
= polar equation.
eurv.
=
curvature.
perp.
= perpendicular.
d. c.
=
differential calculus.
pi.
= plane.
d. e.
=
differential eqications.
pr.
= 2^^'oblcm.
d. i.
=
definite integral.
q. c.
= quadriplanar coordinates.
eq.
=
equation.
rad.
= radius.
ex.
=
example or exercise.
sd.
= solid or ^-dimensional.
ext.
=
extension.
sol.
= solution.
f.
=
formula.
sym.
= sijmmetricalhj.
f. d. c.
=
finite difference calculus.
ta.
= table.
f. d. c.
=
finite difference equation.
t. c.
= trilinear coordinates.
geo.
=
geometrical.
tg. c.
= tangential coordinates.
S- P-
=
geometrical progression.
tg. e.
= tangential equation.
gn-
_
general.
th.
= theorem.
gz-
=.
generalization.
tr.
= treatise\i.e.,moTQ than 50 pages)
h.c.f.
=
highest common factor .
transf.
= transformation.
i. c.
=
integral calculus.
153 —
The
suffix means that three articles under
the same
heading zcill be found in the volume
-2 1 . — Means that one article on the subject xvill be found in each of the four consecutive volumes.
References to the Synopsis stand first, and are tlie numbers of Articles,
not of Pages. An asterisk (*) is prefixed where such numbers ^vill be found.
The unclassified references following a principal title commonly refer to
papers on the general theory of the subject ; but some papers are occa-
sionally included amongst these of which the titles are too long for inser-
tion, and do not admit of abbreviation.
Subjects which might well have been included under the same head-
ing appear sometimes under dili'erent ones, for the folloAving reasons : —
Exigencies of space have decided the insertion of the number of the volume
only of the particular work in question, and a subsequent examination of
the Index of that volume is required in order to find the page. It, thej'e-
fore, became desirable not to change the original title of the pa]>er when
there was danger, by so doing, of making it unrecognizable. When, how-
ever, the same subject appears in two parts of this Index under different
names, cross references from one to the other are given. Some changes,
however, have been made when tlic syiinnyin was perfectly obvious ; for
instance, when a reference to a journal, ])n1)Iisli('(l fifty years ago, is found
under the Invading of " Binary Quantics," tlic aclnal title of tlio ai'ticle will,
in all i)rol)ability, be " Homogeneous Kiiiict ions of Two Variables," and so
in a few otlier iiistiuices.
INDEX.
Abacus of the Pythagoreans : L.39.
AbeUan cubics and symmetrical ei( na-
tions : Q.5.
of class x/(-31) Mo.82.
Abelian eqnations : A. 68: C.95: J.93 :
M.18: Mo.77,92.
Abelian functions : see " H-yper-elliptic
functions."
*Abel's formula for F{x+i7i)+F{x—!y):
2705. Me.73.
*Abel's theorems: 1572: C.94: J.9,24.,
61,90: LM.12: M.8,17: P.81,
83 : Pr.30,34.
cj> {x)+(i> (2/) = ^/^ {xf{y) + yf{x)] : An.
57.
*Abscissa: 1160.
Acceleration: Me.tr 65.
*Algebra: 1—380: A.tr 20.
application to geometry : JP.4-.
foundation, limitations : AJ.6 : CP.7,
8,: Q.6.
history of, in Germany : Mo. 67, 70.
Algebraic: Calculus: N.81.
definitions: C.37.
forms: C. 84,94: M.15.^
coordination of : J. 76.
whoseHessian vanishes identically :
M.IO.
in theory of cubics : M.8.
formulaj: G.12 : Q.5.
functions: A.10,31 : J.92 : L.50,51 :
M.IO : N.62.
applied to geometry : G.22 : M.7.
number of constants : J.64.
as partial fractions : Z.9.
rationalization of : A.69.
representation by : J. 77,78.
resolution into factors : A. 46.
theorems: J.82 : M.1,6.
synthesis : C.I63.
Algorithmic geometry : N.57.
Algorithm : re definition of ( -^ ) :
J.27. ^ ^ ^
of higher analysis : Mo. 75.
of arithmetical functions : G.23.
Alternants of 4th order, co-factors of :
AJ.7.
Alternate numbers : LM.lOo.
Alternating functions : AJ.7 : C.12,
22 : J.83 (.Vandermond's) : Me.
82.
Altitudes, determination of: A.12,19:
Mem.l5: ISr.45.
Amicable numbers : A.70.
Anallagmatic curves and surfaces :
C.87 : N.64 (quartic surface).
Anallagmatic pavements : E.IO.
Analysis : A.l : An..50 : C.3,11,12 :
J.f7: P.14: Q.6,7.
ap to geometry : G.23 : JP.4.
Analytical : aphorisms : A.5 : J.IO.
combination theorem : J.ll.
functions : Ac. 6 : thsAn.82 : La-
grange, trJP.3.
system of, and series from it: An.
tr 84.,.
* geometry : 4001—6165 : A.2,11,38 :
C.6: JP.9: L.72: M.2 : Mem.
13: Z.9: tr 11,12.
theoremsand problems :A.8,52:J.46.
plane and solid in homogeneous co-
ordinates : Z. 15,16.
of three dimensions : CM.4.
metrics : Q.7,8.
theorems: A. 8. treatise: C.13.
Angles : conterminal : Me. 74.
* of a central conin : 4375.
division into n and n + 1 parts : A.70.
of five circles or six spheres : Me. 79.
* of two circles : 4180.
problems on : P.1791.
two relations between five : A.20.
* trisection of: 5325: A.4,34 : C.2,66,
81 : G.15 : Me.72 : N.56,76.
Anharmonics : LM.2,3.
*Anharmonic pencils of conies : 4809
—21.
*Anharmonic ratio: 1052,4648: GM.12.
corresponding to roots of a biquad-
ratic : N.60.
* of a conic: Q.4: of four tangents: 4986.
5 Q
550
INDEX.
Auharmouic ratio {eoiitlnued) :
of 5 liues iu space : Me.76.
of 4 points in a plane : A.l : C.77.
sextic : LM.2,60 : Q.37,38.
systems : cnM.lO.
*Aunuities: 302: A.22: Ac.l : CP.3 :
J.83 : 1.1—24: P.1788, -89, -91,
-94, 1800, -10 : N.47.
Auticaustic (by refraction) of aparabola :
N.83,85.
*Anticlastic surface: 5623,5818.
Aplauatic liues, lemniscates, caustics,
&c. : thL.50: N.45.
Apolarity of rational curves : M.21.
Apollonius's problem : A. 37 : M.6.
Approximation : J.13 : N.66.
algebraic : J.76.
to functions represented by integrals :
C.20o.
of several variables : C.70.
successive : Mem.38.
Apsidal surfaces : Q.16.
Arabs, mathematics of: 0.39,60.
Arbitrary constants : C.15 : L.80 : in
d.e and f.d.e, TI.13.
*Ai'C, area, &c. : quantification of :
1244, 5205, 5874, 6U15.
relations of: G.16 : C.80,94: L.46.
Arcs with a rectifiable difference and
areas with a quadrable difference :
L.46.
*Areas : and volumes in t.c and q.c :
4688 : Q.2.
* approximation by ordinates : 2991 — 7:
C.78: CD.9.
* between 3 lines : 4038 : Me.75.
ext. of meaning : CD. 5.
Arithmetic : A.5,18 : L.59.
ancient : C.71 : N.51.
degenre, ext. of the notion : C.94.
higher : J.85.
history of: C.17.
of Ibn-Esra : trL.4L
of Nicomaque de Gerase : An. 57.
Arithmetico-geometric mean : Mo.58 :
Z.20.
*Arithmetic mean : 91 : CD.6 : of n
({uantities : 332 : L.39.
*Arithmetical progression : 79 : tliL.
39: Pr.lO.
and g. p when n (the number of terms)
is a fraction : A. 35.
when the terms are only known ap-
proximately : C.96.
Arithmetical theorems : A.IO: 0.93,97.:
0D.6: G.7,18: L.63 : of 1. c. m",
N.57 : Genjunne, 0.5.
Ai'ithmetical theory of algebraic forms :
J. 92,93.
Arithmographe polychrome : 0.51,53.
Arithmometer : I. 16 — 18.
Aronhold, theorems of : gzJ.73.
Associated forms : systems of :
gzAJ.l : 0.86 : Op.6.
and spherical harmonics : Me.85.
Astroid of a conic : A.64.
^Astronomical distances : p 5.
*Asymptotes: 5167: A.p.c 15,17: OM.
4 : M.ll : N.68 : thsN.48, and
73.
* of conies : 1 182, 4490, t.c 4683 : tg. c
4904, -66 : Me.71 : Q.3,8.
of intersections of quadrics : N.73.
of imaginary branches of curves :
0M.2.
Asymptotic : chords : A. 12.
* cone of a quadric : 5616 : E.30, g.e 34.
* curves : 5172.
lines of surfaces : A.60,61 : R.84,
law of some functions : Mo.65.
methods: M.8o.
* planes of a paraboloid : 5625.
planes and surfaces : 0D.3.
Atomic theory and graphical represen-
tation of invariants and covari-
ants of binary quantics : AJ.I2.
Attraction : of confocal ellipsoids :
Me.82.
of ellipsoids: OD.4,9: J. 12,20,26,31 :
JP.15: L.40,45: M.IO: N.76 :
Q.2,7,17.
of ellipsoidal shell : J.12 : JP.15 : Q.17.
of paraboloids : L.57.
of polyhedra : J. 66.
of a right line and of an elliptic arc :
An.59 : 0D.3.
of a ring and of elUptic and circular
plates : G.21 : Z.ll.
of spheroids : J.Pi : JP.8 : L.76 :
ML'ni.31 : l?.{Ivory) 12.
of solids of revolution, &c. : An.56 :
0D.2.
solid of maximum : TE.6.
theorems : Q.4,17.
theory of : L.44,6.
*Auxiliary circle : 1160.
Averages : 1.7,9.
*Axes : of a conic : gn.eq4687 : A.30 :
E.36: G.12: Q.q.c4; t.c 5,8,15
and 20 : Me.a.c 64,71 : N.43,48,58:
t.cQ.12.
* construction of : 1252 : Me.66._,.
* en. from conj. diameters: 1253:
A.13,20 : Me.82 : N.67,78.
* of a cjuadric : 5695 : A.30 : An. 77 :
G.9 : J.2,64,82 : N.43,51, en 6S,
69,74.
* rectangular, nine direction cosines
for two systems : 6577 — 8 :
L.44,.
INDEX.
851
*Axis : of perspective or homology :
975.
* of reflexion : 1007.
* of similitude : 1046, 4177.
Axonometry and projective collineation
in space: M.25 : Z.12,21.
Babbage's calculating machine : C.992.
Barycentric calculus and right line con-
struction : J.28.
Battement de Monge : L.82.
Beltrami's theorem : A.44.
*Bernoulli's numbers : 1539 : A. 3 :
AJ.5,7: An.59,77: C.64.,583,81. :
G.9: J.20,21 ,28,58,81,84,85 (first
62),88,92 : LM.42,7,9 : Me.75 :
gzMem.83: K76 : Q.6,22. _
application to series : see " Series."
and interpolation : C.86.
and their first 250 logarithms : CP.12.
and secant series: A. 1,3, 35: C.4,32.
bibliography of : AJ.5.
indeterminate representation of: Z.
19.
new theory of : C.83.
theorems on : E.2,8 : lSf.77.
Bernoulli's series : 1510.
Bessel's functions (see also " Integrals
of circular functions ") : J. 75 :
M.3,4,9,14,16 : Q.20,21.
representation of arbitrary functions
by : M.6.
squares and products : M.2.
tables : Z.2.
Bonnet, two formulae of: G.4.
Bicircular quartics: LM.3,99 : P.77 :
Pr.25 : Q.19.: : TI.24.
focal conies of : LM.ll.
with coUinear triple and double foci :
LM.12,14.
nodal, mechanical en. of : LM.3.
Bifocal variable system : M.16.
Bihnear forms: J.68o,84,86 : L.74:
Mo.66,682,74.
congruent transformation of : Mo.
74.
four variables : G.21 : Mo.83.
relation between two and their
quadricandquartic system: M.l.
reciprocals : G.22.
reduction of : C.78,92.
Bilinear functions : GM.ll.
polynomials : C.77.
trilinear and quadrilinear systems :
E..5.
Billiards, theory of : L.83.
Bimodular congruences : G.21.
Binary and ternary quadratics: N.G4,65.
*Biuary cubics : 1631 : A.17 : C.92 : G.r7 :
J.27,53,41 : Q.1,11.
Binary cubics — {continued) :
automorphic transf. of : LM.14.
and quadratic forms : G.21.
system of two : E.7 : G.17 : LM.13 :
M.7.
resultant : Q.6.
tables and classification of : A.31.
transformation by linear substitution :
J.38.
Binary forms: Au.56,77 : At.65 : G.
2,3,10,160: J.74: M.2,3,20: Q.14.
and their covariants, geo. : M.23.
ap. to anal, geometry : L.75.
ap. to elliptic functions : AJ.5.
ap. to Eulei''s integrals : C.47.
canonical : J. 54 : M.21 .
evectant : Q.ll.
geo. interpretation or ap. : C.78 : G.
17 : M.9,22..
having the same Jacobian : C.94.
having similar polar forms : M.S.
in a cubic space-curve : J. 86.
in two conj u gate indeterminates : C .97.
most general case of linear equations
in: C.99.
(q) groups of: M.23.
with related coefficients : M.12.
transference of, when not of a prime
degree: M.21.
transformation of : M.4,9.
typical representation of : An. 68,69.
Binary homographics represented by
points in space, applied to the
rotation of a sphere : M.22.
Binary nonics, ground forms : AJ.2.
Binary octics : thC.96 : M.17;.
Binary quadrics : C.47 : G.3 : J. 27 :
L.59,77 : M.15,172.
construction of, through a symboli-
cal formula : C.57.
indeterminate, integral sol. : J.45.
for a negative determinant : No.Sl :
C.60 (table): L.57 : M.172,21,2.5.
partition table : AJ.4.
representing the same numbers: L..59.
transformation of: 0.41.
with two conjugate indeterminates :
C.96.
*Binary quantics : 1636 : An.56 : C.52 :
CD.9.
(2h— l)-ic, canonical form of : Q.20.
derivatives of: Q.15.
* discriminant of: 1638 : Q.IO.
reduction of : J.36 : L..52 : Q.7.
transformation of : CM.l : thE.23.
in two polynomials U, V, prime to
each other and of the same de-
gree : N.85.
Binary quartics : and their invari-
■ ants • A.18 : G .14 : J.41 : M.19 : Q.7.
852
INDEX.
Binary quartics — {continued) :
condition for perfect square: E.36.
or quintics, with three equal roots or
two pairs of equal roots : P. 68.
and ternary cubic, correlation be-
tween: An. 76.
Binary quintic : G-.14.
canonical form for : Q.19.
Binary sextics : taAJ.4: 0.64,87,96;,:
G.14: M.2,76,77.
syzygies of: AJ.7.
Binet's function : L.76.
Binodal quartic with elliptic function
coordinates : A J. 5.
*Binomial : coefficients : 283,366 — 7 :
A.1,2: G.14: Mem.24: N.th60,
61,70,85: Z.25.
sum of selected : Mo.85.
* equations : 480 : A.IO : At.65,68 :
C.10,44: G.S-aO: L.57: LM.ll,
12,16.
irreducible factors of : An. 69.
a;'' — 1 = : see " Roots of unity."
equivalences to any modulus : C.25.
* theorem: 126—36: A.8,geo.61: C.
45: CD.7: CM.3: G.12: J.1,4,5,
28,65: Me.71: N.423,47,50,71,78 :
P.16,16,95,96: TI.12.
generalization of : J.l : N.572.
*Binormal : 5723.
Bipartite functions and determinants :
LM.16.
Bipartition, repeated : Mel. 4.
*Biquadratic : equation (see also
"Cubic and biquadratic"): 492
—501: A.1,12,16,23,31,39,40,41,
45,69: A J.l: CM.1,46..,47: G.5
—7,13,21: J.90:Me.62:N.44,582,
59,60,63,78,81,83: Q.7,28: Z.6,8,
18.
cond. for two equal roots : E.44.
and elliptic functions : C.57 : L.58.
reduction to canonical form : G.5.
reduction to a reciprocal equation :
L.63.
solution of: A.51,56 : C.49,82 : L.
73: Q.19: numerical, C. 61: with-
out eliminating the 2nd term,
A.39 : 4 variables, J.27: trigono-
metrical, A.19,70.
and sextic efjs. in the theory of
conies and ([uadrics : J. 53.
* values which make it a s(]Uare :
496: G.7: E.22.
function with four variables: An. 59.
involutions : C.98.
Bi(|uaternions : AJ.7: LlNl.l.
Biternary forms with contiMgredient
variaijles : M.l.
Borchardt's functions : J.82.
*Brachistochrone : 3037, 3044 : L.48 :
Man .31: Mc.80 : Mem.22 : N.
77,80.
Brahmins, trigonometrical tables of the :
TE.4.
*Brianchon's theorem: 4783: A.53 : C.
82: CM.4: G.2: J.84,93: gzN.
82 : Z.6.
and analogues : CD.7 : Q.9.
on a quadric surface : C.98 : E.19.
on a sphere : A. 60.
*Brocard circle and Lemoine's point :
4754c : gzN.85.
*Burchardt's factor tables p. 7 : erra-
tum, A.23.
*Burmann's theorem, d.c : 1559.
Calciilating machine : Pr.37.
Calculus: algebraic, which includes
the calculus of imaginaries and
quaternions: C.91.
of chemical operations : Pr.25.
of direction and position : AJ.6 : M.
pr 6.
of enlargement : A J.2.
of equivalent statements : see"Logic."
of forms (Invariant theory) : CD. 72,
8,9..
of infinitesimals, third branch of :
viz., given i/ and y^, to find a;:
TE.24.
of limits, ap. to a system of d.e : C.156.
of Victorius : Z.16.
of other subjects : see the subject.
Calendar : J.3,9,prs 22.
Jewish: J.f28.
Canal surfaces : A. 1,10.
Cauon-arithmeticus of Jacobi . C. 39,63 :
L.54.
Cantor's theorem : M.22.
*Cardioid: h=a. in 5328, Fig.129 :
A.59,63,68: LM.4: Me.64: N.81.
and ellipses : Pr.6.
*Carnot's theorem : 4778.
*Cartesian oval : 5341—5358 : A.69 : C.
97 : LM.1,3.,,99 : Me.75 : Q.l : Me.
74: Q.12,cnl5.
area of: E.21.
eq. with triple focus as origin : E.9,23.
foci: E.7.
functional images : Q.18.
mechanically drawn : LM.5,6 : Q.13.
])erimeter: E.21.
rectific. by ellip. functions : C. 80,87 :
LM.5.
through 4 points on a circle : LM.12.
witli 2 imaginary axial foci: LM.3.
*(^assinian oval: 5313: Me.77,83: N.57.
analogous surfaces : An.61.
radial of : E.26.
INDEX.
853
Cassinoid with n foci, rectif , of : L.48.
Catacaustic and diacaustic of a sphere :
JP.17.
Catalecticant of a binary quantic : E.
37,38.
Catenary: 6273: Me.64,66,68.
by parabolic trigonometry : Pr.8.
revolving: E.22.
*Cauchy's formula i.c : 2712.
closed curve theorem : Mo.85.
various formulas : C.27io.
*Caustics : 5248-9 : Ac.4 : L.46 : P.57,
67: Pr.8,14,15: Q.l,2,3cn8,9,12.
by successive reflexion from spheres :
J.13.
identity with pedals : Z.14.
of a cardioid : Me.83.
* of a circle : 5248 : CD.2.
of a cycloid : A. 30.
of an ellipse, focus at centre : J.44.
of infinitely thin pencils : J.98.
of refraction at a plane surface : N.47.
radii of curvature : lSr.65.
surfaces and singularities : J. 76.
Cells of bees : N..56 : Q.2.
Cell structure : LM.16.
Centimetre - gramme - second system :
p.l.
Centrals, theory of : J.243.
Centre of: curves and surfaces : L.
46 : Q.IO.
a circle touching two : A.24.
geometrical figures : A.16.
harmonic mean : J.3.
mean distance of curves and surfaces :
N.ths85: Q.33.
mean distance of points of contact of
parallel tangents, est. of th.: L.
44.
* similarity : 947.
* similitude: 1037.
* similitude of 3 circles : 1046 : 4176.
similitude of 2 quadrics each of which
circumscribes the same quadric :
J.31.
Centres, theory of : J.24.
Centro-baric methods in anal, geometry :
J.5-2.
*Centroid : formulas for : 5884—5902,
6015: JP.26: L.43.
and its use in stereometry : A. 39.
of common points of two conies : A.3.
* of circular arc : 6021 : E.13.
of a dice : lsr.63.
* of frustrums : 6048, &c. : A.33.
of a gauche curve after development
on a right line, locus of: C.88.
of algebraic curves and surfaces:
An .68.
of a frustrum of a prism : L.39.
Centroid — {continued) .-
of a frustrum of a pyramid : Me. 79 :
N. 76,.
of oblique frustrum of a cone : E.33.
of a perimeter : A. 51.
* of plane curves : 5887: J.21 : G.12.
* of spherical and other areas : 5898,
6051: J.50: L.39,422.
* of surface and solid of revolution :
5896—9: AJ.3: L.39.
of a trapezium : Q.9.
* of a triangle: 951: A.52,58.
*Cliange of independent variable : 1760 —
1816 : AJ.3 : CM.l : G.2 : L.40,58 :
Q.1,2,10: Z.17.
* from x, y to r, 6: 1768.
* from .V, y, ;; to r, 6, : 1783.
* in a definite multiple integral : 2774 —
9: A.22,41.
in transcendent definite integrals :
C.23.
in the theory of isotropic means : C.34.
Characteristics : E.5 : J.71 : M.6.
of conies : A.l : C.67,72,76,83. : JP.28 :
LM.9: M.15: ISr.666,71.
of conies of 5-point contact : E.27.
of cubic systems : 0.74-2.
of curves and surfaces : C.73.
of quadrics : C.67 : JP.28 : N.68.
relation between two characteristics
in a system of curves of any de-
gree : C.62.
surface groups defined by two : C.79 :
;x = l, v = \, C.78.
Chart construction: Mel.2: ]Sr.60,78,-i.
Chemico-algebraic theory : AJ.l.
Chemico-graphs : AJ.l.
Chess board, ths and prs : A. 56 : E.34,
42,44.
Chessmen, relative value of: E.39 :
Mel.3.
Chinese arithmetic and algebra : C.51 :
N.63.
*Chord : joining two points on a
circle: 4157: A.43,44: E.22.
* of contact for two circles : 4172.
Chronology : J.26.
*Circle: 4136— 90c.c: A.l,3th 4,9,2-5,27,
th47: C.94: J.14,17: Q.19: TE.
th6.
* approximate rectification and quadra-
ture: 6019, &c: A.2,6,geol3,14,
43: J.32: Me.75,85: N.45,47 :
Q.4.
arc of : see Circular arc.
area of segment: 6035: A.27,39,44.
* chord of: 4157—8.
* chord of contact : 1017 : 4138,-72.
* coaxal: 1021—36: A.23.
configuration of : C.932.
854
INDEX.
Circle — {continued) .-
* en. from 3 conditions : 937 : JP.9.
* Cotes's properties : 821: A.ll,ext.to
ellipse 30: P.13 : TI.7.
* cutting? three at given angles : 4185 :
LM.3,5: N.83..
division of: A.27,37,41 : At.l9 : E.l,
31: G.6: C.85,93: J.27,54,56:
N.53,54.
and theory of numbers : J.30,842,87 :
N.56.
ths. on sum of sqs. of perpendicu-
lars, &G. : 1094—8.
* eight circles touching three, en :
4189 : Mel.3 : _Q.5.
eight through 6 points of intersection
of 3 conies : Q.IO.
* equation of: 4136 — 48, p.e 4151 :
Pr.27: TI.26.
* general eq. : t.e, 4691,4761 : tg.c,
4906.
Euler's th. extended to ellipse : A. 51.
five-point th. : E.5.
four pairs of circles through 6 points
common to 3 circles : Q.9.
four points concyclic, condition : A.
44: N.84.
geometry of: A.67 : Z.24.
groups of points on : A. 14.
in tri-metric point-coordinates : Z.27.
* and in-quadrilateral : 733 : CD.9.
lines of equi-difterent powers in two
circles : A. 19.
* of curvature : 1254,5134: A. 31,63: J.
45: p.cN.84.
* polar of x'y' : 4138, — 64.
rectangles of: Z.14.
ring of, touching two fixed : J. 39 :
Me.78.
six points th. : Q.8.
and self-conj. triangle: A.41.
and sphere, geo. : Mo. 82.
system through a point on a plane or
sphere: G.16.
* tangents : 4137—43,4160 : L.56 : P.14.
common to a circle and conic :
cnA.69.
* common to two circles: 953,4171:
en A. 34.
* locus of a point, the tangents from
which to two circles have a given
ratio : geo. 965 — 6.
* three: 997—9,1036,1046—51,4183—7.
* prs.{Gerf/onne) : 1049 : At.l9 : L.46.
tlu'ough mid-points of sides of a tri-
angle : see Nine-point circle.
* tlirougli 3 points : 4156,4738.
through 3 points on a conic: A.2 :
J .39.
touchintr iicoiiic twice : J.5<) : N.65.
Circle — {continued) :
* touching 3 circles, en : 946,1049 :
A.24,26,28,35 : An.68: C.60: Me.
62 : N.63,65,66,84 : Q.8.
touching the 4 circles which touch
the sides of a spherical triangle :
A.4.
and triangle, ths and prs : A. 30, 57 :
LM.15: Q.4.
* two; eq. for angle of intersection:
4180—1.
* theorems: 984—1046: Q.ll : see
" Radical axis " and " Coaxal cir-
cles."
and two points ; Alhazen's pr : A J.4.
Circulants: final expansion of: Me.85.
of odd order : Q.18.
*Circular : arc : length, centroid, &c. :
6019.
with real tangents : Z.l.
gi'aphic rectification and trans-
position of : Z.2.
cubics, involution of : LM.1,7.
chord of curvature of : E. en 36.
and elliptic functions in continued
fractions : CD.4.
* functions: 606: A.17: J.16.
points at infinity : see " Imaginary
ditto."
relation of Mobius : LM.8 : N.76 : Q.2.
* segment : arc, chord and area : 6035 :
^ N.63.
Circulating functions : P. 18.
*Circum-centre of a triangle : 4642 : tg.
eq 4883.
*Circum-circle : of a triangle : 713,
4738: tg.eq4895: A.51,58.
coordinates of centre : 4642.
hypocycloidic envelope of Ferrers :
N.70.
and in-conic : N.79.
* of a polygon: 746—8: A. 19.
* of a quadrilateral : 733 : ISr.79.
Circum-cone of a quadric, locus of
vertex : N.52.
*Circum-conic : of a triangle : 4724 :
tg.cq4892: An.57.
* of a (piadrilateral : 4697 : At. 54.
locus of centre : E.l.
Circum cubic of a complete ((uadri-
lateral: G.IO: Q.5.
*Circum - parallelogram of an ellipse:
4367.
Circum-pentagon of a conic : M.5 : N.67.
Cireum - polygon : of a circle :
746— 8 ': CD.l : Mc.80: N.66.
of a conic : !M.25.
of a parabola : CJ\r.2.
of a eusjiidiil cubic: TiTM.l:!: ditto
([uartic : JiM.14.
INDEX.
855
W
Circum-quadrilateral : oi: a circle :
E.35: N.48.
of two circles : N.C7.
Circum-rhombus of au equil. triangle :
AAo.
Circum-triaugle : of a conic : N.70.
* locus of vertex : -iSOO : E.35.
of a triangle : J. 30.
*Cissoid: 5309—12: A.62,69 : N.43,85.
tangents of : LM.2. ^
Cissoidal curves : A.56.
Clairaut's function and equations: Pr.
25.
Clinant geometry : Pr.l0,ll2,12,15.
Closed ciirves : Me.77 : geo tli QA.^
* and moving chord, Holditch's tb: 5244 :
gzMe.78: Q.2.
ext. to surfaces : Me.81.
quadrature of : A.61 : lsr.43 : Crofton's
tbsA.55: C.65,68: E.36;.
^^B^clvdy, t = 0, t' = being the
tangents from x)j : LM.2.
* rectification of : 5204.
Closed surfaces : JP.21.
*Coaxal circles : 1021—36 : 4161—70 : Q.
52 : reciprocated, 4558.
Poncelet's limiting points: 4165:
thJ.86.
Coaxal conies : Q.lO.
Cochleoid, {;c^ + y~) tan" ' ^ = Tvry : A.70.
*Coefficients : detached: 28.
* differential: 1402.
* indeterminate : 232.
*Cogredients : 1653.
*Collinear and concurrent systems of
points and lines : 967 : A.69 : G.21.
Collineation and correlation : M.22 :
prM.lO.
and reciprocity : M.23.
" gleichstimmigkeit " of, in space :
Z. 24,28.
multiple c. of two triangles : A.2,70.
of plane figures, ground forms : J. 74.
paradox : Z.28.
*Combinations : 94—107 : trA.15 : CP.8 :
G.18 : J..5,13,21.,34,38,th53 : M.5 :
Z.2.
ap. to determinants : JP.28.
complete, i.e., with repetitious : C.92 :
N.42,74.
compound : Man. 79.
* C {n, r) an integer : 366 : L.42.
G {n, r) when n is fractional : A.70.
* C in,r) = C {n-l,r-l) + C {n-l,r) :
102.
C {m+m',p) = 2'^ C {m, r) . C {m,ij—r) :
L.42.
Combinations — {continued) :
^ problem or theorem : 105 — 7 : A.21 :
C.97: CD.2,4,7,8.2: L5 : J.3,45,56 :
L.38:Mem.ll: N.53,73 : Z.15.
of Euler and its use in an eq. : L.39.
of 1,2, ... n, each c. having a sum
>a: G.19,20.
of dominoes: An.73.
of n dice each with jj faces : TE.21.
of n points in space : L.40.
of observations : L.50.
of planes through a system of
points : N.57.
Combinatorial : products : A. 34.
systems : L.56.
Combinatory analysis : A.2,50,70 : J.ll,
22: Mera.50: K80.
Commensurable quadratic divisors : N.
47.
Commensurables : TE.23.
*Commutative law : 1489.
*Companion to the cycloid : 5258.
Complanation formula : A. 48.
Complementary functions: C.19 : J.ll.
Complete functions : C.86 : J.48i.
Complete numbers : Mo. 62.
[ *Complete primitive : 3163.
Complex axes of a quadric : Z.19.
Complexes : L.44,47 : M.2,4.
of axes of a quadric : N.83.
in combinations and permutations :
A.21.
of 1st and 2nd degrees and linear
congruences : Au.76 : L.51 : M.2,
9: N.85: trZ.27.
linear: N.85 : Z.18: of an in-conic of
a quadrilateral : G.21.
of 2nd degree: G.8,17,18: cnJ.93:
M.7 : N.72e.
of 2nd degree with a centre : L.82.
of 2nd degree of right lines which cut
two quadrics harmonically : M.23.
quadratic ray- & web-complexes : J.98.
of nth degr-ee, singularities : M.12.
and congruences, spherical of 2nd
degree, their circles and cyclides :
J.99.
and spherical complexes, ap. to linear
p. d. e : M.S.
tetrahedral in point space: Z.22.
Complex numbers: A.28: C.90,99: G.ll:
J.22,35,67,93: L.54,75,80: M.22:
Mo.70: Q.4.
from the 31st roots of unity : Mo.70.
from the nth roots of unity : J.40 :
Mo.70.
index and base of a power, geo : Z.5.
prime and from roots of unity : J.35 :
taMo.75.
856
INDEX.
Complex numbers — {continued) :
prime and from the 5th roots of unity:
Mo.ta 59.
resolution of A"+B" + C" = 0, and
when n = 5: L.47.
from the roots of unity ; class num-
bers : J. 65: Mo.61, 632,70.
in theory of residues of 5th, 8th, and
12th powers : L.43.
Complex unities : C.96,99 : J.53.
Klein's groups : J.50.
Complex roots of an algebraical eq :
M.l: N.443: oix"= 1, M0.57.
Complex variables, functions of: An.
59,68,712,82,83: G.3,6 : J.54,73,83 :
M.19: Z.82,10.
especially of integrals of d.c : J.
75,76.
Composite functions of a higher order :
G.2.
Composite numbers : for construc-
tion of factor tables : A.45.
groups of : J. 78 : LM.8 : Me.79.
" Compteurs logarithmiques " : C.40.
*Concavity and convexity: 5174.
Concentric circles : LM.14.
3 quadrics, intersection of : E.39.
*Conchoid: 5320: A.55 : N.432.
Concomitants of a ternary cubic :
AJ.4.
*Concurrent lines and collinear points
967-76: A.69.
th. on conic and triangle : E.35.
*Concyclic conicoids : 995 : Q.ll.
*Cone : 1150—59, 6043 : A.16 : L.61 : Me.
62.
and cylinder, superfices, tr : An. 57.
general d.e of : E.18.
intersection of two : N.64.
* oblique: eq5598: J.2 : Me.80.
* sections of: 1150—9.
* and sphere : 5652 : thsMe.64.
superfices of oblique frustruni : J.2.
through m points and touching 6 — in
lines: LM.4.
volume of frustrum : Ac.41 : N.13.
Configuration of 16 points and 16 planes :
J.86.
(3, 3),o and unicursal curves : M.21.
*Confocal conies : 4550 — 8, 5007, tg.e
5005: J.54: LM.12,13 : Me.66,68,
73 : N.80 : Q.IO : TE.24 : Z.3.
* Graves' theorem: 4555: Griffitli's ext.
LM.15.
* tangents of: 4555.
*Confoc'al (luadrics : 5656 — 72: A.3 : CD.
4,5,92: G.16: M.18: thMc.72: Q.3.
relation to curves and cones : CD. 4,9.
volume bounded by three and the co-
ord, planes : A.36.
M.
Confocal surfaces : Me.66.
Conformable figures : A.59 : LM.IO:
19:Z.17.
Congruences : C. 51,88 : th and ap.T.19 :
thMe.75: N.50: P.61.
binomial : AJ.3 : C.61 : expon. to base
3, M0I.4.
classification of roots : C.63.
Cremonian : LM.14.
irreducible : J.40.
and irreducible modular functions :
C.61.
linear: LM.4: of circles in space : C.93.
numerical : An.60 ; multiphcation of,
61.
of 1st degree : A.32.
in several unknowns : L.59.
sol. by binomial factorial : Mem.44.
transformation of modulus : Mel.
2: N.59.
with composite modulus : E.30.
of 2nd degree: C.622: Mem.31 : re-
duced forms : C.74.
of 3rd order and class : LM.16.
higher, with real prime modulus :
An.83: J.31,54,99.
resultant of systems of linear :
and trigonometrical functions ;
x^+'if^l {mod.p): J.19.
aj' = 1 (mod. jj) : J. 31.
Congruent divisors of a number,
A.37.
Conical functions : M.18,19.
*Conical surfaces : 5590 : A.63 :
LM.32: M.3.
through 6 points, locus of vertex : J.92.
*Conicoids : 5599 : A.48 : Q.tg.c 9, q.c
and t.c 10.
50-point : Me.66.
*Conics: 4032—5030: A.l2,5,17,31,32,60,
68: C.83: G.1,2,3,21: J.20,30,32,
45,69,86: M.17,19: N.42,435,44.,,
45,71.>,752,82 : P.62 : Q.8,tg.c9:
Z.18,21,23.
anharmonic correspondents, problem
of 5 conies and 5 lines : N .56.
of ApoUonius : L.58.
* angles coiuiected with : 4375.
arcs similar to: N.44.
* areas of (see also " Sectors ") : 4688,
6097—6121 : N.46 : t.cQ.2.
* auxiliary cii'cle : 1160.
* centre : coordinates of 4402, 4267,
t.c 4733 and 4742 : tg.eci of 4901 :
ths and prs N.45.
* locus of 4520, 5028.
* chords of: 4315,4322; p.c 4337 and
CD.l: Mc.66: see also " Focal."
cutting an ellipse at a given angle :"
E.28.
C.88.
J.19.
of:
Ac.5;
INDEX.
857
Conies : chords of — [continued) :
* intersecting: 1214, 4ol7.
moving round an ellipse : A.43,44 :
* chord of contact: 4124,4281; 4699—
4721.
* and circle, intersection th : 12G3:
A.&9: N.64.
collinear relation to circle : Z.l.
and companion quadrie : An.60 : JP.7.
conjoint lines of : L.oSo.
* conjugate diameters: 1193 — 1213,
4346 ; ths 1278—85 : CM.l : L.37 :
N.42, ths 44,69 : Q.3.
* parallelogram on : 1194: 4367.
relation to ellipse when equal : A. 18.
conjugate points : Q.8.
* construction of: 1245,4822: A.28,43:
E.29: Q.4: N.59,73: with help of
circle of curv. A. 24.
* from conj. diameters : 1253: A. 52.
* contact of : 4527—33 : A.1,60 : C.78 :
Pr.34.
at 2 points : Q.3 : N.74.
* do. with each of 2 conies or circles :
4803-6: CD.5,6: E.31,34.
4-pointic with a quartic : M.12.
* 5-pointic : 6190-1 : C.78o : E.5.,,23 :
J.21: P.59.
with surfaces : 0.91-: : P. 70, 74.
convexity from focal pi'operty : N.56.
criterion of Mobius : J.89.
* criterion of species: 4464 — 77; t.c
4689: 5000.
* curvative: centre and radius of:
4534—49 : geo 1254—66 : thsMe.
73: N.79,85.
* geo on: 1265: A.17.
* chord of: 1259,-64: Q.6 : locus of
mid-point A.70.
as curves in space : tr A.37 : M.64.
* definitions : 1160.
degenerate forms : LM.22.
* diameters : 1214, — 35 : eq 4458 : en
Me.66 : N.65.
* director circle : geo 1217, eq 4693 — 5 :
o.cE.40: LM.13: N.79.
* directrices: 1160: trA.63 : LM.ll.
* eccentric values of coordinates : 4275 :
CM.4.
* eccentricity: 1151,4200.
elementary formula : G.9.
* ellipse and hyperbola : 4250 — 96.
* equations of: 4251,4273; p.c 4336;
tg.c 4663,4870 : J.2.
* general: 4400,4714,4719; t.c 4755
and 4765; tg.c 4664 and 4872;
p.c 4493 : t.c A.51 : CP.4,5 : t.c
G.6,7: Mem.52: N.43,45,65. (See
also " Conies, general equation.")
Conies : equations of — {continued) :
* intercept: 4498.
'^ equations of parabola : 4201 ; t.c
4775; p.c 4336.
* general : 1430,4713 ; t.c 4656 with
4689 ; tg.c 4974 with 5000.
* ay:= A/SS and derived equations : 4697
—4719: Q.4.
* ay = h(3'' or LM= B^ : 4699, 4784 : N.
44: S+L' = 0,&G.: 4707.
«= La"+3II3-+Ny' = 0: 4756, '65 : Me.62.
equation in p and p : Q.13.
^ equi-conjugates, gen.eq: 4491.
formute: J.39 : N.62.
from oblique cone : L.38.
* general equation ; cond. for a circle :
* 4467: t.c 4691 and Me.68 : Q.2 :
from eq. of axes : JSr.67.
* cond. for an ellipse : 4464 ; t.c 4689.
* cond. for a hyperbola : 4468 ; t.c
4689: A.39.
* cond. for a rectangular hyperbola:
4737 ; t.c 4690 ; tg.c 5000.
* cond. for a parabola : 4430 ; t.c 4696,
4735, 4746 and 4776 ; tg.c 5000.
* cond. for two right lines : 4469,
4475, t.c 4662.
generation of : N.75 : Z.23.
by a moving chord of a circle : A.34.
* Maclaurin's method : 4830 : LM.4.
* Newton's method : 4829.
graphic problems : N.80.
Halley's pr : N.76.
harmonically in- and circum-scribed :
Q.18.
intersecting in 4 points : J.23.
intersecting a surface in 6 points :
C.63.
with Jacobian = : M.15.
limiting cases : 4465—77 : Me.684.
* normals: 1171,sd5629— 32 :A.16,24,32,
en 43,47 : An. en 64,78 : C.72,84 :
J. en 48,56,62 : Me.66 : Mel.2 : N.
70,81: Q.8: Z.11,18,26.
circle through feet of: N.80.
cutting off the min. or mas. arc or
area : N.44.
dividing ellipse most unequally :
E.29.
* eccentric angles of the feet of four,
th: 4334.
* equations of : 4286, 4483, 4512.
* intercepts : 4294 : segments ; 4309,
4486.
least distance between two : A.21,38.
number of real : J.59 : N.70,722.
number cut by 8 lines in space : J.
68.
number under double conditions : C.
59.
5 R
858
INDEX.
Conies — {continued) :
octagrara : LM.2.
* passing throngh given points and
touching given lines : en 4831 —
40.
6 points : en 4831, eq 5024: A.27 :
A.9,24,64: An.50: N.57.
4 foci of a conic : Q.5.
4 points: l!i.6Q: Q.2,8: Z.9.
4 points, envelop of: E.28 : Do. of
axes, N.79.
* 4 points, locus of pole of a line :
4770.
* 4 points and touching a line : en
4833 : A.65.
3 points and touching a circle
twice: N.80.
3 points and touching a line : Q.6.
3 points with given focus : en A..54.
2 points and toucliing a line : Q.2.
2 points and touching 2 lines, locus
of centre : G.7.
four such conies, th : Q.8.
1 point and touching 3 lines : Q.8.
parameter of : N.43.
* perpendicular from centre on tan-
gent : 1195, 4366—73.
* ditto from foci : 1178, 4300.
of 8 points : J.65.
of 9 points : A.43 : G.l.
of 9 points and 9 lines : G.7,8.
of 14 points : Me.66.
pencil of : M.19.
and polar, Desarques' th: N.64.
* i^ole of chord joining Xiy^, x.2y-2 : 4326 :
parabola, 4218.
* properties of: 1274: A.4,25,70 : p.cJ.
38.
quadrature of : TE.6.
and quadrics : A.30 : L.42 : N.58.,,66,,
ths 73 : geo interpretation of
variables, 1^.66.
* rectification of: 6071, 6084: P.2 :
TI.16: Z.2.
* reduction of it, u' to the forms
x-+y'+z^ = 0, ax^+^y'+yz- = :
4995.
series of : A. en 67 and 68.
* seven points of: C.94.
* similar : 4522.
six points of : A. 62 : J.92.
systems of : C.62,65 : M.6 : liiM : of 2,
Q.7 : of 4, L.54.
and orthogonal lines : N.71.
aiid quadrics : Z.6.
* tangents or polar : 1167—9, 4280—5,
4790: gn.eq 4478: A.61 : enCD.
3: CM.l,p.eq4: K79.
intercepts on the axes : 4292.
* two at the origin, eq : 4489.
Conies : tangents — (continued) :
* two from x'y' : 4311 : gn.eq 4488,
4965, A.57 : t.e 4680—2, en 1181:
A.53...
* do. for parabola: 4216, en 1232:
ratio of lengths, 1243.
* quadratic for on : 4313 : paral^ola,
4220.
* quadratic for abscissas of points of
contact : 4312 : parabola, 4216.
* subtend equal angles at focus :
1181, 1234 : CM.2.
locus of x'y' : A.69.
* segments of : 4307.
* at the points n, tan ^, ficot(f):
4799.
tangent curves of : Z.15.
* theorems: 1267: A.54 : J.16 : L.44:
M.3 : N.4.55,483,72,84 : Q.4,6,,7 t.c :
by Pascal, Desarques, Carnot, and
Chasles, A.53.
conic and triangle, Q.5.
3 circles touch a eonic in A,B,C and
all cut it in D; A,B,C,D are
concj^clie, J.36.
* three : 4707, 4710 ; in contact, 4803.
* Jaeobian of : 5023.
* touching : a conic and line, eond. :
5017.
a curve twice : J.45.
curves of any order : C.59.
five curves : C.SSj.
* four lines: 4804: locus of centre,
4772, 5028 : locus of focus, 5029 :
N.45,67.
n lines : N.61.
a group of lines, and having a given
characteristic and focus : A.49.
a quintic curve in 5 points, no. of :
'NM.
* two circles twice : 4806.
* two conies twice : 4803.
* two sides of the trigon : 4784 —
4808.
transformation of : G.IO.
* two : 4936-5030 : N.58.
* with common chords or tangents :
4700—5.
* common elements, en : A. 68.
* with common points and tangents:
4701—7 : LM.14 : Z.18.
* at infinity : 4715 — 6.
* common pole and polar, en : 4762.
* condition of touching : 4942, t.c
5021.
* intersecting in 4 points : 4700 : A.
32: atoc, A. 16.
* ])oiiits of interseetiou : tg.e 4973 :
en A.69.
reciprocal properties : E.29,
INDEX.
859
Conies : two — {continued) :
* reduction to x-+y^+z-^=0 and
a.v-+l3if-+yz'-=0: 4995.
* six chords of, eq : 4941.
* tangents of, four: eq 4981 : J.75:Q.3.
* under 5 conditions : 4822— 43 : L.10,59.
6 conditions, en : 0.59.
7 conditions in space : 0.61.
Coniufjate : functions : apLM.12 :
TI.17.
lines of surfaces : CD.9.
* points : 1066, 5184 : in ellipse, A.38.
point-jtair of a conic : Z.17.
tetrahedrons of a quadric, each ver-
tex being the pole of the opposite
side: N.60,88.
triangle of a conic : N.88.
Connexes (1, ?^) corresponding to d.e :
A.69.
in space : Mel. 6.
Hnear: M.15.
of 1st order and class in simple invo-
lution: G.20.
of 2nd order and class : G.19.
analoguein anal. geo. of space: M.14.
Cono-cunei : A. 2.
Constant coefficients, theory : 1.23.
Constant functions and their deriva-
tives : A.15.
*Contact and circle of curvature : 4527,
5188, 5134 : A.30.
^Contact of curves : 5188 : cir. of curv.
A.53 : C.32 : J.66 : P.62 : Pr.ll :
Q.7.
with a parabola : Z.2.
with faisceaux of doubly infinite
curves : C.83.
with surfaces : L.78 : with triangles,
M.7.
Contact : of lines with surfaces : L.
37: Q.1,17: Z.12.
of an implexe with an alg. surface :
C.84.
of quadrics : LM.5.
4-pointic on an algebraic surface : M.9.
of spheres : JP.2.
of surfaces: J.4 : JP.15: P.72,74,76 :
Q.12.
of 3rd order between 2 surfaces : C.74;.
problem: of Apollonius, A.66 : spheri-
cal, At.19.2.
transformation : C.85 : M.23.
Contingence angle of : see " Torsion."
Continuants : AJ.l : Me.79.
*Continued fractions : 160— 87: A.18,33,
66,69 : An.51 : gzC.99 : CM.4 : th
E.30 : G.10,15 : J.6,8,ll4 and ap,
18,53,80 : L.50,58,66 : LM.5 : Me.
77: Mem.aptoi.c9,13:Mo.66:N.
42,tr49,56,66 : Q.4 : geoZ.r2.
Continued iractions— {continued) :
11 1 1 1 1 1 A JO T C5
and :; -— -- : A.42 : J .6.
a+ a+ a+ h+ a+ b +
b h+l
a+a + l+ a-\-2+&c.
h>a+l: Mel.l.
.v+x + l-\- X + 2 +
1 1
and
A.3O2
•2.v + l+2x+3+2x+5 +
■2+LL ^'^^ ^+-x-^= ^•3^'^^-
2+2+ q+q+
ascending: A.60 : Z.21.
* algorithm ]i>i ^= (f-l^n i + ^2'"-2 : 168 : J.
69,76.
do. ap. to solution of trinomial eq:
A.66.
combinator representation of the ap-
proximation : A. 18.
development: Ac.4 : C.87: Mem.9.
for e^ and log (1+^) : CM.4.
for e exp. a + h,v+cx'+ : C.87.
for x-'-.^-'+x-'-x-^'+ . J 27,28.
l—x'''-+x~*—x'^+
for UsinjB+VcosJB+W; U,V,W,
polynomials in x : J.76.
for {m-\-Vn)-7-'p: N.45.
for powers of binomials : CM.4 : Mem.
18.
infinite : A.SSa.
numerical values : Q.13.
Eisenstein's, TE.28: Wallis's, Mem.
15.
periodic : A. 19 (2 periods), 33 : G.16 :
J.53 : K42,43,45,462.
reduction of : J.46.
* reduction of a square root to a : 195 :
A.64: J.31.
do. of a cube root : A.8.
do. of an nth root : A.64.
Contingence, angle of : 5725.
Continuity, principle of : CP.8 : in rela-
tion to Taylor's and Maclaurin's
theorems, L.47.
*Continuous functions : 1401 : A.l 5 :
Ac.5., : C. 18,20, and discontinuous
40, of integrals of d.e 23 : TA.7.
Continuous manifoldness of 2 dimen-
sions : LM.8.
*Contragredients : 1813.
Contraposition : E.29.
*Contravariants : 1814 : G.12, of 6th deg
19.
* of two conies : 4990 and 5027.
*Convergeuts : 160—87 : gzC.98 : CD.5 :
J.37,57,58: N.46.
Convex polygon, intersection of diags :
N.80.
860
INDEX.
*Coordinates, transformation of : 4048 —
61,4871,5574-81.
*Coordinatc systems: 4001—31, 5453,
5501—6 : G.16 : J.5,45,50 : LM.12.
ap. to caustics : An. 69.
* areal : 4013 : Me.80,82, gn cq 81.
axial : N.844,85 : Q.IO.
* biangular: 5453—73: Q.9,8,13.
* bilinear : ap 5341 : A.32.
bipolar : N.82.
bipunctual : AJ.l.
Boothian : see tangential.
* Cartesian : 4001.
dual inversion of c.c and t.c : Q.17.
central: Z.20.
curvilinear : A.34 : An. 57,64,683,70,
73: C.48,54: CP.12 : G.IO, and
i.c 15 : L.40,51,82 : Mem.65 : Q.19 :
on a surface and in space, An.692,
71,73: including any angle, J. 58.
* eccentric values of : 4275, 5638 : CM.4.
elliptic: A.34,40o: J.85 : Q.7 : Z.20.
Puchsian functions of a parameter :
C.92.
hyperelliptic coordinates : J. 65.
* one-point intercept : 4026.
* two-point intercept : 4025.
linear : Z.21.
mixed coordinates : A. 13.
parabolic : C.50.
parallel: N.84,,85.
pedal : Me.66.
pentahedral : Me.66.
of a plane curve in space : LM.13.
* polar : 4003,sd5506 : Me. 76.
polar linear in a plane : Z.21.
quadrilinear : Me.62,64o.
* quadriplanar or tetrahedral : sd5502,
apA.53: Q.4.
surface: C. 65,81.
* tangential: 4019,4870—4915,5030:
Me.81 : Pr.9 : Q.2,8.
* tangential rectangular, or Boothian :
4028.
tetrahedral point coordinates : Z.8.
triaxial : A. 64.
trigonal: G.14 : Q.9.
* trilinear : 4006 : A.39 : Me.62,64 : N.
63: Q.4,6,6 : conversion to tan-
gential, 4876.
trimetrical point : A. 67.
Coplanation : Z.ll.
of central quadric surfaces : Z.8.
of pedal surfaces : Z.8.
Coresolvcnts : Q.6, 10,14 : non-linear,
TN.67.
Correlation of planes : J. 70 : An. 75,77 :
LM.5,6,8,10.
Correlative figures, focal properties:
LM.3.
Correlative or reciprocal pencils : M.12.
Correspondence : principle of : geo
thAn.71 : extC.78,80,ex83,852 :
N.46.
of algebraic figures : M.2,8.
application to Bezout's th C.81 ; to
curves, C.72 ; to elimination, N.
73 ; to evolute and caustic, N.71.
complementary theorem to : C.81.
determination of the class of the en-
velope and of the caustic of a
curve : C.72.
determination of the degree of the
envelope of a curve or surface of
n parameters with n — 2 relations :
C.83,.
determination of no. of points of inter-
section of 2 curves at a finite
distance: L.73 : C.75.
determination of the number of solu-
tions of n simultaneous algebraic
equations : C.78.
determination of the order of a geo-
metrical locus defined by alge-
braic conditions : C.82,842.
forms : M.7.
multiple in 2 dimensions : G.IO.
of curves: C.62 : P.68 : Q.15.
of two planes : LM.9.
of points : LM.2 : C.62 on a curve :
Q.ll on a conic : M.18 on two
surfaces.
for groups of n points and n rays :
M.12.
of two variables (2,2) : Q.12.
of 2nd deg. between 2 simply infinite
systems : An. 71.
Correspondent values, method of : P.
1789.
Corresponding points : in some in-
volutions : LM.3.
on two curves : M.3.
on two surfaces : C.70.
Corresponding surface elements : M.ll.
Cosmography, graphic method: N.82,79.
*Cotes"s theorem: circle, 821; areas, 2996.
analogous ths : CM. 3.
Counter-pedal surfaceof ellipsoid: AJ.4.
Coupures of functions : C.99.
*Covariants : 1629, 4936-5030 : C.80,81,
th90: G.1,20: Q.5,16: J.47,87,
90 : thM.5 : N.59., : ap to i.c C.66.
binary : G.2.
of binary forms : An.58 : C.82,86,87 :
G.17: L.76,79: M.22.
of binary quadrics, cubics, and (juar-
tics : An.65: J.50 : Q.IO.
of binary quantics : E.31 : Q.4,5,17.
of abinaryquartic: J. 53: quintics.An.
00.
INDEX.
861
Covariants — (continued) :
and coiitravariants of a system of
simultaneous forms in n varia-
bles ; to find the number of: C.84.
of quantics : An. 58 : binary, Me. 79.
of a septic, irreducible : C.87.
sextic : G.19.
of a system of binary cubo-biquadra-
tics ; number of irreducible co-
variants : C.872.
of a system of 2 binary quadratic
forms ; number of : 0.84.
of ternary forms : G.19o.
* of two conies : 4989, 5026 : of three,
Q.IO.
Covariants and invariants : An. 60.
of binary forms : An. 58, 59, 61, 83 : 0.
66,69 : reciprocity law, C.86.
of a binary octic, irreducible system :
0.86,93.
of a binary quintic : 0.96.
of a binary sextic : 0.96.j.
as criteria of roots of equations : An.
68.
Oribrum or sieve of Eratosthenes : N.
43,49.
Critical functions : see" Seminvariants."
Criticoids and synthetical solution : E.
9,26,32.
*Cubature of solids : see " Volumes."
Cube numbers, graphic en of: E.43 :
Me.85 : tables to 12000, J.42.
Cube root extraction: A.22,64: J.ll :
N.44,583: TI.l.
*Oube roots, table of (2 to 30) : p.6.
Cubes of sums of numbers : N.71.
Cube surfaces : Pr.l7.
Cubical parabola : Q.6,9.
metrical properties : M.26.
*Oubic and biquadratic equations : 483 —
501 : A.45 : No.1780 : An. 58:
O.geo41,90: JP.85: L.59 : M.3 :
N.79 : Z.8; Arabic and Indian
methods, 15.
Sturmian constants for : Q.4.
mechanical construction : LM.22.
Cubic and biquadratic problems : Au.702.
Cubic classes which belong to a deter-
mining quadratic class, number
of: A.19.
Cubic curves : An.7o : AJ.5,15,27 : 0.
37: thCD.7: CP.ll: G.l.;,2,14,
23 : J.ll,32,34,ths42,63,geo78,90 :
L.13,44,45 : M.15 : t.cMe.64 : Mo.
geo56: ]Sr.50,67,geo723 : P.67,58 :
Pr.8,9: Q.4.,5: Z.geol5,22.
classification of: Q.16.
and conies which touch them : J. 36.
coordinates, explicit functions of a
parameter : J. 82.
Cubic curves — {continued) .-
48 coordinates of: P.Slii.
with cusps : A. 68.
degeneration of : A. 4: M.18,15.
derivation of points : LM.2.
with a double point : M.6 : with two,
M.3.
with double and single foci : Q.14.
Geiser's : J.77>
generation of : G.ll : J.36 : M.5o,6 :
by a conic pencil and a projective
ray pencil, Z.23 : linear, J. 52.
and higher curves : A. 70.
mechanical construction of : LM.4.
number of cubic classes which belong
to a determining quadratic class:
A.19.
nodal, tangents of : LM.12.
of third class with 3 single foci : Q.
14,160.
18 points on : E.4.
inflexion points of : J. 38 : M.2,5 : N.
73,th83,85.
12 lines on which they lie in threes :
E.29.
rational : A.58 : G.9.
referred to a tetrad of corresponding
points : Q.15.
represented by elhptic functions :
JP.34.
and residual points : E.34o.
resolved into 3 right lines : M.14.
and right lines depending on given
parameters : J. 55.
16 cotangential chords of : Q.9.
and surfaces : J. 89.
synthetic treatment : Z.21.
tangential of: P.58 : Pr.9.
tangents to : cnE.25— 7,ths28 : LM.3 :
Q.3.
with a double point or cusp : M.l.
forming an involution pencil : LM.
132.
their intersections with cubics or
conies : 0.41 : TI.26.
through 9 points : O.C!i36,37 : L.54 :
two cubics do., OD.62.
through 8 points : Q.5.
through 2 circularpointsat 00 : cnZ.14.
transformations of : 0.91.
*Oubic equations: 483—91 : A.1,3— 7,11,
22, 25o, 32, 37,41,42,44,, prs47, 68 :
An,55: 0.num46,85: CD.4,6 : E.
35: G.12,16: J.27,56,90: LM.8 :
M.3 : Mem.26 : N.45,th52,th56,64,
66,703,75,78,81,84: TE.24: TI.7.
See also " Cubic and biquadratic
equations."
and division of angles : A. 15.
in a homographic pr of Ohasles : 0.54.
INDEX.
Cubic equations — {continued) :
irreducible case: A.30,39,41,42: No.
1799: AJ.l,: C.58: J.2,7: L.79:
N.67 : in real values, A. 49 : by
continued fractions, A. 2.
roots : condition of equality : E.30.
definition of: thA.31.
geometrical construction of: C.44,
45: Arab, m.s.s, J.40j.
integral : N.75o.
power-sums of : 0.54,55.
sq uares of diiSerences of : An. 56 : Q.3.
* solution of: Cardan's formula:
484: A.14,40,gz22 : Me.85.
Cockle's : CM.2,3.
* trigonometrical : 489 : A.19 : N.61,
07,71 : TE.5.
by differences of roots : J.42.
by continued fractions : A.10,39.
by logarithms : C.66.
by a; = — +/t: C.60.
z
* mechanical : 5429 or C.79.
numerical : C.44.
Cubic forms (see also " Binary cubics "):
thsJ.27.
ternary: J.28,29 : JP.31,32.
quaternary : transf J.58 : P.60 : Pr.lO :
division into five, J. 78.
Cubic surfaces : Ac. 3, 5 : thsAn.55 : C.
97.,98 : G.22 : J.63,65,68,69,88,89 :
M.6 : Mo.56 : N.69 : P.69 : Z.20.
classification of : M.14.
double sixers of : Q.IO.
with 4 double points : M.5.
"gobbe": G.14,17,21.
hypcrboloidal projection : G.2.
27 linesof: C.52,68,70: J.62 : L.69 :
M.23 : Q.2.
27 lines and 45 triple tangent planes
of: An.84: and 36 double sixers
Q.18.
locus of centre of quadric through 8
points : Me.85o.
model of: CP.12.
polar systems : M.20.
properties of situation : M.S.
in quaternions : AJ.2.
reciprocal of Steincr's surface: N.72o,
73.
singular points of : P. 63.
tri|)le tangent planes : 00.40.
Cubo-biquadratic eqs., no. of irreducible
forms: 0.87.
*Ourvature: 1254—8: 5134,5174: A.l,
28,43 : J.81 : Me.62,,f6-|.,72,75 :
N.th60,69: Q.t.c8,12.
* circle of: 1254—5, 5134: A.cn30,37 :
J. 08.
Curvature — (continued) :
* at a cusp : 5182 : N.71.
* at a double point : 5187 : Q.3.
dual, evohite and involute: Q.IO.
of an evolute of a surface : 0.80.
of higher multiplicity (Riemann) :
Z. 20,24.
* of higher order : 5188—91 : M.7,16.
of third order : 0.2G.
of intersection of 2 quadrics : An.63.
mean : 0.92.
* at a multiple point : 5187 : 0.68.
of orthogonal lines : JP.24.
* parabolic : 5818.
of a plane section of a surface : 0.78 :
Z.17.
spherical : A.25.
*Ourvature of surfaces : 5818—26 : A.4,
20,41,57: An.fandths61,64: 0.
th 25, 49, 60;,ths66,67,68,geo74,84 :
J. 1,3,7,8: JP.13: L.44,72o: p.c
Me.71 : Mel.3 : Q.12 : Z.27.
* average, specific, integral, &c. : 5826
—30.
axis of curvature of envelope of a dis-
placed plane : 0.70.
approach of 2 axes of finite neigh-
bouring curves : 0.86.
circular and spherical: see "Tortuous
curves."
constant : J.88 : G.3 : mean, 0.76, L.
41,53; neg.,O.60,M.16; pos.,G.20;
total, 0.972.
Euler's theorem : gzO.79.
Gauss's thO.42: analogy M.21: Q.
16.
ap. to aneroid barometers : 0.86.
indeterminate : CD. 7.
and inflexion : trA.19.
* integral : 5826.
and lines : An. 53, 59 : L.41.
mean = zero throughout : Mo. 66.
and pencils of normals : 0.70.
and orthogonal surfaces : P. 73.
of revolution : L.41 : Z.21,22.
skew : Z.26.
sphere of mean curvature of ellipsoid :
A. 43.
*Ourvcs (sec also "Curves algebraic"
and " Curves and surfaces ")
5100: A.2,1 6,32,66: An.53,54v-
O.geo72,91 : J.14,31,34,,63,64,,70
L.38,44,ths57 and 61 : M.16 : N
p.c61, 71,77,803 en from p.c.
from Abel's functions, p = 2 : M.l.
Aoust's problem : A. 2, 66.
arcs of, compared with lengths : JP
23.
of " aliineamento" : G.21.
analytical method : L]\l.9,16.
INDEX.
Curves — {continued) :
whose arcs and coordinates are con-
nected by a cjuadratic equation :
J.62.
whose arcs are expressible by elliptic
or hyperelliptic functions of the
1st kind: Z.-25.
argument of points on a plane curve :
LM.15.
bicursal : LM.4,7.
with branches : imaginary, CM.1,Q.7:
infinite, Q.3.
2 characteristics defining a system
of algebraic or transcendental
curves : C.78.
least chord through a given point :
A.23.
class, diminution of : N.G?.
closed : see '" Closed curve."
of 2-point contact with a pencil of
curves : M.S.
of 3-point contact with a triply infinite
pencil of curves : M.IO : 4-point
do., LM.8.
whose coordinates are functions of a
variable parameter : Me. 85 : ellip-
tic, SM : N.68.
cutting others in given angles or in
angles whose bisectors have a
given direction : C. 68,83.
and derived surflices : An. 59, 61.
derived from an ellipse : A.IO.
determination from their curvature :
P.83,84.
from property of tangents : A. 51.
determination of the number of curves
of degree r which have a contact
of degree 9i<'nir, with an wt-tic,
and which satisfy | r (r+oj—ii
other conditions, and similar
problems : C.GSg.
defined by a differential equation : C.
81,90,93,98: L.81,82.
do. algebraic and an analogous
space theorem : L.762.
^;,+0oy = 0, Pr.l5 : p oc r~, Mem.
24 : p" = yl sin co, N.76.
diameters of : L.49 : N.71 : and sur-
face, C.60.
eq obtained from tangent : N.45.
whose equations are : y = \/x, A.14,
16: v^ — u (tt— !)'(«— i«) C«— 2/)j
a; and ?/ constants, C.93.
* r (^ = a sin (/), A.48 : y =ir(i«). 2323.
linear functions of the coordinates :
N.65.
equidistant, tangents to : cnZ.28.
whose evolute and involute are equal :
C.84.
extension to space : C.85.
Curves — {continued) :
a family of: N.72.
four, with two common points : Q.9.
generation of : geoJ.58,71 : M.18.
by intersections of given curves :
Z.14.
by collinear ray-systems : Z.19.
geometrical : A.37 : two laws, 0.84.
relation to harmonic axes : C.734.
" gobbe": of zero kind, G-.ll : rational,
G.9,12.
higher plane : A.70 : L.61,63.
homofocal : N.81.
defined by intersecting conies : C.37.
intrinsic equation : CP.8 : Q.5.
joining two points : pr L.63.
with multiple points : C.62 : L.69.
with three of higher degrees, en
An.58.
n-tic with m.p of li— 1th order : C.
80: K76.
network of : C.67.
pencils of: A.65 : of 3rd order, Z.13.
p+P2^ = S4r: E.ll.
with a constant polar sub tangent : IST.
62.
with several " points d'arret " : N.
60.
in a power-series of sines : J.3.
* of pursuit : 5247 : C.973 : N.83.
of " raccordement " : JP.12.
rational: A.56 : G.thl5,16: M.9,18.
generation of : C12.
reciprocal of: J. 42.
rpd.
J 11
of section: A.43.
of a series of groups of points, ths
G.73a.
with similar evolutes : Me. 66.
* singularities of: 5187: Au.71 : C.78
80 : CP.9 : J.64 : JP.7 : L.37,45
LM.6 : M.8— 10.,16 : N.50,80,8l3
Q,2,7 : higher singularities, J.64 :
L.70.
of the species 1 : C.973.
sextactic points of : P. 65.
on surfaces : see " Surface curves."
systems of: An.61 : G.13 : Mo.82 :
theory, C. 632,94.
and surfaces : A. 73 : Ac. 7 : L.65.
tangential polar eq of : Q.l.
theorems or problems : A.prl,3l3.prs
37 and 42: G.L: J.l : M.14: Q.3.
re arc CP and chords GP, PM, CM
Mem.lO.
to describe curves which shall have
equal arcs cut off by a fixed pen-
cil of lines : Mem.lO.
J.l.
864
INDEX.
Carvca— {continued) :
re lines drawn at all points of a
curve at the same inclination to
it: C.74..
tracing apparatus : LM.4.
transformation of : CD. 8 : LM.l :
scalene Q.13 : M.4,20,21 : of 1-i-
ics which cut a quartic in the
points of contact of its double
tangents : J.52.
and transversals : J. 47.
under given conditions : P. 68.
Curves algebraic (see also " Curves and
surfaces ") : C.99,ths60 and 80 :
CM.4 : G.1,4,5 : J.12,47,59 : N.50 ,
81,cn83.
represented by arcs of circles : JP.20.
with axes of symmetry : N.80.
of 2nd class and 2ud order : G.l.
of 3rd class and 4th order : G.4.
of 3rd class and curves of 3rd order :
J.38 : L.78.
of 4th class with a triple and a single
focus: Q.20,.
of 6th class : Ac. 2.
of class n and order m, two laws :
C.85.
common points, a system of : J. 54.
generation by right lines : J.42.
and homothetic conies, ths : J.63.
lemniscatic : An. 58.
manifoldness of : M.IO.
* mechanical construction of an «-tic :
5407: LM.7.
with a mid-point : J. 47.
number of points of contact : C.82.
number of intersections : C.76 : M.15.
projective involution : M.3.
remarkable group of : M.16.
species determined : M.23.
symmetrical expression of constants :
Q.5.
theorems : two metric, M.ll : Mac-
laurin's, N.50.
Curves and surfaces : M.8 : N.59 :
gnthsC.45 and G.4 : algebraic,
An.77 ; J.49 and L.55.
" arguisianc ": G.12.
curves having the same principal
normals and the surface which
the normals form : C.852.
of same degree, a common property :
G.8.
satisfying conditions of double con-
tact: C.89.
Curvilinear angles, ths : L. 44,45.
Curvilinear triangles : A.Gl.j : N.45.
Curvital functions : C.60.
(Jurvo-graph : A.l.
*Cusps: 5181: Mem.22 : Q.IO.
Cusps — [continued) :
construction of 8 cusps of 3 quadric
surfaces when 7 are given : J.26.
* keratoid: 5182.
* ramphoid : 5183.
Cyclic : curves : A.37 : Z.cn2,26,27.
functions : A. 09, and hyperbolic 37.
interchanges (higher algebra) : Man.
62.
projective groups of points : M.13,20.
number of do. in a space transf. :
C.90.
surfaces : Z.14.
s^'stems : C.76.
Cyclides : N.66,70 : Pr.l9 : Q.9,12.
reducible: LM.2.
and sphero-quartics : P. 71.
*Cycloids : 5250 : A.13 : N.52,82.
and trochoids on surface of sphere :
Mem.22 : Q.19.
surface of, th of Archimedes : Me.
84.
Cycloidal curves : Z.9.
Cyclosis in lines : LM.2.
Cyclotomic functions : C.9O3:
*Cylinder, frustrum of : 6048.
circumscribing a torus of revolution:
C.45.
and cones, intersection by spheres,
ths : J. 54.
and hemisphere : P.12.
Cylindrical functions : A. 56 : An.73 :
M.5,16 : and d.i M.8.
of 1st and 2nd class : M.l.
/ (.!') analogous to the spherical func-
tion P" (cos 6) u : M.3.
representing a function of 2 variables :
M.5.
*Cylindrical surfaces : 5591 : LM.32.
quadratua-e of : A. 9.
Cylindroids : At.19,39 : Me.80 : Z.25.
Decimal fractions : approximation
by : N.51.
error in addition of non-terminating :
C.40: K56.
repeating: A.16,33,56 : G.9: Me.85:
Mel.5 : N.42,49,74.
- where w is one of the first 1500
primes : A.3.
Definite integrals : sec " Integrals."
Deformation : of conies : Z.26.
of a cache-pot* N.81.
of a one-fold hyperboloid: E.30.
of surfaces : C. 68,70 : G.16 : jr.22 :
L.60.
*Do Moivre's theorem : 756: A.6,n.
Demonstrations, reduction to simplest
form: C.83.
INDEX.
865
prs
Derivation : applied to geo
An.54.
of analytical functions, gz : G.22o.
of a curve : An. 52.
Derivatives : see " Differential coeffi-
cient."
* Arbogast's: 1536: CD.6 : 1.12: L
82 : extMe.78: P.Ol : Pr.ll : Q.
4,7.
Schvrarzian : CP.IS^.
Descriptive Geometry : An. 63 : L.39 •
N.52,,56.
*Detached coefficients : 28.
Determinants : 554 : A.44,56,65 : apAn.
57 : J.22,tr5l2,72,73,74,89 : C.86 :
CP.8: G.1,4,8,9,10: L.84: LM.
10: Me.62,78,794,83 : N.51,69,ap
70: Pr.8: Q.8 : TE.28: Z.16.
and algebraic " clefs " : 0.36.
of alternate numbers : LM.ll.
application to : algebra and
geometry, A.5l,50,5o.
contact of circles and spheres : N.
60.
cylindrical surfaces : A.58.
equations : Q.19.
geometry : J. 403,490,77.
surfaces of revolution : A.58o.
arithmetical: G.23 : LM.IO : "Me.78-
Pr.l5.
of binomial coefficients : Z.24.
catalogue of papers and treatises : Q.
18.
of Cauchy ("aleph"): G.17n.
combination of : CD. 8.
combinatory analysis of : C.86.
* composite : 555 : J.88,89.
* compound: 555: AJ.6 : LM.14 : Me.
82.
in conies : J.89,92.
with continued fractions : J.69.
cubic : G.6 : LM.13 : and higher, 11.
cycle of equations : G.ll.
of definite integrals : L.52 : Z.ll.
development of: An.58 : N.85 : in
binomials, G.IO: in polynomials,
G.13,15: and ap to resultant of
2 eqs, G.21.
division problems : A.59.
double orthosymmetric : Z.26.
and duadic disynthemes : AJ.22.
elements of : G.10,15.
equation in which Opg = a^p -. C.41.
of even order, analogy between a class
of: J.52.
of figurate numbers : G.9.
functional: CD.9 : J.22,69,70,77,84 :
M.1,18 : Me.80 : Q.l : of binary
forms, C.92 : of a system of func-
tions, L.51.
Determinants — {continued):
function in analysis for a certain de-
terminant of n quantities : C.70.
gauche (a^, = —a,,,) -. C.88,89o : CD.9 :
J.32,38,50,th55 : L.54.
involving ^1, &c. : Q.15,16,17.
of lower determinants : J.61.
of minors of given determinant: C.86.
* minor : 554 : G.l.
* multiplication of : 562,570- A 14 59 •
L.52.
number of terms in : LM.IO.
partial: C.97.
persymmetric : Me.82.
with polynomial elements : Me. 85.
of jjowers : AJ.4.
quadratic forms of : J.53,89 : L.56 •
K52.
ditto of negative dets. : J.37 : L.60 •
M.22: Mo.62,75.
of rational fractions : Me.82.
resolution into quadratic factors of a
det. formed from two circulants :
Me.82.
of the 16 lines joining the vertices of
two tetrahedrons : J.62.
of sixth order : Me.84.
* signs of the terms : 557 : E.29 : Me.
80.
skew: Q.8,18.
of squares of distances of points : Q.
11.
Sylvester's det. and Euler's resultant:
An.59.
symmetrical : G.l : J.82 : M.16 : th
Me. 85: Q.14,18.
and Lagrange's interpolation : LM.
13.
ap to a pr in geo : Z.20.
of nth. order and «.— 1th power x
sq. of a similar determinant :
AJ.4.
theorems and problems : AJ.3 : An.
pr60: G.2,4,6o,12,16 : J.pr66,pr
84 : L.51,54 : M.13 : Me.79 : N.65 :
Q.l,pr2,15 : Z.7,prsl8.
transformation of: An. 73: G.10,fl6:
of product, L.60.
unimodular, en : Z.21.
for verifying a system of d.e : 0.23.
with a diagonal of zeros : Me. 73.
Developable cylinders, motion of : Man.
84.
Developable surfaces : A.69 : M.18 : Me.
17: Q.6.
circumscribing given surfaces : Z.13,
15.
circumscribing 2 quadrics: C.67,ths
54,gz63 : CD.5.
5 s
INDEX.
Developable surfaces — (continued) :
of a conical screw : A. 69.
edge of regression : L.72.
of first 7 degrees : J.64.
througii a given curve which develops
into a circular arc : L.56.
through a gauche curve : C.97.
mutual : J.19.
quintic: C.54 : CD.6 : G.S..
of surfaces having principal lines of
curvature plane : C.36.
Development of tortuous curves : prs
Mem,
Diacaustic of a plane : N.75.
Diagonal scales : LM.6.
Diametral curves : CM.2.
of constant sectional area, prs : N.43.
Didon, proposition of : C.86.
Differences : and differential-quo-
tients : A.49,53 : N.69.
equations of mixed : JP.6 : N.85.
parameters of functions : C.95.
*Differential Calculus: 1400—1868: J.
11,,12,133,14„15,,16 : Me.66,: Q.4.
reciprocal methods : CD. 7,8.
*Differential coefficients or differential
quotients or derivatives (see also
" Differentiation ") : 1402, 1422—
46: Pr.l2.
of algebraic functions : Mel.l.
of log X and a' : A.l ; of x" and a'' : N.
63.
of {fix)]" : M.3 ; of y exp {z') : A.22.
calculated from differentials : AJ.16.
* of a composite function : 1420 : nth,
G.13.
* of exponentials and logarithms : 1422
—7 : A.11 : N.50,52,85.
* of a function of a function : 1415 : A.9 :
in terms of derivative of inverse
function, Mem. 57.
of a function of two independent
variables : 1815.
of irrational functions : P. 16.
of products whose factors arc con-
secutive terms of a series : Me. 31.
ratio to the function at the limit oo :
J.74.
successive or of «th order : 1405, 1460
—72, 2852—91 : A.l, 4,7 : An.57 :
G.18: M.4: Z.3.
* of a sum, product, or quotient: 1411.
independent repres. of: M.4.
* of a function of a function : G.13 :
« = 4, 1419.
of functions of several variables :
C.93.
of a logaritliniic! function : A.8.
of a product : 1460, 1472.
and summation symbols : J.33,ths32.
Differential coefficients of nth order —
[contimced) :
* of (a^'+a-^)": 2860; ^, 1467.
* (l-a;2)"-J (Jacobi): 1471, A.4 ;
y(a2_feV), A.3.
* -i_:U69;-^, 1470, J.8;
1+X^ 1 + .X'2
* ia+bx+cx-)'\ 2858;
{x''-+ax+b)-"\ A.8.
* tana: A.12 ; cos"'.c, A.9; ^^^^ ax,
1461, N.62.
* ^^" icS, 2862 ; sin-'«, 2854—5 ;
cos '
* tan-^aj, 1468, apN.9.
* e"-', e'"'y : 1463—4 ; e^'ic"*, CM.2.
* e""' : 2861, A.30 ; e"' cos bx, 1465.
* .c"-! logce : 1406 ; e^^^'^cos {x sin a),
2856.
of nth order with fc = in the result :
* tan-'..', sm--\c, (sin-'fc)2, 2865—9 ;
* ^°' »tsin-'.v, ^^'mcos-ifts 2871-7;
sui ' sm
* (l+cc2)±f ^"^ mtau-'x, 2883-7 ;
^ ' '' cos
* -^^, ofe'-'-cosbx, 2889—91 ;
e^— 1
sin (x or i/)-|-cosa; i o
l + 2y cos x+y^
*Differential equations (D.E.) : p.460,
3150 — 3637: A.1,52,67: AJ.4 :
An.502: C.8,15,23,29,42,54,70,83 :
CM.3: E.9: J.1,36,58,64,66,74,76,
76,78,86,91 : L.38,52,56: LM.4.10 :
M.8,12,25 : Man.79 : Me.81 : M^m.
30 : Mo.84 : N.72,80 : Pr.7 : TI.13 :
Z.4,16,27.
Abel's theorem : J.90.
algebraic : An. 79 : C.86.
with algebraic integrals : J. 84..
approximate solution : C.5.
by equations of differences : L.37.
* by Taylor's theorem : 3289.
of astronomy: C.9,29 : P.4.
asymptotic methods : C.94: Q.5.
Bessel's numerical solution : Z.25.
* Complete primitive : 3163—6 : J.25.
no. of constants : CP.9.
with complex variables : Mo.85.
of a conic : E.38.
continuous and discontinuous integ-
rals of : C.29.
for a conical pendulum : A.84.
relation between its constants and
the constants of a particular
solution: C.92.
INDEX.
867
D. E. — {continued) :
of curves having the same polar sui'-
face : An. 76.
* depression of order by unity : 3262 — 9.
with different, total integrals : L.84.
of dynamics: 0.5,26,40..: CD.2 : G.l,
4: L.37,49,52,5.53, 72,74: M.2,17,
25: Mel.4: Pr.l22: P.54,55,63.
ap. to engineering : JP.4.
and elliptic functions : L.49.
elliptic: G.19: M.21.
elliptic multiplier : M.21.
* exact : 3187, 3270—5 : G.12 : C.1,10,11.
of families of surfaces : Me. 77.
with fractional indices : JP.15.
of functions of elliptic cylinders : M.
22.
general methods : L.81.
* generation of : 3150.
geometrical meaning of : Q.14.
* homogeneous: 3186, 3234, 3262—8:
C.13 : CM.4 : J.86.
hyperelliptic : J.32,55 : Mo.62.
of hypergeometrioal series : J. 56,572,
73.
integrability of : Z.12 : immediate,
C.82.
whose integrals satisfy relations of
the form Fl(px^ = fx Fx -. C.93.
whose integrals are satisfied by the
periodicity modulus of elliptic
integrals of the first kind : J.83.
integrating factors : pp468 — 471,
3394: C.68.,97.
ofPdx+Qdy'+Ech: Q.2.
integration : by Bessel's function :
Me.80.
by Gamma function : TE.20.
* by definite integrals : 3617—28 :
C.17 : J. 74.
by differentials of any index : C.17 :
L.44.
by elimination : CP.9.
by elUptic functions : An. 79,82 : C.
41: JP.21.
by separation of operative symbols :
Z.15.
* by series : 3604—16 : C.10,94 : LM.
6: Me.79: Q.19: TI.7.
by theta-functions : C.90.
irreducibility of: J. 92.
isoperimeters, pr : Mem. 50.
of Lame : J. 89.
of hght : M.l.
in linear geometry : M.5.
of motion : C.55 : of elastic solids,
Q.13 : of fluids, CP.7 : of a point,
C.26.
with integrals " monochrome and
monogene " : C.40.
D. E. — {continued) :
'• Parseval's theorem : 3628.
and p.d.e of first order : J.23.
particular integrals of : CM.2 : alge-
braic, C.86.
relations of the constants : C.93 :
J.IO: JP.6.
in problem of n bodies : An. 83.
of perturbation theory : Mem.83.
with quadratic integrals : J. 99.
for roots of algebraic equations : P.
64: Pr.l3.
^ rule for equivalence of two solutions
3167.
* singular solution of : 3169-78, 3301—
6, 3401-3 : C.19,94 : CM.2 : JP.
18 : M.22 : Man.83,84 : Q.12,14.
of sources : A J. 75.
of a surface : G.2.
satisfied by the series l±2r^ + 2q'^±
2q^+&c. . 2Vq+2V(f+2i/T'+
&c. : L.49 : J.36.
satisfying Gauss's function F (a,^,y,a•) :
L.82.
synectic integrals of : C.40.
and tortuous curves : L.53.
transformation of : An. 52 : CD. 9 : in
curvilinear coords : J. 85.
D. E. linear : A.28,35,40,41,43,45,46,53,
59,65,69 : Ac.3 : AJ.7 : An.50,85 :
At.75 : C.7o,293,58,73,84,88,903,9l3,
92o,94: CD.3,4.2,9: CP.9,10 : G.
15 : J.23,24,25,40,42,55,63,70,76,
79,80,81,83,87,88,91,98 : L.38,64 :
M.5,11,12 : Me.75 : P.48,50,51 :
Pr.55,18,,193,20 : Q.8: Z.3,7,9.
without absolute term, condition of
solutions in common: C.95.
with algebraic integrals : C. 96,97 : J.
80,90: M.21.
determination of arbitrary constant :
At.65: q.l9-2.
argument & parameter interchanged
in the integral : J.78.
bibliography of : AJ.7.
with "coefficients that are algebraic
functions of an independent varia-
ble : C.92,94.
* with constant coefficients : 3238—50 :
An.64: CM.1.V2: E.34 : JP.33 :
L.42: N.47,84.
with periodic coefficients : C.91,92 :
doubly periodic : C.902,92,982 : J.
90.
with rational coefficients, algebraic
integrals of : C.96 : JP.32,34.
with rational coefficients, upon
whose solution depends the quad-
rature of an irrational algebraic
product : C.9I3, 922.
8G8
INDEX.
D. E. linear — {continued) :
with variable coeflficients : C.92 :
J.66,68,76: L.80,81.
•which connect a complete function of
the 1st kind with the modulus :
C.86.
homogeneous: Ac.l : J.90 : Mo.82.
integrating factors of : C. 97,98.
integration by Abelian functions, C.
92 : J. 73 ; by finite differences,
Q.l ; by series, J. 76.
which admit of integrals whose loga-
rithmic differentials are doubly
periodic functions : L.78.
whose particular integrals are the
products of those of two given
linear d.e : A.41.
irreducibility of: J. 76.
Landen's substitution, geo : J. 91.
Malmsten's theorem : J.40.
singular solution : J. 73,83,84.
transformation of : C.91,96.
* n variables, 1st order : 3320—32 : C.
14,15 : G.13 : J.20,80 : L.38.
n variables, 2nd order : L.37 : 2 varia-
bles, C.70.
n variables, any order : Mem. 13.
* Pdx+Qdy+Edz = 0:P,Q.,Bmwo\Yir\g
X, y, z, 3320 ; geoM.16 ; Z.20 : P,
Q, B, integral functions of x- only,
Q.19n: P = (rta;" + 6a'''-' + &c.)"",
Q, B similarly with y and z, Q.20o.
* Xd,( + Ydy + Zdz + Tdt = : condition
of being an exact differential,
3330.
iedt+ydx + zdy + tdz = : A.30.
*D. E. of first order, linear : p4G7 : C.86 :
G.13 : algG.18 ; M.23.
* exact: 3187.
* homogeneous: 3186.
integration by a particular integral :
C.86.
reduction to a continued fraction of a
fraction which satisfies a: C.98.
* separation of variables : 3185: CM.l.
* Mdx+Ndy = : 3184 : N.74,77.
* {ax + by + c) dx + { ax + h'y + c')dij ^0:
3205,p471: L.59.
{ax + hy + c)'>dx + {a'x + h'y + c'ydy : A.
64.
^^^L = o. r^Q being quartics in
vP \/Q
X, y : C.66 : LM.8 : ME.79.
/(;,;) dx . f(y)dy ^ . / (,.•) of 1st dcg..
F{x) "^ F{y) ' F(-..')of5thdcg.-
C.92.
D.E. of first order, linear — (contimied) :
P,d'C+P,dy+Q {xdy-y dx) = ; P„
P..> being homogeneous and of the
2)ih deg. in a-, y ; Q homogeneous
and of the ^th deg. : 3212.
Pi, Po, Q dif!erent linear functions
of X, y : C.78,83 : L.45
yr-\-Py = Q, where P, Q
only: 3210.
2/.+-P!/ = Qy": 3211.
yy -\-Py+Q- Mem .11.
y^+y~ =
J.24.
nvolve X,
where P,
{P+2Qx+Bx
Q, B are functions of x : Mem .11.
yr+a + hy+y^ = 0: J.25.
y,V{m+x)=\yym—x: A.42.
y, =f{y) : J.9 '; y. =f{x, y) : An.73 :
y.+fix) sin y+F{x) cos y+cfiix) = :
L.46.
* '!(;,+6zt2 = ca''" (Riccati's eq.) : 3214:
A.40: C. 11,85: (m = -6)E.7:
ext28 : JP.14 : L.41 : P.81 : Q.7,
11,16.
allied eqs : L.51 : Me.78 : Q.12.
sol. by continued fractions: Mem.18.
by definite integrals : J.12.
transformation of : Me.83.
*D. E. of first order : 3221—36 : A.29 :
C.40,45,66 : M.3.
two variables : An. 76 : J.40 : Mem.
62 : N.50 : singular solution, J. 88.
* Clairaut's equation, 9/ = j),v+/(p) :
3230: CM.3i: Me.77.
integration by second order d.e: A.46.
homogeneous in x and y : 3234.
reduction to alinear form with respect
to the derivatives of an unknown
function : C.87.
related transcendents : Ac. 3.
separation of variables : CD. 9.
* singular solution : 3230 : A. 56, 58 :
CP.9: J.48: Me.73,77.
* solution by differentiation : 3236.
* solution by factors : 3222.
transf. by elliptic coords : J.65.
verified by a recipi'ocal relation be-
tween two systems of values of
variables : C.15.
dx-+d)i'^ =z ds~ and analogou.s eqs :
L73.
* adx + hdy = ds: 3287.
dx"~+dy"~+dz^ = ds^: L.48.
INDEX.
D. E. of first order — (contwued) :
die^ + d>f~ + dz"- = X (£Za2 + f7/iJ2 + f?y2): L.
60.
F {u, u,) = : C.93.
* xct>{p)+yylr{p) = xip)- 3226.
D. E. of second order, linear : A.29,
32.55,64 : An.63,79,823 : C.82o,90,
91,93o,97 : J.51,74 : L.36 : Me.14 :
M.ll : Mo.64 : Z.5.
with algebraic integrals : C.90 : J.81,
85: L.76.
witli doubly periodic coefHcients :
Ac.2.
homogeneous : M.22.
integration by Gauss's series : Z.19.
transformation of : An. 62.
* y" = a: 3288.
y" = Py : C.9.
* y"^cfiy = Q : 3522,'5 : geoMe.66 :
Q = cos nx, 3626 : Q = 0, 3623—4.
y" = Ay {a+2bx+cx^)-" : L.44.
* y" — ax+hy. 3281.
xy" = y : Z.2.
(l=Fa;2) y"± my = 0, &c. : OM.32.
x^{y"+q^) =p{p-l) y : CM.2.
1/" = {h+n {n+1) lc^sn"x} y (Lame's
eq.) : C.85.
qj" = ^o+#i+i/02+&c., when (p^&c.
are trigonometrical series : C.98.
,/' = 2/(e-+e--)-2: L.46.
y" + ax'^y =/(m) : E.6.
y" = ay+yj/' {x) : A.45.
* y"=f{y): 3257.
* y" =f{x, y) (Jacobi) : 3286.
y" + ^y'-^ry = : 0.86,90 : Q.19.
y" = xhj' — nxy : A.63.
xy" -\-my' -\-nxy = : L.45,78.
xy"+y'+Aie"'y = : C.39.
x^'+rxy' = (bx'^+s) y : An. 51 : CD.
5.
x^y"+2xiy'+f{y) = : A.28,30.
y"+f{«^)y'+F{y)y'"~ = 0: 3284: L.42.
* {a+lx) y"+ {c + dx)y'+ {e+fa)y = :
A.58.
{a+bx")x?ij"+ (c+ex») xy'+ if+gx")
y = Q (Pfaff) : 3598 : J.2,.54 : and
like eqs., Z.2,3 : with h = 0, A.38.
a;2 (a—bx) y"— 2x {2a— bx) y'
+2 {2>a-bx) y = 6a3 : A.28,30.
^y"+ y'+y {x+A) = 0: Me.81,84.
xy"+y' + y{^c-A)=d.^^^^:Me.
82.
a.2y"_ 2xi/+2y = xhjf-^^ : A.28,30.
D. E. of 2nd order, Wwe^r—icontirtucd) :
si/" + (r+gr.')iy' + (2' + '*'^' + W''-'~) ^J = ^ :
A.23: Z.8,9.
* (1— a;2) y"— xy' + q~y = : 3282.
* (1 +ax^)y"+axy'±q^y = 0: 3283,3694.
2x {l-x^)y"—y'' +n{n+l)y = 0: Q.18.
X (l-a-) y"+ i-ix) 7/'+-;« 1/ = : Me.
82: Q.17.
im+x) {n+x)y" + {m—n)y'
-\Hm+x-fy = 0: A.42.
{ynx^+nx+p) y"+ (qx+r) y'+sy = :
JP.13: Z.4.
\fiy"-\-A\y'-\-Biiy = 0, fiy"+A'Ky'
+B'Kfiy = 0, and ixy"+AXij'+BiJL
= 0; with \^a+lx+cx'^ and
^ = b-\-2cx: A.423.
dr {{x—x^) yr}—xy — : L.54.
* y"+Py^+Qy + B = 0, P, Q, B being
functions of x : 3280.
Py"+Qy'+By = 0: Ac.l.
zy"2.+azyy"^.+f{y) = : Me.71.
D. E. of second order : Ac.l : An. 79 :
JP.29 : C.67,69,80,91 : J.90: L.39:
LM.n,12,13,16: Z.16.
with algebraic integrals : C.82.
derived from hnear eq : Me. 73.
with elliptic function coefficients :
Ac.3o.
iutheneighbourhoodofcriticalpoints:
C.87.'
polynomials which verify : Ac.6.
solution by definite integrals : A.27.
by factors : C.68.
by ChaUis's method, and application
to Oalc. of Yariations : A. 66,66.
yy" = hp+^Py"- ^■'^^■
Myy"+Ny'"-=f{x): N.79.
* y"+Py' + Qy'^ = 0, P,Q functions of
X : 3276.
* y"+Py'+Qy"' = 0: 3278.
* y" + Py'-^+Q.y"' = 0: 3279.
* y" + Q,j'-2-^B = 0: 3277.
J^j,-f — I^-|-I= 0, where I is Bessel's
function : J. 56.
of third order, linear: C.88-2: Q.
7,8,14 : M.24.
-y'": JP.16: Ut=^U3.r: C.3.
"— y = 0: Z.8.
= 3ma;V' + 6wi(M+2)ay + 3m(/i+2)
{lM+l)y: A.42.
= x"'{Axhj"+Bxy' + Cy): A.68.
third order: An.832: C.98o: M.23.
of higher order, linear : 3237 — 60 :
A.65 : C.972 : J.16 : M.4 : Q.18.
D. B.
y'-
xy
y'"'
y'"
D. E.
*D. E.
870
INDEX.
D. E. of liigher order, linear — [cont.) :
of ordei's p and vi-\~j), th : 0.43.
yi^ = xyx—y: A.l.
* 2/- =/(.'■): 3256.
i/„j. ^ a""i/ : L.39.
i/„^=(a+i3,iOy : J. 10.
9^'"*yiiu=^ u"'>j : A. 32.
x^'"y2,nx = ij : A.12.
a;?/,,^: := y : A. 26.
x"*y„x=±y : by definite integrals:
C.482,49 : J.57.
y„r = a;"'i/ : by definite integrals : J. 19.
a3"''*'i/(2„,+i)x =±2/ : by Bessel's func-
tion : M.2.
y,,^ = x"'y + A + Bx+ac^-+ . . . +N.C" :
Z.IO.
y,,^ = ^a;'»;2/,+Bc<!"'-'?/ : A.28,38.
x-"yn^ = Axy +By: A.33.
^«j-— a'?/ = y-2x+nh'V"ij : by definite in-
tegrals : J.17.
y„^ = Ax^y'i^+Bxy^+Cy: AM: M.3.
y„jc = J.a!'"2/'^+-Ba;"'~'i/j,+ Gx"'--y : A.
29,30,33,38.
a;i/nx + oy(n-i)x = hxy : J.2 : Z.IO.
a;y„j: — ^(„_i)j,4-i?^a;3'i/ := : A.40.
xynx+^in-i)^ = « i^y^+l^y) ■■ A.86.
■Axy„x+By(n-i)x=^ x"'{Axy:c+By) : Z.8.
«^,nx + q"'X^y =P ip — l) y{m-2)x : J.2.
= *'" (a.e2;y,, + 6.,-7/,+C?/) : A.38.
* i/,u-|-«i^/(»-i'.+ ... + «„y = : 3239 : A.
4o:
* ditto =f{x): 3243, 3516: a, ... a,.
functions of x, 3237 : J. 39.
n fractional and. all lower orders
integral : L.36.
{ch+ary=f{x):CM.A.
* ditto = e"-' : 3528 : ditto = sin mx -.
3529.
* {f+qxy'ynx-\-ai (25 + (2a0"-'!/(«-rix+...
+ any =f{x) : 3250 : with 2^ = 0,
C.96.
a,n,ny{m*n)x-\- ■ ■ . +(«m + .'') y ,nx + ...
+ aoy = : A.47.
x»-\a+hx) y„,+x"-%c+d) y,n-^>.. + ...
+ ty = 0: J.39.
y 3x -\- my j:+ 71 yT+])y = q : L.44.
*D. E. of higher order : 3251—69.
* y,.. = I'Mv/(„ .;,): 3258.
* ynx = F{y^n-2,.): 3260.
Bynr+Q = 0, where P, Q arc func-
tions of X; y, and the first n — 1
derivatives of // : J. 31.
D.E., simultaneous system of: An. 69,
82,84: C.10,43;,47,92o : CM.l: LM.
14: Me.13,80: Pr.12.
Harailtonian : Q.14.
integration and inversion of the in-
tegrals : C.23.
Jacobi's : CD.3.
* method of multipliers : 3353.
numbtr of arbitrary constants : Me.
* reduction of order : 3350.
redaction to a P.D. eq : C.44.
theorem of Abel : C.24.
theorem analogous to Lagrange's
in the Perturbation theory : L.
tlieorem of a new multiplier : J.27.
transformation and integration of :
L.45.
* Xit = ax + ly and y->t = cx + dy : 3354
and a similar example.
*D.E., simultaneous linear: 3340—59:
AJ.4: C.9,92: E.5: N.66,84.
Pfaff's method : C.14 : J.2 :
transformation of: J.98.
* ^^ = '^^ = 1:3346: Q.14.
jt dx dv
dz
3347.
Pi—xP Pi—yP Ps—zP
* xt+P(ax+by) = Q and y, + r{cx+d>i)
= B: 3348.
* txt + 2(x—y) = t and fyt + ix + i>y) = t^-.
3349.
* equations in a', x-u, x.u, &c. . (//, y^, y^t,
&c.: 3357.
* homogeneous in x,y,z...aud their
second derivatives only : 3358.
*D.E., simultaneous first order : 3340 —
49: C.43: J.48: Pr.62.
*D.E., symbolic methods: 3470—3636:
CD.l : P.61 : Q.3,172.
* F{de)u=Q: 3515.
* Hix+a-u = Q and similar : 3522.
* exceptional case of the inverse pro-
cess : 3526.
* reduction of an integral of the i!th
order : 3530.
* ax"'i/,nx+hx"y„. + &c. = Q : 3531.
* ay,„i-\-hy„«-\-&c.=j'{e'',s'u\6,cos6): 3535.
* a7ri"z + hni"z + &c. = '^^l + ^^i+&c. :
3540.
* F{7r)a = Q,: 3541.
* Kcduction to the form (n" + .l„n"-'-|-
...+An)u = Q, whore II — ^[dx +
Ndy + &c. : 3546.
INDEX.
871
D.E., symbolic methods — {continued) :
* F {xch, ydy, ...)u = ^Ax"'y" ... : 3558.
* to transiovm{a+hx + ...)u„x+{a' + b'x
+ ...)'W(„_i)x+ + ...&c. = Q intothe
symbolic forin : and tlie con-
verse : 3571, 3573.
* tt+ai0 (D) e,u+&c. = U: 3575.
* to transform m+<P {D) e'^it = C:
3579—80.
* to reduce a homog. eq. to the form
y„,+cpj = X: 3585.
*DifBerential expressions : 1407 : prAn.
85.
algebraic : An. 79 : M.9, by homog.
coords.
transf. of: J.85 : Q.I62 : Mo.69.
Differential : formuljB, theory of : L.
62.
functions, theory of: C.60.
expressions, linear : J.85.
parameters of functions : C. 66, 78.
quadratic forms : An. 8-4 : transfor-
mation of, A. 16.
* resolvents of alg. eqs : 3631 — 7 : An.
83 : C.91 : LIVl.1,9,14 : M.geo4,18 :
Me.75,82: Man.65,84 : Q.6,11.
* of y"— ny"-' +{11— l)x = 0, &c. :
3633—6: Man.65.
of^'»+%'-+cfc = 0: Q.17.
of 12 iy^+ay^+xy) = cfi : Me.82.
Differentiants in terms of differences of
roots of parent quantities : AJ.l.
*Dlfferentiation : (see also ' Differential
coefficients ') 1402-82 : CM.l.
* formula: 1411—72: An.59 : CD.2 :
CM.l : Pr.9.
by the method of " Eates " : Me.75.
general, i.e., with any index fractional
or imaginary : An. 58 : AJ.3 : CD.
3.,4,5 : J.12 : JP.13 : L.55 : N.84 :
Q.3,4: TE. 14,15: Z.16 : change
of independent variable, JP.15.
* under the sign \ -. 2253 : A.17.
successive : A.20 : Q.12.
when the function becomes infinite :
C.88.
Digits :- calculus of : Sbouimsky's
th : J.30.
frequency of in numbers : AJ.4.
origin of: L.39.
Diopbantine analysis : see " Partition
of numbers."
Di-polar geometry : Z.27.
*Direction ratios and cosines : 5511 — 14.
Directive algebra : N.68.
Directrix: of a conic: 1160: gn.eq
A.25: E.36: Me.80: t.cQ.13.
of in-parabola of a triangle : Me.80.
of a curve : A.20 : J.2.
of a parabola : gn.eqE.29.
of a qnadric : N.74,75.
Discontinuity : in curves : CM.4 :
Z.26.
in fractions : Man. 48.
in maxima and minima: CD. 3.
Discontinuous functions : A. 7 : C.153,
28: G.19: J.7,10: LM.6 : Man.
48: TI.21.
Discriminant : 1627, 1638—9, 1644 : Ac.
1 : J.90 : LM.2 : M.12 : N.59 :
Pr.l4: Q.10,11.
of an algebraic d.e of 1st order in 4
variables, and of its complete
primitive : An.84.
of alg. eqs., resolution into factors :
M.24.
of alg. functions : J. 91.
applied to conies and quadrics : A.
58.
of binary quantic : Au.56.
of a binary sextic : An.68.
of a quartic : ISr.83.
of a ternary quadratic form : Me. 68.
*Discriminating cubic : 1849, 5693 : G.
16 : J.26,71.
* proof of real roots : 1850 : A.29.
Displacement : theory of : N.82.
of plane figures : C.80 : N.73.
of an invariable dihedron : Me.85 :
infinitesimal ditto, C.84.
of an invariable figure : C. 51, 523,66,
922 : JP-26 : L.74,75.
of a figure, two of whose points slide
on two curves : C.82.
of a solid : L.40 : determination of
the normals to the lines or sur-
faces described : C.62.
of a system of points : C.78.
virtual displacement : J. 11.
infinitesimal : of an alg. surface : C.
70.
of bodies only defined by 4 coordi-
nates : C.73.
of a parallelogram : C.97.
*Distance: between 2 points: 4034 — 5,
t.c4601 : Q.7 : sd5508, 5510: some
relations, G.9.
correspondences for quadric surfaces :
LM.16.
of a point from a line and from a
plane : A. 57.
* of a point from a plane in a given di-
rection : 5559.
relatioiis : Z.27.
*Distributive law : 1488.
872
INDEX.
Divisibility : C.96 : E.8 : M.39 : N.67,74.
of decadic numbers : Z.22.
of numbers of the form 23'"+ 1 : Mel.Ss.
of a quotient by the powers of a fac-
torial : C.94.
of {x+yr + {-x)" + {-!ir ■■ Me.79.
Division: prJ.47 : prL.oO.
* abridged: 28: arithJ.31 : N.452,463,
52,54,57,81 : algN.42.
by 73 or 47 ; rule for remainder: E.22.
* effected by determinants : 581.
Foui-ier's rule : N.52.
of planes and spaces : J. 1,2.
of a rectilineal figure and of a spheri-
cal polygon : J.lOj.
* of a right line into equal parts : 950.
and transformation of plane figures :
A.4,.
of trapeziums, pyramids, and spheres :
A.11.
of triangles: A. 11, 17.
Divisors : of an integer, number of :
374: C.96: N.68.
of integral rational functions : Mem. 57.
* Newton's method of : 459.
of a pol3'nomial with commensurable
coefficients : N.75.
rational, of 2nd and 3rd degree: N.45.
* sum of : 377: Ac.f4 : L.56,57.
sum of powers of: L.58.
of;c2+^//2: L.49.
Double algebra : LM.15.
Double function, laws of change of
higher order : A. 21.
*Doablc points : 5178 : )i-tic with
hi (?i— 3), C.60 ; with I (n — 1)
(7i,— 2),M.2; Clebsch'sthsof these
quantics, C.84: n-ticwith^vu — 1)
(;/i— 2)— 2 double points, L.80.^
of plane curves in cubic space : Z.28.
in a locus defined by alg. conditions :
C.88.
of a pencil of curves : An. 64.
of plane curves in cubic space : Z.28.
in the projected intersection of 2 quad-
rics : ISr.84.
of tortuous curves : M.3.
Double relations : A. 60.
Double tangents : An. 51 : J. 49 : P. 59 :
Q.4: Z.21.
of a Cartesian : E.30.
of an >t-tic : M.7 : number = In (ii— 2)
(?i2— 9), J.40,63; N.53.
of a quartic : C.37 : J.49,55,68,72 : M.
1: N.67: P.61: Pr.U : with a
double point, M.4,6 : reciprocity
of 28 double tangents,
to the surface of centres of aquadric :
C.78.
Drilling, shape of liolc : Pr.35.
Dual relation between figures in spa«e :
J.IO.
Duplication of the cube, appro.x. : Pr.20.
*e (see also " Expansion ' ) : 151 : N.67,
68-:: geo meaning E. 4: N.55.
combinatorial definition of : A.I2.
* incommensurable: 295 : Cil.2 : L.40 :
Me. 74.
and TT, numerical th : Q.15.
e'"''-^*'-^'* in fractious : L.8O2.
e~'^\ &c. : CP.6.
gI>x2>gA^. E.37.
*e" : 766 : AJ.7 : in transformations :
CM.4 .
e"^''': A.33.
*Edge of regression : 5729: tg.eC.71.
* radius of curvature of : 5742.
Eisenstein's theorem : G.16.
Elastic curve : C.18,19: JP.34.
Elementary calculation : N.45.
*ElimiLiants : 583, 1626.
and associated roots : LM.16.
of two cubics : J. 64.
degree of: G.ll, two eqsl2 : J.22,31.
*Elimination : 582—94: A.23 : Ac.6,7 :
C.12,87,90: CD.3.,,6: CM.3 : G.
15,17 : J.34,43,60 : JP.4 : L.41,44 :
LM.ll: M.5,11: N.42,453,46,80,
82,83.. : Q.7,thl2 : Z.23.
problems : C.S4,97 : J.58 : M.12 : Q.
8,11 : in metrical geo, A. 63.
*Elimination of x between two equations :
686—94 : C.12 : CM.2 : " J.16,27 :
M0.8I : N.43..,76o,77.
* by Bezout's method: 586: A.79 : J.
53: Me.64: N.74,79.
bv cross multiplication : CM.l.
* by the dialytic method : 587 : N.79.
in geodetic operations : Z.3.
* by h.c.f : 593 : JP.8.
by indeterminate multipliers : CM.l.
* by symmetrical functions : 688.
degree of the final equation: J.27:
L.41.
Elimination : ap to alg. curves : M.4.
ap to in- and circum -conies of a poly-
gon ; At.63.
calculation of Sturm's functions : C.
80.
* of functions: 3163: C.84,87 : Me.73,
76.
with linear equations : At.63.
with linear differentials : L.36.
with )) variables : CP.5.
resultants, comparison of : J. 57 : and
interpolation, J. 57.
transformation and canonical forms :
CD.6.
tND^X.
8?3
*Ellipse (see also "Conies"): 1160,
4250: cnr245: geoQ.9.
theorems : A.30,47.prs49 : N.76 : Mc
Oullagh's, N.72.
eq. r+r' = 2a : A. 2.
equal chords : tg.eB.22.
of maximum surl'ace : N.65.
* as the projection of a circle : 4921 :
N.75-:.
* rectification of : 6083—96: A.3,22,27,
30: At.39: graphic: M^l.l : N.
43: TB.4 : Z.6: when e is very
small, TI.9.
representation by a circle : An.70.
* quadrature of : 6108,6113; t.c4688 :
A.46 : of sector 6098, A.20 : of
segment, 6103.
and triangle : thQ.4.
*Ellipse and hyperbola : 1160, 4250 : A.
24,28.
theorems : A.23 : 0M.3 : N.85.
sectors: TE.14.
*Ellipsoid : 5600, 6132 : A.28 : thCD.2 :
prC.20 : L.38 : cnM.20 : P.9.
centro-surface : CP.12 : LM.3.
cubature of some derived surfaces :
A.12.
* and enveloping cone : 6664 — 72 : Q.6.
generation of (Jacobi) : CD. 3.
* of gyration : 5930 : of inertia : 5925 —
'39.
and plane of constant segment, th :
E.32.
* of revolution : 5604 : area, ]Sr.42.
a locus in space : Q. 16,170.
-* quadrature of : 6143 : J.17 : Z.l : of
zone, A.22.
* volume: 6144,-8: A.46.
Ellipsoidal geometry : A. 10: LM.4.
surfaces : G.17.
*Elliptic functions : 2125: A.1,122,16,21,
35,48: trAc.5,6.7,ths7o : An.61,84 :
C.46,506,90,96,97 : CD.2,32,5 : CM.
3o : E.23 : G.I4 : J.2,34,44,6,8,16,26,
27,30.^32,35,37,39,46,72,83., : JP.
25 : L.55,56,61 : LM.7,10,29 : M.
3,ll,12,pr25 : Me.79,80,81,822 :
Mo.81,823,83o,85 : N.773,784,792 :
P.31,34,76,78 : Pr.6,9,10,12,232 :
Q.11,17,19 : Z.22,11,27.
of first kind : A.12,21 : C.16 : J.93 :
L.43.
with complementary moduli exten-
sion of a theorem of Lagrange :
An 832-
normal forms of 3rd and 5th de-
gree : M.172.
replaced by one of second kind : J. 55.
r^presentedbygauchebiquadratics:
C.83.
Elliptic functions — [continued) ;
of first and second kind : CIO.
as functions of their amplitude : JP.
14.
representation in a simple form :
Z.21.
series by which they arc computed :
J.16,17.
of second kind : J.93.
mechanical representation : Me.75.
reduction to first kind from same
modulus : A. 56.
of second and third kind, expression
by 6 function : Z.IO.
of third kind: C. 94,96: CD.8 : J.14,
47 : LM.13.
addition of: A.47,geo64 : AJ.7 : C.59,
78: J.35,41 , 44,54,880,90 : LM.13:
M.17: Me.80,84: Q.18: Z.l.
of 1st kind, Z.26 : 3rd kind, Me.81.
2nd kind by q series : Me. 83.
application of : C.857,865,892,90,93,946.
to algebra : J. 7.
to arithmetic : C.98 : L.622.
to confocal conies : Z.72.
to geometry : G.12 : J.38,53.
to in- and circum-circles of a poly-
gon : L.45.
to rectification : L.45.
to spherical conies : Z.22.
to spherical curves and quadrature :
An. 50.
to spherical polygons with in- and
circum-circles : L.46.
to spherical trigonometry : Q.20.
^ approximation to : 2127-32 : P.60,62.
arithmetical consequences : Ac. 5.
arithmetieo-geometric mean : J.58,85,
89.
arg sn a and (arg sn a)", as def. inte-
grals : Q.19.
in complex regions : Z.282.
development of : 2127—32 : J.81 : 1st
and 2nd kind, C.92 : with respect
to the modulus of X (a), /x {k) and
their powers, C.86.
development of an imaginary period
when the modulus is small
enough: An.70.
differentiation by periods and invari-
ants : J.92.
discriminant of modular equations :
M.8,9.
double substitution : J. 15.
am — - cos
"^^ am— sm —
Xa; A^am — xdx: J. '37.
Jo TT
5 T
874
INDEX.
Elliptic runctious — (continued) :
eqs. for the division of : Mo. 75.
formulae: AJ.5 : J.15,36,.50: LM.13 :
Me.78,80,85 : Jacobi Mel.l : Q.16,
19 : from confocal conies, LM. 14 ;
differential, Me.82 ; for sn, en, dn,
of u + v + w,M.e.82.
Galois' resolvent : M.18.
geo. problems : M.19.
geo. properties : L.43,45 : P.52,64.
geo. representation : A.22,61 : An. 60,
61 : At.53 : C.19,2l3 : J.63 : L.44,
78 : in solid geo, M.9 : of 1st kind,
An..53: C.70: CD.l : JP.28: L.
43,45„46,78 : of 3rd kind, A.24.
identities : Me.77.
imaginary periods : AJ.6.
infinite double products, A. 14 : with
elliptic functions as quotients,
J.35.
inversion of : J.4 : JP.34 : L.69.
KE'+E'E-KE' = iTT : Me.75.
otiK: Me.85.
modular equations : 0.47^ : J. 58 :
LM.9,10: M.12.
modular functions : A.11,13: G.12 :
J.72,83: L.40.: M.17,18: difEeren-
tiation for modulus of am, LM.
13 : expansion in powers of modu-
lus, J.41 : formulas, L.64: relation
between the modulus and the
invariant of a binary quar tic, Z.18.
multiplication of : 0.88.: J.14,39,41,
74,76,81,86,882: M.8: Mo.57,83,3:
and division, Z.7o : formulae, trA.
36 : C.59 : J.39,48 : complex, M.
25: Mo.62: Q.19,2U: mod.-Vi,
J.48.
periodicity moduli of hyper-elliptic
integrals as functions of a para-
meter : J.71,91.
subsidiary, pm («, h) : LM.15.
products of powers : Mem. 71.
quadriquadric curve: M.25.
g-forraula for sin am: LM.ll.
<ir-serics : and f85o: for ~- + ^^ coeffs.:
Q.21.
reduction of: An.64: in canonical
forms, J. 53.
relations : A. 67 : J.56 : between E'
{1c) and F' {k) : J .39.
representation by power series: J. 54.
representation of quantities by sin
am{u-i-w, h) : J.45.
series : C.95.
sn 8m, on 8m, dn 8u in terms of sn u,
tables: Pr.33.
sn.cn, and dn of u + v + ui -. LM.13.
Elliptic functions — (continued) :
spherical triangle of : Q.19.
and spherical trigonometry : Q.17.
substitution of 1st order: J.34.
and theory of numbers : L.58.
transformation : An. 573,58,60 : Ac.3 :
C.49,f79,f80,82 : CD.3,5 : J.3,34,
35,f65,55,87,88,89 : LM.9,11 : M.
14,19.„22 : Me.83,tr84 : Pr.27 : Q.
13,20:
of 1st kind : A..33.
of 1st and 2nd kind as functions of
the mod : L.40o.
of 3rd order : J.60 : Me.83.
of 7th order, square of mod : J.12 :
LM.13.
of 11th order: At.5.
of the orders 11, 13, 17, and 19 : J.
12,16.
cubic: C.64: Q.13.
and division: J.76 : M.25.
of a double integral, &c. : Me.75.
Hermite's ; tables : J. 72.
Jacobi's : LM.153,16 : J.87.
linear : J. 91.
modular, of Abel : ap to geom : C.
58 : to conies, C.79.
modulus of in a function of the
quotient of the two periods : An.
70.
pertaining to an even number : J.
14.
quartic : Q.12.
by roots of unity : J.6.
of rectangular coordinates : LM.15.
and of functions in theory of Cate-
nary : A.2.
triple division of and ap. to inflex. of
cubics : A.70.
Weierstrass's method : AJ.6.
*Emanents : 1654.
Empirical formula^ calculation of : Me.
73.
Engrenages : L.39,40.j.
" Ensembles," theory of : Ac.24.
*Envelope : 5192 : A.24,prs56 : C.45,86.
p.d.eOM.4: G.ll : M.84: Me.64,72: N.
44,-59,68,74.
application to jierspective : A.9.
class of (Chaslcs), th : C.85.
from ellipse and circle : LM.15.
* of a carried curve: 5239.
of conies, theorems : N.45.
of chords of a conic: N.48: subtending
a constant angle at the focus,
CM.3.
of chord of a closed curve : E.28 : cut-
ting of a constant area, E.31.
of curves in space : L.83.
* of a curve with 11 parameters : 5194.
INDEX.
875
'Envelope— (contimicd) :
of directrix of a parabola : E.34.
of geodesies : M. 14,20.
imaginaiy, of the conjugates of a plane
curve : C.75.
of pedal line of a triangle : Q.IO : do.
of in -triangle of a circle, Q.8,9.
of perpendiculars at extremities of
diameter of an ellipse : lSr.46.
of a plane : C.35.
of planes wliich cot a quartic gauche
curve of the 2° in 4 points of a
circle : An.71.
of planes perpendicular to radiants of
an ellipsoid at the surface : An.
69 : Pr.9.
of plane curves : G.11,12 : singulari-
ties of, LM.2.J.
of polars of a curve : J. 58.
of a quadric : Q.ll.
of a right line : N.63,79,83 : Q.13.
cutting two circles harmonically :
K85.
sliding on two rectangular axes :
N.45.
of a Simson line : E.29,o4.
of a sphere : C.67 : J. 33 : touching 3
spheres, IST.OO.
of a surface : CM.l : M.5 : degree of,
N.60.
of a surface of revolution : L.65.
of tangent of 2 variable circles : N.
51.
Enveloping asymptotic chords and
polars : A. 14, 16,17.
*Enveloping cone : 5664 — 72.
of an «-tic surface : CD. 4.
* of a quadric : 6697 : th of Jacobi, J.
12 and CD.3.
of a twisted hexagon, locus of vertex :
A.IO.
Enveloping line of class cubic : invo-
lution th, E.29.
Epicycloids : J.l : Mem.20: K45,46,60:
TE.24: thsZ.16.
centre of curvature : N.69 : plane and
spherical, JP.14.
double generation of : lsr.69.
reciprocal polar of : geoE.19.
*Epi- and Hypo-cycloids : 5266—72: LM.
4 : Z.18.
and derived curves : Z.17.
tangential properties of : absPr.34.
*Epi- and Hypo-trochoids : 6262 — 5 :
LM.4.
Equality and similarity of figures : J.52.
*Equations (see also " Linear equa-
tions ") : 50-67, 211-222, 400-
694: A.6,18,57,58,60,61,65o,67: tr
Ac.32 : AJ.6 : Au.51,54o : C.44,
'Equations— (coyifinued) :
47,69,62,68,91,97,995: CM.3 : CP.
4: G.l : J.13,16,34: L.67.^,69 : M.
14.21: Me.76: Mo.79,f80': ISr.67,
68,t_hs55,67,and80: P.1799.
(For Binomial, Biquadratic, Cubic,
Cubic and biquadratic, Linear,
Quadratic, Quiutic, and Trans-
cendental equations, see those
headings. Other kinds will be
found below.)
Abel's properties : C.91.
algorithms for solving : M.3.
whose coefficients are rational func-
tions of a variable : J. 74.
of degree above the 4th not soluble :
j.83.
whose degree is a power of a prime :
An.61 : C.48 : L.68.
* derived: 424 — 31: A.22 : in d.c,
1708—12.
developments : An.61.
differential operators in : LM.14.
Eisenstein's theorem : LM.7.
extension of theoi-y of : C.58.
fundamental principles or theorems
A.1,11: C.96,97: L.39,40: J.23.
Galois' theory : C.60 : G.12 : M.18,23.
of geometry : C.68 : homogeneous,
N.64.
generic : Q.4,o.
Hariot's law of : J.2 : extC.98.
homogeneous, reduction of a princi-
pal function which verifies a
characteristic homog. eq. : C.IS^,
14.,.
identical: J.27.
impossible : Man. 51.
in geo. mean of roots : ]Sr.45.
in quotients of roots : N.45.
in Slims of the G («,2) roots of another
eq. : N.43.
insolubility of quintics, &c. : J.l.
irrational : Man. 51.
* with integral coefficients : 503 : C.24:
J. 53 : complex, J. 53.
irreducible: An.51 : Mo.80 : of prime
degrees, AJ.7.
* linear: see " Linear equations."
* miscellaneous : 214.
numerical: C.10,123,32,78,81 : G.13 :
J.IO: L.36,38,41,83.
and commensurable quadratic fac-
tors : L.45.
of ttth degree with two real roots :
C.98. '
from observations : A.21.
* with only one positive root : 411 :
C.98.
of payments : A.34,36 : CD.l : CM.2.
876
INDEX.
Equations — (continued) .-
* reciprocal: 466: A.44: C.I62: of a
quartic, N.66.
reduction of : C.97 : CD.6 : to recipro-
cal eqs., A. 35.
relation to linear d.e and f.d.e : L.36.
roots of: see " Roots of equations."
* simultaneous (see also " in two or
three variables ") : 59, 211, 582 :
C.25 : LM.6 : thsN.48 and 81 :
quadratics, N.60.
deducible the one from the other :
C.22.
of the form a;"'+?/"' + ^"' = « : N.46.
* solution of : 45,54,59,211,466—533,
582: A.64: trAn.52: 0.3.2,53,62,
643: J.4,272,87: Mo.56,61.
* by approximation : 506 — 533 : A.30 :
Ac.4: C.ll,17.45,60,79o,82 : E.4:
G-.8 : J.14,22 : Me.68 : N.51, 62,783,
80,84: No.58: P.5 : Q.3: TI.7:
Z.23.
* Horner's method: 533: ^P.19.
* Lagrange's method : 525 : C.91.
* Newton-Fourier method : 527 — 8 :
AJ.4: G.2: Me.66: N.46,56,60,
69,79.
Weddle's method : Z.7,8. ^
by continued fractions : J. 33.
by definite integrals : Me.81 : P.64 :
Z3.
by diminishing the powers of the
roots : C.41.
by elimination of integers: N.70.
by geometry : C.87.
by imaginary values : J.20.
by infinite series : J.33.
by interpolation : C.5.
by logarithms : C.95.
the one by the other : C.72 : L.71.
by radicals : C.58 : Q.15.
by series : An. 57 : C.49,52 : J. 6 :
Mem.33.
by transcendents : An.63 : Q.5.
* by trigonometry : 480: A.l.
a nonic eq. which has this charac-
teristic : A given rational sym-
metrical function 6 [a, (3) of two
roots, gives a third root y, such
that a = e ((3, y), (3 = 6 (y, a),
y = <9 (a, [3) : J.34.
symbolic, non-linear : 0.22.
systems of: 0.67: G.11,18: LM.2,8 :
Q.ll: M.19: Z.14,18 (see also
" Linear equations.")
transformation of : C.6k
* in one varial^lc : 45 — 58, 214 — 16, 400
— 550 : approx A.20.
graphic solution : C.65.
(C-'"-i— :o_/,- = 0: An.59.
Equations — {continued) ;
x'^—pn'-^q^O : number of real
roots : C.98.
g,2n_j_^^n_|_^-^n _ Q j^^^j dcrivativcs :
N.652.
^»-i^x"-- + ...+l =0 irreducible
if n be a prime : L.56.
ax-'" * " + hx'" * " + ex" + d = 0: G.14.
(.<•— 1)!+1 = x"': L.56.
(,,;•_„;-) ^ (a.) = 0: N.82.
(14-a')*"(l + t'i') = when x is small:
A.2.
* in two variables : 59—67,211,217—8 :
A.20,25: 0M.2: J.14: N.473,48,
63: Pr.8: Q.18.
of any degree with a variable para-
meter: L.59.
implicit : Mem. 30.
numerical solution : Z.20.
x^+if^ = a and x~y+xy~ = h : A. 48.
in three variables : gn.sol, 60 : A.l, 64 :
N.47: M.37: byacubo-cycloid,0.69.
* {y—c){z—h) = a^, sym in x,y,z:
219.
* y"+z^+y3 = a^, &c., sym : 220.
* x^— yz:= a", &G.,sjin, and x = cy-{-hz
&c., sym : 221—2.
x-yz = iaV {(l-i/2)(l-z2)}, &c.,
sym : A.36.
ax + hy + CZ = I, a 'x + h'y + c'z = V,
x~+y'^-\-z~ = 1, by trigonometry :
A.6.
*Equiangular spiral: 5288: Me.62.,: N.69,70.
Equilateral hyperbolic paraboloid and
derived ray-system : Z.23.
Equimultiples in proportion : Gl.
EquipoUences, method of : ^.692,702,73;,
743.
Equipotential curves : Me.82 : Pr.24.
Equipotential surfaces : G.20 : geoJ.
42 : M.8.
of ellipsoid : L.822.
Equivalence of forms : 0.88,90 : JP.29.
Equivalent representation: Z.23.
E((uivalcnts, theory of: A.44.
Eratosthenes' crib or sieve : N.43,49.
Error in final digit of decimals : C.40 :
Me.74: N.56.
Errors of observation : A.18,19 : An.58:
0.93 : JP.13 : N.56 : P.70 : TE.24.
Errors of constants : Mo.83.
*Euclid, enunciations: p. xxi.
axiom 11 : J.l ; 1.47 : new proof, 0.60.
II. 12 and 13: Mc.80 ; VL7: Q.ll
new proof, Q.9.
XL, &c., Me.71 : XI.28 : A.IO.
XIL, &c., G.9; criticism on : Q.7,9.
INDEX.
877
Euler's algorithms : A.67.
*Euler's constant : 2744. : Pr.15,16,18,19,
20, Table 27.
and Binet's function : C.77 : L.75.
Euler's equation : N.72 : integration of
it by the lines of curvature of a
ruled hyberboloid, N.75.
Euler's equations of motion solved by
elliptic integrals : Q.l^.
Euler's formula for (!+'»)": L.44.
*Euler's integrals: 2280-2323: A.41 :
Ac.1,2: An.54: 0.9,17,94,95,th96:
J.15,21o,45 : fL.43 : Me.83 : Z.9.
* B{l,m): 2280: An.69 : G.9.
r (?i) : see " Gamma function."
ap. to series and functions of large
numbers : JP.16.
sum formula and quadratic residues :
An.62.
Euler's numbers : AJ.5 : An.77 : 0.66,
83 : J.79,89 : prsL.44 : Me.78,80.
Evectant of Hessian of a curve: E.
32.
*Even and odd functions : 1401 .
*Evolute: 6149 — 59, 5165: An.53,61 :
C.30: Q.3,11.
analogous curves : L.76.
* of a catenary : 5159.
of a cubic curve : Q.ll.
of a cycloid : A. 30.
of au evolute, in inf. : L.59 : Me.80.
* of an ellipse: 4547,4958: C.84: N.52,
63,.
and involute in one : L.41.
of the lima9on, rectif. and quadr. of:
E.40.
of negative focal pedal of a parabola :
E.29.
oblique, direct and inverse of differ-
ent orders : C.85.
* of aparabola : 4549,4959 : Q.5 : N.65.
rectification and quadrature of : A.4.
of surfaces : C.74.
of symmetrical bicircular quartics :
Q.18.
* of a tortuous curve: 5731: A. 25.
* angle of torsion of evolute : 5754.
integrable equations : L.43.
*Evolution : 35.
E {k) = integral part of x : O.50 : L.57.
*Ex-circle of a triangle : 711,953 :
4749: A.54: thN.60.
locus of centre, th : Q.9.
*Expausions of a function in a series
(see also " Series " and " Sum-
mation ") : A.31 : An.7 : thsAJ.
3 and 4: C. 7,13,17,20 : CM.4:
J.90 : L.38,46,76 : M.16 : Mel.3 :
Mem.33 : N.82,83 : num, Q.3 :
Z.2.
Expansions of a fi;nction in a series —
* of circular functions : 2955 : A. 11 :
CM.3: J.43: L.36: Q.12: of
imag. arcs, J.6.
coefficients of : gn form, C.85 : gn
property, J.41.
connected with a 2nd order d.e : C.5 :
L.36,37,.
of denominators of convergents :
C.46: JP.21.
of exponentials : J.80.
* of explicit functions : 1500 — 47.
extended class of: C.82 : approxi-
mating to functions of very large
numbers, L.782.
of faculties of the variables : Mem.31.
of implicit functions : 1550 — 73.
of Jacobian functions : Au.82.
of Legendre's functions, X„ : An. 75.
with limits : C.34.
of another function : 1559 : C. 95,96.
of periodic quantities : C.52,.53 : JP.ll.
of powers of the variable : At. 57 :
0.19 : L.46.
of powers of a polynomial : 0.86 : J.53,
88..
of powers of another function : Mem.
33: N.74.
within a given interval according to
the mean values of the function
and of its successive derivatives
in this interval : O.90.
by Bessel's function : J.67 : M.10,17 :
Z.l.
* by binomial theorem : 125.
* by factorials : 3730.
* by generating functions : 3732.
* by indeterminate coefficients : 232,
1527—34: A.3.
by logarithmic method: 0.92.
* by Maclaurin's th : 1524.
by a series : 0.93,95.
Expansion of : alg. functions : 0.89
Z.45.
Eisenstein's th : J.45.
n, alg. functions from 11 eqs : G.ll.
[l + ax)" in an integral series: A.65.
(l_a;)(l_a;3)(l_a;3) ... ; 0.92 : J.21 :
L.42.
(l-a-)(l-;e2)(l-:ei)(l-K8) ... : Me.80.
(l+a;)(l + 2.o) ... (1 + ^^=1 A-) : 0.25 : J.
43.
{l + ax+bxZ+...+le"-Y^ : AJ.6.
{(a;-a,)2+... + (,-«-a4)3}-i : 0.95,.
(l+aa;+&«2)»: Q.18.
{l—2ax+x^)-i : L.372.
(1— 2aiB+a2a2) ^ . 0.863.
878
INDEX.
: J.40.
Expansion of — {continued) :
{l-ax~hx^)-" : J.43.
{x—z)'" in powers of ::^—l : C.86.
(x+yY"^: CM.3.
nth derivative of \/{a~—h'X~) : A.4o.
' l-s/{i-it) y
X
y =
in powers of t, wlien
1566.
l + y(l-a;2)
Bernoulli's numbers : 1545.
circular functions : J. 24 : Q.5.
an arc in linear functions of sines or
tangents of fractions of the arc
in g.p : L.43.
powers of arc in powers of sines : J.ll.
n : 2931—2, 2945, 2960—2 : Me.78.
powers of TT : Me.78: ir^, 858: tt"',
Me.83.
sin 6 and cos 6 in powers of 6 : 764,
1.531 : A.5,29 : C.16.
sin"^ and cos"^ in sines or cosines of
multiple arcs : 772—4 : A.24,55 :
0.12 : CD.3: J.1,5,14: K71 : TI.7.
sin nd and cos nd in powers of sine or
cosine : 758, 775-79, 1533 : 0.82 :
CM.2 : Me.76 : Mem.13,15,18 : N.
732,83 : Q.4 : convei'gency of the
series, J.4.
cos nO in powers of cos 6 : 780 : Q.
12.
sin-'a;: 1528,-64: J.25:N.74: re-
mainder, Z.lo.
cos"«.i; : A.ll.
tana;: 1525,2913,-17: A.16 : 0.88:
N.57.
cot a;: 2911,-16: 0.88: Q.17.
sec X : 1526 : A.16 : 0.88 : J.26 : N.57 :
Q.17.
coseca; : 2914,-8.
tan-^c : 791.
tannO: 760.
ft^sin a' in differences of sin x : 3749.
^"^ nx I cos .c" : A.4.
cos '
(1 — ^cos 0)"' := 2a„cos2ft^ : A. 21.
{a'^+b^—2ah cos i^)exp.— (m+^):TB.5.
(rt + ^ cos <|) + c cos <?)')"" in cosines of
multiples of >P and <p' : J. 15.
cos /c cos"' (cos 0) -fa) : 0.15.
sin
(^ + ^i + ... + ^.-0: A.34.
7/ in powers of x wlicn x =
796,1558.
LL"J/
Expansion of — {continued) :
* do. when x .
log ■'/
s\n{i/ + a)-
1570.
^cot^ in powers of sin2^: Q.6.
cot-'(m— 1)— cot-' (»t+l) : A.47.
differential coefficients by f.d.c and
tlie converse: J.16.
elliptic functions : A.19.
aiidof their powers : 0.83: cos amx,
L.64.
equations : L.60.
exponential functions : N.82.
j_
e = limitof (H-,v)-1590:A.3,23:Q.7.
e- : O.30 : N.48.
X e"" — 1
e^±T' e^+1'
e"'-' in powers of ye'"-' : 1571.
(ffle^- 1)-': At.57.
1539, 1543—4.
2961
e exp
cos hx : 798.
Z.3.
e'"-' ^bxdx, and summation of
Jo cos
the series : J. 41.
e exp. sin a;: 1629.
e exp. asiii-'«: 1535.
e exp. \og{z+x sin?/) : 1557.
e exp.—(p{x,y,3...) : 0.58-2.
a; exp.[.(' exp.[a;exp, &c. : J.28.
fractions : 248-
functions :
Al{x),Ali{x), ALix) (Weierstrass's func-
tions) in powers of the modulus :
0.822,85,86 : L.79,.
'-, by Taylor's th : 0M.4.
■IV ;
f{x+h) (see Taylor's th) : 1500—9,
1520: Abel's th, 1572; Stirling,
1516 ; Boole, 1547 : AJ.3.
f{x) (Maclaurin) : 1507: 3759.
f{x+h,y+h): 1512,1521.
f{x,y) : 1516, 1523 : Me.3.
/(O) (Bernoulli): 1510.
<P{a+hx + cx-^ + ...) (Arbogast): 1536:
OD.1,6.
-— • (Oaylcy) : 1555.
''{"+•''</>(.'/)} in powers of .(• (La-
grange) : 1552 : liaplace's th>
1556.
INDEX.
879
Expansion of — {continued) :
*= f{y) in powers of ^//•('/) (Burmann) :
1559.
*= /{^|'-H.«)} andrl^-'OO: 1561—3.
^ </) (eO (Herschel) : 3757.
* it,.„ : 3740 ; A"26 : 3761.
*= A"itx and A"itj : 3741 — 2.
*= in differential coefficients of u :
3751.
i^ A".i3'" and A»0'" : 3743—4.
* Unx in differences of a : 3752.
^ i(.j:cIm in terms of Uo,Ui,ih, &c. : 3778.
a function of a complex variable : M.
19.
a function of a function : AJ.2,3.
functions of infinitesimals : G.\2.
a function of a rational fraction : At.
65.
a function of n variables : C.6O4 : J.
66.
a function of y,y' in ascending powers
of x,x when y = z-\-x^{y) and >/ =
z'+x'0{i/') as in 1552: J. 48.
holomorphic functions : M.21 : by
arcs of circles, C.94.
^ implicit functions : 551,1550: L.81.
integrals : A.l : of linear d.e, An.71 :
of log X, A.4.
"= logarithms : 152—9 : N.82.
^ log(l±a;),log|±i|;
&c. : 155—9.
* log y and log y" in powers of a ^ when
yi—ay+b = 0: 1553—4.
* \og{a+hx+cx"-+...): 1537.
* log(l+2acosa;+a2): 2922.
* log (1+91 cos a;): 2933.
* log 2-^ (I): 2927.
* logrd+a;): 2294, 2773.
higher integrals of log x : A. 4.
numbers : M.21.
* a polynomial : 137 : Z.26.
a quartic function : A.35.
Exponents : N.57 : P.1776.
reduction for d.i : C.16.
Exponential, th : 149 : IS]'.52.
functions : P. 16.
replaced by an infinite product : C.99.
Exponentials, successive of Euler : L.45.
Factorials: calculus of: L.67 : N.
60: Pr.22: Q.12 : Q.f8.
geom. i.e(l+a;)(l+ra')(l + r2^)...: C.
17.
* notation : 94, 2451 : Q.2.
Factorials — {continued) :
reciprocal : C.17.
treatment by limits : J. 39.
1, , 2^,3:5... 7i": Me.78.
^ n\ = T{l+n) : 2290.
approx. to when n is large : C.9,50 :
J.25,27 : L.39.
5i ! = w"e- V(2H7r) (Stirling): Q.15.
1.3.5
CM.3.
theorem : 339.
2.4.6 ...
G (», r) when n = a+i[^ : J. 43.
Factors: 1—27.
in analysis of integral functions : M.
15.
application to rotations to indicate
direction: J.23.
^ of composite numbers : 274: J.ll.
complex : C.24.
equal, of integral polynomials: C.42 :
L.56.
^ of an equation : 400 : J.3 : condition
for a factor of the form x" — a",
A.56,63.
irreducible, of an integral function ac-
cording to a prime modulus p
C.86.
linear, resolution into : N.822.
of polynomials and geo.ap : J.29,89.
product of an infinite number of : A.
59.
cos ^ cos -^ COS -^ ■ • • : N.70.
radical, of numbers : C. 24,25.
o? Ax^+By^+Cz^, th of Lagrange:
AJ.3.
oU^-fgif = ±1 : A.33.
of {x + y)n—xn—y" : thQ.15,16o.
* of iL'2"— 2.-c''^"cosn^+i/-": 807.
of «"— 2iicos nd+x-'' : CP.ll : Me.76.
{l-x){l-xZ){l-x^)...: C.96.
* tables of (Burchardt's) p.7 : to 4100,
J.46.
geo. properties : J.22.
transformation of: A. 57.
of 100... 01: Me.79.
Faculties, analytical : J.7,11,334,35,40,
44,51.
coefficients of : A.9,11 : At. 75.
divisibihty of : A.48.
numerical, of 2nd order : Mem. 38.
series : Z.4.
*Fagnani's theorem: 6088: A.26 : LM.
6,13,23 : Z.l.
curves having Fagnanian arcs : LM. 1 1 .
stereometric analogy : Z.17.
880
INDEX.
Faisccaux : of binary forms having
the same Jacobian : C.93.
of circles : C.76.
of conies : Z.20.
curvature relations : Z.15.
formation of : C.45 : CM.3.
intersections of : N.72 : degree of the
resulting curve, J. 71.
of lines and surfaces : N. 53,83.
plane : N.53 : defined by a first order
d.e, C.86.
of tortuous cubics in connection with
ray-complexes : Z.19.
Fan of Sylvester : E.33.
Faure's theorems : G.1,19 : and Pain-
vin's, lSr.61.
Fermat's theorems : of (N''~^—l)
H-p : 369 : A.32 : AJ.3 : J.8.
oi x"+y"^ z" being insoluble when
n is an odd prime, &c : An. 57:
C.gz84 and 965,91 : J.40,87 : TE.
21.
analogous theorem : J.3.
case of « = 14 : J.9.
and periodic functions : Mo. 76.
x+y = D, «3-f 7/3 = d3: Mem.26.
of the semicircle :' A.27,30,31 : gzA.31.
method of maxima : C.5O2.
Feuerbach's : th of the triangle, Me. 84 :
circle, A. 59.
Fifteen girl problem : E.34,35 : Q.8,9.
" Fifteen" puzzle : AJ.22.
*Figurate numbers : 289: A. 5,69.
*Finite differences, calculus of: 3701 —
3830: A.13,18,24,63: C.70 : J.ll-:,
12,133,14.:,15.,,16: Me.82 : Mel.5 :
Mcm.l3 : N.G9 : thsP.16,17.
ap. to complex variability : An.82 :
ap. to i.eq, An. 50.
* first and ?ith differences : 3706.
A2«=0: An. 73.
* A"0"' : 3744: Q.5,8,9: Herschel's table,
N.54.
A"0"'H-n(m), table of: CP.13.
8uq, 8^u„, &c., in a function of Au^„
A2«„, &c. : N.61.
* A"« in successive derivatives of «:
3761 : N.73.
Al'' and Bernoulli's numbers : An. 59.
hu'^
AUj. — — AUx-\ -^ AHj: — &C.
Ac. 5.
A sin a! and Acos.i': CM.l.
Finite difference ecpiations : AJ.4
An.
59: CD,2: C;M.1,3,4: CP.6 : JP.
6: L.83: P.60: Pr.lO.
of integrable form : C.54.
of mixed differences : Q.IO.
Finite ditt'erence eqs. — {continued) :
of the kind M,,j,= Mr-!/,r + y : CM.4.
linear: AJ.7 : Au.50 : At.65: Q.l.
first order, constant coefficients :
C.8.
determination of arbitrary con-
stants : A.27: At.65: G.7.
integration to differences of any
order : J. 12.
with variable coefficients : 0.17.
partial:
constant coefficients : C.8.
linear of 2nd order : C.98.
of physics : C.73.
Finite differences : exercises : No.
44,47.
formula;: CD.9 : Q.2.
sura and difference: J..58.
of functions of zero : TI.17.
n [n, r) value of : Q.9.
integrals : C.39,57 : JP.42 : L.44.
expressed by definite integrals :
An.53.
2e'i/ : A.6 : No.44.
inverse method : C.74 : P. 7.
involving I/l : Me. 78.
of powers converted into d.i : JP.17.
Fleflecnodal planes of a surface: Q.15.
Flexure : AJ.2.. : Me.2.
of ruled surfaces : An. 65.
of Slices : LM.9.
of spherical surfaces : Me. 77.
^Fluctuating functions : 2955a : LM.5 :
M.20: TI.19.
Fluents: P.1786.
of irrational functions : P. 16.
*Focal : chords of conies : 1226, 4235,
4339.
circle of conies : Mel.2.
* distances: 4298: N.64.
pedal of a conic : N.66.
*Focal properties: of conies: 1163,
1167—9, 1181, 1286-8, 4298-
4306, 4336—45, 4378, 4382, 4516,
4550—58, 4719 — 21, 5008 — 16:
CD.7.
of curves: CD.7.
of homographic figures : N.71.
* of a parabola : 1220, 1223-6, 1230—4,
4231, 4235—8 : G.22.
of a quadric surface : An. 59 : N. 58.
Focal quadrics of a cyclide : Me.85 .
Foci: J. 64: N.42,44,.53,85 : Q.2.a.c9.
* of conies : 1160: trA.2.5,63,64,cn69 :
gzC.22andL.39: CP.3 : N. 69,74,
78,81.82: t.cQ.8,13,12and45: gen.
cq, N.48.
analogous points in higher plane
curves: J. 10.
INDEX.
881
Foci : of conies — (continued) :
* coordinates of: 4516.
eq. of: LM.ll: o.cE.40.
exterior : N.79.
* to find them : Q.25: fromgn.eq,5008.
through four points : N.SSa.
* of four tangents : 5029 : N.83.
locus, a cubic : M.S.
negative : A. 647.
under three conditions : Q.8.
of curves: C.82; )ithclass,86: N.59,79.
of cones : N.79.
of differential curve of a parabola :
A.58.
of in- conic of an H-tic, locus of: E.21.
of lines of curvature of an ellipsoid :
Z.26.
of quadrics : N.42,66,74,75,78.
of quartics : J.56.
of the section of a quadric by a piano :
N.64,70.
by another quadric : N.47.
of surfaces : C.74 : of revolution, N.
74.
*Folium of Descartes : 5360 : N.44.
Forms, theory of : M.18 : of higher de-
gree, Mo.83 : Pr.38.
reciprocity principle : An. 56.
Formulas: G.15,19.
for log 2, &c. : Me.79.
in the Fund. Nova : Me.76.
* of reduction in i.c : 1965: Me. 3.
Four colors problem : AJ.22.
Four-point problem : E. 5,6,82.
Four right lines not 2 and 2 in same
plane: J.5.
Fourier-Bessel function : J.69 : M.3.
*Fourier's formula in i.c : 2726 — 42:
CM.3: J.36,69: L.36: M.4 : Me.
73: Q.8: gzZ.9.
ap.tocalculationof differentials : J.13.
*Fourier's theorem : 518 : 528 : An. 50,
75: J.13: M.W.: Me.77,82,83.
ap. to a function of a complex varia-
ble : M.21.
Fractions : AJ.S^ : G.9,prl6 : J.88 : L.IO.
continued, decimal, partial, vanishing,
&c. : see each title,
number expressible by digits If n :
C.96.
reduction to decimals : A.1,25.
transformation into decimals : A.ll.
*Frullani's formula: 2700: LM.9.
Fuchsian functions : C. 927,93,943,95,96.
Fuchs's theorem on F{:<,y,y.) = : C.99.
Functional equations : CM.3 : J.90 :
TB.14.
f.<p.v=fa: C.88.
/.</>.« = H-/c: C.99.
(p .fx = F. (px, to find <p : Mem.31.
Functional e(j nations — (continued) :
<p.v + <i>y = <p(xfy+yfx) : J.2.
=fix . <P\y+f-2x . <p2y+&c. : J.5.
<p,+^y = irf ^-^'''+f^;-^y : J.46.
J '/(•'■> 0) <P (a.'+^) dd = F{x), to find
Pr.8.
'P'>i-'p'--t-j = &C.: Q.15.
C,i} -\- (.1
/(,.)=/(sin^): C.
Functional images in ellipses : Q.17.
in Cartesian ovals : Q.18.
Functional powers : Mem. 38.
symbols : Q.4.
Functions: A.28 : AJ.6 : xVn.79 : C.43,
91 : CP.l : J.16,prs71,74,84,87,
91 : L.45 : Me.7 : Mo.80,81 : P.15,
16,17,62: Pr.lla: prsZ.26.
algebraic, alternating, analytical, cir-
cular, circulating, conjugate, con-
tinuous, curvital, cyclotomic, de-
rived, discontinuous, elliptic, even
and odd, exponential, Fuchsian,
gamma, generating, hyperbolic,
implicit, infinite, imaginary, in-
tegral, irrational, irreducible,
isotropic, iterative, monodrome,
monogenous, monotypical, non-
uniform, periodic, polyhedral,
quantitative, rational, representa-
tive, transcendental, trigono-
metrical : see the respective
headings.
analogous : to algebraic functions :
C.89.
to circular functions : C.84.
to Euler's : C.89 : M.19.
to functional determinants : J.75.
sine and cosine : Q.16.
to modular functions : Ac.2 : C.93.
connected by a linear eq. : C.17.
condition oif{x,y) being a function
of (?)(.(•, ?/) : A.21.
development of: see "Expansion."
defined by d.e : JP.21,28.
differing very little from zero : L.74.
errors of geometricians: J. 16.
expressed by other functions, remain-
der : C.98.
fractional : J.8 : the variable being
the root of an equation, ]Sr..56.
from functional equations : M.24.
from Gauss's equation : C.92.
with lacuna : C.96.
Lagrange, tr : JP.5,7.
5 u
INDEX.
Functions — (continued) :
linear: C.90.
with linear transformations inter so :
M.19,20.
whose logarithms are the sums of
Abel's integrals of the 1st and
3rd kind : C.92.
with non-interchangeable periods :
M.20i,21,25.
number of values of : C.48.
do. through permuting the variables :
JP.10.,18: L.50,60.
of two variables : Ac.3 : C.90,962.
made constant by the substitution
of a discontinuous group : C.97.
which arise from the inversion of
the integrals of two functions :
0.922.
whose ratio has a fixed limit : G.5.
f{x,y) aiich that f{zf{x,y)} is sym-
metrical : J.l.
of three variables satisfying the d. e,
AF = 0: Ac.4.
of three angles, th.re 1st derivatives :
J.48.
of 4 and 5 letters : L.56.
of 4, 5, and 6 letters : L.50.
of 7 letters : 0.57,95^.
of 6 variables which take only 6 diffe-
rent values through their permu-
tation, not including 5 symmetri-
cal permutations : A.68.
of n variables : C.21 : Mo.83 : with
2n systems of periods, 0.97.
analogous to sine and cosine : Q.16.
number of values : J.85. : do. by
permutationof the variables : O.2I4.
obtained from the inversion of the
integrals of linear d. e with
rational coefficients : 0.90-2 : J.89.
of an analytical point, ths : 0.952.
of a circular area from a given inte-
gral condition : Z.26.
of imaginary variables : 0.32,48 : JP.
21 : L.58,593,60;„6l3,62 : LM.geo
8.
of large numbers, approx. : C.2O3.
of a real variable, connexion with
their derivation : M.23,24.
of real arguments, classification ac-
cording to their infinitesimal
variation : J. 79.
of the species zero and unity : 0.95.
of a variable; analogous to the poly-
nomials of Lcgc^iidre: 0.95.
allied to Pfafi'iatis ': Q.16.
rationally connected : L.59.
with recurring derivatives : LM.i :
TE.24.
Functions — (continued) :
which relate to the roots of the equa-
tion of division of a circle or of
n»-l = 0: J.17.
representation of : C.923 : M.17 : one-
valued, Z.25.
approximate : Z.3.
by an arbitrary curve : M.22.
by Bessel's functions : M.6.
by definite integrals : Ac.2.
by elliptic functions : An.82.
by Euler's sum-formula: J..56.
by Fourier's series : Mo.SSj.
by graphic methods : A.2 : imag.,
J.55. _
by infinite products : Z.24.
y =1 e''-^''X, constant and r a positive
integer : A.42.
2/ = .«"e^'': A.52.
reproduced by substitution : 0.19.
resolution into factors : Ac. 66 : 0.19,
30: OP.ll: J.18.
satisfying the eq. aF=0: 0.96.
singularities of : M.19.
whose successive derivatives form an
arith. prog. : An.71.
systems of: Mo.78 : of two inter-
connected, 0.98.
of two systems of quantities, cor-
relative and numerically equal
0.98.
which are neither rational norreduci
ble to irrational algebraic ex
pressions : O.I82.
which are of use in elliptic functions
and logarithms : No. 58.
which take a given value in a given
position : An.82.
which have no derivative throughout a
certain interval : An. 77.
which vanish with their variables :
TI.I62.
x": An.63.
I (,(•')* U and so on, and the corres-
ponding inverse function : J. 42.
arising from V(4!—2xz+z-): J.2.
</>(„■) = "-^^+4 :LM.9.
ex -j- a
f(u,z), ?i being an implicit function
of an imaginary variable z : Pr.42.
/(,(•), formula of analysis : J. 53.
/(,.) = 0, y =f (.-•). th. re <p (y) : E.36.
y}r(x)^d,]ogr(x): 2743.
yjr (a) of Jacobi : J.93.
Q(x): Ac.2.
INDEX.
883
Functions — (continued) :
Bessel's, IM = — Tcos (.<• cos 6) cl6
~ 23" "*" 23. 42 "" 22.42.62 "^
Cauchy's numbers ; N-kj,p
{e"'—e-'")>'du.
cosine integral ; Ciq = '- dx :
taP.70. '" "*'
Diri chiefs function,
F{x) = 2
Z.27.
elliptic ;
jaj'-r^-i 5 (rc^O {B {.'•>')}''^'ch : J.23.
* Euler's ; B {I, m) : 2280.
expon-integral ; i/i g s — — dx- : A.
K
10 : taP.70.
siu-" CO cos (e cos w) fZco : An.
70.
E (ic) : Mel.6.
tan-'a-
dx = l-
0~ 0~
= -915965,594177 ... : Mem.83.
r (ic) : see " Gamma function."
Jacobi's
(A).(-i,e.p2;::-t'.(^):
C.592,60: L.47,50.
Laplace's Y^ -. M.14.
log-integral ; Lig"
I log X
i«(l+,,)^,,_-^ +
-&c.
Legendre's Xn : see " Legendre."
P, where e esp ( — ^^) f^* = ^-T^ •
Q.IO. "
P"(cosy), 7i = 0: G.22.
Pf«^^,.c): Z.14
Va'^'-y' /
n (s) = I x'-e-^'dx ; >|^ (2) = d, log n (r) :
Q-1- ^^.^^^
sine-integral ; Siq=\ ' dx -. taP.
70. ^° '"^
2 1 e exp i-z^) F{z)dz = 0: C.93.
Functions — (continued) ;
E (p,<P) d<P
^(^''^)"joy{l-^3sin3^}-
X, Y, &c., such that ,SZYcZ(r = and
and that any function can be ex-
panded in theformaX+^Y+&c.:
LM.IO.
*Gamma function, T (n) : 2284 : A.4,6,61 :
An.69 : 0.35,92,96 : J.35,82,90,
ap57: L.42,46,52,55 : Q.9 : Z.l,
25.
application of this and other trans-
cendents : C.86.
of a complex : Me.84.
* curve y = T (x) -. 2323.
* deductions: 2286—2316: A.IO.
derivatives of : Q.6.
of equidifferent products : J. 36.
of an infinite product : J.39.
* = limt. of 4nT : 2293 : A.30.
* logarithm of : 2294, 2768 : C.9.
* numerical calculation : 2771.
* as a definite integral : 2768.
71, negative : CD.3.
* numerical calculation of : 2317.
* the function r/^ (*) = fL log r (.r) : 2743
—70.
reciprocal of : Z.25.
reduction of : J.40.
* transformation of : 2284, 2318 : J.57.
T(n + 1) = V (27r) e-"n"^Hl + e) (Stir-
ling): C.5O2.
*r (m) r (1-hO =
2313: Ac.3.
H,)-^^{'^'-^)-
2316:
*r(,,') r ia-+
L.55,56.
T(x) = P{x) + Q(x): Ac.2.
2 ^, : G.6.
T(x)
Gauche cubics : C.82 : J.60 : N.623.
3rd class, theorems : J.58.
number of common chords of two :
An. 70.
through five points : N.83.
through six points, en : 'B.QQ.
Gauche curves : C. 70,77,903.
Mo.82: thsN.53.,.
classification of : JP.32.
on a cubic surface : C.62.
of a developable surface, singulari-
ties : An.70.
differential invariants of : JP.28.
884
INDEX.
Ganche curves — (confinued) .-
intersection of two surfaces having
common multiple points, singu-
larities of : C.80.
on a one-fold hyperboloid : An.l : C.
52,53.
metric properties of, in linear space of
n dimensions : M.19.
representative curve of the surface of
principal normals of : C.86.
of the zero species : C.80.
Gauche helicoids : rad. of curv. : N.45.
in perspective : JP.20.
Gauclie : in-polygons of a quadric :
C.82.:.
perspective of algebraic curves : C.80.
projection : N.65.
quadric : IST.G? : and orthogonal tra-
jectory of generatrices : thN.48.
quartic : A.62 : C.82 : L.70.
9 points of, 7 points of a gauche
cubic and 8 associated points :
C.98.
iinicursal, a class of: C.83.
surface: N.61.
sextic curve : C.76.
surface: JP.17: L.37,72.
deformation of : C.57.
•which can be represented by a p.f.d.e
of the 2nd order: C.61.
Gaussian periods of congruent roots
corresponding to circle division :
J.63.
*Gauss's function : see " Hypergeometric
function."
Gauss's theorems : J. 3.
*General methods in anal, geometry :
4114.
General numerical solution of any
problem : LM.2o.
*Generating functions: 3732: J.81 : N.
81: Pr.5
and ground-forms of binary quantics
of first ten orders : A.J.23,3.
do. of binary 12-ic and of irreducible
syzygies of certain quantities :
AJ.4.
of some transcendental series : At. 55.
for ternary systems of binary forms,
ta : AJ.5.
♦Geodesies : 5775, 5837—55 : A.39 : C.
402,41 ,96,p.c97: CD.5 : G.19 : J.
50,91 : M.20 : Mc.71 : N.45,G5 : Q.
1,5 : Z. 18,26.
of cubic surfaces, loci : CD. 6.
curvature: 5816: C.42,80 : L.12.
on an ellip.soid: M.20.
* radius of: 5776,5846.
duals of : Q.12.
* equations of; 5837.
Geodesies — {coidinued) :
flexure of: C.66.
forms from en of their polar systems :
Z.24:.
* geometry : 5855.
problems: A.8 : An.65: CP.6: ^.hX:
L.49 : TK.73.:.
on a quadric or ellipsoid: C.222: CD.4:
J.19: L.4..41, 44.48,57: M.35 : N.
76.
Joachimsthal's theorem : J.42.
and corresDonding plane curves
C.50.
and lines of curvature : L.463 : N.
82.
* fd constant along such : 5842.
shortest lines : .7.26.
* throTigh an umbilic : 5850.
on a right cone : A. 69.
* radius'^of torsion of: 5848: HE'P^.^-
P = 0, Me.75.
sections : Z.2.
* shortest lines on surfaces : 5838 : J.
20.
on a spheroid : J.43.
triangles : Mo.82.
best form of : N.55.
reduction of arc of a small one : An.
50.
Geodesy, spherical problems : A.252,63.
representation of one surface upon
another: An. 70.
Geodetica, integration of its eq : An.
53.
Geography, comparative : A. 57.
♦Geometrical conies: 1150 — 1292 (see
Contents p. xviii) : G.l : Me.62,
64,71,73 : Q.IO.
Geometrical : constructions : LM.2.
definitions : J.l.
dissections and transformations : Me.
75.
drawing : A.23.
figures, general affinity of: J.12.
forms : G.l : of 2nd species, G.3,4.
* mean : 92 : CD.8 : Pr.29.
approx. to by a series of arith. and
harm, means : N.79.
paradoxes : Z.24.
* progression: 83: A.pr2,6 : G.U : N.
"54.
a property of 1,3,9,27... : A.33.
proportion, theory of pure: A.62.
quantities and algebraic eqs : C.29.
reckoning (Abziililcndcn) : ]\r.l(>.
relation of the 5th degree : i\r.2.
relations, ap of statics : J.21 ...
signs : Z.14.
* theorems and problems : 920 — 1102,
INDEX.
885
Geometrical — {continued) :
theorems : J.ll : L.46 : N.T^a : Law-
son's, Man. 13.
method of discovering : J. 8.
from a principle in alg. : LM.ll.
problems: At.25,32.
transformations : A.32.
and ultra-geometric quantities : C.52,
55.
Geometry : of the Ancients : At.22.
comparative, ap. to conies : N.653.
of derivation : An. 543.
* elementary : 920—1102 : J.6,10 : A.2 :
N.623.
principles of: A.40 : 0.56: G.11,14,
20: LM.16: Me.62: Z.20.
enumerative : Ac.l.
higher : A.20 : No.73 : prsA.55.
instinct of construction : N.56 : Q.2.
der Lage : Z.6.
linear : A.27 : ap. to quadrics, M.IO.
linear and metrical : M.5.
of masses : JP.21.
and mechanics, on their connection :
L.78.
organic, of Maclaurin : L.57.
plane, new anal, foundation : M.6.
plane and solid, analogies : L.36 : N.
653.
plane and spherical, ths : Mem. 15.
of position : J.50 : Q.l : analTE.9.
theorems: An.55: J.31,34,38,41o :
TE.28.
5 points in space : CM. 2.
in lieu of proportion : CP.IO.
of space, abstract : Pr.14,18 : aphor-
isms, J. 24-2.
* of three dimensions : 5501 — 6165 : N.
63.
Glissettes : problem : Q.ll.
centre of curv. : JP.21 : I'ad. of curv.,
L.45.
Golden section : A.4.
Goniometrical problems : Q.7,15.
Graphic calculus : C.89.
Graphs (Clifford's) : LM.IO.
application to binary quantics : LM.
172.
to compotmd partitions : AJ.96.
Grassman's life and works : M.14.
Greatest common measure : see "Highest
common factor."
Greatheed's theorem, D.C : CM.l.
Grebe's point : A. 58.
Green's theorem, &c. : J.39.44,47 : TE.
26.
*GrifRth's theorem (Conies) : 6096.
Ground figures, single and double rela-
tions : J.88.
Groups: AJ.l: LM.9: M.13,20,22.
cyclic, in Cremona's transf. of a plane
An.82.
in a quadratic transformation : An.
82.
discontinuous : C.94.
of linear substitutions : Ac.l.
of finite order contained in a group of
quadratic substitutions : C. 97,98.
of finite order contained in the semi-
cubic groups of Cremona : C.99.
Fuchsian : AC.I3.
formed from a finite number of
linear substitutions : C.83.
of interchangeable elements : J.862.
introduction to the theory of : Me.62o.
Kleinean: C.93.
of many-valued functions : Man. 62.
modular eqs. (Galois) : M. 14,18.
non-modular : Man. 652.
of points G'4 on a sextic with 5 double
points : M.8.
primitive: C.72,78,96 : L.71.
for the first 16 degrees : C.75
degree of, containing a given sub-
stitution : J. 79.
(P)36o (n)36o of the figure of six linear
complexes of right lines two and
two in involution : An. 83.
principal, classification of: C.73.
of substitutions: C.67,84,94: M.5:
isomorphism of, G.16.
of 168 substitutions and septic equa-
tions : M.20.
transitive : G.22 : J.83 : N.84.
*Guldin's theorems : 5879 : Me.85.
Harmonic axes : of curves : C.743.
of a system of right lines and planes :
G.4.
Harmonic centre for a system of 4 points
in relation to a given pole : Z.20.
Harmonic division : of a conic : G.IO.
of a quadric : G.IO.
* of a right line by a circle and chord of
contact : 948.
Harmonic: hexhedron and octahe-
dron: Z.18.
* pencils and ranges : 933, 4649 : Q.6.
* of 4 tangents to two conies, locus of
vertex : 4984.
* points, system of four : 1063 : N.51.
polar curves : A.50 : M.2.
* progression or proportion : 87 : ext
of th, A.31,43,tr41 : C.43 : Me.82 :
N.85 : Z.3,14 : sum of, Pr.20.
divergency of : A.l.
* section by a quadric and polar plane :
5687.
Harmonics in a triangle : A.57.
INDEX.
Hermitc's cp function, linear transf. of :
M.3.
*Helix : 5756 : AM.
conical : N.13,53 : rectif. of : N.45.
on a twisted cone: A.IG.
relation with cycloid : C.51.
♦Hemisphere, volume, &c. : 6061.
Herpolode of Poinsot : C.99.
Hesse's surface, &c. : Z.19.
*Hessian : 1630 : J.80 : curve, M.13.
covariant of binary quintic form : M.
•21.
of a quaternary function : Q.12 :
cubic, Q.7.
of a surface : nodes of, J. 59 : con-
stant of, M.23.
Hexagon : thN.65.
Pascal's : see " Pascal."
in space : J. 85,93.
♦Hexahedron : 907.
Higher algebra : An. 54 : N,66.2 (Serret) :
Q.45.
Higher : analysis : A.25 : G.14.
arithmetic : J.6,9 : N.81.
geodesy: Z.19: trZ.lS..
geometry : A.IO: N.57 : Z.6,17.
planes : A.47.
variation of simple integrals : Z.22.
*Highest common factor : 30 : A.3 : M.
7.2-. N.42,44,452.
of 2 complex numbers : no. of divi-
sions : L.46,48.
of 2 polynomials : CM.4.
remainder in the process : C.42.
Holditch's theorem : see " Closed
curve."
Holomorphic functions : C.99 : G.22.
development in series : C.94 : M.21.
Homalographic projection : N.61.
Homaloidal system, 7i-tic surface and
an (n — l)-ple point : G.13.
Homofocal : conies : thISr.492 : loci
relating to parallel tangents, C.
62,632.
quadrics: C.50 : L.th51,60 : N.th64,
79: Pr.332.
j)araboloids : A.35.
and conjugate surfaces, tr : Z.73.
common tangents of : C.22 : L.46.
quartic surfaces, triple system of, in-
cluding tlie wave surface : N.85.
sphero-conics : L.60.
surfaces, and fx ain-i' + v shi~i" = a~ :
C.22.
Homogeneity of formulae : C.96 : thsN.
49.
Homogeneous coordinates: G.13,8: Z.15.
metrical relation : G.ll.
Homogeneous functions : see " Qiian-
tics."
♦Homogeneous products, II{n,r) : 98 —
9: Q.6,9,10.
♦ and sums of powers : 538 : E.39,40.
nomographic division of three tangents
to a conic : Mcl.2.
nomographic figures : threeC.94 : thQ.
3: N.58,68,pr61.
corresponding points, th : L.45.
focal properties : LM.2n.
relation of roots : N.73.
♦nomographic : pencils : 4651.
♦ systems of points : 1058 — 73.
on quadric scrolls : Q.9.
theorem of a conic : N. 48,49.
transformation: N.70: of angles,Q.14.
Homography: Me.62 : ^N.60 : Z.21.
and perspective : N.69.
and rotations, correspondence of : M.
15.
Homological polar reciprocal curves :
ET.44.
♦Homology : 975 : G.3,8 : N.44 : E.24.
conic of: C.94.
of sets: Q.2.
♦ of triangles : 975 : Me.73.
♦nomothetic conies : 4523 : N.64,th68.
with the same centre : C.66.
♦Horograph : 5826.
♦Hyperbola : theorems : A.27,46 :
CD.l : N.424.
♦ with asymptotes for coord, axes :
4387: Me.73.
♦ asymptotic properties : 1182.
♦ conjugate : 1160.
♦ construction : 1247, 1289.
eccentric circles : A.44.
♦ quadrature of.&c. : 6118 : A. 25,26,27 :
N.44 : TI.7 (multiple areas).
♦ rectangular : 4392 : Z.26 : under 4
conditions, A.3.
♦ segment of : 6118 : N.61.
♦Hybcrbolic arc, rectification of: 6115 :
J.55 : P.2,11,59.
♦ Landen's theorem : 6117 : LM.ll,13o.
♦Hyperbohc functions : 2180 : A.19 : G.
15 : Mem.30 : N.64.
analogy with the circle: An. 51.
ap. to evolution and solution of eqs. :
A.38.
construction of tables of: J. 16.
generalization of : A.35.
♦Hyperboloid : 5605 : J.85 : Me.66.
theorems : geoG.4 : J. 24.86.
♦ one-fold : 5605 : of rotation, A. 70 :
L.39 : M.18 : N.58.
parameter of a parabolic section of :
N.75.
two-fold: 5617: A.18,ths27.
conjugate: CD.2.
equilateral and of revolution : Ac.5.
INDEX.
887
Hyperboloid — (continued) :
^ generating lines of : 5607.
and relation to ruled surfaces : Z/23.
of revolution : N.72.
Hyperboloidic projection of a cubic
" gobba " : An.63.
Hypercycles : C.Q-is.
Hyperdeterminants : CD. 9 : J.342,4'2.
Hyper-elliptic functions : A.16 : AJ.Ss,
7 : An.70 : C.40s,62..,67,92,94,97 :
CD.3 : J.2.5,27,30,40,47,52,54,75,
76,81,8.5: L..54: M.3,11,13 : Q.15,19.
of 1st order : J.12,16;,35,98.
containing transcendents of 2ud
and 3rd kind : J.82.
multiplication of : Ac.3 : M.17,20.
transformation of: Ac.3: (jj = 2),
M.15.
transf. of 2nd degree : M.9 : Mo.66.
transf. of 3rd degree : M.1,193.
transf. of 5th degree : M.16,17,20.
of 3rd order (p = 4) : M.12.
of 1st order and 3rd kind : J.65,68,88.
of 1st and 2ud kind : An..58.,: J.93 :
in series, M.9.
of 3rd kind, exchangeability of para-
meter and argument : J. 31.
of «th order, algebraic relations : C.
993.
Gopel's relation : An. 82.
addition theory : M.7.
addition th. for 1st order in a system
of coufocal quadrics : M.22.
approximation to : P.60,62.
choice of moduli : C.88.
division of: C.68,98: L.43 : M.l.
bisection: C.7O2 : trisection, An. 76 :
M.2.
generalisation of : C.84,98.
geo. representation : L.78.
inversion of : C.99 : J. 70.
in logarithmic algebraic functions :
M.ll.
and mechanics : J. 56.
periodicity moduli : A.68 : An.70.
periodic: J.32 : of the 1st class, LM.
12 ; with four periods. An. 71.
with quartic curves, 4 tables : M.IO.
reduction of, to elliptic integrals :
Ac.4: C.8.55,93,99 : J.55,76,79,86,
89 : M.15 : TI.25.
transformation of 2nd order, which,
applied twice in succession, pro-
duces the duplication : C.88.
transformation: M.7,prl3.
of two arguments, complex mult, of:
M.21.
Hyper-elliptic ^-functions, alg. charac-
teristics : M.25.
Hyper-Fuchsian functions from hyper-
geometric series of two variables :
(J.99.
Hyper-Fuchsian gi'oups : 0.98^.
Hyper-Jacobian surfaces and curves :
LM.9 : P.77 : Pr.26.
Hyper-geometric functions or series :
291 : A.55,57 : J.15o,75 : M.3 : Q.
16: Z.8,26,27.
as continued fractions : 291 — 2 : J.66.
of two variables : C.90o,91,95 : L.82.,
84.
extension of Riemann's problem :
C.90.
of «th order: C.96 : J.71,72 : M.2.
and Jacobi's polynomials : C.89.
square of : J.3.
Hyper-geometric integrals : J. 73 : Z.22.
*Hypocycloid : 6266.
with 2 cusps : Z.19.
with 3 cusps : J.64 : Me. 83 : N. 703,75.
Hypsometric tables of Bessel : Pr.l2.
*Icosahedron : 907 : M.12,25 : and star
dodecahedrons, Z.18.
Icosian game : Q.5.
*Imaginary : quantities : 223 : A.20,
22: 8 square, A J.4 : C. 18,24,25,
882,94 : JP.23 : N.63,64 : P.1,6,31.
ap. to primitive functions of some
derived functions : jSr.63.
* conjugates : 223 : modulus of, 227.
* logarithm of : 2214: LM.2.
curves : Q.7.
exponents : A.6.
integrals of d. e : C.23.
prime factors of complex numbers
formed from the roots of irreduci-
ble rational equations : Z.IO.
transformation of coordinates : Q.7.
variables: 0.96,.: polygons of, C.92.
V\/a-\-ib in the form x+iy : A. 55.
tan-' it+ir]) in the form x+iy : A.49.
<P (.'.', y)+H (■'■, y) = F {x + iy), to deter-
mine <p and 4' : A.IO.
* geometry : 4916 : A.32,,61 : C.61 :
CD.7,8: J.55,70: M.ll: Me.81:
N.70o,72„: TE.16.
of Lobatschewski : G.5: J.17 : N.
683.
of Standt : M.8.
use in geometrical drawing : J.l.
* circnlar points at infinity : 4717, 4918,
4935 : tg.eq4998, 5001 : Me.68 : Q.
3,8,32.
* coordinates : 4761 : C.75 : Man. 79 :
homog.Q.18.
elements in geometrical constructions,
and apparent uncertainty there-
from : Z.12,
INDEX.
Imaginary — (confinued) :
* lines throus^h imaginary points: 4761,
4722— :J.
problem, Newton-Fourier : AJ.2.
* tangents through the focus of a conic :
_ 4720—1, 5008 : A.22.
variables, generating polygons of a
relation between several : JP.30.
xc-\-iy = VX-\-iY, and the lemnis-
catic coordinates of the nth order.
Implexes of surfaces : C.SO..
*Implicit functions : of one independ-
ent variable, f {xij) ; values of (p,-,
<p-% <P,, : 1700—6.
* the same when \p {x,y) = is also
given: 1718—9.
* JJr, y>r, ijix when <P (.«,?/) = : 170 7—
16 : 1/,,^, An.58.
* 'P- («,Z/-2), 'Pir (n,y,z) : 1720—1.
* 'Pi {■'', y, ''■>$) when. 3 eqs. connect
^••,y,^,l: 1723.
* of two independent variables :
y,-: when (p {x, y,z) = : 1728.
* 'P.i'i'yz), (P'iT, <pxz, when ^i{x,y,z)
= 0: 1729—32.
* (p (if, y, z, ^, rj) when 3 eqs. connect
^e,y,z,^,r,:l7Sb.
* of n independent variables : 1737 :
An.58.
trausf. into isotropic means and trig.
series: C.38.
transf. into explicit functions : C.38.
defined by an alg. eq. : C.47.
determined by the infinitesimal cal-
culus : C.34.
*Incrcment : 1484.
Incommensurable : numbers : JP.
15 : N.43.
limits of numbers : N.81.
lines : ]Sr.44: in ratio a/3 : 1, A.3.
*Indeterminate coefficients : 232, 1527 :
A.3 : J.5.
*Indetcrininate equations (see also
" Numbers " and " Partition ") :
188—94: C.10,th78,88 : G.5 : J.9.,.
Mem.44 : N.44.,45,pr57,69o,71,78",
81,prs81,85 : TE.2.
ap. to a geo. problem : Mem. 20 : Z.20.
impossible class of : N.63.
linear: JP.13 : L.41 : N.43: P.61 :
Z.19.
* with 2 unknowns : 188—93 : A.3,7 :
J. 42 : L.63,69 : M6m.31.
* with 3 unknowns : 194 : G.2.
with n unknowns : C.94 : N.52.
.«i+2,C2+...-f »i,(;„ = w : G.l.
Indeterminate equations — (continued) .-
and congruences : Pr.ll.
quadric : J.45 : in n unknowns, N.84.
quartic : geo.cnL.63.
quintic : J.3.
quadratics in two unknown integers :
x^-mf- = ±1 : A.12,52 : E.23,28 :
J.17 ; by trig. : L.64-66 : N.78.
x'—ay^'^b: C.69 : L.37,38 : Mem.
28.
x^~aif = ± 4, a= 5 (mod 8) : J.53.
x^ + y^= (0,2+6=)*: C.36: An.53.
a;2 + 2/^ = : geoA.55.
(n + 4)x'-ny- =4: N.83.
ax- + bx = y": L.76.
(IX- + bx + c =^ ?/- : G.7.
2x^ + 2x + l = y-. N.78.
ax" + bxy + cy'^ = : geoC.9.
;<;2 + nxy - ny'^ = 1 : N.83.
a;^ — 2/2 = «?/ impossible: N.46.
ax^ + bxy + cy- + fk + c?/ +/= : 0.87 .
quadratics in three unknown integers:
X- + y" + z^^=Q : geoA.55.
x2 + y- = 22 : A.22,33 : E.3( » : G.19 :
solution prior to Diophantus,
C.283.
x^ + ay- = z^: N.78.
ax^ + by^ = z^: G.8
x^ + ay"- = z: N.78.
x^±ay' = 4z: N.72.
x"' + a{x + bf^y: N.78.2.
(x- + ky-) z^=ax + blcy : J.49.
(a, b, c, d, e,f'^xyzY = t : Pr.l3.
«2 + 7/2 + 16z2=«2: Mel.4; =4h + 1,
L.70.
quadratics in four unknown integers :
x"- + y"-±z"-:= t- : CM : N.48 ; x- + 2y-
+ Sz^ = t, L.69.
7y2 = 22 + ^(2+/3)-: N.78,.
quadratics in five unknown integers
X- + by'^ + cz- + dir = u, with the follow
ing values of b,c,d: 1,1,1 ; 2,3,6
L.45: 1,1,2; 1,1,4; 1,1,8; 2,2,2;
2,4,8; 4,4,4 ; 3,4,12 ; L.61 : 1,2,4
1,4,8; 2,2,4; 2,4,4; 2,8,8; 4,4,8
4,4,16; 4,16,16; 8,8,8; 8,8,16
8,16,16; 16,16,16; L.62 : 2,3,3
3,ci,3a; L.66: 1,3,3; L.6O3, 63
1,1,3; 1.2,6; 2,2,3; 2,4,6; 4,4,
12; 1,1,12; 2,2,12; 1,4,12; 1,
3,4; 3,4,4; 4,12,16; 3,6,6; 3,3,3;
3,3,12; 3,12,12; 12,12,12; L.6 3:
INDEX.
889
Indeterminate equations — (contimwd) :
1,1,5; 2,3,6; 5,5,5; L.64 : 1,5,5;
1,6,6; 1,9,9; l,n,oi; 2.n,2n; L.65,
59 : with c = ab, 0.42 and L.56.
ax" + hij^ + cz" + dt~^it,with. the follow-
ingvakies of a, 'b,c,d: 2,2,3,4 ; 2,3,
3,6; 3,3,3,4; 1,2,6,6; 2,3,4,4;
L.66: 2,2,3,3; L.65: 3,4,4,4;
3,4,12,48; L.63.
a;3 + 2/2 + z2 + /2 = 4^t: L.56.
ax^ + hy^ + cz^ + dt^ + exij +fzt = u, with
the following values of a, h, c, d,
e,f: 1,2,-2,2,1,2; L.63: 1,2,3,3,
1,3 ; 2,2,3,3,2,3 ; 1,1,6,6,1,6 ; L.64 :
2,3,2,3,2,2 ; 2,5,2,5,2,2 ; L.64 : 1,1,
2,2,0,2; 1,1,1,1,0,1; 1,1,1,1,1,1;
2,2,3,3,2,0; 1,1,3,3,1,0; L.63; 3,5,
10,10,0,10; 2,3,15,15,2,0; 2,3,3,3,
2,0; L.66.
rc3 + 27/3 + 2z3 + 3^2 + 2ijz = u: L.64.
2*3 + 3?/3 + 3^3 + 3i3 + 2ijz = 11 : L.66.
x'^ + y^ + z^ + 2u^ + 2uv + 2v- + t- = w :
L.64.
a;3 + 1/3 + 22;3 + 2zt + 2t^ + oio^ + 3v"-=iv :
L.64.
2 (a;3 + xy + y~) + ii {z" + 1^ + ifi + v~)=iv :
L.64.
xy + yz + zt + tu^v : 0.62; : L.67.
y^=^xl+xl+ ... +Xn : G.7.
quadratics in seven unknown inte-
gers :
x^ + ay^ + hz^ + ct~ + du^ + ev~ =^ lo, with
the following values of a, h, c, d, e :
4,4,4,4,4; 1,4,4,4,4; 2,2,4,4,4; 1,1,
4,4,4; 1,2,2,4,4; 1,1,1,4,4; 1,1,2,
24; 1,1,1,1,4; 4,4,4,4,16; L.65:
1,1,1,1,1; 1,1,1,1,2; 1,1.1,2,2;
1,2,2,2,2; 2,2,2,2,2; 2,2,2,2,4; 3,3,
3',3'3': L.64.
higher degrees :
cubic : AJ.2.
aj3 = y3+a: ]Sr.78..,83: with a = 17,
N.77.
a;3+2/3 = azi : N.78o,80.
a!3+2/^+s3+.it3 = 0: A.49.
a«4+&i/4 = 22: C.87,91,94: N.79:
a =1,1 = -5, L.79.
a;4±2"'i/* = ^3 and similar eqs : L.53.
x^-\-axh/-\-rj^ = s3 : Mem.20.
ax^+bx^y"-+cyi+dx^y + exy^ =fz^ :
C.883.
a;5-|-i/5 = az5 . L.43.
xT-\-y1 = Z-, impossible : 0.82^ :
L.40,.
Indeterminate equations — {continued) :
n-tic solution by alg. identities :
C.873.
a;»-|-y" = 3" impossible if 11 >2 {For-
mat's last th.) : A.26,58 : An.64:
C.24,89,90,,98 : J.17.
x'^"—y~" = 2x": L.40.
ax"'+by"' = cz"': 1^.7%
a;'" = '2/"+l' impossible : N.50,70,71-
fc2— ay3 = 2»: C.99.
simultaneous :
fc = «2; fe + l=2u3; 2,c + l=3i(;3:
N.78.
,c~ + a = y~; y~—a:=z~: An.55: C.78.
xZ+x+2 = y3; xZ-x-2 = z"-: N.76.
ax + by + cz=0 ; Ayz+Bzx-\-Cxy
= 0: A.28.
,^2 + ^2-22 = D ; a;2_y2 + j;2 = q ^
— .f2 + l/3 + a3 = □ : E.20.
.c2+a«?/-f 1/2= D ; 7jZ+ayz+z^^ a ;
zZ+azx+x^=a : E.20,21.
x+y + z = a ; a;2+2/2+a2 == □ ;
fc3+i/3 + s3=a: E.17.
six eqs. in nine unknowns : N.50.
exponential, x'J =y'' : A.6 : Z.23.
a--&!' = l: N.57.
*Indeterminate forms : 1580 — 93 : A. 26 :
AJ.l exponential : J.l : Me. 75 :
N.48,77 : -, A.21 ; Z.l : — ; L.41,
CO
* 42 : N.46 -.-^^ when * = 00 , 1592:
0", J.ll,12:'0exp0^ J.6.
* with two variables : 1592a.
*Indeterminate multipliers : 213, 1862,
3346: N.47.
*Index law : 1490.
Indian arithmetic, th : L.57 : calculation
of sines, N.54.
*Indicatrix: 5795: C.92: Me.72: N.74.
* a rect. hyperbola, condition : 5824.
* two coinciding lines : 5825.
* an vimbilicus : 5819.
to determine its axes : L. 78,82.
determination of a surface from the
indicatrix : A. 59.
*Indices : 29 : N.765,778,78.
in relation to conies : !N'.722.
of functions, calcixlus of : J P. 15.
*Inductiou : 233 : C.39 : G.15 : L.48.
*Inequalities : 330—41: A.1,24.
in integrals: Mel.3 : f.d.c.Mem.59.
{lny~>n": N.60.
5 X
890
INDEX.
Inequalities — {continued) :
a*>a; : A.14.
if 8.2-1-,/ = 22, x"' + y"^ I 2'"*: A.20.
geo. mean of n numbers < arith.
mean : 332 : N.42.
Infinite : equalities : M.IO : prG.22.
functions : An. 71 : J. 54.
from gnomonic projection : Mc.CG.
linear point-manifoldness : M.lo,17,
20,212,23.
point-mass : M.23.
products: J. 27: N.69.
value expressed by r functions :
Ac.3.
exhibiting circular arcs, logarithms
and elHptic functions of the 1st
kind : J.73 : Ac.4.
use of in mathematics : C.73.
*Infinitesimal calculus : 1407: M.11,18.
Infinitesimal geometry : An. 59 : C.82.
of a surface, formula : G.13.
Infinity, points at on alg. surfaces : C.
59.
*Inflexional tangents : 5789 : A.35.
of a cubic curve : E.30 : J.38,58.
Inflexion curves : Z.IO.
*Inflexion points : 5175: CM.4 : J.41.
of cubic curves : J.28 : axis, E.31.
Horse's equation : N.81.
Inscribed figures :
In-circle : of a quadrilateral, locus of
centre: A.52.
* of a triangle : 709, 953, 4747—50 : tg.e
4889: CM.l.
* in-centrc : 709, t.c4629, tg.e4882.
In-conic : of a circle : thJ.91.
four of a conic : prJ.39.
of a developable quartic : An.59.
of a polj'gon : M.25.
of a quadric : J.41.
* of a quadrilateral : tg.e4907 : N.63 :
four, N.56.
* of a triangle : 4739—46, tg.e4887 :
A.2: N.50; max, A.8 : Q.2 : lat.
rect., E.34.
In-cubic of a pencil of six lines : Q.9.
In-hcxagon : of a circle : A. 22.
* of a conic : 4781 : N.57,82.
In-parabola of a triangle : CD. 7.
In-pentahcdron of a cubic : Ac. 5 : M.5.
*In -polygons : -of a circle: 746 : CM.l :
J.35 : N.50.
regular of 15,30,60,120, &c., sides :
A.62.
do. 9 and 11 sides : LM.IO.
do. 17 sides : TI.13.
do. four of 30 sides : N.78.
do. 5, 6 and 10 sides, relation : A.40,
43,45,48.
In-polygons — (contimied) :
two stars, one double the other :
A.61.
of a circle and conic (Poncelet) : G.l.
^ of a conic : 4822 : thsN.47 : cnTN.69.
*= with sides through given points,
en : 4823 : An.5l2.
semi-regular: N.63.
of a cubic (Steiuer) : M.24.
of a curve : Q.7.
of a polygon, th : CD. 5.
of a quadric with sides thi'ough given
points : LM.22.
In-quadrics : of a developable : Q.IO :
quartic, An.59.
6 of a quadric, 2 touching 4 : An. 69.
*In-quadrilateral : of a circle : 733 :
A.5: cnE.21 : area, N.44: P.14.
* of a conic : 4709.
of a cubic : N.84.
In-sphere of a tetrahedron: A.61.
In-spherical quadrilateral : N.49.
In-square : of a circle : J.32.
of a quadrilateral : A.6.
In-triangles : of a circle : P. 71.
with sides through given points :
J.45 : N.44.
of a conic : J.7 : Maccullagh's th, N.65.
with given centroid : &.23.
similar : A.9.
of a triangle : thsQ.21.
two ( Stein er's " Gegenpunktc ") :
J.62.
In- and circum-circles : of a poly-
gon : N.45.
distance of centres : A.32.
of a quadrilateral : Fuss's prMel.3.
* of a triangle : 935 : A.38.
* distance of centres : 936, 4972 : eq,
4644.
In-and circum-conics : of a pentagon:
N.78..
of a polygon : J.64,70 : regular, Z.14.
of a quadrilateral : 60 theorems, N.
45.,: N.76.
of a self -conjugate triangle : Me. 81.
* of a triangle: 4724, 4739: An.52 :
G.22,.
In- and circum-heptagons of a conic :
A.3.
In- and circum -pentagon : of a circle :
A.22,43.
In- and circum - polygons (see also
"Regular polygons"): of a
circle: L.16: N.80 : P.ll : Q.H.
* sum of squares of pcrps., &c. : ths
1099.
dillercncc of perimeters, ths : N.433.
of two circles, respectively : C.53 :
G.21 : L.78.
INDEX.
891
In- and circura -polygons — [continued):
of a conic : A.4 : ellipse An. 52 : An.
57 : J.64 : N.67,84
of two conies : C.90.
of a curve : 0.78.
of a homonymous polygon : A. 50.
In- and circum-quadrics of a tetra-
hedron : eqsN.65.
In- atid circum-quadrilaterals : of a
circle : A.48.
of a conic : 4709 : and pentagon ths,
N.48.
In- and circum-spheres : of a tetra-
hedron : N.73.
of a regular polygon : A.3'2.
* of a regular polyhedron : 910.
In- and circiim-triangles : of a circle
(Castillon's pr) : Q.3._
equilateral, of another triangle : Me.
74.
and square of an ellipse : A.30.
* of two conies : 4970 : N.80.
* envelope of base : 4997.
respectively of two conies having a
common pole and axis : CD. 4.
*Instantaneous centre : 5243.
*Interest : 296—301 : and insurance, A.
26.
Integrability of functions : An. 50,73 :
C.28: J.59,79: JP.17: L.49,.
criterion for max. and min. values of
a primitive : An. 52.
*Integral calculus: 1900—2997: A.ext
18,26: Euler's, A.20 : 0.14,42:
Newton, 0D.8 : G.19 : L.47 : Me.
72,74,75: Mem.18,36.
paradoxes : 0.44.
* theorems, &c. : 2700—42: A.45 : 0.
13 : L.geo.ap50,56 : Me.77 : Mem.
prsl5,30.
Integral functions : 0.88,89,98 : 0.4,22.:
h.c.f of G.2.
with binomial divisors : J. 70.
and continued fractious : An.77.
reciprocal relation of : A. 67.
*Integrals or Integration : 1908 : A.1,2,
^ 4,5,6,10,23: Ac.1,32,44: O.90 :
0D.9: 0P.3: J.2,4,8,17,25,36,39,
61,92 : JP.9,10,11 : L.39 : M.6,16,
73,75 : Mem.31 : P.14,36,37 : Pr.
7,39: Q.11,13: Z.7,ll,15,18o,222,23.
* approximation to : 2127, 2262, 2991 :
A.9,14: 0.97: 0M.2: G.3: J.l,
16,18,37,48 : L.80.
* Gauss's: 2997: 0.84: M.25.
by the principle of Abel's derivative :
J.23.
* by diiierentiating under the sign of
integration : 2258.
by elliptic functions : G.ll : L.46.
Integrals or Integration— (cow^iuwed) ;
from orthogonal surfaces' theory :
L.38.
by Pfaff's method : A.47.
by series : Me. 71.
by substitution : A. 18.
by Tchebyehef's method: L.74 : M.S.
comparison of transcendents : Me.79 :
Pr.8.
complex, representing products and
powers of a definite integral : J.
48.
connected with trinomial integrals :
L.55.
convergency of : M.13.
definite : applied to Euler's, &c. :
J.16.
with finite diff"erences : J. 12.
from indefinite : J.41,51,52.
whose derivatives involve explicit
functions of the same variable :
0.12.
determination of functions under the
signf: JP.15.
^ difference between a sum and an
integral: 2230: G.9.
division into others of smaller inter-
vals : A.4.
* eight rules for definite integration :
2245.
equations for obtaining functions as
integrals : J. 79.
expressible only by logarithms : An.
76.
extended independently of the con-
ception of differentials : A.61.
formulas of: A.l : J.18,19 : M.4: Me.
76: Mo.85: N.85: failure of f,
0M.2.
and gamma-function : LM.12 : Z.9,12.
higher, of composite functions : A.20.
with imaginary limits : 0.23: J.37.
use of imaginaries in : M.14.
inverse method: 0M.4: OP.4,5: L.78.
involving elliptic functions : Pr.29 :
Q.19.
* limits of: 2233— 44: L.74.
multiplication of : Pr.23.
number of linear independent of 1st
kind: An.82.
of alg. differentials by means of
logarithms: Mo.57 : An.75: 0.
9O2: J.12,24, 78,79: Mo.84: N.81
(see " Integrals ").
of algebraic surfaces: O.9O3: J.26 :
- octic. An. 52 : cubature, 0.80.
* of circular functions : 1938—97 : 2453
-2522: No.1799: LM.4: M.6 :
Mem.9.
892
INDEX.
Integrals or Integration — {continued) :
sine and cosine : G.6 : M.ll.
* of exponential and logarithmic func-
tions : 2391-2431: E. 17,18.
* of circular, logarithmic and expo-
nential functions : 2571 — 2643
(see "Integrals ").
of a complex function: A. 66: th of
Cauchy, Ac.84.
* of a closed curve : 5204 : C.23 : E.28 :
Z.17.
of differentials containing the square
root of a cubic or biquadratic :
Me.57.
* of discontinuous functions : 2252 :
C.23 : LM.6.
of dynamics : L.52,55,.58.
of explicit functions, determination of
algebraic part of result : An. 61.
* of functions wbicli become infinite
between the limits : 2240 : J.20 :
JP.ll : Q.6.
of infinite relations : M.14.
of irrational alg. curves by loga-
rithms : An. 61.
* of irrational functions : 2110—20 :
AJ.2 : An.56 : C.32o,89 : L.63,64 :
Mem.30.
* limits of : 1903,-6, 2233, 2775.
* for quadrature of curves : 5205 — 11:
0.68,70 : circle, J.21,23 : JP.27.
triple integrals : J.59.
* of rational functions : 2021- 32, 2071
—2103 : L.27 : N.73.
* of rational fractions : 1915 : CD.3 :
Mcm.33,: N.72.
of total differentials : 0.99-2.
of transcendental functions : JP.26.
of two-membered complete differen-
tials : J.54.
periods of : C.36,38,75o : G.753 : JP.274.
* principal values of infinite definite :
2240: A.68.
propei'tics by elliptic coordinates :
L.51.
quotient of two d.i of the form
^dxdy...d:: : J. 67.
reduction to elliptic functions : An.
60 : LM.12 : Me.77,78.
residues of : JP.27.
Eicmanu's of first kind : An. 79.
singular values of : A. 11.
* successive : 2148 : L.62 : 2nd order,
M.20 and Z.ll.
* summation of : 2250 : J.47 : JP.12,21.
tables of definite, by B. de Ilaan, note
on: C.47.
and Taylor's theorem : Me. 84.
theorems : L.48 : P.55 : Q.10,12.
Integrals or Integration— (co7i^inuef?) .-
* transformation of: 2245— 62: A.IO :
CM.4 : J.f 16,22,36 : L.36 : Mel.3 :
Q.l.
* variation of arbitrary constant : 2247 :
J.33.
whose values are algebraic : J. 10 :
JP.14 : L.38.
ALGEBRAIC PTJNCTIONS. Indefinite :
unclassified: An.75 : C.gOo : J.12,24,
78,79: Mo.84: N.81.
* simple functions of x^±a^ : 1926—37 :
x"\ A.4 : v/(a3-a;2), A.38 : J—-„,
N.82:
V{1-
An.
fractions involving a binomial surd :
2008—19.
1
(l+fc)V(2a;2-l)
Mem.l3.
US.
{x" — a) \/{x" — h)
^^il±^ : 2015 : L.80 : Z.8.
(l±a'g)g
(l±«2)(H-ta33+fc4)5
Mcm.lO.
A.3.
(a;3+8) v^(x3-l)
* J^ and deductions : 2021—8.
r<;»±l
* a;'" {a+ hx")'' : 2035—60 : A.30 : Mem.l 1 .
* —1— : 2061—5 : J.36.
x~+a~
1 ... 1
: A.40;
2007.
{x—aYix—b)"' '"' x"{x—l)"
* functions of {a+hx-\-cx"~) : 2071-80:
2103—9.
A.55.
l/{a + hx+cxiy'
* functions of {a+hx^^+cx-^) : 2081—6.
* functions of {a + hx" + cx"') : 2086—
2102.
rational algebraic functions of irra-
tional ones :
* integrated by rationalizing : 2110—
20.
reducible to elliptic integrals, viz. :
rational functions of n/X^, \/X-i, and
^/A'n :
* where Xi is a quartic in x -. 2121
—41.
* F{x, ^X,) : 2121 : LM.8. : L.57.
* ^-'-l., 2133— G: J.3G: LM.14.
yx,
INDEX.
893
Integrals : algebraic — {continued) .
"^^^ : L.64 : Mel.3.
J—: C.59: CD.l: E.36 : J.IO.
-^M_ . C.51.
Fix)^X,
2141 : J.17 : Man.79.
^/X3
F{^X-j): L.57.
^ : Me.82.
{i-x^Y
± , &c., reduced to
(a;3-l)v/(x3-fc3)'
Jacobi's functions : Q.18.
_/M : J.32.
V± (l-a^')
sundry: CM.2 : L.47.
Limits to 1 :
Euler's : see " Euler's integrals."
* £b'-i (l-a!)'"-i . 2280 : A.40 : C.16 :
J.11,173.
Integrals : algebraic— (co?i/nn(fd) ;
* x'-'^fUcix-^V} (Cauchy): 2712;
A.9.
A.12: E.41.
l+2.rcos<^ + i
deductions from this involving in-
te grals of the forms
J— clc and n , a.35.
-^° •' " sin 077
Other limits :
P F{.c)dx [" F{x)dx . 03^8 _o
J.,./(l-ro^)' ] (.c-c)" •
I'
* (a— ia')'a;"'-it7i-c: A. 35.
, ,^^"" ^ , fZr'-=/(«): geoN.85.
. 1— 2fc coso+fc-
x'-'^+x"
2341 : 0.55.
* similar forms: 2342—4, 2352, 2356
—67.
1— as 1 — a;"
: 2367 : A.IO.
a;"' (1-a;)''
{l+axY
: L.59.
{l-xY
: L.57.
{l + y(l + «^c)f'^^^^
^--Ml:-ajr:i : L.56-7
{a+hx+cx'i)^''^
: J.42.
{a-hxy"{\-xy'"x'
Limits to CO :
J^,&c. : 2309—12, 2345—55 : A.38 :
l±x
J.24: L.41 : Q.12: Z.19.
^ : 2364.
(x^ + a^)"
X'^x-'--. Me.83.
-, 2?i > « > : N.48.
CIRCULAR FUNCTIONS. Indefinite:
* sin X, sin-'iK, &c. : 1938 — 49 : sin":c, &c.,
1954—7 : cos'-K, N.74.
1 1951: J.9; ^
!+«;"+»■
&c.: A.16.
l+tc'
^i^\ and 12 similar : A.35.
a+tcosa;' """"' """' (a+6cosrc)»'
1958 : Me.80.
* products and quotients of sine and
cosine and their powers : 1959 —
80, 2066—70 : A.49.
* binom. fuucs. of sin and cos : 1982 — 92.
* ditto of tangent : 1983, 1991.
^ J^^^siuascos^fcl^ . i994_7. a.12:
acosa;+& sin fc + c
J.19,32.
* (a + 26cos«+ccos2a.-)-i: 1993;
* — ^T. 2029.
«-»— 2*" cos Ji^+1
* . F(cosx) . ^c)gQ
{ai+bi cos x){a.2+b.2 cos x)...&c.
a function of sine or cosine in a ra-
pidly converging series, and suc-
cessive integration of it : J. 4,15.
^-^'^- : A.ll.
(1—fc sin (^) v/ (1—^2 cos-0)
(sinama;)-": J.81.
v/{l— A;2 sin2 i {a + x) sin^ * [d-x)} :
J.39.
?iji:^ .. A.17 : with m = 1, G.7.
894
INDEX.
Integrals: circular. Indef. — [cont.):
* sin/vv. .,. (Fourier): 2726—42:
X or sin a;
A.38..
Limits to ~ :
* sm"x: 2453-5/2458,2472: E.29 :
n = i, E.28.
* tan2"'-ia;: 2457.
* sin" aj cos^ a; : 2459 — 65.
* CDS'' X ^^^ nx : 2481, 2484—92 : L.43.
cos
sin ,e 2
cos" ' - X sin nx : 2494.
a~ cos^x-\-h^ sin^a;
—2501, 2344.
and similar : 2496
jra ; and
cos^-'-'aJsin"-' ^^^
sin''~- a; sin^^a; : 2585 — 8.
X cot ax : A.34 : No.l9 ;
a^tani7r(l— M: 5340.
Limits to TV :
sin" a; cos'' a; : 2459.
nx: 2474—82,2493.
P^^ a;
V COS / cos
siir"xd„x/(cosa!) : 2495.
cos n {x—a sin x) : L.41 : to 2it, C.39.
cos {a-\-nx) cos (i+jja;) cos {c-\-qx) : C.
54.
X sma;
-: Pr.25: with a = l, 2506
1 + a^cos^x
and Q.11.
X sin a; or sin a; sin7'« or cosi'a;
: 2623 -9.
1— 2a cosa!+a2
^in^''a;orco3y^a; , Q.n , lM.11 :7i=^l,
(1 — 2rt cosa;4-«~)"
L.74.
fcl^(sin« cos a;): JP.27.
al)out 250 integrals with limits chiefly
from to TT, some from to go :
Pr.252,26,27,29,30,31 ,32,,;53.
Integrals : circular. to oo — [cont.):
f- a;±"' ^^^hx: 2579-81.
cos
cos hx
[■" cos a'.
clx: A.IO.
x+a '
^^°"^ ; 2510 : A.30 : E.26 : Z.5 : 7i = 2
X"
and 3, 2511—2.
cos f/.e — cos pa; nr-ic) r
a; or a;2
sin ax sin 6a;
and similar : 2514 — 22.
Limits to CO:
cos2aa; sin^a;
2722—5.
V'aj
cos X-, cos
(aa;)2: 2507-9: Q.
12: 2602.
(^"M'a;: Q.13 ; •'^'" a;" : Me.7
Vcos/ ^ ' cos
(sinaa;, cos 6a;), reduction of: J.15.
' (sin ax, cos hx) _ y r>o 95
2572 : L.49 : Z.7,
* 9211±^: 2.573: A.10,11,59: J.33: Me.
rt2-(-;c-
72,76: Z.7,8.
~''" '"' : 2575 : A.IO : J.33.
a2+a;3
-i'^^^^: L.40: Q.18.
{a~+x^)"
d)(n) ^^^rx
^°" : A.ll: with (/.(„•)=«;"
a"+x?
L.46: 4. (,(•) = tan-' cv', A.ll.
(«,r)2
a3+fc2
(l+rc2)sin6
Ac. 7.
1— cos
and similar : Z.13.
and related integrals :
: Z.7,8.
1 or a; sin ex
(l+a!2)(l— 2acosc,f + a2)
2030-2.
* ton_^^ 2503; Itan-'-^ tan"' ',
fc(l+a'2) i.;2 a b
2504.
,^ tan'' ga- — tan-' h .v c)rr\f
lim. to 27r ;
Other limits :
Jo a +
6cosa;-|-c sin-.c
A.55.
lo 1 + ,,;
L.69.
d.c, deduced from (2116)
INDEX.
895
Integrals : circular — {continued) .
^ p tan~^ a«(Za5
Jo i«.
r.
ev/(l-a:3)
F{x)
2502.
dx, where F (.c) is a rational
X"
integral circular function : CM. 3.
sine-integral, &c. : sec " Functions."
EXPONENTIAL FUNCTIONS. Indefinite :
e',a^: 1924; x'"^"'', 2004.
e^{<?.(«)+<^'(.^)}: 1998.
eexp(ia;2): E.34.
X being a rational integ. funct. oE x -.
e-'X : J.13 ; e-^, Mem.83 ; e exp
(-a'2), Q.l.
cexp(-a;3) _ ^^ ^^^^ ^^^^^ ^ ^^o ^^.
V{a+hx"^)
-T- (c + tlic2) : L.52.
Limits to ca :
* e-*'' K" : 2284—91 : see " Gamma
function."
* e exp (— /^.c2) : 2425 : evaluation by a
continued fraction, J.12.
* other forms : 2426—31, 2595, '8, 2601.
* eexp (-cc2-^) : 2604-5: L.56.
Z.6; ditto X x"\ L.46.
1+x^
and the same X a- : A. 10.
a;3 — a~
e exp (af«")--Z?K± «'«"): Q-i8.
Other limits :
expon-integral : sec " Functions."
Tc exp (-a-2) [ = Erfc $ (Glaishcr)] :
Me. 76.
P e-'''=F{x)dx: C.77,.
fc^" e exp [ — a;3— iL.
jp(re*')e-"*': A.15.
C.12.
LOGARITHMIC FUNCTIONS. Indefinite :
log a; : 1950 ; a;'" (log xf ", 2003—6.
a;"'log (1+a;), &c. : A.39.
F (a;) log a;: G.12.
Integrals : logaritumic. Indef.—
{continued) :
^'"^'^■^•' , type of several: Mem.18.
Limits tol :
* log-^: 2284; a;^(log^]", 2291.
* involving log a'- or log(l ± a) : 2391 —
2403, 2416—22.
* logll±«). 2416: C.59: J.6: L.43,44.
l+a;2
logfc
: 2636 : Q.12 ;
log a;
re' (log re)"
a; ± 1
and a similar form : 2030-
and many cases, J.34.
* J_ log l±r : 2403 : L.73.
X 1 — X
''"' ^''-^' : Z.3 ; ^!— ^, A.37.
log X log X
about 540 expressions chiefly formed
from log(l±«) or log (1 ± a„)
or log (1 +a'-|-a;3) orlog (1 -(-fc2+a;^),
joined to a single factor of the
form a-'" or (1 ± a'") or ' ,„
with integral values of m and n ■■
A.39,40.
about 280 expressions, nearly all
comprised in the form
£.
x'-{l-x")''{logxY,
with integral values of m, n, 2'>, 1,
and t : A.40.
about 130 expressions of the forms,
X'-{1-X")" (logCc)2«'-3,
_p
fc"' (1— K") « log a;, and
X (X — 1 ) j^g ^^^ with integral
x'' — 1
values of vi, n, p, q, and r : A.40.
* Limits to CO: 2423—4.
Other limits :
* l^^iln^) . 2408—12 : L.73 : Zl.
X
* - log ^^', limits to ^2-1 : 2415.
X l—x
log-integral : see " Functions."
circular-exponential FUNCTIONS. In-
definite :
* e"'' I ^^^Yhx: 1999; e^ sin" ft; cos" »,
^cos j
2000.
INDEX.
Integrals : circular-exi'Onential-
(continued) :
{e" sina;)""^ : L.74.
Limits, to cc :
* e""* sin rx
2571,2591.
* e-"^ sin hx : 2583 : to 1, Mem.30.
* c-'-^o'" ^''^ bx : 2577, 2589 : J.33 : Z.7.
cos
* e-«.r /sin V"^. 2608-11: A.7 :
\cos /
* with limits-^ to ~, 2612.
2 2
sin Hi.i;, &c. : 2593—2600.
* c-*^ cos ma; sin" a; : 2717—20.
* c exp (— a2.i;2) ^°^, 26.i;, and similar :
2614—8: Z.1,10.
* c cxp j - (fc3-|- ^^ \ cos ^ I X &c. :
2606: Q.l;
- : 2619.
CIllCULAR-LOGAIlITnMIC FUNCTIONS :
* x^i'^op;^)'" . 2033.
a;2»_2a5"cosw^+l
Limits io 1 :
* ^"^ (m loff x) I log *, and similar : 2641
cos ^ i^ I I o '
—3.
log(l-2.ccos.?>+a:-)^ ^^^^ &c. : A.34.
X
, l-2a;cos<?>+i«g^^ 13 ^^^-.^^^ ._ l.73,.
° a;(l— a;2)
log sina; log cosw, «fcc. ( limits Oio —\:
E.22. ^
Limits to — :
* log sinfc : 2635 : CM.2 : E.23 : Q.12 :
to rnr and similar integrals, A.
16.
* log(l + cco3^c) . 2633 : to TT, 2634.
cos a;
tana; log coseca; : E.27.
l og(l+»'-^ sin^a;) . -j^^q
^{l—lc"sin~x)
Integrals : cuicuLAii logakithmic—
{continued) :
loo; ^"^ X and log ^(1— />;- sin^ x), each
" cos '^
with the above denominator : J.92.
Limits to TT :
* log (l-2a cos cc+a2) : 2620—2 : L.38 :
Q.ll.
* cosr.Blog(l — 2acos*-|-a-) : 2625: A.
13.
* X log sin x : 2637 ; x log sin2 x, 2638.
Limits to <X) -.
cos ax , c, All
iiog2ca;: A.ll.
h^+x^
^ log (1—2 a cos cx+a~)
l+.i;2
Other limits :
r-a + tcosa;^^^^^ A.53.
Jott — ocosa;
F (cos nx) log sin ''- dx : Z.IO.
I cos-'"~'^.B log tan x dx -. A. 16.
Jo
I '" ic"' log (1—2 a cos x+x^) dx, and
Jo
similar: J.4O3.
exponential-logarithmic functions :
2631.
e"''— e-
-log (.<;'- -|- tt3) (l,i;, &c. : J. 38.
1;
"'e'-'- log (l-2rtC0S a'+a2) dx: J.40.
MISCELLANEOUS THEOREMS :
* formula? of Frullani, Poisson, Abel,
Kummer, Cauchy, &c. : 2700—13.
* |^y'(«,)=/(&)-/(a): 1901-3: A.16.
* C' <p{nx)-<t> {hicl ^^^ . j^jg_8i . „ _o^ 2700.
* H.O{A"-')}^"^'<^= 2001-2.
riiM.dx {6 of Taylor's th) : LM.13.
[ F{x)dx, approx. to: C.97.
lfle-I{)dx = U\y)dr-GDA.
f ^f,,.^}J\.tS:!2(lx\ and similar:
" Me.75,76,77.'
{/(a;) <f) («)(?.«: L.49.
{' avdxlf udx\\dx: AJ.7.
Jo ■'o •'0
INBEX.
897
Integrals : miscellaneous
{continued) :
f {^i)ndu: Z.25.
{u + Jc),,,da = {-iy' j {ti),.2du:
A.38.
\ P,n (x) P„ (x) dx (P,, X = Legendre's
coeff.) : Pr.23.
du
uU^iu)
{x + p){x + qYdd
= log^ia) + G: J.ll
pos integer :
C.78.
I ■ <?> (sin 2x) cos x dx
J" n
= (p (cos2 x) COS X dx : A.21 :
L.532.
J/(a;) sin anxd (log .-) : Mo.85.
r°° r , COS ,T sin" , c p
I $ . a;^ o .1', transi. of:
Jo r2+a;2 sin cos-
J.36.
integrals deduced from
s"+Ps' + Q2 = 0, and y" +
iP+2B) y'+ {Q+B {P+R) + B'}
y = 0: J.18.
ln{x,'ij)dx: J.61:Mo.61: Q.7.
* if f{x+iij) = P+iQ, th : 2710.
[Upwards of 8,000 definite integrals have
been collected and arranged in a 4to
volume by D. Bierens de Haan ;
Leyden, 1867 (B.M.C. : 8532. ff.)]
*lntegrating factors of d.e : pp.468— 471,
3394.
*Integrator mechanical : 6450.
Intercalation : CM. 3.
*Intercepts, to find : 4115.
*Interest: 296: N.48,61,64.
Interpolating functions : C.ll.
*Interpolation : 3762 : A.32,61,62,70 :
AJ.2 : C.19,48,68,92 : J.5 : L.37,
46 : Me.78 : K.59,76 : Q.7,8.
of algebraic functions, Abel's th : J. 28.
Canchy's method : A.2 : C.373: L.53.
by circular functions : N.85.
by cubic and quintic equations : C.25.
formute: J.2 : 0.99.,: Mo.65.
* Lagrange's : 3768': J.1,84 : N.57,61.
Newton's: N.57,61,71.
for odd and even functions : C.99.
* and mechanical quadrature : 3772 :
A.20.
Interpolation— (cojiiiimed) :
by method of least squares: C.373:
Mem.59.
* by a parabolic curve : 2992 : C.37.
Stirling's series : Me. 68.
and summation : I.llo,12,14.
tables: I.II3.
of values from observation : Mel.2 :
tr, Mem. 59.
Intersection : of circles and spheres :
L.38.
* of 2 conies : 4916 : CD.5,6 : 1^.6Q.
* of 2 curves : 4116, 4133 : CM.3 : J.15 :
L.54: by rt. lines, Me.80.
* of 2 planes : 5528.
of 2 quadrics : 0.62 : N.684.
of right line and conic : see " Right
line."
of successive loci, ths : N.42.
of surfaces : J.15 : L.54 : by rt. lines,
Me.80.
^Invariable line, plane, conic,and quadric :
5856-66.
*Invariants : 1628: An.542: C.853 : E.42 :
G.1,2,15: J.62,68,69: L.55,61,76 :
M.3,5,17,19 : Me.81 : N.58,59.,69,
70 : P.82 : Pr.7 : Q.12 : Z.22.
of binary cubics : An.65.
of binary forms : of 8th deg., C.84 :
G.2 : M.5 : simultaneous, M.l.
of higher transformations : J.71.
superior limit to number of irre-
ducible : C.86.
of a binary quadric : M.3.
* of a binary quantic : 1648 : E.40 :
Me.79 : of two, 1650.
of a binary quartic : M.3 : Q.IO.
of a binary quintic, table of irreduci-
ble : AJ.l.
of a bi-ternary quadric : J.57.
* of a conic: 4417: 4936—5030.
* of two conies : 4936 : N.75.
of three conies : Q.IO.
* of particular conies : 4945.
of a correspondence : G.20.
differential : M.24 : of given order
and degree belonging to a binary
10-ic, C.89.
of d.e linear: C.88 : of 4th order, Ac.
3.
and covariants of f{x^,y~) relative to
linear transformation : G.17.
of linear transformations : M.20.
mutual relation of derived invariants :
J.85.
of an orthogonal transformation : J.
65: LM.13.
of a pair of homog. functions : Q.l.
partial : LM.2.
of points, lines, and surfaces : Q.4.
5 t
898
INDEX.
Invariants — (continued) :
of a quadric : J.80 : of two, M.24 : Q.6.
of a quintic : of 12th order, Q.l : of
18th order, C.92 : J.59.
related to hnear equations : C.94.
of sixth order: G.19.
skew, of binary quintics, sextics, and
nonics, relations : AJ.l.
of ternary forms : G.19.2.
transformation theoi'era : M.8 : Me.85.
Inverse calculus of differences : N.51.
•^Inverse equation of a curve or inverse
method of tangents : 6160 : No.
1780: J.26: Mem. 9,26.
*Inversion: 1000, 5212: An.59 : C.9-i,pr
90 : LM..5 : Me.66 : thsN.61 : Pr.
34 : problems by Jacobi, J.89 :
geo. ths, Q.7.
formulge: An.53: Lagrange's, J.42,54.
of arithmetical identities : G.23.
* of a curve : 6212 : (:^.4 : J.14 : Pr.l4.
* angle between radius and tangent:
6212,6219: E.30.
of 2 non-intersecting circles into con-
centric circles : E.39.
* of a plane curve : 6212 : G.4: Pr.l4.
of a quadric: J.52,76 : Q.ll.
of a system of functions : An. 71.
and stereographic projection : E.35.
*Involutc : 6149,— 63,— 66.
* of a cu-cle : 5306 : C.26 : successive,
E.34.
and evolute in space : CD.6.
integrals of oblique : C.86.
* of a tortuous curve : 6753.
*Involution : 1066 : A.56,63 : gzAn.69 :
At.63: CD.2: thCP.ll: thE.33:
G.10,20: J.63: N.63,64,65.
and application to conies : A.4,5.
of a circle : Me. 66.
of a cubic space and the resulting
complex : Z.24.
of higher degrees : C.993 : JM.72 : of
3rdand4th, An.84: Z.19.
of numbers, machine for : P.16.
of n-i\(i curves : C.87.
relation between a curve and an 7i-tic,
the latter having a multiple point
of the n — 1th order: C.96.
pencils with problems in conies and
cubics : N.85.
of points on a conic : N.82.
of 6 lines in space : C.52.
of right lines considered as axes of
rotation : C.62.
* systems of points in : 4S26, 4828.
ditto, marked on a surface : C.99.
Irrational fractions: decomposition of,
J. 19 : irreducibility of, Mem. 41 :
rationalization of, A. 18.
Irrational functions : M.4 : of the 2nd
degree, C.98.
Irreducible functions with respect to
a prime modulus : C.70,90,93 : L.
73..
Isobaric : calculus, N.85 : homog.
functions, G.22 : algorithm, N.84.
Isogonal relations : A.60: Z.18, 20.
do. represented by a fractional func-
tion of the 2nd degree : M.18.
representation of x = '^X and
V cX"-\-d
transformation of plane figures: N.69.
Isomerism, pr : AJ.l.
Isoperimeters, method of: N.47, 74,82.
problems: J.18: M.13: Mel. 5.
triangle with one side constant, and
a vertex at a fixed point : C.84.
Isoptic loci : Pr.37.
Isosceles figi^res : C.87 : JP.30.
Isothermp, families of: Z.26.
Isotropic functions : 0.260,272.
Iterative functions : L.84.
*Jacobian: ths 1600— 9 : AJ.3 : thZ.lO.
* of 3 conies : 5023 : LM.4.
* formulEe : d.C.1471 : J.84: : Mo.84.
function : one argument, G.2.
of several variables : Mo.822.
modular eq. of 8th degree : M.16.
and polar opposites : Me.64.
sextic equation : Q.18.
system, multiplicator : M.r2.
Jacobi-BeruouUi function : J. 42.
Kinematics: A.61 : AJ.3: G.23: L.
63,80: LM.thl7: N.82, ths 83
and 84.
of plane curves : A.56 : N.82 : caus-
tics, Z.23.
paradox of Sylvester : Me.78. .
of plane figures : Mel.26 : N.78,802 :
of a triangle, N.63.
of a point : N.49,82 : baryccntric
melhod, Mel. 5.
of sliding and rolling solids : TA.2.
Kinematic geometry: of space: J.
90.
of similar plane figures: Z. 19.2,20,23.
Knight's move at chess : C.31,52,74 :
CD.7: CM.3: E.41: N.64: Q.IU.
Knots : TE.28 : with 8 crossings, E.33.
*Kummer's equation, i.e. : 2706.
rational integrals of : M.24.
an analogous eq. : C.99.
Kummcr's 16-nodal quartic surface :
C.92 : J.84.
figures of : M.18.
lines of curv. of : J.98.
INDEX.
899
Lamp's equation: An.70 : C.86,90,91,
ths92.
Lame's functions : C.87 : J.56,60,62 :
M.18.
*Lagrange's theorem (d.c) : 1552:
C.60,77: CD.6: 0M.3: Mel.2:
gzC.96 and Me.85 : gzQ.2.
*Lambert's th. of elliptic sector: 6114:
of a parabolic sector, A.16,33,48 :
Me.78: Q.15.
*Landen's th. of hyperbolic arc: 6117:
E.21.
Laplace's coefficients or functions : see
" Spherical harmonics."
Laplace's equation: and its analogues,
CD. 7 : and quaternions, Q.l.
Laplace's th. (d.c) : 1556.
Last multiplier, Jacobi's th. of : L.45.
Lateral curves : A.58.
*Latus rectum : 1160.
*Law of reciprocity : N.72 : d.e,3446.
ext. to numbers not prime : C.90.
*Least divisors, table of, from 1 to 99000 :
page 7.
Least remainder (absolute) of real
quantities : Mo. 85.
Least squares, method of : A. 11, 18, 19
AJ.l : 0.34,37,40 : CP.8,11
G.18: J.26: L.52,53,67,75
Me.80,81: Mel. 1,4: gzZ.18.
Legal algebra (heredity) : N.63.
*Legendre's coefficient or function, X„:
2936: C.47,91: L.76, gz79 :
Me.80: Pr.27. _
and complete elliptic integral of 1st
kind: Me.85.
rth integral of and log integral of:
Me.83.
product of any two expressed by a
series of the same functions :
Pr.27.
Legendr
■e's symbol ( — ) : Mel. 4,5.
Leibnitz's th. in d.c : 1460 : N.69 : a
formula, Mo.68.
*Lemniscate : 6317: A.55,cn3: At.51 :
thsE.4: L.46,47: Me.68 : N.45.
chord of contact, en : Z.12.
division of perimeter: 0.17: L.43 :
into 17 parts, J. 75 : irreducibility
&c. of the partition equation,
J.394.
tangents of: J.14 : cnZ.12.
Lemniscatic geometry: Z.2I2: coordi-
nates, Z.12 ; of nth order, J.83.
Lemniscatic function : biquadratic
theorem, multiplication and
transformation of formula), J.30.
Lexell's problem : LM.2.
*Lima(?on : 5327 : 0.98 : N.81.
Limited derivation and ap. thereof ;
Z.12.
Limiting coefficients : 0.37.
Limits : theory of, Me.68.
of functions of two variables : M.ll.
of 1 +
ir
when a; ^ 00 : L.40 :
N.85: Q.5.
Life annuities : A.42 : en of tables, P.
59.
Linear : associative algebra : A.T.4.
construction : Man.51.
coordinates in space: M.l.
dependency of a function of one vari-
able : J.55.
dependent point systems : J.88.
forms : L.S4 : with integral coeffi-
cients, J. 86,88.
function of n variables : Gr.l 4.
U' = F- where U, V are products of u
linear functions of two variables :
0D.5.
geometry, th : M.22.
identities between square binary
forms: M.21.
systems, calculus of: J P. 25.
Linear equations : A.5l2,70 : Ac.4 : C.
81,th94. : G.14 : J.30 : JP.29 : L.f
39,66: Mo.84o: N.51,75,80o: Z.
152,22.
analogous to Lame's : 0.98.
with real coefficients : M.6.
similar : N.45.2.
solution by roots of unity : 0.25.
* systems of : 582 : A.10,22,52.57 : G.
15: 0.81,96: L.58: N.462.
in one unknown : 0.90 : G.9.
of nth. order : J.16.
* standard solution : 582: Q.19 : gen.
th, A.r22.
symbohc solution in connexion with
the theory of permutations: 0.21.
whose number exceeds the number of
variables: N.46.
Lines : alg. representation of : 0.70.
" de faits et de thalweg " in topo-
graphy : L.77.
generated by a moving plane figure :
0.86.
of greatest slope : A.29 : and with
vertical osculating planes, 0.73.
loxodromic : J.ll.
six coordinates of : OP. 11.
*Lines of curvature: 5773: A.84.,37 :
Au.53: 0.74..: 0D.5: L.46: M.2,
3o,76: N.79: Q.5.
of alg. surfaces : Z.24.
* and conies, analogy : 5854 : Me. 62.
900
INDEX.
Lines of curvature — {continued) :
dividiug a surface into squares : C.
74: LM.4: Mo.88.
of an equilateral paraboloid : N.84.
of an ellipsoid: A.o8,48 : An. 70: ths
CD.3: CM.2,3,4: JP.l : N.81.
comparison of arcs of, by Abel's th :
An.69.
and of its pedal surfaces : Q.12.
projection of : Z.2.
rectification of : An. 73.
generation of surfaces by : J. 98 : N.
77.
and geodesies of developables : L.59.
* near an umbilic : 5822 : A.70 : Q.IO.
* osculating plane of : 5835.
* plane, condition for: 5843: An.68 :
C.36,,42,,96 : G.22 : Me.64.
plane or spherical : An.57 : C.46 : JP.
20: L.53.
* of a quadric : 5833—4 : C.22.,,49,51 :
G.ll: J.26: Me.l: N.63o:'Pr.32:
TI.14.
* i^d constant along it : 5836.
projected from an umbilic into con-
focal Cartesians : E.19.
* quadratic for ?/x, giving the direc-
tions : 5810.
of two homofocal quadrics : L.45.
of quartic surfaces : C.59 : L.76.
of ruled surfaces : C 78.
spherical : C.36o,42,.
and shortest distance of 2 normals
one of which passes through an
umbilic : L.55.
of surface of the 4th class, correlatives
of cyclides which have the circle
at infinity for a double line : C.92.
of the tetrahedral surfaces of Lame,
&c.: C.84.
and triple orthogonal systems : M.3.
*Linkage and link work : 5400 — 31 : E.
28,30 : Me.75 : N.75,78.
* 3-bar: 5430, E.34; 4-bar, Me.76.
conjugate 4-piece : LM.9.
* for constructing : an ellipse: 5426.
a lemniscate : AJ.l.
* a lima^on : 5427 : Me.76.
x" and x'" : AJ.l and 3.
* root of a cubic ecj nation : 5429.
* Hart's: 5417: LM.6,8,.
* Kempe's : 5401 : Pr.23.
* Peaucellier's : 5410 : E.21 : LM.6 : N.
82.
the Fan of Sylvester : E.33.
* the Invertor': 5419.
* the Multiplicator : 5407.
* the Pentograph : 5423.
* the Plagiograph : 5424.
* the Planimetcr : 5452.
Linkage and linkwork — {continiicd) :
* the Pro])orliunator : 5423.
* the Quadiuplane : 5422.
* the Reciprocator : 5419.
* the Reversor : 5407.
* the Translator: 5407.
* the Versor-invertor : 5422.
* the Yersor-proportionator : 6424.
Lissajons' curves : A.70 : M.S.
*Lituus : 5305.
Loci, classification of: C.83,85 : P.78 :
Pr.27.
Locus of a point : the centre of a circle
cutting 3 circles in equal angles :
N.53.
the centre of collineation between a
quadinc surface and a system of
spherical surfaces : A.05.
dividing a variable line in a constant
ratio : gzAJ.3.
of intersection of common tangents
to a conic and circle : ISr.63,79.
of intersection of curves : CM. 2.
ditto of two revolving curves : N.64.
on a moving right line : L.49.
on a moving curve : Mem.18.
the product of 2 tangents from which,
to 2 equal circles is constant :
An. 64.
at which 2 given lengths subtend
equal angles : A.68.
whose sum of distances from 3 lines
is constant: A. 17,46 : from 2
lines, N.64 : from lines or planes,
A.192,prs and thsol.
whose distances from 2 curves have a
constant ratio : An. 58 : or satisfy
a given relation, A.33 symbolic f.
Locus: of pole of one conic with
respect to another : N.42.
of remarkable points in a plane tri-
angle : A.43.
of vertex of constant angle touching
a given curve: N.61.
of vertex of quadric cone passing
through 6 points : N.63.
*Logarithmic": curve, 5284: quadra-
ture, N.45.
integral: A.9,,19 : J.17 : Z.6.
inimerical determination : A.U.
of a rational differential : J. 3.
parabola : CD. 7.
potentials : M.3,4,8,13,16 : Z.20.
rational fractions : A.6.
systems : A.14.
transcendents : P.14.
waves : LM.22.
*Logarithms: 142: P.1792, 1787, 1806,
17 : TE.26.
and anti-logarithms, en : I.12.„24.
INDEX.
901
Logarithms — {continued) :
* calculation of : 688: A.24,27,42: LM.
1,6 : Me.74 : N.51 : Pr.31.2,32 :
TE.6,14 : TI.6,8.
Huygben's method : 0.662,680.
and circular functions from definite
integrals : A. 65.
common or Briggean : A.24.
constants in integral : J.60.
of diiferent orders of numbers : L.45.
higher theory of : trA.15.
impossible : CM.] .
natural, or Napierian, or hyperbolic :
A.25,26,57.
of commeusurable numbers or of
algebraic irrationals : C.95.
base of: see " e."
* modulus of : 148 : A.3.
of negative numbers : No.l784.
new kind of : J. 70.
powers of : CM.2.
of sum and difference of 2 numbers :
A.45.
with many decimal places : N.67.
of 2, 3, 6, 7, 10 and e all to 260 decimal
places : Pr.27.
of 2, 3, 6, 10 and e to 205 places : Pr.
6,20.
* of primes from 2 to 109 : table viii.,p.6.
tables of log sines, &c. en : Q.7.
Logic, algebra or calculus of : A. 6 : A J.
3,72: CD.3 : M.12 : Man.71, 76,823:
Q.ll.
of equivalent statements : LM.ll.
Logic of numbers : AJ.4.
Logocyclic curve : Pr.9 : Q.3.
Longimetry applied to planimetry : J.52.
Loto, game of : L.42.
Loxodrome: eqA.21 : N.eio : Z.5.
of a surface of revolution : N.74.
of cylinder and sphere : A.2.
of elhpsoid and sphere : A. 32.
of paraboloid of rotation : A.13.
Loxodromic triangle upon an oblate
spheroid : A. 27.
Lucas's th : G.14 : analogous f., G.13.
Ludolphian number : Mo. 82.
Lunes : J. 21.
*Maclaurin's th. (d.c) : 1507: A.12 :
CM. 82: J.84: N.70.
symbolic form : CM. 4.
Maclaurin-sum-formula : J. 12.
Magical equation of tangent : Q.6.
Magic : cubes, Q.7 : CD.l. -
cyclovolute : TA.5,9.
parallelopiped : A.67.
rectangles : A. 65, 66.
squares: A.21,57,66 : CD.l: CM.4:
E.8 : J.44 : Me.73 : Pr.15,16 : Q.
6,10,11 : TA.5,9.
Malfatti's problem, to inscribe 3 circles
in a triangle touching each
other: A.15,16,20,55: J.]0,45o,76,
77,84,89 : LM.7 : M.6 : P.52 : Pr.
6 : Q.l : TE.24 : Z.21.
Malfatti's resolvents of quintic eqs : A.
45.
Malm's surfaces, th : J.84,88.
Mannheim, two theorems : G.8.
Martin's measure of distance : A.19.
Matrices : E.42 : LM.4,16 : thMe.85 :
P.58,66 : Pr.9,14.
(«;^^) and function /(,) = 5^J:
Me.804.
Cayley'sth: LM.16: Me.85.
equation, px = xq : C.992.
of 2nd order : linear eq., C.992 : quad-
ratic, Q.20.
of any order : linear eq., C.994.
notation of : J. 50.
per symmetrical, th : E.34.
product of : G.5,11.
roots of a unit matrix : C.94.
whose terms are linear functions of ;» :
J.60.
*Maximum or minimum : 58, 1830 : A.4,
13,22,35,49,53,60o,70 : C.17,24 : J.
48 : JP.25 : Me.l,geo5,72,76,81,
83 : N.43 : Z.13.
* problems on: 1835—40, 1847: A.2,
geol9,38,39: geoL.42 : Mem.ll,
a paradox, N.63.
of an arc as a function of the abscissa :
J.17.
* continuous : 1866,
of a definite integral : Z.21.
discontinuity in : CD.3.
distances between points, lines, and
surfaces, geo : At.65.
duplication of results : Me. 80.
ellipse which can pass through 2
points and touch 2 right lines :
A.14.
elliptic function method : Mel.6.
of figures in plane and in space : CM.
3: L.41 : J.242: Z.ll.
* functions of one variable: 1830 : ditto.
with an infinity of max. and min.
values : J.63.
* functions of 2 variables : 1841 : Mem.
31 : Q.5,6 : Lagrange's condition,
CM.2.
* functions of 3 variables : 1852 : CD.l :
prs 1860—5.
* functions of n variables : 1862 : L.43 :
Mem. 59 : Q.12 : symmetrical,
Mel. 2.
902
INDEX.
Maximum or minimiim — [coniinucd) :
of iu- and circnra-polj-gon of a circle :
A.29,30 : do. of ellipse, and analo-
gous th. for ellipsoid, An. 50.
indeterminates : CM. 4.
in-i)olygon (with given sides) of an
ellipse: A.30.
by interpolation, f : A. 25.
method of substitution : A.23.
of multiple definite integrals : Mel.4.
planimetrical groups of: A. 2.
of single integrals between fixed
limits : J.54,69 : M.25.
of the sura of the distances of a point
from given points, lines, or
planes : J. 62.
of the sum of the values of an integral
function and of its derivatives :
L.68.
solids of max. vol. with given surface
and of ini!i. surface with given
vol.: C.63.
* of j i^ (.V, y) ds, &c., to find s : 3070—2.
* o?llF (.e, y, z) dS, &c., to find 8 : 3078
—80.
Maximum : ellipse touching 4 lines :
A.12.
ellipsoid in a tetrahedron : Z.14.
of a factorial function : Me. 73.
polyhedron in ellipsoid: A.32.
of a product : N.44.
of a sphere, th : N.53.
* solid of revolution : 3074.
tetrahedron : in ellipsoid, A.32 :
whose faces have given areas, 0.
54,66 : E.62a.
* volume with a given surface : 3082.
of
A.362 : of ^x, &c., A.42.
ofax + hy + &c., when x" + y~-\-&c.=^'\. :
N.46.
Mean centre of segments of a line cross-
ing three others : A.40.
Mean distance of lines from a point : Z.
11.
Mean error of observations : A. 25 : C.
37; : TI.22.
in trigonometrical and chain measure-
ments : A.46 : Z.6.
Mean proi)ort.ionals between two lines :
A. 3 1,3 4.
Mean values : 0.18,20,23,26,27.. : L.67 :
LM.8 : M.6,7 : Z.3.
of a function of one variable : G.16 :
of 3 variables, C.29.
and i)robabilities, gco : 0.87 : L.70.
♦Measures of length &c. : page 4.
exactitude of : Z.6 : do. with chain,
Z.l.
Mechanical calculators : C.28 : 1.16 :
P.85.
for " least squares " : Md.2.
Mechanical construction of : curves :
M.6 : N.56.
Oartesian oval : AJ.l.
conies : An. 52 : three, N.43.
ellipse: A.65 : Z.l.
lemniscate : A.3.
conformable figures : AJ.2.
cubic parabola : N.58.
curves for duplication of roots : A. 48.
(a3-a'2)/7/ : E.18.
sm-faces of 2nd order and class : J. 34.
Mechanical : division of angles : Q.4.
measurement of angles : A. 61.
* integrators : 5450 : 0.92,94,95 : Pr.24,.
for Xdx+Ydy: Me.78.
involution : AJ.4.
* quadrature : 3772 : A.58,59 : J.6,63
gzA.66 and 0.99.
solution of equations : Me.73 : N.67.
linear simultaneous : Pr.28.
cubic and biquadratic, graphically :
A.l.
Mensuration of casks : A. 20.
Metamorphic method by reciprocal
radii : N.64.
Metamorphic transformation : N.46.
Metrical : system : E.30.
properties of figures, transf. of : N.
582,59,60 : j.4.
properties of surfaces : AJ.4.
*Meunier's theorem : 5809 : gzC.74.
Minding's theorem : Quaternion proof :
LM.IO.
Minimum : theory of : L.56: prM.20.
angle between two conj. tangents on
a positive curved surface : A.69.
area : J. 67.
of circum-polygon : 0D.3.
of a hexagonal " alveole," pr : N.
43.
circum-conic of a quadrilateral : A.
13: An.54.
circum-tetrahcdi'on of an ellipsoid :
Z.25.
circutn-trianglc of a conic : Z.28 : of
an ellipse, Z.25.
curves on surfaces : J. 5 : see " Geo-
desies."
distance of 2 right lines : G.4 ; of a
point., ths : A. 8.
ellipse through 3 points and ellipsoid
through 4 : L.42.
ellii)soid, tli : Mo.72.
N. G, F. of a binary septic : AJ.2.
INDEX.
003
Minimum : theory oi— {continued) :
numerical value of a linear function
with iutegral coefficients of an
irrational c[uantity : C. 63,54.
jDerimeter enclosing a given area on a
curved surface : J. 86.
questions relating to approximation :
Mel.2 : Mem.59.
* sum of distances from two points :
920-1.
sum of squares of distances of a point
from three right lines : Z.12.
sum of squares of functions : N.79.
Minimum surfaces: eqA.38 : G.14,
22: J.81,85,87,ext78: Mo.67,72 :
projective, M.14 : metric, M.15.
algebraic : M.3 : lowest class-number,
An.79.
not algebraic and containing a succes-
sion of algebraic curves : C.87.
arbitrary functions of the integral eq.
of: C.40.
between 2 right lines in space : C.40.
generation of : L.63.
representation of by elliptic functions :
J.99.
of a twisted quadric : At. 52.
limits of (Calc. of Var.) : J.80 : on a
catenoid, M.2 : determined by
one of the edges of a twisted
quadrilateral. Mo. 65.
variation of surface, capacity of : Mo.
72.
Minimum value of
r ^{A+B.^ + Cx'~+&c.)d.e: N.73.
of j ■^/{dx~+d)j -+...) when the varia-
bles are connected by a quadric
equation : J.43.
Models : LM.39 : of ruled surfaces,
Me.74.
Modular : equations : An.79 : of 8th
degree, 59: C.483,493,66 : M.1,2 :
Mo. 65 : see also under " Elliptic
functions."
degradation of : M.14.
factors of integral functions : C.24.
functions and integrals : An.51 : J.
184,193,20,21,23,25 : M.20.
indices of polynomials which furnish
the powers and products of a
binomial eq : C.25.
relations : At. 65.
Modulus: of functions, principal:
C.20.
of series : C.17.
* of transformations : 1604: A.17.
*Momental ellipse : 6953,
*Momental ellipsoid : 5925, 6934 ; for a
plane, 6936.
♦Moment of inertia : 6903 : An.63 : At.
43 : M.23.
* of ellipsoid : 6150 : CD.8 : J.16.
by geometry of 4 dimensions : Q.16.
* principal axes : 5926, 6967, 5972 : At.
43.
* of a quadrilateral : 5951: Q.ll.
of solid rings of revolution : Q.16.
* of a tetrahedron : 5957.
* of a triangle : 5944 : Me.4 : Q.6 :
polar, N.83.
* of various lamina and solids : 6015 —
6165.
Monge's theory " des Deblais et des
Remblais " : LM.14.
Monocyclic systems and related ones :
J.98.
Monodrome functions : C.43 : G.18.
Monogenous functions (Laurent's th) :
Ac.42: C.32,43.
Monotypical functions : C.32.
Monothetic equations : C.99.
Mortality : A.39.
" Mouse-trap " at cards : Q.152.
Movements : JP.15.
elliptic and parabolic : JP.30.
groups of : An. 692-
of a plane figure : thAn.68 : JP.20,
28: LM.3.
of an invariable system : C.43.
of a point on an ellipsoid : AJ.l : J. 64.
relative : JP.19 : of a point, L.63.
of a right line : C.89 : N.63.
of a solid : JP.21.
transmission of and the curves result-
ing : JP.3.
of " ahnlich - veranderlicher " and
" affin-veranderlicher " systems :
Z.24 and 19.
*]\[ultinomial theorem : 137 : Me.62^
Multiple-centres, geo. theory : L.45.
Multiple curves of alg. surfaces : An. 73.
Multiple Gauss sums : J. 74.
♦Multiple integrals : 1905, 2826 : A.64 :
An.62: C.8,11: thsCD.l : thE.
36: J.69: L.39,43a,45.:,46,th48,66:
LM.82: Me. 762: Z.13,3.
* double : 2710, 2734—42 : A.13 : Ac.5 :
An.70 : J.272 : G.IO : L.682 : Mem.
30.
approximation to : J.6.
Cauchy's theory, ext. of : C.762.
* change of order of integration :
2775: A.19.
expressing an arbitrary function :
J.43.
reduction of : J.46 : Z.9.
residues of: C.75.i.
904
INDEX.
Multiple integrals, double — {continued):
{x—y-)clxd]i_
o;
V{(«'-!>')(c'-«2)(!.'-./)(o'-r)}
= f: L.38.
same with log of numerator : L.50.
jx' —■x){dy dz —d z dy')j\-sym
= 4»i7r : CM.
transf. of [ [
n
d(pd\lr
J.20.
7r2Jo Jo A
II:
\/ {sin^u— sin^ cj) cos2^/^)'
cos ix cosjx dx dy
v/{l+a2 + 2a(^C0Sa + j/C0S?/)V
C.96. ■•
jj F {a + hx + c)j) dx dij : A.37.
II F' {x+iy) dx dy : J.42.
J^;^,dtdu: C.96.
G {u, t, z)
r r</) {ax"' ± %") «"-• 7/'-i dx dij : J.
"37.
evaluation : A.6 : by Fourier's th,
CM.4.
expansion of : Q.8.
Frullanian : LM.15,
limits of : LM.16.
reduction of : An. 57 : L.41,39.
by transf. of coordinates : C.13.
\"F (fr+7y2+ ...)<p {ax + % + ...) dxdy
...: L.57.
of theory of attraction : CD. 7.
transformation of : 2774 : A.IO : An.
63 : N0.47 : CM.4 : Mel.2 : Mem.
38: Q.4,12.
an indef. double : J. 8^,10.
a triple integral : 2774—9 : J.45.
ijidxi + ... +yndx„ : LM.ll .
triple: 2774: A.30 : J.45.
which are unaltered in form by trans-
formation of the variables: J. 15,
91.
^jj...dxdydz...: Q.23.
Ill
'y"
dxdy d:: ... with
equations
different limiting
2825: CM.2: L.51.
some other integrals evaluated by r
functions: 282G— 34.
jj\F{ax-\-hy + cz,a'x + h'y+c'z,
a"x+h"y+c"z) dx dy dz, limits
± 00 : A.30.
\l
Multiple integrals — {conthmed) :
volume integral of
\{j e exp ( — x~ — y~ — z~), x''y^ z dx dy dz -.
N.54.
Jlj---<P («''•" + ^.V" + &c.) x"y'J . . . dx dy . . .
with limits to go in each case
(Pfaff) : J.28.
dx du ... 1 J.
■{(«-,,.)=+(;;-!,)=+...}•• "y*"
continuous functions : TI.21.
do. with n z=— and with a numerator
<'-">^(S + |i + -)^CM.3.
\\ ---F {ic, y,z...) PQ dx dy dz . . . where
r = il-xr-'{l-y)''- ...
arising from (2604), viz :
Te exp (—x^- -\ dx : Pr.42.
*Multiple points: 5178: CM.2: thG.15:
Me.2: Q.2,6.
on algebraic curves : An. 52 : L.42 :
N.51,59,81, at 00 643.
on two curves having branches in
contact : C.77.
on a surface : J.28.
*Multiplication : 28 : J.49 ; abridged, N.
79. _
by fractions : Me.68.
Multiplicator equations : M.15.
Multiplicity or manifoldness : J.84-,86 :
thAc.5: Z.20.
Music: B.273,28: Pr.37.
Nasik squares and cubes : Q. 8,1 So.
Navigation, geo. prs.of use in : A.38.
Negative in geometry: No. 1792.
Negative quantities : At. 55 : N. 443,67 :
TE.1788.
Nephroid: LM.IO.
Net surfaces: J. 1,2: M.l : any order,
An.64.
quadric: J.70,82 : M.U.
quartic : M.7.
and series : C.62.
trigonometrical : Z.14.
having a 3-poiut contact with the
intersection of two algebraic sur-
faces : G.9.
Newton, autograph m.s.s of : TE.Ti.
*Nine-point circle: 954,4754: A.41 : E.
7,30,th35,pr39 : G.l,ths4: Mc.64,
08: Q.5— 8.
INDEX.
905
Nine-point circle — {continued) :
an analogous circle : A. 51.
* contact with in- and ex-circles : 959 :
Mc.82: Q.13.
and 12-point sphere, analogy : N.OS.
Nine - point conic of a tetraliedron :
Me.71.
Nonions (analogous to Quaternions) :
0.97,98.
Non-uniform functions : 0.88.
Nodal cones of quadrinodal cubics : Q.
10.
Node cusps : Q.6.
Nodes, two-plane and one-plane : M.22.
*Normals: 1160: 4122—3,5122: A.13,
53: LM.9: p.eMe.66: Z.cn2and3.
of envelopes : Me.80.
* plane of a surface : 5772.
* principal : 5722 : condition for being
normals of a second curve, 0.85.
of rational space curves : J.74.
section of ellipsoid (geodesy) : A.40.
* of a surface : 5771, 5785 : 0.52 : 0D.3 :
0M.2 : L.39,47,72 : M.7.
coincident : L.48.
transformation of a pencil of : 0.88-2.
*Notation (see also " Functions ") :
A,B,G,F,G,H: 1642.
* A.P., G.P., H.P. : 79,83,87.
* {a,hc3): 554.
(n) {-), (Jacobi's function (see
" Functions ").
* a'' or a"l» ; 2451.
* a+\J^^ : 160.
* B^„, Bernoulli's nos. : 1539.
* G {n,r) or 0,,,,- : 96. Otherwise
G {n,3) = number of triads of n things,
&c.
T'^) = rth coeff . of nth power of (1 +x) :
also Jacobi's function (see " Func-
tions ").
* D : 3489 : chAu; &c. : 1405.
* dh/,dx\%&c.: 1407.
* d juvw) . -^gQQ_
d{xyz) '
*A: 582,1641, 3701; a', 1645.
*E : 902, 3735.
*d: 150, 1151. A
e esp 83 H or e
Notation — [continued] -.
*<|) (a/3y) = u : 4656 ; "I' (X/iii/) or U, 4665.
*Gu,: 3732.
*H (n, r) : 98.
*/: 1600.
iV.(T.-F= numerical generating func-
tion.
iV=6(modr) signifies that N — h is
divisible by r.
* Ijz, = %(») = » ! : 94, 3713.
* TT as operator : 3500.
* P {n, r) or P,,,, = n"') : 95. Also,
P {n, r) = number of trijDlets of n
things, &c.
* if/ {x) or Z'{x) = clr log r (,c) : 2743.
* B, r, Ta : 909—13.
* /Sm, 8,n p : 534 ; 8n, 2940.
* sin-i, &c. : 626 ; sinh, &c., 2180.
* 2: 3781—3.
* iin : 3499.
fi (;n) = sum of divisors of n.
Y or ^ ( — - ) = integer next
%
*/(»!): 400, 1400 ;/->(?«), 430.
*/' (33) /" {x) : 424, 1405.
I Y = integer next > —-.
( — ]=rthcoetr. of (1+*)".
■4; = not less than ; ';^ = not greater
than.
(-^) = denominator to be stated after-
wards.
( X ) and ( I ) : 1620.
algebraic : 0P.3.
for some developments : 0.98.
continuant = contd. fraction determi-
nant.
median = bisector of side of a triangle
drawn from the opposite vertex.
subfactorial : Me. 78.
suggestions : Me. 73.
*Numbers (see also "Partition of," and
" Indeterminate equations") : 349 :
A.2,16,26,68,59 : Ac.2 : AJ.4,6 :
C.f 12,43,44o,454,f60 : OM.l : G.16,
32 : J.93,39,404,48,77 ; tr,273,28and
29o : JP.9 : L.37— 39,41,45,586,59,
60 : LM.4 : Mem.22,24; tr(Euler),
30: N.443,62,79: Q.4: TE.23.
ap. of algebra, JP.ll ; of r function,
No.81 ; of infinitesimal analysis,
J.19,21.
formulas : L.64;,65.2.
relation of the theory to i.c : 0.82.
5 z
906
INDEX.
Numbers — (continued) :
approximation : to \/N, E.17 ; to
functions oflartre numbers, C. 82.
binomial eqs. with a prime mod : C.62.
cube : Q.4.
cubic binomials : fc'^ii/^ ; C.6I0.
determined by continued fractions :
LM.29.
digits, calculus of, th : J. 30.
digits terminating? a power: A.58 :
N.46.
Dirichlet's th. 2^
(§)=a»0:L.57.
Dirichlet's f. for class numbers as
positive determinants : L.57.
division of : A.26: J. 13: Mel.3 : Pr.
7,10 ; by 7 and 13, A.25,26 ; by
ma!2+>M/, Mem.l5: P.17,88 : Q.
19,20.
divisoi's of i/2+yls2 when^ = 4« + 3
a prime : J.9.
divisors arising from the division of
the circle : L.60.
4))2. + l and 4j/i+3 divisors of a num-
ber : LM.16.
factors of: Mem.41.
Gauss's form : L.56.
integral quotients and remainders :
An.62.
large, analysis of : A.2 : C.2,29.
method with continuous variables :
J.41.
multiples of : C.2.
non-pentagonal th : J.31.
number of integers prime to n in
n\ = <p {n) : L..57.
odd: A.lo : and prime to all squares,
0.67.
Pellian equation : prA.49 : LM.15 :
sol. by ell. functions, Mo.63.
perfect : C.81 : N.79.
polygonal, Fermat's th. of : P.61.
polynomials having determinate prime
divisors: C.98.
powers of, 12 theorems : N.46.
prime to and < N : A.3,29 : E.28., :
J.31 : N.45.
prime to and < the product of the
iirst n primes : A. 06.
prime with respect to a given ar.p :
C.54.
pi-ime to the radix having multiples
made up of repeating digits :
Me. 76.
l)roducts of divisors of: Q.20.
quadratic forms of: Mem. 53.
rational linear functions taken with
respect to a ])rime modulus, and
connected substitutions : C.48;j.
Numbers — [continued) :
representation of by forms : C.92 ; by
infinite products, A.l.
square having prime factors of the
form 4?; + l : N.78.
squares of: J.84 : M.13: Pr.63,7.
three in ar.p : N.62.
sums depending upon E [x) : L.6O0.
sums of digits : Me.66 : TE.16.
sum and difference of two .squares :
thsN.63.
* sums of divisors : 377 : Ac. 6 : G.7 :
L.63.2: Mel.2: Mem.50.
* sums of powers of (see also " Series "):
276, 2939: An.61,65 : thCD.5 :
Me.75 : N.42,56,70 : Q.8.
of cubes: An. 65: L.66 ; of the odd
nos., A.64.
of n primes : lSr.79.^ ; 4th powers,
A.54.
of squares : A.67.
of uneven orders : Mo. 57.
symmetrical functions of: Q.7.
systems: Z.14; history of, by Hum-
boldt, J.4.
theorems: A.7,10,20,49 : An. 70: C.
25,,43,83: CM.2: G.8 : L.48,52 :
N.75; Cauchy's, gzC.63; Eisen-
stein's, J.27,50,83 : LM.7 : Q.6,6.
r^P n,P
Gauss's on X = '- — -J- : C.98.
x—y
Lagrange's arithmetical : A.47.
^y + f/" in terms of jj(/: N.75.
on2"±l: C.85,86: Me.78.
2, biquadratic character of : 0.57,
66: L.59.
on {n+\)"'—n"': N.44 ; p; (*i), L.69.
* on n'—n (w— l)'--)-, &c. : 285 : A.30.
on 2m positive numbers : N.43.
on F{m)-\-E
P(1)+:Q.C
on the greatest product in whole
numbers of given sums : J. 57.
on an odd sum of 12 squares : L.60.
on products of sums of squares : G.2.
on 1- squares : N.57.
on 2, 4, 8, and 16 squares : Q.17.
on <['[a)-\-<p{a')-\-, &c. = n, where a,
(t', etc., arc the divisors of «: Om.3.
Numeration, ancient decimal : 0.6,8^.
Numerical approximations : N.42a,53.
Numerical functions : L.57 : Me.62.
simply periodic : AJ.l.
sums of, approximately : 0.96.
which express for a negative deter-
minant the nunjber of classes ol" a
((uadratic form, one at least of
whoso extreme coefficients is odd :
0.62..
INDEX.
007
Obelisks : A.O.H.
Oblique : bevilled wheels, en : J.2.
* coordinates : 4050, 5511—9 : fN.54
cyclic surface : TI.9.
and oscillating circle of a conic : G.22.
Octahedron function : Q.IO.
Octahedron, centroid of: LM.9.
Octic equations : G.7,10 ; and curves,
M.15.:.
Octic surface : G.13 : M.4 : Q.14.
*Operative or symbolic calculus: 1483,
3470—3628: AJ.4 : C.17: 0.20^:
J.5,59 : LM.123 : Me.82,85 : P.37,
44,60—63: Pr.l0o,ll,,12o,134 : Q.
4,5,8.
applications : G.19 : Me.82.
algebraic : TE.14 ; ap. to geometry,
CM.1,2; ±,CM.3.
expansions: Pr.l42.
formuIa3 : C.393.
* index symbol : 1485 : CD.H.
integration : CD. 3 : exMe.76.
representation of functions : C.43 :
CD.2.
seminvariant operators : Q.20.
on the symbols .v-', log/,ft', sin x, cos x.
sin-'ir, cos'^a; : A.9,11.
theorems : A.57 : CD.80 : LM.ll : Q
32,15,163 ; from Lagrange's series
Q.16; from Trpu — pmt = pu, CD. 5.
*Operators : d,, 1405; e'"^ 1520—1, Q.9o
* {ch—m)-\ &c. : 3470—85 ; gz of 3474,
C.43.
* D (D-1) ... (D-n+l) : 3489.
* Tr = xd^+yd,j+&G.: 3500.
expansions and fcrmuloB for :
* F {xd,) U, where U = f{.i') = a + bx+
cfc3+ : 3486.
* /(D)wv, 3494: tif{D)v, 3495, with/
as above.
D"f{xD)U: E.36.
* {cp (D) e'"Y Q : 3491.
e^^^F (,.■), &c.: E.34.
e^' ^P (,.•) = '^^ F (,«+l) : E.86.
nF{a,x): Q.13.
"-"'*y{a v'v±h^/iwi)}, &c. : C.96.
f{x + hD).l: E.39.
* yjr {dr+y) (f^y^ = (p {d,j+x) ylry : 3498.
* reduction o{ F {tti) : 3503.
* F{n)U and F'^ (tt) U : 3509—10.
* G{ii,vi)u„lml: 3514.
transformation of Vdj. . Ud^..., &c: G.
21.
Orthocycle: Q.17.
Orthogonals, algebraic system of : C.69.
*Orthogonal : — ^circles : 4170, 4182—4 ;
of in- and circum-circles of a tri-
angle, Q.18.
circle and conic : E.7.
coefficient system : A.2,61.
coordinates : C.60 ; curvilinear, JP.
26.
conies : N.84 ; families of, A. 63.
curves : J.35 : N.52,81.
system from logarithmic repre-
sentation: Z.16.
* lines of a triangle : 4633.
lines and conies : C.72.
* projection : 1087.
in metrical projective geometr}- :
GM.14.
of a circle into an ellipse : A. 2.
of a triangle: E.30,31,36,37.
substitution: J.67 : M.13: Z.24.
surfaces : C.29, spheres 36,54^,59,72,
79,87,thsl7 and 21 : J.84: JP.17 :
L.43,44,46,47,63 : N.51 : P.73o :
Pr.21: Q.19.
cubic eq. for : C.762.
with elliptic coordinates : 0.53 :
J.62.
and isothermal: C.84: JP.18 : L.
43,49.
systems: C.67,754 : M.7 ; condition,
J.83 ; quadric, J. 76 ; parallel,
M.24.
triple : A.55 — 58 : An.63,77,85 :
C.alg58,67 ; cychc, G.21,22 : J.84 ;
quartic, 82 : Z.23.
trajectories: L.45 : Me.80 : Z.17.
of circles : Me. 85.
of circular sections of an ellipsoid :
L.47.
of a moveable plane : Pr.41.
of a moveable sphere : C.42.
of a surface : Mem. 20.
Orthomorphic projection of an ellipsoid
on a sphere : AJ.3.
Orthomorphosis of a circle into a para-
bola: Q.20.
Orthoptic : lines of a conic: A.57.
loci of : LM.13 : Pr.37 ; of 3 tangs.
to a quadric, E.40.
surface of a quadric : J. 50.
Orthotomic circles : Me.64,66 : Q.2.
*Osculating : circle : 5724 : L.39.
of conies : A. 70: N.60.
of a family of curves : N.70.
of a parabola, ths : N.66.
of quadric curves lsr.43.
of tortuous curves : N.81.
* cone : 6727 : angle of, 5752.
conic : L.39 : triply, A.69 : Z.19.
of a cubic curve : J.68: : Z.17.
908
INDEX.
Osculating — {continuecl) :
curves : Q.ll.
helix: N.71.
line of a surface : C.82 : J.81.
parabola : N.81.
paraboloid: JP.15: N.82: of aquadric,
L.38.,.
* plane : 5721, eq 6733 : and radii of
curv. at a multiple point of a
gauche curve : An. 71 : C.68.
of a tortuous curve : C.96 : J.41,63.
sphere : Mem. 20 : N.70 : of curve
of intersection of two surfaces,
en : C.83.
of two curves having a common
principal normal : LM.16.
surfaces : C.792, degree of 98 : L.41,
80 : of quadrics, N.60.
Oval of Cassini : see " Cassinian oval."
Oval of Descartes : see " Cartesian
oval."
Pangeometry : G.5,15.
*Pantograph : 5423 : Mem. 31 : TE.13.
Paper currency : A.42.
Pappus, prs in plane geometry : A.38 :
Z.5.
*Parabola: geo. 1220— 44 : anal. 4200—
39: eqCM.2: K424,54,70 : geo
0M.4 : Me.71 : cnl249 : thsN.60,
63,71,76,802.
circum-hexagon and triangle: CM.l.
* circle of curvature of : 1260: A. 61.
* chords of : 1239, 4224.
* two intersecting : 1242.
determination of vertex and axis :
N.58.
* eq. deduced from eq. of ellipse: 1219.
focus and directrix: N.49.
* focal chord : 4235—9.
* latus rectum : 1222 : Me.75.
* normal, length of : 4233 — 4.
plane and spherical : A.60.
* quadrature of : 1244 : A.32.
* and right line : see " Right line."
* segment of : 6078 : A.26,29.
solid generated by it : N.42.
sector : E.30 : N.57 : Lambert's th,
J.16.
in space : A. 8.
* tangents : sec under " Conies."
* tlirough 4 points, en : 4837 : J.26.
* triangle of 3 tangents : 1237,-68 : A.
47: Me.75.
trigonometry of : CD.8.
♦Paraboloid : 5621, 6126—41 : N.Gl : Q.
13.
* generating lines of : 5624.
of eight lines : C.84.
* elliptic: 5622: A.ll : L.5S : P.96.
Paraboloid — {continued) :
* quadrature of: 6127: An.55.
* segment of : 6127-33 : A.29.
* hyperbolic : 5623 : A.ll.
* of revolution : 6134.
Paradoxes of De Morgan : J.l 1,3,12,, 13,
16.
Parallels : A.8,47 : At.51 : J.11,73 : Mel.
1,3 : Mem.SO.: Z.21,22 : Thibaut's
proof, A. 15.
in analytic geometry : A.44.
Parallel curves : J.55,ths32 : LM.3 : Q.
11 : Z.5.
closed : A. 66.
* of ellipse : 4960 : A.39 : An.60: N.44<>:
Q.12.
Parallel surface : C.54 : LM.12.
of surface of elasticity : An. 57.
of ellipsoid: A.39: An.50,60: E.17 :
J.93.
Parallelogram with sides through four
given points : A.39.
Parallelogram of Watt : A.8 : L.80.
*Parallelopipeds : on conjugate diame-
ters : 5648.
diagonals, &c. : CM.l.
equality of : A.4.
analogues of parallelograms: LM.2 :
Me.68.
on a spherical base : N.45.
system of: LM.8.
Partial differences: question in analysis:
J.16.
*Partial differential equations (P.D.E.) :
3380-3445 : C.34,11,16,78,95,96 :
thsCD.3 : J.58,80,prs26 : JP.7,10,
11: L.36,80,83: M.ll: Z.6,8,18.
*P.D.E., first order : 3399-3410 : A.33,
tr50 : An. 55,69 : C.14c,53:„545 :
CD.7: CM.l.,: J.2,17: tr JP.22 :
L.75: M.9,ll,th20: Z.22.
* com])lete ]U"imitive connected with
any solution : 3405.
* derivation of the general primitive
and singular solution from the
complete primitive : 3401.
* derivation of a singular solution from
the differential equation : 3403.
with a general first integral : Me.78.
integrntioii by: Abclian func-
tions : C.94.
Cauchy's method: C.81.
* Charpit's method : 3399.
Jacobi's first method: C. 79,82 : and
ap. to Pfaff's pr, J. 59.
Jacobi-Hamilton method : M.3.
Lie's method : ]\r.6,8.
Weilcr's method : M.9.
* law of reciprocity : ;M46.
and Poisson's function : C.9I2.
INDEX.
909
P.D.E., first order — {continued) :
simultaneous: C.68,76 : L.79 : M.4,5.
* singular solution : 3401 — 3 : J.66.
systems of: A.56 : M.11,17.
theorem of Jacob! : C.45.
3 variables : J. 64.
* n variables : 3409 : A.22 : J.60 : LM.
10,11.
with constant coefficients : Mel. 5.
integration by calc. of variations :
C.14.
z=px + qii-\-F{li,q): Z.5.
as" 1/* z''^'" q"=A: CM.l.
(l+P,+ ...+Pn'^d.,d,jYZ=Q: Z.13.
*P.D.E., first order, linear: 3381—95:
reduction to, C.15 : J.81 : Me.78.
* Pz.,+Qz,j = B: 3383: extension to
n variables, 3384.
L{px + qij-z)-3Ip-Nq+B = :
C.83.
a(yUz—^ii';) + l>{zUx—xUz) + c{xu,j—yHr)
= 1:"Q.8.
* Ff,+Qf,+ ... +BL = 8: 3387.
* z,= L^ ; 2,. — fc3+y2 : 3390-1.
* {x-a)z, + {y-h)z,j = c-z: 3392.
* x2+y2+z^ = 2ax: 3393.
* simultaneous : 3396 — 7 ; ex. 3398 :
J.81.
u = Vy and My =. — Vr-. J. 70.
*P.D.E., second order : 3420—45 : A.33
O.54„70.„78,98 : JP.tr22 : L.72
M.15: Me.76o,77: Mel.3 : N.83 ;
P.46 : transf. of, C.97.
in two independent variables : trA
64 : trNo.81 : C.92 ; transf. of, 97
M.24.
in 4 and 6 variables : Mem. 13.
* Er+8s+Tt=V, Monge's method
3423: CM.3: N.76: Q.6.
* Br+Ss+Tt+U{rt-s^) = V: 3424
3434—40: J.61.
* Br+Ss + Tt+Pp+ Qq+Zz = U:
3442.
T+t = 0: A.2 : CM.l: J.59,73,74
L.43.
r+t+h2z = 0: M.l.
* r-aH = 0: 3433.
* v—aH = (a'.,i/), &c. : 3565.
x{r-aH) = 2np: E.13.
r-a"-y-H = 0: B.27,28.
r=q'""t: C.98.
T = q: J.72.
P.D.E., second order — (continued) :
dx {p s\nx)-\-t-\-n {n + \)z sin2 « = :
L.46.
r (l + r/) = ^l+y2). J.58,.
qh--AH+ ^r^„q = 0: E.22.
V
dx {px)-\-dy [qx) = : Z.28.
construction of explicitly integrable
equations of the form s = zX {ie,y) :
JP.28.
MIMI^'A
: C.81.
{f{x)+F(y)y^
a2tZxylogX±X = 0: L.53.
s + Pp + Qq+Z=0: Me.76.
* s+ap + hq+ahz=r: 3U4.
{ax+by+c) s+a\q + h,jip = : A.33.
(a;+i/)2 s+a {x+y)p + h {x+y) q+cz
= 0: A.33,38.
z2 {zs—pcj)2 + q = F {ij) : A.70.
rt—s^: geoQ.2.
* P = {rt—s2)"Q„ Poisson's eq. : 3441.
(1+r) t + {l + t)r-2pqs = : An.53o.
* q (1 +q) r+p (1 +p) t-{p + q + 2pq)
s = 0: 3432.
4s2 + (r— ^2 = 47-3 ; approx. integn.,
C.74.
As+Bq + ^ {r,p, q, x, y, z) = : C.93.
(Iog2)., + «3 = 0: C.36.
«,, = Br '^ where t = \ ., '''I' ., .
r J \^{2Br^ + A^)
L.38.
* ihi: +U2,j+U2z = (see also " Spherical
harmonics ") : 3551, 362G, J.36 :
Mo. 78.
* «.• + ^^y+^t; = ^B^/:3: 3552.
* aUx-\-huij-\-cuz-=xyz, &ic.: 3554.
* xu2, + au^—q^xu = Q: 3618.
* a^{u-2r-\-u-2„^-u-:.:)=.U2t: 3629: C.7 :
LM.72.
integration bv change of variables :
C.74j.
P.D.E., third order, two independent
variables: LM.8: N.83.
P.D.E., fourth order : AAtt = : C.69.
P.D.E., any order : No.73 : C.80,89 :
M.11,13.
X-Znx=^Zny: Z.7.
two independent variables : C.75 :
CP.8.
910
INDEX.
P.D.E., any order — [continued) :
any number of functions and inde-
pendent variables : C.80.
and ap. to physics : JP.13.
of cylinders : Me.77.
and elliptic functions : J.36 : hyper-
ell., J.99.
integration by definite integi'als : An.
59 : C.94 : L.54.
of dynamics : C.5 : J.47.
Hamiltonian: M.23 : Z.ll.
of heat : L.48 ; of sound, L.38.
integrated in series : C. 15,16.
of Laplace: G.23.
linear: An.77 : C.13,00 : CD.9., : CM.
2: J.69: JP.12o: L.39.
of orthogonal systems of surfaces :
Ac.4: C.77.
with periodical coefficients : C.29;(.
of physics : L. 72,47.
P.D.E., simultaneous : C.92,th78 : LM.9 :
M.23 : Z.20.
linear : J. 65.
P.D.E., system of: C.18,74,81.
a'"Zmt = 'A-™z,„^ : A. 30,31, by 2 = e'"/(.v.).
Zn, = .C'%„,„), + Fl(l/) + ^«i^2 (!/)+ ... +
fc-i^„,(y) : A.51.
Az„t^{d2^-^diy^- ...Yz = ^: C.94.
dz = Edx+Kdij+Ldp+Mdq+Ndr+
&C.: J.14.
Partial differentials of — ^ — = J. 11.
*Partial fractions: 23.''>, 1915: A.30,66 :
C.46,49.,,783: CM.l : G.2: J.1,5,9,
10,11,22',32,50., : JP.3 : L.46 : Mem.
9: N.45.2,64,69 : Q.5.
Partition of numbers (see also " Nura-
bei's" and " Indeterminate eqs.") :
AJ.2,5,6 : An. 57,59 : At.65 : C.80,
86,90,91: CP.8: J.13,61,85: M.
14: Man.55: Me.78,79 : Mem.13,
geo.ap20,44 : Mcl.l : N.69,85 : P.
50,56,58: Pr.7,8: Q.li;,2,7,15 : Z.
20,24.
by Arbogast's derivatives : L.82.
of complex numbers in Jacobi's th :
C.96.
by elliptic and hyper-elliptic func-
tions : J. 13.
formula of verification : Pr.24.
into 2 ,s{|uares : An. 50,52, 54: C.87 : J.
49: LM.8,9: N.54,78,algG5 : odd
squares, Q.19.
into 3 squares : J.40 : L.59,60.
into 4 squares : C.99 : L.68 : Pr.9 : Q.l.
4 odd, or 2 even and 2 odd : Q.l 9,20.
into 5 squares : C.97,98 : J.35.
Partition of numbers — {continued) :
into ten squares : C.60 : L.66.
into p squares : C.39,90 : L.61 : N.
54.
and an integer : L.57.
into the product of two sums of sqs. :
L.57.
into parts, the sum of any two to be a
sff. : Mem.9.
into 2 cubes : L.70.
into sum or difference of 2 cubes :
AJ.2.
into 4 cubes : lSr.79.
into maximum «th powers : C.95.
into 10 triangular numbers: C.62.
formation of numbers out of cubes :
J.14.
2 squares whose sum is a sq. : E.20 :
N.50.
3 squares, the sum of every two being
asq. : E.17.
4 squares, the sum of every two being
a sq. : E.16.
3 nos., the sum or diff. of two to be a
sq. : Mem.l8.
2 sums of 8 sqs. into 8 sqs. : Me.
78.
a sum of 4 sqs. into the product of 2
sums of 4 sqs. : 'ri.21.
n nos. whose sum is a sq. and sum of
sqs. a biquadrate : E.18,22,24.
a quadi'ic into a sum of squares : N.
80,81.
ttS— ?n« into 3jj2-(-r/3 : N.49.
a square into a sum of cubes : N.67.
a cube into a sum of cubes : E.22,
23.
into 4 cubes : N.77 ; into 3 or 4
cubes, A.60.
7il~— 9)ui~ or its double into 2 cubes :
N.81.
3 nos. whose sum is a cube, sum of
sqs. a cube, and sum of cubes a
s(i. : E.26.
5 biquadrates whose sum is a square
E.20.
of n into 1, 2, 3, &c. different num-
bers : E.34.
of pentagonal numbers : C.96.
a series for the : AJ.6.
tables, non-unitary : AJ.7.
theorems : AJ.6 : C.40,96 : Me.76,80,
83 : pr. symm. functions, G.IO.
Partitions : in theory of alg. forms :
G.19.
numl)cr of for n things : E.IO.
in planes and in space: J.l.
Sylvester's theorem : Q.4.
trihedral of the X-ace and triangular
of the X-gou : Man .58.
INDEX.
911
*Pascars theorem : 4781 : AJ.2 : CD.3,
4: CM.4: J.34, 41, 69,84,93: LM.
8: Me.72: N.44,52,&2: Q.1,4,5,9 :
Z.6,10.
extension of and analogues in space :
C.82,98: CD.4,5,6: G.ll : J.37,75 :
M.22 : Me.85.
ap. to geo. analysis : A. 18.
on a sphere : A.60.
Steiner's " Gegenpunkte " : J.58.
Pascal lines : B.30.
Pedal curve : A.35,36,52 : J.48,50 : M.6 :
Mc.80,81 : Q.ll : Z.5o,21.
circle and radius of curvature : C.84 :
Z.14.
of a cissoid, vertex for pole : E.l.
of a conic : A.20 : LM.3 : Z.3.
central : A.9 : Me.83 : N.71.
foci and vector eq. : LM.13.
negative central: E.20,29 : TI.26.
negative focal : E. 16,17,20.
nth and n — 1th : E.18.
of evolute of lemniscate : E.30.
inversion and reciprocation of : E.21.
of a parabola, focal and vector eqs. :
LM.13.
rectification of difference of arcs of :
Z.3.
which is its own pedal : L.66.
Pedal surfaces : A.22,35,36 : M.6 : J.50 :
Z.8.
connter: AJ.5.
volumes of: C.55: A.34 : An.63 : J.
62: Pr.l2.
Pentagon, ths : A.4 : J.5,56 : N.53.
diagonals of: A.57.
Pentagonal dedecahedron : A.25.
Pentahedron of given volume and mini-
mum surface : L.57.
Periodic continued fractions : A.62,68 :
C.968: J.20,33,53: N.68 : Z.22.
closed form of : A.62.
representing quadratic roots : A. 43.
with numerators not unity : 0.96.
Periodic functions : A. 5 : J.48 : ISr.67 :
gzO.89: Mo.84.
cos a- — I cos 3x+l cos 5« : CD. 32.
doubly: C.32.,40,70,90 : J.88o : L.54.
of 2nd kind : 0.90,98 : gzL.83.
of 3rd kind : 0.97.
monodromic and monogenous :
0.40.
with essential singular points : 0.89.
expansion in trig, series: N.78.
4- ply, of 2 variables : J. 13.
2?i-ply, of n variables : Mo.69.
multiply: 0.57,680.
integrals between imaginary limits:
A 23
real kind of: Mo.66,84.
Periodic functions — {co7itmued) :
of 2nd species : M.20.
of several variables : 0.43: J. 71.
in theory of transcendents : J.ll.
of 2 variables with 3 or 4 pairs of
periods: 0.90.
with non-periodic in def. integrals :
0.18.
Periodicity theory : M.18.
Periods : cyclic, of the quadrature
of an algebraic curve : 0.80,84.
of the exponential e-"^: 0.83.
of integrals : see " Integrals."
law of: 0.96^.
in reciprocals of primes : Me. 733.
*Permutations : 94: Al. : 0.22: 0D.7 :
L.39,61 : LM.15 : Me.64,66,79 : N.
44,71,763,81 : Q.l : Z.IO.
alternate : L.81.
ap. to differentiation and integration
A.21.
of n things : 0.95 : N.83 ; in groups,
L.65.
of Sq and 2q letters, 2 and 2 alike :
N.74,753.
number of values of a function
through the permutations of its
letters: 0.20,21,46,47: L.65.
successive (" battement de Monge ") :
L.82.
with star arrangements : Z.23.
*Perpendicular from a point : upon
a line : length of : 4094, t.c4624 :
eq4086, t.c4625 : sd5530,
* upon tangent of a conic : 4366—73.
* upon a plane : 5554.
* upon tangent plane of aquadric : 5627.
* ditto for any surface : 5791 — 3.
Perpetuants : AJ.7.,.
*Perspective : 1083 : A.692 : G.3 : thsL.
37: Me.75,81.
analytical : A.ll.
of coordinate planes : 0M.2.
* drawing : 1083 — 6.
figures of circle and sphere : A.57.
isometrical : OP.l.
oblique parallel : Z.16.
projection : A.16,70.
relief: A.36,70 : K57.
* triangles : 974 : E.29 : J.89 : M.22,16 :
in a conic, Al.
Petersburg problem : A. 67.
Pfaff's problem: A.60: 0.94: J.61,82 :
M.17 : th of Jacobi, J.57.
Pfaffians, ths on : Me.79,81.
TV (see also " Expansion of") : 0.95 : E.
30 : N.42,45.
calculation of: A.6,18: E.27 : G.2 :
en J.3 : Me.73,74 : N.50,56,66.
by equivalent surfaces : N.48.
912
INDEX.
IT : calculation of — {continued) :
b}'' isopcriracters : N.46.
by logarithms : N.56.
to 20U decimal places : 3.27; to 208,
P.41 ; to 333, A.2I2 ; to 400, A.22 ;
to 500, A.25 ; to 607, Pr.6,11,22.
formulai foi', or values of : A.12 :
J.17 : L.46 : M.20.
TT = 3 + -^ appros. : Me. 66 ;
= - log i, J.9.
functions of : p.6 : A.l : C.56,74.
7r-i : E.27; to 140 places, LM.4.
hyperbolic logarithm of : LM.14.
powers of tt and of tt"' : LM.8.
* incommensurable : 795.
series for : Q.12 : TE.14.
theorem on tt and e : Q.15.
n(,«)=l(l+.^)... {l + {n-l)x]: A.12:
J.43,67.
n {x) and imaginary triangles and
quadrangles: A. 51.
Piles of balls and shells : ISr.72.
Pinseux's theorem : Mo. 84.
Plane: J.20,45 : Mem.22.
* equations of : 5645, q.c 5550, p.c 5552 :
Z.l.
* under given conditions : 6560 — 7'^.
* condition for touching a cone : 5700.
* ditto for a quadric : 5635, 5701.
cond. for intersection of two planes
* touching a quadric : 5703.
figiires, relation between : A. 55 : J. 52 :
M.3.
kinematics of : Q.16.
and line, problems : CD.2 : J.14.
motion of : JP.2 : LM.7.
point-systems : J.77 ; perspective,
Z.17.
representing a quadric: N.71.
*Plane coordinate geometry : 4001 — 5473.
Planimeter: A.58 : Mel.2,3 ; Amsler's,
5452,0.77; Trunk's, A.44; polar,
A.51 : N.80.
Planimetrical theorems : A.37,60.
Pliicker's complex surfaces : M.7.
Pliickcrian characteristics of a curve
discriminant: Q.12.
Pliickcrian numbers of envelopes : C.78o.
Point-pair, absolute on a conic : Q.8.
harmonic to two such : Z.13.
Point-plane system : M.23o.
I'oints : in a plane, relation between
four: A.2,26.
* tg.c(i of two : 16G9, 41113.
on a circle and on a sphere : N.82.
of equal parallel transversals: A.61.
Points — (continued) :
four, or lines, ths : CD. 8.
at infinity on a quadric : N.65.
roots in a closed curve : N.68.
in space, represented by triplets of
points on a line : LM.2.
systems: JP.9 : M.6,25 : N.58.
of cubic curves : Z.15.
three : coordinates of, N.42 ; pr, A.8.
*Polar: 1016,4124: A.28: J.58: gzLM.2 :
Me.64,66: N.72,79.
* of conies : 4762 : thsN.58.
of cubic curves : J.89 : L.57 : Mel. 5 :
Q.2.
curves, tangents of : N.43.
* developable : 6728.
inclined: N.59.
* line of two points with respect to a
quadric : 5685.
plane: Q.2 : Z.22.
* of a quadric : 5678, 5687 : Au.71 ;
of four, LM.13.
of a quartic : L.57.
of 3 right lines : A.l.
* subtangent : 5133.
Polar surface: of a cubic: J.89;
twisted, Z.2.32,24.
of a plane: C.60: N.66.
of a point : N.65.
tetrahedron : J.78 : N.65 : Z.13_.
of a triangle : A. 59 ; perspective. J.
65.
Pole: 'Of chords ]oining feet of nor-
mals of conic drawn from points
on the evolute : N.60.
* of the line Aa+/^^-fi^y: 4671.
* of similitude : 5-587.
*Pole and Polar : 1016, 4124.
Political arithmetic : trA.36— 38.
Pollock's geo. theorems : Q.l.
Poloids of Poiusot : CD. 3.
Polyacrons, A-faced : Man.62.
Polydrometry : A. 38,39.
*Polygoiial numbers: 287: Pr.l0,ll,12„
13.
Polygonometry : thsAn.52 and J.2,47 :
Mem.30.
Polygons (see also "Regular poly-
gons"): An.cn53,63: JP.4,9 : N.
74: Z.ll ; theorems: A.1,2 : C.
26o: prsCD.62: Mel.2: N.58.
* area of : 748, 4042 : J.24 : N.48,52.
articulated and pr. of configuration,
tr: An. 84.
centroid : N.77.
of circular arcs, en : A.3 : J. 76.
classification : Q.2.
lUvision into triangles : A. 1,8 : L.38i,
?,%.: LM.13: rr.8.
of even number of sides : LM.l.
INDEX.
913
Polygons — (continued) :
family of : N.83.
maximum with given sides : J.26.
of n + 2n sides, numbers related to:
AM.
of Poncelet, metrical properties: L.79.
semi-regular: JP.24. ; star, A.59 and
L.79,80.
sum of angles of : A. 52 : N.50.
Polyhedral function (Prepotentials) :
CP.13..
*Polyhedrons : 906 : C.46o,60— 62 : J.3 :
JP.4,9o,15,24 : L.66 : Man. 55 : N.
83: P.56,57: Pr.8,9,ll2,12 : Q.7 :
Z.11,14 ; theorems : E.39— 42 : J.
18: N.43; F+S = E+2: 906:
A.24: E.20,272.
classification of : C.51,52.
convex: angles of, C.74; regular, A.
59.
diagonals, number of : ]Sr.63.
Euler's theorem : J.8,14 : Mem.lo :
Z.9.
minimum surfaces of : A.58.
maximum : regular, C.6L ; for a given
surface, M.2 : Mel.4.'
regular: ellipsoidal, 0.27; self-conj.,
A.62 ; star, A.62 : thsO.26.,.
symmetrical : J.4 : L.49.
surface of : A.53 ; volume, J.24 : ]Sr.52.
Polj'uomials : geometry of, JP.28 : th
determined from its partial differen-
tial : A.4.
product of two : lSr.44.
system of : L.56 .
of two variables analogous to Jacobi's:
A.16.
value when the variable varies be-
tween given limits : 0.98.
Polyzonal curves, v/?7+ v/F+= : TE.
25.
Porisms : L.59 : P.1798 : Q.ll : TE.3,4,9.
of Euclid: C.29,48,56— 59 : L.55.
of two circles : Me.84.
Format's fourth : A.46.
of in- and circum-polygon : Me.83 :
P.61.
of in- and circum -triangle : LM.6,9 :
Q.I.
Poristic equations : LM.4,5.
Poristic relations between two conies :
LM.8.
Position, pr. relating to theory of num-
bers : Mel. 2.
Potential: thsAn.82 : 0.88: G.15 : J.
20,32,63,70,th81,85: M.2,3: N.70.,:
Z.17.
of a circle : J.76.
Potential — [continued) :
ofcyclides: 0.83: J.61 ; elliptic. Me.
78.
cyclic-hyperbolic, tables : J.4,63,72,83,
94.
of ellipsoids: J.98 : Me.84: Q.14 ;
two homog., J.63,70.
elliptic: J. 470.
of elliptic disc, law, ?-"^ : Q.14.
Gauss's f. and theory : Z.84.
gz. of first and second : L.79.
history of : J.86.
of homogeneous polyhedra : J.69.
Jellett's eq. and ap. : Q.16.
Newton's : M.11,13,16.
one-valued: J.64.
p.d.eof: 0.90.
Poncelet's ths : Z.3.
a related integral : L.45.
of a i-ight solid: J.58.
of a sphere : Me.81 ; surface of, Me.
83 : Q.12 : Z.7. _
surfaces : J. 54 ; conicoids, &c., 79.
vector : Me.80.
Pothenot's problem of the sphere : A.
44,47,542.
Powers : angular functions, &c. : J.72-
and determinants, relation : Z.24.
of negative quantics : Me. 73.
of polynomials : JP.15.
Power remainders : M.20.
Prepotentials : P. 75.
Prime divisors of quartics : J.3.
Prime factors of numbers : J.51 : N.71,75.
Prime-pairs : Me. 79.
*Primes : 349—378 : A.2,19 : A.J.7 : fAn.
69: C.thsl3,49,50,f63,962: G.5:
J.thl2,thl4,20 : L.52,54,th79 :
LM.2 : M.21 : Me.41: N.46,56: Pr.
5 : Q.5 : up to 109, M.3.
in ar.p : Z.6.
calculation of : J.IO : in 1st million,
M.25.
in a composite number : 0.32.
distribution of, ap. of recurring series :
0.82.
division of a prime 4n+l into sum of
squares: J.50 : ditto of 8n+3,
7n+2, and 7ii+4, J.37.
even number = sum of 2 odd primes
{'^): E.IO: N.79.
Fibonacci's problem : LM.ll.
of the following forms and theorems
respecting them: 4-lc + l, ih+S ;
L.60: 6A;-fl; J.12 : 8fc+l ; L.6I3,
62: 8k + 3; L.58,60,61 4,622 : 8fc+5
or 7 ; L.60,61 : 12^ + 5 ; L.61,63 :
16fc+3; L.61: lQk + 7; L.60,61:
16;;; + 11; L.60— 62: 16^ + 13; L.
61 : 20fc+3, 20&+7 ; L.63,64 :
6 A
914
INDEX.
Primes — {continued) :
247v+l, 5, or 7 ; L.6I4 : 24/,; + ll
or 19; L.60: 24?^: + 13; L.61 :
40h + S; L.61: 40A: + 7or23; L.
603,613: 40A;+11 or 19; L.60:
40A-+27; L.6I3: 120/.: +31, 61, 79,
or 109; L.61: 168/^+43,67, or 163 ;
L.63 : vi^ + hn"- with A- = 20,36,44,
66, or 116, and n an odd number;
L.65 : 4Hi-2+5?i2 with vi odd ; L.
66.
general formula for : C.63.
of a geo. form, limit of : C.74.
irreducibility of l + a' + . ..+,';''"', when
jj is a prime : J.29,67 : L.50 : N.49.
law of reciprocity between two, ana-
logue : J.9.
* logarithms of (2 to 109) : page 6.
* number infinite : 357 : Mo.78.
number within given limits : A.64 :
C.953: M.2: Z.5.
number of digits in their reciprocals :
Pr.222,23.
number in a given quantity : Mo.59.
product of n, th : N.74.
* relative : 349—50, 355,-8, 373 : J. 70:
L.49.
tables of : from 108 to 100001699 : of
153 of the 10th million :
of cube roots of to 31 places : Me. 78.
of sums of reciprocals and their
powers : Pr.33.
M^a'V for a prime or composite
modulus : Q.9.
totality of within given limits : AJ.4 :
L.52.
transformation of linear forms of into
quadratic forms : C.87.
on that prime number X for which the
class-number formed from the
Xth roots of unity is divisible by
X: M0.74.
Primitive numbers : thC.74.
Primitive roots : C.64 : JP.ll : fL.54 :
taJ.45.
of binomial eqs. : thsL.40 : N.52.
of primes : J.49.
product of, for an odd modulus : J.31.
of unity: C.92.
Abel's theorem : An. 56.
divisors of functions of periods of :
C.92.
period, Jacobi's method : C.70.
of primes : J.49 : and their residues,
Mc.85.
sum of, for an odd modulus : J.31.
table, for primes below 200: Mora.
38.
table of, for primes from 3 to lOl :
J.9.
*Principal axes of a body : 5926, '60,
'72,'77: J.5: JP.15: L.47 : N.46.
Prismatoids : A. 39 : volume of, Z.23.
Prismoids : A. 39.
Prism : volume of, A.6.
*Probability : 309: A.U19,47 : C.65,97 :
CP.9: E.274,30: G.17: 1.15: J.
26.,302,33,34,36,42,50 : L.79 : N.
51,73: Z.2.
theorems : Bernoulli's : LM.5 ; P.62 ;
TE.9,21.
problems : A.61,64 : CD.6 : E. all the
volumes: G.16 : L.37 : Me.4,6 :
Q.9.
de I'aiguille, &c. : L.60.
in analysis : J. 6.
on decisions of majorities : L. 38,42.
duration of life : CM. 4.
rouge et noil- : J.67.
drawing black and white : L14 : L.41.
errors in Laplace, p. 279, and Poisson,
p. 209: CP.6.
games : A. 11 : head and tail ; C.94.
* of hypothesis after the event : 324 :
Mel. 3.
local : P.68 : exE.7.
notation : LM.12.
position of double stars : Pr.lO.
principal term in the expansion of a
factorial formed from a large
number of factors : C.19.
random lines : Pr.l6 : E. frequently.
* repeated trials : 317—21 : C.94.
statistics : L.38.
testimony and judgment : TE.21.
Products : continued : Me.772.
of differences : thQ.15.
of 4 consecutive integers : N.62.
* of inertia: 5906.
infinite, convergency of : A. 21 : transf.
of, C.17.
of linear factors : C.9.
of n quantities in terms of sums of
powers : Me. 71.
of 2 sums of 4 squares (Euler) : Q.
16,17.
systems of: L.56.
*Progressions : 79—93 : An.64 : C.20 :
G.62,7,11 : of higher order, G.122.
with n ■= a fraction : N.42.
*Projection : 1075, 4921 : A.3,6,12 : prsJ.
70: Q.21,prl3.
and new geometry : A.l.
central : G.13 : derivation from or-
thogonal, A.62.
central and parallel, of quadrics into
circles : A. 52.
* of conies : 4921—35 : A.66: J.37,86.
of a cubic surface : M.5.
INDEX.
915
*Pi'OJection — (continued) :
of curves : J.66 : Z.lSo: loci of cen-
tres, A. 6.
on spheres : Q.14.
tangents to : cnL.37.
of a curved surface on a plane : J.67.
of an ellipsoid on a plane : J. 59.
of figures in one plane : A.l.
gauche : 'N.66.
of Gauss: Mel.2.
* map : Mercator's, 1093 : A.50 : G.18 :
JP.24: N.793.
* of one line on another : 5529 : on a
plane, A.6.
* orthogonal: 1087.
plagiographic : A.8.
of ruled quartics : M.2.
of shadows : N.56.
of skew hyperboloid : Me. 75.
of solids : Man.84.
* of the sphere : 1090 : AJ.2.
* stereographic : 1090: A.30— 32 : L.
42,54.
of surfaces on a plane : M.5.
of surface of tetrahedron on a sphere :
J.70.
* of two rectangular lines : 4934.
Projective : correspondence between
two planes and two spaces : G.22.
equations of a surface, relation to
tg.e: E.2.
figures on a quadric : A.18.
generation of alg. surfaces : A. 1,2 : of
curve forms, An. 14: J.54: M.26
Z.18.
geometry : A.8 : An. 75 : At.75..,78
G.r2— 14,17: J.84: M.17,18: ths
N.77 : ths of Cremona, G.13.2,14
thZ.ll.
loci and envelopes : G.19.
and P.D. eqs. : A.l.
point series : J.91.
and perspectivity of higher degree in
planes : J.422.
*Proportion : 68 : A.8 : G.6,13 : TE.4.
Pseudosfera : G.IO.
Ptolemy's theorem : A.2,67 : At.l9 : J.
13: LM.12.
ext. to ellipse : A.30 : inverse of, A. 5.
Pure mathematics, address by H. J. S.
Smith, F.R.S. : LM.8.
Pyramid : triangular : A.l, 3,21,28,32,
36 : J.3 : volA.14.
vertices of : A.2.
and higher w-drons : prsA.9.
and prism sections, collineation, &c. :
A.9.
Pythagorean theorem: A.11,17,20,24:
gzJ.26 and N.62.
spherical analogue : A.44 : N.52.
Pythagorean triangles : taA.l : E.20.
^Quadratic equations : 45 : A.24o : with
imaginary coefficients, A.8.
graphic solution : Me. 76.
real roots of : J. 61.
solution by continued fractions : L.40.
by successive approximation : N.74.
Quadratic forms (see also " Quadric
functions"): trA.15 : Ac.7 : C.
85 : J.27,f39,54,56,76,86 : L.51o,
73: M.6,23: Mo.682,74,: thsAn.
54: J.53: M.20 and Z.16,19.
Dirichlet's method : Mo. 64.
having one at least of the extreme
coefficients odd : L. 67,69.
Kronecher's : L.64.
multiplication of : An. 60.
number of the genera of : J. 56.
number which belong to a real deter-
minant in the theory of complex
numbers : J. 27.
odd powers of sq. root of 1 — 2t]U+t]":
Mel.5.
positive : A.II2.
reduction of : J. 39 : L.48,56,57.
relation, anal, investigation : Z.14.
Quadratic loci, intersection of : AJ.6.
*Quadratrix: 5338.
tangent, en : lSr.76.
of a curve : 0.76^.
Quadrangle : prA.55 : 0.95.
of chords and tangents : A.2.
differential relation of sides : Me.77.
dualism in the metric relations on the
sphere and in the plane : Z.6.
and groups of conies : A.l.
of two intersecting conies, area :
LM.8.
metrical and kinematical properties :
C.95.
*Quadrature: 6871-83: A.26 : An.SOa
CD.ths5,9: E.6: J.34 : L.54
Mem.24,41: N.42,f55,64: N.75
TI.l.
approximate : C.95 : N.58.
of the circle : Me.74 : Pr.7,20.
Cote's method : lSr.56.
with equal coefficients, f. : C.90.
Gauss's method: A.32 : C.84,90o:
J.gn55,56.
from integrals of differentials in two
variables : M.4.
* Laplace's formula, f .d.c : 3778.
of a small geodesic triangle. Gauss's
th: J.16,58.
* by lines of equal slope : 5881.
quadrics: CD.l : L.63 : formed by
intersecting cylinders, An.65.
sphero-conic : J.14.
916
INDEX.
Quadratures — (continued) :
which depend upon an extended
class of d.e with rational coeffi-
cients : C.92.
*Quadric cones : 5590,6618,'64;97 : N.66.
locus of vertex : C.52.
through six points : C.52.
Quadric functions or forms : A.13i2,38 :
C.44,55,78,892,95 : J.47.: JP.28,
arith 32, positive 99 : algL.74 :
Mo.58 : see also " Quadratic
forms."
bipartite: P. 58.
in coefficients and in indeterminate
complexes : J.24.
equivalence of: C.93.
in n variables : Me.2 : disappearance
of products, N.55.
quaternary : An. 59 : L.64 : M.5,13 :
whose det. < 0, C.96 ; and corres-
ponding groups of hyperabelians,
C.98.
reduction of : C.91,93,96 : G.5 : to
sum of squares, Mel. 5.
represented by others : C.93.
transf.of: C.86 : CD.4 : G.l : LM.16 :
N.66: Q.17.
reciprocal : J. 50.
invariability in number of pos. and
neg. squares : J. 53.
with two series of variables : C.94.
♦Quadric surfaces : 5582—5703 : A.32,4,
12,16,45,56: An.52,geo60: At.51 :
C.76: CD.3: G.13,14: J.1,18,38,
f42,63,69,89 : JP.6 : L.39,43,50 :
M.2,23: geoMe.74: N.56,57,58..,
593,60,61,77 : Z.5,cnl3.
theorems : An.54 : CD.5 : J.54,85 :
L.43 : Mo.79 : N.632,64o : Q.2,4.
problems: An.61 : C.60: J.73 : N.
58: Q.IO.
analogy with conies : Me. 72.
anharmonic section of : G.12.
bifocal chords of : CD.5.
* central equations : 5599 — 5672.
* central sections : 5650 : area, 5650 :
axes, 5651.
locus of focus : N.66.
* non-central sections : 5654.
* centre : N.75 : area, 5655.
* centre, coordinates of : 5690: A. 16.
* circular sections : 5596, 5601, '6,'19 :
C.43: CD.l : CM.l: E.30: J.47,
65,71,85: N.51.
common enveloping cone : G.6.
* condition for a cone : 5699.
* conjugate diameters : 5637 : N.42,61 ;
parallel, G.1; rectangular sys-
tem, L.58.
Quadric surfaces : conjugate diame-
ters— (cori^i)ut«?) :
* parallelopiped on them : vol. 5648 :
sum of squares of areas of its
faces, 5645 : do. of reciprocals of
the perpendiculars on its faces,
5644.
* sum of squares of their reciprocals :
5643 : ditto of their projections
on a line or plane, 5646.
construction and classification by
projective figures : A.9.
correlation of points and planes on :
CD.5.
* cubature of : 6126 — 65 : A.14.
* diameters: 6677,-88.
* diametral plane : 5636 ; gn.eq, 5689.
of constant sectional area : N.43.
director-sphere of : Q.8.
duplicate : CD. 6.
* enveloping cones of : 5664—72, 6697.
equation between the coefTs. : J.45.
without foci : L.36.
focales, a property from the theory
of : L.45.
* general eq. : 6673 : CD.5 : CM.4 : LM.
12,13: M.l: Me.64.
* condition for a cone : 5699.
condition for a sphere : Q.2.
coefficients : A.l.
* generation: 5607—24, N.47,75 ; Ja-
cobi's, J.73.
homofocal : L.60.
indices of points, lines, and planes,
theory: N.704.
* intersection of two : 5660, C.64. : G.
6 : M.15 : Q.IO ; ruled, N.83 ; with
a sphere, ths, N.64.
parameter representation of: M.15.
tangent to : LM.13.
* intersection of three : 5661, J.73 :
TN.69.
loci from : A.27.
* normals of : 5629—32 : C.78 : J.73,83 :
N.63,78 : Q.8.
oblique coordinates : N.82.
* polar plane of : 5681 — 8.
Pliicher's method : L.S8.
of revolution : N.72,81 ; through 5
points, 66 and 79.
self-reciprocal : M.25 : Z.22.
sections of : J. 74.
similarity of two : CD. 8.
with a "■ Symptosen-axe": A.60,61.
system of: Q.15; reduc. and transf.,
L.74: Z.6.
* tangents : 5677 : G.12.
* tangent planes : 5026,-78 : CM.l :
cnJ.42: LM.ll: N.46.
INDEX.
917
Quadric surfaces — {continned) :
locus of intei-sect. of three : LM.15.
throusjh 9 points or uuder 9 condi-
tions : A.17 : CD.4., : J.24,62,68 :
L.58,59: Q.8: Z.25.
under 8 conditions : C.62.
through 12 points : G.17,22.
through a twisted quintic curve : E.
38.
transformation of two, by linear sub-
stitution into two others, in which
the squares only of the variables
remain : J.12.
principal axes : see "Axes."
principal planes of : L.36 : N.67,71 :
Z.24.
and their umbilics : 0.54.
volume of segment : A.27 ; of oblique
frustum, E.19.
Quadricuspidals : L.70.
Quadrilateral: A.6,48 : Me.66 : thG.5:
prsN.43.
area: N.48; as a determinant, N.74.
between two tangents to a conic and
the radii to the points of contact :
A.53o.
bisectors, a property : lSr.75.
* and circle, geo. : 733 : Q.6.
* complete : 4652 : A.24,69 : At.63 ; mid-
points of 3 diagonals, collinear,
Me.73.
* and conic : 4697 : ]Sr.75,76.
* and in-conic : tg.c 4907 : Q.ll.
en. with given sides and equal diags. :
A. 5.
convex: A.66; area, Q.19.
Desargues' theorem : Z.24.
plane and spherical : A.2 : Z.6.
and quadrangle: A.l.
right-angled: A.2,3.
with sines of angles in given ratio :
A.2.
* sum of sqs. of sides : th 924.
th. extended to 3 dimensions : J.56.
Quadruplane : LM.14.
*Quantics: 1620: C.47: J.56: LM.6 :
N.48: Cayley P. 54,56,58,59,61,
67,71 and 78: Pr.7>.,8,9.,ll,15,17,
18,23 : thsCD.6 : J.53 : K53 :
Q.14; Cauchy, Pr.42.
derivatives, relation between 1st and
2nd: E.39.
derivation from another by linear
substitution : 0.42.
of differentials : J.70,71 : Mo.69.
index symbol of : CD. 5.
integration of a rational : C.97.
in linear factors : C.50: L.ths61 : Q.6.
transf. of axi+bij"+cz~-\-dw^,hj linear
subst. : J.45.
Quantitative function, transf. of : CD. 3.
Quartic equations : see " Biquadratic."
Quartic curves : An. 76,79 : At.52 : C.37,
ths64,65,77,98 : CD.5 : G.14,16 :
J.59: M.thl, 4,7,12: prN.56.
aU+6m = 0: M.l.
and Abel's integrals : M.ll.
binodal, mechanical en. of : E.18.
characteristics of a system : C.75.
chord of contact, eq. : M.17.
classification by inf. branches : L.36.
with cuspidal conies : M.19.
with 3 cusps of 1st kind : An.52.
degenerate forms : LM.2.
developable reciprocated : Q.7.
with a double hne : A.2 : C.75.
with a double point : M.19 ; two, C.97. ;
three, Q.18; three of inflexion,
N.78.
with double tangents : J.66.
and elliptic functions : J. 57,59.
of 1st kind and intersections with a
quadric : An.692.
o-eneration of C.45 : J.44 ; 3rd class,
J.66 and Z.18.
16 inflexion points of 1st species of :
trZ.28.: E.32.
and in-pentagon, th : M.13.
oblate: C.74.
parameter representation of : M.13.
penultimate : Me. 72.
with quadruple foci : Q.18i;.
rectification : C.87i.
and residual points : E.342.
and secants : M.12.
singularities of : L.75: M.14.
synthetic treatment of : Z.23.
through which one quadric surface
only can pass : An.61.
trinodal : thE.30.
unicursal twisted : LM.14.
Quartic surfaces : A.12: C.70 : G.11,12 :
LM.3,19: M.1,7,13,18,20 : Mo.66,
72: N. 70,873: Q.10,11.
containing a series of conies : J.64.
with a cusp at infinity : LM.14.
with double conic: A.2: M.l, 2,4 :
Mo.68.
with eq. Sym. det. = : Q.14.
generated by motion of a conic : J. 61.
Hessian of : Q.15.
and 2 intersecting right lines : M.3.
with 12 nodes : Q.14.
with 16 nodes : J.65,73,83,84,85,86,87,
88: Mo.64; principal tangent
curves of, M.23 and Mo.64.
Steiner's : C.86 : J. 64.1.
with a tacnode at infinity at which
the line at infinity is a multiple
tangent : LM.13.
918
INDEX.
Quartic surfaces — (continued) :
with triple points : M.24.
Quaternions: C.86,98.>: CD.4 : G.20 :
M.11,22: Me.62,,64,81 : P.o2 : Q.
(J: TE.27,28: T1.21.
ap. to linear complexes and congru-
ences : Me.83.
ap. to tanf^ent of parabola: AJ. la-
elimination of afSy from the conditions
of integrability of Suadp, &c. :
TE.27.
equations : C.98 ; linear, QQo : of sur-
faces, Joachimstahl's method,
E.43.
vp<pp=0: TE.28: qQ-qQ; = 0,
Me.8.>j.
finite groups : AJ.4.
f. for quantification of curves, sur-
faces, and solids : AJ.2.
geometry of: CD. 9.
integration ths : Me.85.
transformations : Man.82.
*Quetelet's curve : 5249.
Quintic curves : cnM.25.
Quintic equations : AJ.6i,7 : An.65,682
C.46..,48,50,61,62a,73,80.,85 : J.59
M.13".,14,15: P.64: Pr.ll : Q.3
TI.19 : Z.4.
auxiliary eq. of : Man. 152 = P-61 : Q-3.
condition of transformability into a
recurrent form : E.35.
irreducible : AJ.7 : J. 34.
functions of difference of roots : An.
69.
reduction of : Q.6.
resolvents of : C.63o.
whose I'oots ai'e functions of a varia-
ble: Q.5.
solution of : An.79 : J.59,87 : N.42 : Q.
2,18.
Descartes' method : A. 27.
Malfatti's : An.63.
in the form of a symmetrical deter-
minant of four lines : An. 70.
Quintic surfaces : LM.3.
having a quintic curve : An. 76.
Quintics, resolution of : Q.4.
Radial curves : LM.l : of ellipse, Q.18.
of conies, catenary, lemniscate, &c. ;
E.24.
Radiants and diameters of a conic : C.
26.
*Radical axis (see also " Coaxal Cir-
cles ") : 958, 984—99, 4161 : LM,
2 : Me.66.
of symra. circle of a triangle : A.63.
of two conies : Q.15.
♦Radical centre : 997.
♦Radical plane : 5585.
Radii of curvature of a surface : A.ll,
55: Q.12: Z.8.
principal ones : L.47,82 : M.3 : Me.80 :
N.55.
*Radii of curvature of a surface : 5795—
5817 : A.11,55 : Q.12 : Z.8.
* ellipsoid : 5831.
flexible surface : L.48.2.
* principal: 5814—6: L.47,82: M.3:
Me.80: N.55.
constant : Me.64.
* for an ellipsoid : 5832.
equal and of constant sign : C.41 :
JP.21 : L.46,,50.
* Euler's theorem : 5806.
one a function of the other : An. 65 :
C.84: J.62.
product constant : An. 57.
reciprocal of product : An. 52.
sum constant : An. 65.
sum = twice the normal : C.42.
♦Radius of curvature of a curve : 5134 :
A.cn4,9,31,33 : CD. 7 : J.2,45 : ths
M.17: N.62,74: q.c and t.c Q.12 :
Z.3.
absolute : CM.l.
* circular : 5736 ; ang. deviation, 5746.
* of conies : 1259 : A.9 : CM.l : J.30 :
L.36 : Me.66 : Mel.2 : N.45,682.
at a cusp or inflexion point : N.54.
in dipolar coordinates : Me.81.
of evolutes in succession : N.63.
of gauche curves : l^.QQ.
of a geodesic : L.44 : on an ellipsoid,
An.51.
and normal in constant ratio : N.44.
* of normal section of (.'', y, x) = :
5817.
* of a parabola : 1261,4542.
of polar curves : A. 51 : CM.2.
of polar reciprocal : N.67.
of projection of a curve: N.61 ; of
contour of orthogonal projection
of a surface, C.78.
* of a roulette : 5235 : N.73.
transf. of properties by polar recipro-
cals : L.66.
of tortuous curves : Mem.lO.
of circum-sphere of a tetrahedron in
terms of the edges : X.74.
♦Radius of gyration : 5904.
Radii of two circles which touch three
touching two and two : A.55.
Radius vector of conic : J.30 : N.47;.
Ramifications : E.30;,33,40,pr37,th27.
sol. by a diophantine eq. : Mo.82.
Randintcgral : J.71.
♦Ratio and proportion: 68; compound,
74.
of a'' : h'' : geo.cnN.44.
INDEX.
919
Eatio and Proportion — {continued) :
of differences of geo. quantities : O.40.
* limits of: 753.
* of segments of lines and triangles,
geo.: 929—32.
* of two distances, geo. : 926 — 8.
Rational : derivation, cubic curve :
AJ.33.
divisors of 2nd and 3rd degrees : N.
45.
functions, development of : AJ.5.
Rationalisation of: alg. fractions:
A.13,33,35.
alg. equations : A.13 : CD. 8 : J.14 :
P.14: TL6.
alg. functions : A. 69.
a series of surds (Fermat's pr.) : A.35.
Rational functions : of n elements :
M.14.
infinite form systems in : M.18.
Ray systems (see also " Congruences"):
J.67: L.60,74: Me.66: N.60,6l3,
622 : Z.16.
of 1st order and class and linear pen-
cils : J.67,692: Z.20.
1st and 2nd order : Mo. 65 : M.15,17.
2nd order and class : J. 92.
3rd order and 2nd class : J.91.
6tli order and 2nd class : J.93.
2nd class and 16 nodal quartics : J.86 :
Mo.64.
complex of 2nd degree and system of
2 surfaces: M.21.
and refraction theory: Q.14.2 : TI.15
—17.
forming a group of tangents to a sur-
face : Z.18.
infinite geometry of : Z.17.
*Reciprocal polars : 4844, 6704 : A.36 :
gzE.24: J.77: LM.2: N.48,49:
num.fG.21.
*Reciprocal : of a circle: 4845.
* cones : 5664, 6670.
* of a conic : 4866 — 8.
* of a quadric surface : 5717 — 8.
radii: M.13.
relations: J.48,79,90 : M.19,20.
* spiral: 6302.
* surfaces : 6704—19 : J.79 : M.4,10 : P.
69.
curvature of : L.77.
degree of : CD.2 : TI.23.
ofMonge: C.42.
* of quadrics : gn 6705 ; central, 5706.
of surface of centres o a quadric :
Q.13.
of the same degree as their primi-
tives : Mo.78.
theorems on conies and quadrics : L.
61.
Reciprocal — (continued) :
transformation, geo. : L.71.
triangle : th Q.l ; and tetrahedron,
Q.l.
Reciprocants : LM.172.
Reciprocity: anal., A.7 : geo., CD. 3.
*Reciprocity law: 3446: AJ.l,th2: C.
90., : J.28.,,39 : LM.2 : Mo.58 : d.e,
3446 and A.33.
in cubics : M.12.
history of: Mo.75.
for power residues : C.84; quadratic,
C.24,88: J.47: L.472 and Mo.80,
84,85 ; cubic, in complex numbers
from the cube roots of unity, J.
27,28.
quadratic F^ system of 8th degree :
J.82.
supplementary theorem to : J. 44,56.
*Rectangle : M. I. of, 6015.
*Rectangular hyperbola: 4392: Me.62,
66,72: N.42,65.
*Rectification of curves : 6196 : A.26 :
Ac.6: An.69: CR.95 : CD.9 : G.
11 : J.14: L.47: N.ths 53,64.
approximate : M.4 : Mel.4.
by circular arcs : C. 77,85 : L.50 : Mem.
3O2.
by elliptic arcs or functions : J.79 :
Mem.30.
by Poncelet's theorem : C.94.
mechanical : Z.16.
on a surface : Mem. 22.
*Rectifying: developable : 5727.
* line : 5726,-61 : N.73.
* plane: 6726.
* surface : 6730.
Reflexion: from a revolving line:
TE.28.
from plane surfaces : A.60.
from quadric surfaces : J. 35.
Refraction cui've : A.51.
Regie a calcul : C.58 : N.69.
*Regular polygons: 746: A.21,cn24,39 :
L.38 : M.cn6,13 : N.42,44,47.
convex : Me. 74.
eqs. oE and division into eqs. of lower
degrees, tr. : A.46.
in and circum : N.46 : Q.2.
in space : Me. 76.
spherical: N.60,67.
star: J.65 : Me.74 : N.49.
funicular : N.49.
6-gon : M.83.
7-gon: A.17.
7-gou and 13-gon : M.6.
8-gon : A.6.
12-gon : complete, C.96.
920
INDEX.
Eegular Polygons — (confinued) :
17-gon: A.6: J.cu2-t,75 : N.74; and
division of the circle, A.42.
eqs. for sides and diagonals : A.40.
♦Regular polyhedrons : ' 907 : A.llo,47 :
Pr.:34: Me.66: Q.15.
* relation of angles : 9U9 : Me. 74.
volumes by determinants : A.57.
Related functions : M.25.
Relationship problems : E.35,38i.
Relative motion : N.66.
Rents: A.40.
Representative functions : M.18.
Representative notation : Q.6.
Reproduction of forms: C.97.
Reptation : ]Sr.54.
Residues: A.26: C.122,13,32,ap32.,,41,44,
49: CM.l: J.25o,31,89: L.38 : Mel.
4: N.46i,M(?m7U.
ap. to infinite products : C.17.
ap. to integrals whose derivatives in-
volve the I'oots of alg. eqs. : C.23.
ap. to reciprocity law of two primes
and asymptotes : C.76.
of complex numbers : Mo.80.
primes of 5th, 8th, and 12th powers :
J.19.
biquadratic: C.64 : J.28,39 : L.67.
cubic : A.43,63 : C.79 : J.28,32 : L.76.
quadratic : Ac.l : J.28,71 : Q-1.
ext. of Gauss's criticism : Mo. 76.
of primes, also non-residues : L.42 :
Mel.4.
and partition of numbers : J.6I2.
quintic: C.76: Z.27.
septic: C.80.
of 9exp.(9exp.9) by division by
primes : A.35.
Residuation in a cubic curve: Me. 74.
Resultant alg. : M.16 : ext C.583.
and discriminants and product of dif-
ferences of roots of eqs., relation :
Me.80.
of two equations : J.30,50,53,64 : M.3 :
P.57,68.
of two integral functions : Z.17.
of n equations : An. 56.
of covariants : M.4.
of 3 ternary quadrics : J.57o : N.69.
ReversiVjle symbolic factors : Q.9.
Reversion of angles : LM.6.
Rhizic curves : Q.ll.
Rhombus : quadrisection by two rect-
angular lines : Mem. 11.
circumscribing an equil. triangle : A.
45.
Riemann's surface : LM.B : M.6,182 :
thsZ.12.
of 3rd species : M.17.
new kind of: M.7,10.
Riemann's surface — {continued) :
irrationality of : M.17.
Riemann's function : A.68 : J.83: M.21.
ext. to hyper-geo. -functions of 2 vari-
ables : C.95.2.
^-formula, gz : Ac.3.
*Right-angled triangles : 718 : prs A.2.
with commensurable sides : E.33.
Rio-bt cone : Me.72,73,75,76.
Right line : A.49,57 : fQ.15 : t.c Me.62.
and circle : ths N.56.
coordinates of: G.IO.
and conic : Q.7 ; en. for points of sec-
tion, A.59,66 : N.85.
* quadratic for abscissaj of the points :
4319.
* tg.e of the points : 4903.
* condition of touching: 4315,4323,
t.c 5017.
* drawn from x'l/' across a conic : quad-
ratic for the segments in an el-
lipse, 4314; parabola, 4221 ; gen.
eq., 4494; method, 4134.
* joining two points, coordinates of
point dividing the distance in a
given ratio : 4032, t.c 4603, 5507.
* tg. eq. of the point : 4879.
* joining two points and crossing a
conic : quadratic for ratio of seg-
ments in an ellipse, 4310 ; para-
bola, 4214 ; gu. eq., 4487, t.c 4678 ;
method, 4131.
* constants, relations between : sd.
5515.
* coordinates of, relation between the :
4897.
* and curve : 4131 — 5 : ths. in which
pairs of segments have a constant
length, C.836; a constant product,
C.82.,83;; a constant ratio, C.83;
ths. in which systems of 3 seg-
ments have a constant product,
C.832.
crystallography : A.34.
* equations of : 4052—66, p.c 4107, t.c
4605—8 ; sd5523, q.c 5541.
geometry of : A.64 : thsJ.8.
* at infinity : 4612—4, tg.c4898.
* cond. for touching a curve : 4900.
pencils of: C.70: L.72 ; quadruple,
J.67.
and plane : prs CD.l and CM.2 : t.c
and q.c Q.5.
pole of: t.c 4671: tg.e 4674.
* and quadric : 5676 ; harmonic divi-
sion, 5687.
and (luadric of revolution : N.82.
six coordinates of : CP.ll.
INDEX.
921
Right line — (continued) :
system of : At.68 : G.9,10,16.
of 1st degree, G.6 ; of 2nd, G.7.
in space : G. 113,12 : L.46.
and planes, geo. of 2nd kind : At.
65.
* three, condition of intersection: 4097:
t.c4617.
* three points lying on, cond. : 4036,
t.c4615.
* through a point : 4073, 4088—9, 4099,
t.c4608.
* condition: 4101.
* and perp, or paral. to a given line :
sd5538— 9.
* through two points : 4083, sd5637 ;
t.c4616, 4789 ; p.c4109 : on a
conic, equation of, ellipse 4324,
parab. 4225.
through four lines in space: A.l :
CM. 3 : Gergonne's pr. J.2 and
N.17.
* touching a surface : condition, 5786.
* quadric : 5703.
planes or points through or on given
points, lines, or planes, number
of such: Z.6.
* two: angle between them: 4112;
sd5520, 5553 : CP.2 : N.66.
* bisector of the angle : 4113, sd5540,
q.c5543.
* cond. of parallelism : 4076 ; t.c4618 ;
sd5531.
* cond. of perp. : 4078 ; t.c4620 : sd
5532.
* cond. of intersecting on a conic, gn.
eq : 4962.
* cond. of either touching the conic :
4964.
* cond. of intersection : sd5533.
cooi-dinates : 4090, t.c4611.
* shortest distance : sd5534— 6.
drawn to the points of section of a
right line and conic, eq. of : A.69.
* through origin, eq. of: 4111.
under given conditions : C.73,74 :
under four, C.68.
*Right soUd : M. I. of, 6018 : thA.34.
Rodrique's th. : Me.80,84o.
Rolling cones : L.53.
Rolling and sliding solids : geo thsC.46.
Rosettes : lSr.48.
Rotation : CM. 3 : LM.32 : infinitesimal,
C.78.
of system of lines drawn through
points on a directrix, modulus
of: C.21.
Roots of algebraic fractions : N.46.
*Roots of an equation (see also "Equa-
tions"): 50,402 : A.14 : CM.2 :
CP.8: E.36: J.20,31: N.42,56:
P.1798,37,64: Q.1,6,6.
of a biquadratic, en : N.44.
by parab. and circle : N.87.
* commensurable : 502 : N.45,th57.
limits to the number : N.59.
* common: 462: C. 80,88 : N.55,69.
as continued fractions : CM.3.
continuity of : N.76.
in a converging series : C. 23,38.
of cubic : L.44 : N.42.
of cubic and biquadratic : An. 55 : L.
55.
as definite integrals : J.2.
y"-xy"-'-l= : Me.81 : P.64.
as determinants of the coefficients :
A.69,61.
* discrimination of : 409 : A.46.
* equal : 432—47 : CD.5 : E.33 : Mel.l :
P.1782 : Q.9,18.
with equal differences : G.loo.
existence of: A.15 : CD.2 : CP.IO :
E.36: G.2: J.5,44,88: LM.l : Q.
11 : TI.26.
expanded in power series : J. 48.
of the form a+Vb+^^c+ : N.45.
forms for quadrics, cubics, and quar-
tics: Z.24.
* functions of the roots of another eq. :
425,430 ; products in pairs, Q.13.
* squares of differences : 641 : ap,N.
50: E.40.
as functions of a variable parameter :
C.30.
functions of : similar, L.54 ; relation
to coefficients, TE.28.
geo. en of: JP.IO.
in g.p : N.89.
in a given ratio : J. 10.
* imaginary : 408 : geo.cn,A.15,45 : C.24,
86—88: JP.ll: L.50: N.46,47,
682: approx. N.45,53 : Q.9.
between given limits : A. 21 : L.44.
Newton's rule : Me.80 : N.67 : Pr.
13.
Newton-Fourier rule : Q.16.
* Newton-Sylvester rule : 530 : C.
994 : LM.l : Me.66 : Pr.l4 : Q.9 :
TI.24.
* incommensurable : 506 (see " Sturm's
th.")
infinite: N.442,45.
in infinite series : A.69.
* integral, by Newton's method of
divisors : 459.
least : M.9 : TE.28.
* limits of : 448 : C.58,60,93 : geo CP.
12 : N.43,45,59..,72,802,8L
6 B
922
INDEX.
Roots of an equation — {continued) :
uumbei" between given limits : A.l :
G.9: J.52: L.40.
the eq. containing only odd powers
of X : N.63.
* Eolle's th. : 454 : AJ.4 : N.44 : ext
L.64.
RoUe, Fourier, and Descartes : A.l.
number satisfying a given condition :
C.40.
product of differences : Me.80: P. 61.
* of a quadratic: 50 — 3.
of a qnartic and of a Hessian, rela-
tion : E.34.
ofquintics: C.59,60 : LM.14 : TI.18.
rationalization of : P.l 798,14.
real: A.36,58: C.61 : J.50 : JP.IO :
N.50 : P.57.
of a cubic: K72: Z.2.
Fourier's th. : N.44.
developed in a series : L.78 : N.56.
limits of :_J.1:_N.53,79.
series which give the number of :
Z.2.
to find four : An.55.
* rule of signs: 416—23: A.34: 0.92,
98^,99; : N.43,46,47,67,69,79.
separation of: A.28,70 : J.20 : N.680,
72,74,75,802.
by differences : N.54.
for biquadratics : A.47.
for numerical : 0.89,92 : G.6.
simultaneous eqs. : 0.5.
* squares of differences : 541 : CM.l :
N.42 44 : Q.4.
* sums of powers : 534: E.38 : th J.9 :
N.53,75: gzMe.85: Q.19.
in sums of rational functions of the
coefficients : Ac. 6.
surd forms of : CM.3.
* symmetrical functions of: 534: A. 16:
AJ.l : An.54..,55,60 : 0.44,45 : G.
5,11 : J.19,54,81 : Me.81 : N.48,
50,55,66,84: P.57: Pr.8 : Q.4:
TI.25.
do. of the common roots of two eqs. :
N.60: Z.15.
do. of differences of roots : C.98.
which are the binary products of the
roots of two eqs. : An. 79.
with a variable parameter : 0.12.
which satisfies a linear d.e of 2nd
order: 0.94.
*Roots of numbers: 108: A.17,26,35:
0.58: E.36: Me.75: N.61,70.
* square root: 35: 0.93: N.452,46,61,
70.
* as a continued fraction : 195 : A. 6,
12,49: 0M.2o: L.47: Mc.85:
Mem.lO: TE.5: Z.17.
Roots of numbers — (continued) :
to 25 decimal places : Me.77.
* cube root : Horner's method : 37 :
A.67.
of 2 to 28 decimal places : Me.76,78.
and sq. root, limit of error : N.48.
fourth root : A. 30.
11th root as a fraction : A. 46.
*Roots of unity : 475—81 : 0.38 : J.40 :
L.38,54,59: Me.76 : N.43o : TE.
21 : Z.22.
cubic roots, alg. and geo. deductions :
0.84.
function theory : Z.22.
23 roots, composition of number 47 :
J. 56, 56.
*Roulettes: 5229: Ac.63 : 0.70: 0P.7 :
J.65 : L.80,81 : N.56 : TE.16 : Z.28.
areas of, and Steiner's transf. : E.35.
generated by a circle rolling on a
circle: JP.21.
by focus of ellipse rolling on a right
'line: A.48.
by centre of curvature of rolling
curve : L.69.
Ruled surfaces: An.68 : 0D.8 : G.3 :
J.8: L.78: N.6L
areas of parallel sections : Z.20.
and guiding curve : A.18.
of minimum area : L.42.
octic with 4 double conies : O.60.
P. D.eq. of: Me.77.
quadric : Me. 68.
quartic : A.65 ; with 2 double lines,
A.65.
quintic : J.67.
represented on a plane : 0.803.
of species, p z=0 -. M.5.
symm. tetrahedral : 0.62.
torsal line: M.17.
transformation of : 0.88.
*Scales of notation : 342 : J.l : L.48,5ry,
10ry,20ry: Phil. See. of Glasgow,
vol. 8.
Screws: TI.2o.
Scrolls : A.53 : CD.7 : OP.ll : J.20,67 :
M.8 : cubic, M.l : P.63,64,69 :
Pr.12,13,16 : Z.cn28.
condensation of: LM.13.
cubic on a quadric surface : Me.85.
flexure and equilib. of : LM.12.
ruled : A. 68 : :; = mxy~, A. 55.
tangent curves of : M.12.
♦Sections of the cone : 1150.
♦Sectors and segments of conies and
conicoids : 6019 — 6162 : G.l :
thsZ.l.
Secular eq. lias real roots : J.88.
♦Self-conjugate triangle : 4765, 4967 :
G.8 : N.67 : Q.5,10.
INDEX.
923
Self -conjugate triangle — {continued) :
* of 2 conies : Me.77 ; of 3 conies, 5025.
and tetrahedron in couics and quad-
rics : Z.6.
Self - enveloping curves and surfaces :
Z.22.
Self -reciprocal surface : Mo. 78.
*Self -reciprocal triangle : 1020.
Seminvariants : AJ.7 : E.tli42,6 : Q.
19—21.
critical and Spencian functions: Q.4,6.
and symm. functions : AJ.6.
Septic equations : Mo. 58.
*Series : (see also "Summation" and
" Expansions ") : 125 — 9, 149—
59, 248—95, 756—817, 1460, 1471
—2, 1500—73, 2708—9, 2743—60,
2852-64, 2880, 2^11—68, 3781,
3820 : A.4,52,9,14,1S,23,60 : No.39,
472: C.29,pr92: CP.9 : G.IO : J.
3,17 ,34,38,th53 : L.tli56,8l2 : Me.
64: N.59,th62,70 : Q.3 : Z.15,16,
23.
Useful summations :
a,'2 ir3
■■■=-'^Th-
156.
«-f.f-
..=log(l + aO
155.
-f+f+
■•■=*'-S
: 157.
-f+f-
.. =tan-i«: 791.
<+s+-
.. =e^-l: 150
«-s+s-
..=1-1:
«+i;-^
e^— e-'
••~ 2 ■
"'-3T + 5-!-
.. =sin:e: 764
a^ ari ^
2! U!^6!^
2 ■
a;2 x^ ,x.^
2! 4!"''6! ■
. =1— cosa;: 765.
1^+2''-!- ... +nP: 2939: A.65:
Me.78.
p = 1,2,3, or
4 : 276 : A.64 :
B.34.
p_2P+3''-..
. &1P_3P + 5P_
J.7.
2 »«*" : A.27.
2(a+»)V: N.56.
2(a„ + &,X)i«*-
« : Z.15.
l_«-3 + («-
-4) (n-5)
2.3
J.20.
Series — (continued) :
a±nb+C{n,2)c±&c.: J.31.
^ n'- —n {n—l)'+ C {n, 2) {71— 2Y —...:
285; r = 7i,CM.l.
l+?.:+f#^W...: J.37;with
6(6 + 1)
:»3 = 1, J.2.
N.59.
{a+nd)''
deductions from
«■"' '«" \ -. — i«"'
^7^''^.)'' ' ~'^{a + l)'rrnl''
LM.9.
l±l + l±...andl±3l + |-±...:
2940—4 : A.41 : LM.8 : Q.7 : with
n = 1,2... 8; 2945 ; E.:',2,39 ; G.IO ;
N.79 ; Z.3. Note that by (2391),
, _ l^oc
32^52 — - Jol+a;2
: Mem. 11: with a?. = 1, J.5.
dx.
: A.61.
•, A.41
v7 -,'13
77+ Vol — ••• = ^•^'*' ^^^^
J.5.
n ! a" - '
2(-l
A.50.
(a + 2H)(l-a-)
- : Q.6.
a„a;t"' : 2709.
(-1)"
« + l!
: A.26.
: L.60.
1.3... 2)1—1
n ! 2" (2»-l-l)
^+ (2_l)(5v_i)
(rr-i)(r-^-i)(r/^-i)(y---i) ^2
(2-1) (r/-l) (2^-1) (r/^1-1) '
+ ...: J.32,70.
A.35.
an'"-\-ain"'~^-\- ... +a„
924
INDEX.
Series — (confinned) -.
2 K„ — — - , /3 a pos. integer < a,
Kn = the general term of some
recurring series : C.86.
2 -^ : A.34.
n = 1^
2_',- E fniiy. C.oO.
* 2 (eo'j2,i + l(^) /n^ J.54; with I- = 2,
2960-1, J.8.
2 sin:^ (2/i + l) (/> / (2;i + l)^ : E.39.
^(cos«^)'"'°"/«^ '-^-d
2
* 2
(«"eo^^<^)/«= 2922-3.
7^2 2!? , L.40.
* 2?^^illii^:2962: M.5.
a3+%2
2 ^„ cos" ^ . "^ 7i^ : Z.l.
sm
(a+«/3): 800: Q.3 ;
COS
* 2c"^j;^(a+n^),783;
dfe „ c" sin . , „. f,oo
^,7!cos(«+'^^)'^88.
2/(«)^j^%i^: J.42: L.52.
2 if (7>i,«)f«"^"^«0: J.41.
2 |-^ tan ^^ : A.44.
2/(7ia3) : L.61. 2 An cj) {n) x", J.25,28.
from J X (1— a-)" clx : A.47.
10.
from [ ' cos2 x log I±MhJI' ^.^ : G.
Jo 1 — sin;<!
of Abel : C.93 : N.85.
ap])lication : thA.48 ; to arith, G.7.
in a.p : see "Arithmetical progres-
sion."
analogous series : N.69.
a.p and g.p combined : A. 9.
with Bernoulli's nos. : An. 53 ; and bi-
nomial coefficients, A.23.
Series — (conlinned) :
binomial (see " Binomial theorem") :
analogous series : E.35 : J. 32 : N.
82 ; with inverse binomial co-
efficients, Me.80.
coefficients independently deter-
mined: A. 18.
whose coefficients are the sums of
divisors of the exponents, sq. of
this series : Me. 85.
combination : A.26.
*= convergent: 239: A.2,6,8,14,26,41,67,
69: No.44: C.10,llo,28o,40,43o: J.
2,3o,ll,13,16,22,42,4.5,76 : L.39—
42: M.10,20— 22: Me.64: N.45,
46,67,69,70,: P.87 : Z.10,11.
and of d.i with a periodic factor :
L.53.
power-series : A. 25.
representing integrals of d.e : C.40o.
representing functions : M.5,22.
in Kepler's problems : Ac. 1799.
multiplication of : M.24.
and products, condition : M.22.
whose terras are continuous func
tions of the same variable : C.36
with constant ultimate differences
Pr.5,.
converted into continued fractions
J.32,33 : Mem.9 ; into product
of an infinite no. of factors, J.12
L.57 : N.47.
in cosines of multiple angles : C.44
Mem. 15.
and definite integrals : L.82 : Man.
46.
derived : A.22 ; from tan" ' 6, A.16.
developed in elliptic integrals of 1st
and 2nd kind : An. 69.
= difference : 264 : A.23,24.
differential transf. and reversal of:
Pr.7.
and differentiations : A. 10 : J.36.
Dirichlet's f. for 2 / - ) ^- : C.21 : L.
46. \ 2' I "
discontinuous: CP.6 : L.54 : Me.78,
82 : N.85.
divergent: A.64 : No.68 : C.17.20:
CP.8,10 : J.l 1,13,41 : M.IO : Z.IO.
division of : AJ.5.
double: C.63.
doubly infinite : CD.6 : M.24.
ext. of by any parameter : A. 48.
factorial : 268 : Mora.20 : N.67 : TE.
20.
of fractions : L.40.
Fourier's: A.39 : C.91,92,96 : CM.2 :
Z.27.
of Gauss and Heine : C.73 : G.9.
INDEX.
925
Series — {continued) :
*" Gregory's : 791.
harmonic periodic : J.23,25.
of Hermite, a th. : E.29.
from infinite products : Me. 783.
integration of infinite : A.3.
irrationality of some : J. 37.
involving two angles : L.74.
Klein's higher : An. 71.
of Lagrange: C.23,343,522 : L..57 : N.
76,gz86 : Q.2 ; remainder, C.53.
an analogous series : C.99.
of Lambert ; S
A.IO: An.68
J.9 : Z.6.
Laplace's (d.c) : C.68.
of Laplace's functions: SY,,, C.882 ;
2» Y„, C.44.
in Legendre's function X„ : An. 76 :
0.44.
of Leibnitz : J.89.
limits of : A. 20 : Me. 76 ; remainders,
C.34 ; by the method of means,
J.13.
from log(l+ft'), (l+fp)" and e^ by in-
termitting terms in the expan-
sions : A.21.
modular : C.19.
multiple : 0.19.1 ; " regulateur " of,
0.44.
neutral: CP.ll.
obtained by inversion from Taylor's
series : Mem. 11.
of odd numbers : A.64.
a paradox : Me. 72.
periodic, critical values of : OP. 8.
of polynomials : 0.96.
of posterns : G.6.
* of powers (see also " Numbers ") :
277: 0.87: G.2: cubes, L.64 and
65: M.23: Mo.78: Q.8: Z.l.
approximate fractions : J.90.
of a binomial : Mem. 13.
in a convergent cycle, constants
in : M.25.
like numbers : N.71,77.
or multiples of 3 : A.27i:.
of terms in ar.p : L.46.
products of contiguous terms of :
Mem. 18.
of reciprocals : Q.8.
* recurring: 251: doubly, An. 57 : J. 33,
38 : Me.66 : Mem.24,26 : N.84 : of
circles and spheres, N.62 : f,Z.14.
represented by rational fractional
functions ; J.30.
* reversion of: 661: J.52,54: LM.2 :
TL7.
of Schwab : N.59.
Series — (contimied) :
self-repeating : CP.9.
of spherical functions : An. 75.
of Sterling, for transformation : J. 59.
with terms alternately positive and
negative : 0.64.
whose terms are the coefficients of
the same power of a single vari-
able in a multiple integral : 0.20.
in theory of numbers : O.89o.
transformation of : 0.69 : J. 7,9 : into
a continued fraction, Mem.20,
Z.7; of ^lf{x,t)dt and others,
0.13.
in a triangle problem : A.64.
trigonometrical (see above) : A.63 :
Ac.2: 0.95,97: M.4— 6,16o,17,22,
24 : J. 71, 722 : representing an
arbitrary function between given
limits, J.4 ; conversion in mul-
tiples of arc, L.61 ; symbolic
transf. of, Q.3.
triple: G.9.
two infinite, multiplication rule : J. 79.
*Seven-point circle : 4754c.
Seven planes problem : N.56.
Sextactic points of plane curves : Pr.
13,14.
Sextic curves : ax'^ + byi-\-c^ = 0, Q.16 ;
mech.cn, LM.2.
bicursal : LM.7.
and ellipse, pr : J.33.
Sextic developable : Q.7,9.
Sextic equation : 0.64: M.20.
irreducible : J. 37.
solution when the roots are connected
by(a-/3)(6-y)(c-a) +
(a-b) {(3-g) iy-a) = : J.41.
Sextic torse : An. 69.2.
Sextinvariant to a quartic and quart-
invariant to a sextic : AJ.l.
*Shortest distance : between two
lines : 5634 : A.46 : G.6 : N.49,66.
between two points on a sphere : A.
14 : N.14,67,68.
from the centre of a surface : A.63.
of a point from a line or plane : N.44.
Shortest line on a surface : A.23,37,64;
in spheroidal trigonometry, A.40.
Signs: OP.2,11: J.12 : Me.73 ; {=),
Me.76; (±), OD.6,7: Me.85 : N.
48,49.
Similarity: of curves and solids:
A. 13.
Similarly varying figures : LM.16.
Simson line of a triangle : E.29.
*Simpson's f. in ai-eas : 2992 : 0.78.
Sines of higher orders : 0.914,92.^ ; ap.
to d.e, 0.903,91.
926
INDEX.
Sines, natural, limit of error : N.433.
Sin^<^-^: geo.N.75.
4
Qm-' {.v + iy) : Q.15.
Sine and cosine : extension of mean-
ing: A.31: C.863.
* in factors : 807 : A.27 : J.27 : L.54.
of infinity: CP.8 : Me.71 : Q.ll.
of multiple arcs (see also " Expan-
sions ") : CM.4 : TI.7.
* of particular angles: 690; 3'',6^...to
90°, N.53.
sums of powers : An.l.
* tables, formation of : 688 : A.66 : N.
422.
values near and 90° : G.9.
* of{a±h): 627; A.6,21,36.
Six-point circle of a triangle: Me. 82,
83,.
Six points on a plane or sphere : LM.2.
Skew surfaces : see " Scrolls."
Sliding rule: LM.6.
*Small quantities of second order : 1410.
Smith's Prize questions, solutions : Me.
71,723,-4, 77.
Solid angle: A. 42; section of, No.l9.
Solid harmonics : Me. 80.
Solid of revolution : A. 60,67.
between two ellipsoids : A.2.
* cubature and quadrature of: 5877 —
80 : A.68 : N.42.
Space homology : G.20.
Space theories: An.70: LM.14 : P.70 :
Z.17,18.
absolutely real space : G.6.
continuous raanifoldness of two di-
mensions : LM.8.
space of constant curvature : An. 69,
73: J.86: M.12.
Pliicker's " New geometry of" : G.8:
L.66: P.ll: Z.11,12.
Grassmann's " Ausdehnungslehre " :
AJ.l: CP.13: M.7,12: Z.24 ; ap.
to mechanics, M.12.
non Euclidean or w-dimensional : A.
6,29,68: ths64: A J.4,5 : An.71 :
C.75: G.6.10,12,23 : M.4,5.,-7 :
Me.th.s68,72 : Pr.37.
3-dim., J.83; 4-dim., J.83, M.24; 6-
dim., G.12.
angles (4-dimen.) : A. 69.
areas and volumes : A.69 : CD. 7.
bibliography of: A J. 1,2.
circle: G.12,16,18.
conies : AJ.5.
curves : C.79 : M.18.
Fcuerbach's points : G.16.
hyperboloid : Z.13.
Space theories — {continued) :
imaginary quantities : Z.23.
loci (anal.) : C.24.
planes (4-dimen.) : A.68.
plane triangle : A.70.
point grou])S : thsAc.7.
polar s and alg. forms : J.84.
potential function : An. 82,83.
proiection; M.19 ; 4-dim. into 3-dim.,
AJ.2.
quadric, super lines of (5-dim.) : Q.12.
quaternions : CP.13.
regular figures : AJ.3.
reversion of a closed surface: AJ.l.
representation by correlative figures :
C.8L,.
simplicissimum of nth. order : E.44.
screws, theory in elliptic space : LM.
15,16.
21 coordinates of : LM.IO.
Sphere : geo,C.92 : ths and prs M.4 :
q.cMe.62.
and circle : geo,A.57.
en. from 4 conditions : JP.9.
cutting 4 spheres at given angles :
An..51 : N.83.
cutting a sphere orthogonally and
touching a quadric, locus of cen-
tre : TI.26.
diameters, no. of all imaginable: A.24.
* equation of: 5582.
5 points of: J.23: N.84.
illumination of: Z.27.
kinematics on a : LM.r2.
sector of (eccentric) : A.65.
small circle of : Me.85.
touching an equal sphere : E.31,32 ;
as many as possible, A.56.
4 spheres, pr. : L.46.
4 touching a 5tli : At.l9.
8 touching 4 planes : E.19 : N.50.
16 touching 4 spheres : J.37 : JP.IO :
Me.cn82 : Mem.lO : N.44,47,65,
66,84: Z.14,.
* volume, &c. of segment and zone :
6050: A.3,32,39: An.57 : P.l.
*Spherical : areas : 902.
catenaries : J.33.
class cubics with double foci and
cyclic arcs : Q.15.
conies : thQ.3 ; and quadrangle, Q.13 ;
homofocal, L.60.
coordinates : CD.l : CM.l,ap2 ; ho-
mogeneous, G.6.
* curvature: 5728,'40,'47 : thE.34.
curves : A. 35,36 : Mem.lO.
of 3rd class with 3 single foci : Q.
17.
of 4th cla.ss with quadruple foci :
Q.18.
INDEX.
927
*Spherical : curves — (continued) :
of 4th order : J.43.
with eUiptic function coordinates :
J.93.
equidistant : An. 71 : J. 25.
and polars : No.63.
rectification of : An. 54.
elHpse: t.cQ.8.
quadrature, &c. : L.45 : N. 484, 54.
epicycloid : G.12.
excess: Mel.2 : cn,N.46 : f,Z.6.
of a quadrilateral : Me. 75.
figures, division of : J.22.
geometry -. G.4 : J.6,8,132,thsl5 and
22 : M.15 : N.48,585,59 : Q.4 : TI.8.
harmonics or " Laplace's functions " :
An.68o: C.86,99: CD.l : CM.2 :
J.26,56,60,geo 68,70,80.82,90 : L.
45,48 : LM'.9 : Me.77,78o,8o: P.57 :
Pr.8,18: Q.7.: Z.24.
analogues of : J. 66 : LM.ll.
and connected d.i : Q.192.
as determinants : Me. 77.
and homogeneous functions : CM.2.
and potential of ellipse and ellip-
soid : P. 79.
and ultra-spherical functions : Z.12.
P" (cos y), n = QO : J.90.
f Pi^i"'(^/x, &c. : Q.17.
•'-1
hl^ by continued fractions : JP.28 ;
£ Qn Qn- : P.70.
loci in spherical coordinates : TE.122.
oblong : An.52 ; area, J.42.
polygonometry : J.2.
polygons in- and circum-scribed to
small circles of the sphere, by
elliptic transcendents : J. 5.
quadrilateral : A.4,40 : th,E.28 : N.45.
surface of : A.342,352.
quartics : foci, Q.21 ; 4-cyclic and
3-focal, LM.12.
representation of surfaces : C. 68,75,
94^,96 : M.13.
surface represented on a plane : Me.73.
triangle: A.9,ll,20,ths50,65: E.f30 :
J.10,pr28: JP.2.
ambiguous case : Me. 77,852.
angles of, calculated from sides :
A.51.
by small circles, area : N.53.
* and circle : 898 : A.29,33.
cos(^+B+C'), f. : Me.72.
and differentials of sides and
angles : A. 10.
and ex-circles : 898 : E.30.
graphic solution : AJ.6.
Spherical : triangle — {continued) :
and plane triangle : A.l ; of the
chords, A.33 : An.54 : Z.l.
*" right angled : 881 : A.51 ; solution
by a pentagon, A. 11.
of very small sides : N.62.
two, relations of sides and angles :
A.2.
*= trigonometry: 876: A.ths2,ll,13,28,
37: J.prs6,132: LM.ll: N.42 :
Z.16.
d.e of circles : Q.20.
* cot a sin b : Me.64 ; mnemonic, 896 :
CM.3.
derived from plane : A.26,27.
* formulae : 882 : A.5,16,24,26 : N.45,
46,53 : graphically, A. 25: ap. in
elliptic functions, A.40.
geodetic reduction of a spherical
angle : A.5I2.
* Cagnoli's th. : 904.
* Gauss's eqs. : 897 : A.13,17 : J.7,12
LM.3,13.
Legendre's th. : C.96 : J.44 : L.41 :
M.l : N.56 : Z.20.
* Llhuillier's th. : 905 : A.20.
* Napier's eqs. : 896 : A.3,17 : CM.3 :
LM.3,13.
* Napier's rules : 881.
supplement to, and geodesy : A.36.
*Sphero-conics : 5655a: tg.c,Q.8,9o: Me.3:
Z:6,23.
homofocal: C.50.
mechanical en : LM.6.
Sphero-conjugate tangents : An. 55.
Sphero-cyclides : LM.16.
Spheroidal trigonometry: J.43: M.22.
Spheroidic transformation f. of Bessel :
A.53.
*Spheroids : 5604, 6152 ; cubature, 6158,
A.2.
Spieker's point : A. 58.
Spiral : A.28 : L.692 : N.79 : Z.14.
* of Archimedes : 5296 : A.65,66.
conical : N.45.
* equiangular : 5288.
* hyperbolic : 5302.
* involute : of circle, 5306 ; of 4th order,
C.660.
Squares : J.22.
whose diagonals are chords of given
circles : A.64.
maximum with given sides : J.25.
maximum in a triangle : J. 15.
whose sides and diagonals are
rational : J.37.
whose sides pass through 4 points :
A.43.
64 transformed into 65, geo. : Me. 77.
sum of three : G.16 ; of four, L.57.
928
INDEX.
Standards of length : Pr.8,21.
Statistics : Z.26.
Steiner's ths. and prs. : A. 53 : J.13, 14,16,
18,23,25,36,71.73 : N.48,562,593,62 :
Hexagon, A. 6 : Ray - systems,
CD.6.
Stereographic projection : A.39 : JP.
16: L.46.
Stereograms of surfaces : LM.2.
Stereometry: JP.I4 : thsA.10,31,43,57 ;
J. 1,5.
multiplication : J.49.
quadric and cubic eqs. and surfaces :
J.49o.
Sternpolygon and sterupolyhedron : A.
13.
Stigmatics : see " Clinant."
Stewart's geo.ths. : CM.2 : L.48 : TE.
2,15.
Striction lines of conicoids : Z.28.
Stropboids : AJ.7 : N.752.
*Sturm's iuuctions: 506: A.62 : C.36,
62,68., : G.1,20 : J.48 : L.46,48,
67: Mo.58,78: N.43,46,52,54,66,
67,81: Q.3.
and addition tb. for elliptic functions
of 1. St kind: Z.17.
ext. to simultaneous eqs. : C.35.
and H. C. F. : Pr.6.
and a quartic equation : A. 34.
and tbeir reciprocal relations : Mo.
73.
remainders : J.43,48.
tables: P.57 : Pr.8.
ap. to transcendental eqs. : J.33.
ap. to transf. of binomial eqs. : L.42o.
Subdeterminants of a symra. system :
J.93: M.82.
Subfactorial n: Me. 78.
Subinvariants= seminvariants to bin-
ary quantics of unlimited order :
a:j.5.
♦Subnormal: 1160.
♦Subsidiary angles : 726, 749.
Substitutions : A.62 : C.ths66 and 67,
74,76,79 : G.9,10.,,11,14,19 : L.65,
72: M.13.
ap. to functions of six or fewer vari-
ables : C.21.
ap. to linear d.e : C.78.
by approximation, of the ratio of the
variables of a binary quantic to
another function of the same
degree : C.80.
canonical forms of: L.72.
and conjugate substitutions : C.2l5,22.
of the form (r) = e {r"--+ar~) :
M.2.
linear: C.98: J.84 : M.19,20:.
Substitutions : linear — {continued) :
of a determinant : An.84.
and integral : M.24.
powers and roots of : C.94.
reduction of : C.90 : JP.29.
for reduction of elliptic functions
of 1st kind : An.58.
successive : A.38.
which transform quadric functions
into others which contain only the
squares of the variables : J.57.
of n letters : geo. for n := 3,4,5,6, and
mystic hexagram : An.83.
no. of in a given no. of cycles : A. 68.
permntable amongst themselves :
C.2I2.
of equidistant numbers in an integral
function of a variable : N.51.
of systems of equations : N.81.
of six letters : C.63.
a th. of Sylvester : A.J.I.
which admit of a real inversion : J. 73.
which do not alter the value of the
function : 0.21;.
which lower the degree of an eq. in
two variables and their use in
Abclian integrals : C.15.
Sum and difference calculus : A.24i:.
Sum of squares of lines drawn from a
point to cut a curve in a given
angle : J. 11.
♦Summation of series (see also
"Series" and "Expansions"):
3781 : A.IO— 13,26.,.30,55,62 : No.
84: C.87,88: J.IO',31.33: LM.4,
7o: Mem.20,30:P.1782,1784— 7,—
9i,— 98, 1802,— 6,— 7,— 11,— 19 :
Pr.l4.
* approximate : 3820 : J. 5 : 1.24.
of arcs : A. 63.
Bernoulli's method : J. 31.
Cauchy's th. : M.4.
G {n,r) products of a, a + h, ... a +
{n-l)h: Q.18.
by definite integrals : A.4,6,38 : J.17,
38,42,46,74: Man.46 : Mem.ll.
of derivatives and integrals : C.44:.
* by differences : 264 : Mem.30.
by differential formulae : L.31 : Mem.
15.
by ^ (.(•) : 2757.
formulaj : A.47 : An. 55 : J. 30.
Maclaurin's, C.86., : f{A,D)DF{n),
Q.8: Lagrange's, J.34 : P.60 :
Vandermondc's, L.41.
of terras of a high order : C.32.
Kummer's method : C.64.
Lejcunc DirichlcL's : CD.9.
periodic : J. 15.
selected terms : Me.75.
INDEX.
929
*Summation of series — (continued)
of sines and cosines : J.4.
* theorem re
£x/(a.,Z;)&
2708.
of transcendents in alg. diiferentials :
J.19.
trigonometrical and infinite : Me. 64.
Superposition: TE.21,2.3.
* of small quantities : 1515.
*Supplementary : angles : 620.
* chords: 1201.
cones : thN.48.
* curves: 4917—20: Man.54.
Supplement integrals : J. 98.
*Surds: 108: N.47.
quadratic (see " Boots of numbers.")
V {a~+b^), V (a~—h-) approximately :
J.13.
* V(a±Vh): 121: A.3,13 : J.17,20 :
Mem.lO: N.46,48.
* l/(a± Vh) : 122 ; y (a± ^^h), 124.
^Surface of centres : 5774 : A.68., : An.
67: 0.70,71-.: E.30: L.41,58:
LM.4: M.5,16.
curvatures of the two ; relation : C.
74,79.
the two focal conies of a system of
homofocal quadrics : CSs.^.
of a quadric : Q.2.
bitangents of : Q.13.
model of : Mo.62.
principal axes of : N.48.
Surface curves : A.39,60 : An.54 : C.21,
80,: L.66: G.19 : J.2: N.54,84.
an an alg. surface : An. 63.
on cubic surfaces : M.21.
curvature of : lSr.65.
on a developable : An. 57 ; the oscu-
lating plane making a constant
angle with it, L.47.
on an ellipsoid : An.51 : CD.3 : geoL.50.
groups of rational : M.S.
of intersection : M.2 : N.68 ; of 2
quadrics, A. 16.
multiple : G.l^j ; singularities of, M.3.
on a one-fold hyperbola : CD. 4.
on quadrics : N.70.
rectification of : A.36.
relation to their tangents : C.82.
on surfaces of revolution : Z. 18,28.
and osculating sphere : 0.73.
*Surface or surfaces : 5770 : A.14,f32,41,
59,60,62 : An.51 — 3,55,60.,61,65,
71: At.57: C.17,33,37,49',ths58,
61,64,67,69,80,86,99.,: G-.3,21 : J.
9,thsl3,58,63,64,85,98 : JP.19,24,
25,33: L.44,47,51o,60 : M.2,4,7,9.,,
cnl9: Mo.82,833,84 : N.53,65o,68,
72: TI.14: Z.74,8,20.
*Surface or surfaces — [continued) :
^' + ?f+^_!^l: A.35;
a h c
a = h = c = \,K21.
areas of: G.22 : K52.
argument of points on : LM.16.
complex : M.5 ; of 4th order and
class, M.2.
whose coordinates are Abelian func-
tions of two parameters : 0.92 :
M.19.
of corresponding points : M.4.
* cut orthogonally by spheres : 3393 :
0.36.
deficiency of : M.3.
* definitions: 5770.
determined from two surfaces of cen-
tres : A.68.
Dirichlet's problem : An. 71.
Dupin's theorem : 0.74 : 0M.4 : Q.12.
doubly circumscribing an «-tic sur-
face : J.54.
of elliptic cone : An.51.
of equal slope : 0.98 : lSr.65.
* equation of: gn5780 : A.3 ; for points
near origin, 5803.
of even order : A. 70.
families of : O.70 : Me.72.
flexure of : J.I82.
Gaussian theory of: LM.12 : N.52.
generation of : 0.94,97 : G.9 : J.49., :
L.56,83 : M.18.
implexes of : 0.79,82.
of minimum area : 0.57 : J.8,13 •
L.59: Q.14.
of nth order : cnM.23 ; 2nd, 3rd and
4th, Mel.6.
of normals : ]Sr.59.
whose normals all touch a sphere or
conical surface : JP.4 ; do. for
surface of revolution, JP.5.
number under 9 conditions : C.62.
octic of zero kind : G.12.
order determined : Me.83.
one-sided : A.57.
parabolic points of: 0D.2.
and p. d. e : C.13 : CD.2 : Z.7.
and plane curves : J.54,72 : M.7.
and point moving on it : L.76,77.
and point at 00 on it : J.65.
relation of in Eiemann's sense: M.7.
representation of: J.83.
on a plane : An. 68,71, 76.
one upon another : An. 77.
of revolution : 0.86 : G.4.
* areas and volumes : 5877, — 9 : A.48.
of a conic about any axis in space :
JP.23 : L.63.
of constant mean curvature : L.41o.
6 c
930
INDEX.
*Surface or suviaccs— (continued) :
meridian of : a lemniscate, G.21 ;
generation of, C.85.
meridian and contour curves of:
Z.21.
oblique : Q.O.
passing through a given line, tan-
gent plane of : N.8-i.
in perspective : JP.20.
of reg. polygon about a side, vol. :
A.67.
quadric : A.55 : L.60.
shortest line on : A.38.
superposable : C.80: IST.Sl.
Riemann's symmetrical; and perio-
dicity modulus of the related
Abelian integral of the 1st kind :
Z.28.
ruled, with generators part of a linear
complex : 0.84.
ruled octic with 5 quartic curves :
C.61.
screw, parallel projection of : cnZ.18.
section of, homogeneous eq. : LM.15.
self-reciprocal : LM.2 ; quadric, E.36.
sextie of first species : M.21.
singularities of: A.25: An.79 : CD. 7:
CM.2 : J.72 : M.9 : N.64 : Q.9.
cubics : M.4; and quadrics, A. 17.
16 singular points : 0.929.
solutions by infinites of 3rd order :
geoO.802.
Steiner's: LM.5,14 : M.S.
touching a plane along a curve:
0D.3: 0M.2.
touchinga fixed surface always : G.20.
transformation of: M.19.
and transversal, th. : N.49.
trapezoidal problem : Z.14.
two series of: prsAn.73.
web-system : J. 82.
which cuts the curve of intersection
of two alg. surfaces in the point
of contact of the stationary oscu-
lating planes : L.63.
Surveying : N.SOj : geo. th.A.37.
Symbolic geometry : (Hamilton), CD.
1-4.
Symbolical language : E.282.
Symmedian line : E.42 : N.83— 5 : Q.2O2.
*Symmedian point : 4754c.
Symmetrical : conies of triangles :
A.59.
connections by generating functions :
J.53.
* expressions : 219.
figures : J.44.
functions: 0.76: J.69,93,98 : LM.13:
Q.20 : Z.4.
Brioschi's th. : 0.98.
Symmetrical — (continued) :
of the common sol. of several eqs. :
An. 58.
multiplication of : Me.85.
of any number of variables : 0.82.
simple and complete : M. 18,20.
tables of : AJ.5 ; of 12-ic, AJ.5.
points: of 1st order: A.60 : cnA.64.
of tetrahedrons : A.60.
of a triangle : A. 583.
products : P.61 : Pr.ll.
with prime roots of unity : Me.85.
tetrahedral surfaces : Z.ll.
Symmetry ; plane and in space : N.47.
Synthesis : 0.18.
*Synthetic division : 28 ; evolution,
Me.68.
*Syntractrix : 5282.
Tables : mathematical ; Sect. I., con-
tents, p. xi.
of Bernoulli's nos., logarithms, &c.,
calculation of: J.2.
for empirical formulae : AJ.5.
in theory of numbers : Q.l.
of e^, e"-^, logioe-*', logioe'-"': CP.13.
Tac-loci: Me.83.
Tamisage: LM.14.
*Tangencies (circles, points, and lines) :
937 : Pr.9 : Q.2,8.
Tangential eq. with the intercept and
angle of inclination for coor-
dinates of the line : P.77.
*Tangent cone at a singular point : 5783.
*Tangents: 1160: cnA.4.33 : J.562,73:
K42: Z.23.
and contact-point in loci and en-
velopes : 0.85 : cnISr.57.
conjugate (and Dupin's th.) : An.60.
construction of: M.22 : N.80.
double: Q.3.
* eq. of ; to find it : 4120,-32; Me.66.
faisceaux of : N.50.
cut by lines at a constant angle : A.43.
locus of intersections of three : LM.
15.
at a multiple point, en. : JP.13.
* and normals : 5100; relations, A.51.
parallel: N.45.
* segments of : 4307 ; equality of, O.8I4.
* of a surface : 5781 ; at singular points :
5783: M.11,15.
*Tangent planes : 5770,-82 ; p.c5790 :
0M.4: N.45.
* and surface ; intersection of : 5786
—9 : L.58.
to equidistant surfaces : cnZ.28.
triple: 0.77.
tan 120° : P.8,18.
tan X as a continued fraction: Z.16;
do. tan^i.c, Me. 74.
INDEX.
931
tan nx : i. in N.53.
tan-i (ic+iy): Z.14.
*tanh S : 2213.
tantochrone: L.44 : An.63.
*Taylor's theorem : 1500,— 20,— 23 : A.
8,13: AJ.4, extl: C.58 : CM.l :
J.ll: L.37,38.45,o8,gz64 : M.2U
23: Me.724,73,75 : Mem.20: N.
52,63,70,743,79: TA.7,8 : Z.gz 2
and 25.
analogues of: J.6.
convergence of : C. 60,74— 6 : J.28 :
L.73.
Cox's proof: CD.6.
deductions : G.12.
for an imaginary variable : Me. 79.
kinematic meaning of : J. 36.
reduced forms of : C.84.
* remainder : 1603 : An.59 : C.13 : J.17 :
K60,63: Z.4.
a transformation of : C.78.
Terminology : LM.14.
Ternary bilinear forms : G.21.
Ternary cubics : AJ.2,3 : C.56,90 : CD.l :
J.39,55 : JP.31,32: L.68 : M.1,4,9.
in factors : An. 76 : Q.7.
as four cubes : LM.IO.
parameter of the canonical transf. :
LM.12.
reduction to canonical form: C.81.
transformation of : J. 63.
Ternary forms : At.68 : G. 1,9,18.
order of discriminant : LM.3.
with vanishing functional deter-
minants : M.18.
Ternary quadric forms : At. 65'. : C.92,
94,96 : G.5,8 : J.20,40,70,77 ; transf.
of, 71,78 and 79 : L.59,77 : P.67.
and corresponding hyperfuchsiau
functions : Ac.5.
indefinite: J.47 ; with2 conjug. inde-
terminates, C.68.
representation by a scpiare : An. 75.
simultaneous system : J.80.
table of reduced positive : J.41.
Ternary quartic forms : C.563 : An. 82 :
M.17i,20.
Terquem's tb. : Q.4.
Tesselation pr. : LM.2.
Tesseral harmonic analogues : CP.13.
Tetradrometry, differential f . : A.34 :
Z.5.
Tetragon, analogue in space : CD. 7.
Tetrahedroid : J.87.
*Tetrahedron : 907 : A.3,16,23o,51,56 :
J.65: LM.4: Me.62,66^82: Mel.
2 : N.pr 74,80,81 : Q.5 : geo Z.27.
determined from coordinates of ver-
tices : J.73.
Tetrahedron — (contimied) :
of given surface : J.83.
with opposite edges ; equal, N.79 ;
at right angles, Me.82.
with edges touching a sphere : ISr.74.
and four spheres : LM.122.
homologous : J.56.
and quadric : CD.8 : thN.71 ; 2 quad-
rics, Q.S.
6 dihedral angles of, eq. : N.46,67.
theorems : A.9,10,31 : N.61",66o : Q.3,5.
two: M.19.
* volume : 5569 : A.45..,57 : LM.2 : N.
67: Z.ll.
volume and surface in c.c and g.c :
A.53 : CD.8 : N.68.
volume and normal, relation : N.54.
Tetratops : A.692.
Theoretical value function : J. 55.
Theta-f unctions : Ac.3 : C.ths90 : J.61,
66,74: M.17: Me.ap78: P.80,82:
thsZ.12.
analogues of: C.93.
addition theory : J.88,89 : LM.13.
ap. to right line and triangle : A. 3.
argument : J.73 : 4thM.14 : M.16.
characteristic of: complexAJ.6: C.
88-:: J.2S.
th. of Riemann : J.88.
as a definite integral : Me. 76.
double: LM.9 : Q. transf. 21.
and 16 nodal quartic surface : J.83,
85,87,88.
formula of Riemann : J. 93.
Jacobian, num. value : M.7,11.
modular integrals : trAn.52,54: J. 71.
multiplication of : LM.l : M.17.
quadruple : AJ.6; : J.83.
reduction of, from two variables to
one: C.94.
representation of : M.6 : Z.ll.
transf. of: A.l : An.79 : J.32 : L.80 :
M.25...
linear: Me.84 : Q.21.
triple: J.87.
in two variables : C.92.. : J. 84 : M.
14,24.
Theta-series : constant factors of, J.98 :
Me.81.
«-tuple : J.48.
^,, = a2 (c-a')/ {c (c-a-)-62} : Q.15.
*Three - bar curves : 5430 : LM.7,9 :
Me.76.
triple generation of: Me.83.
Toothed'wheels : cnTE.28.
Topologv with tables : M.19,24.
Tore: section of: G.IO : N.59,61,642,65.,
circular, 56 and 65.
and bi-taugent sphere : N.74.
932
INDEX.
Toroid : A.8 : rectif. &c., A.9 : N.44.
Toroidal functions : T.Sl : Pr.31.
Torse: sextic : CP.II2 : Q.U.
circumscribing? two quadrics : Me.72.
depending on elliptic functions : Q.14.
and curve: Q.U; and sphere, G.1'2.
*Torsion : A.19,62,65 : J.60 : angle of,
5725; of involute, 575:3; of evo-
lute, 5754.
* inflected, infinite and suspended :
5739 : N.45.
constant : L.42o.
*Tortuous curves : 5721 : Ac.2 : C.19,43,
58,cn62 : G.4,5,21 : J.16,59,93 :
JP.2,18: L.50,,th51,522 : M.5,19,
tli25: Mo.82: N.60: No.79: Q.
4,6,7.
* approximate coordinates of a point
near the origin in terms of the
arc : 5755,
* circular curvature of : 5722 : An. 60 :
CD.9 : J.60 : JP.152 : Q.6.
* locus of centre of do. : 5741, 5748
—50.
with circular and spherical curvature
in a constant ratio : L.51.
en. upon ruled cubics and quartics :
C.53.
with coordinates rational functions of
a pai'ameter : M.3.
cubic : J.27,802 : M.20 : Z.27.
* definitions: 5721.
determined from relation of curva-
ture and torsion : A.65.
cubics : An.58,59 : C.46 : L.57 : of
3rd class, J. 56 ; four tangents,
M.13.
and developable surfaces : CD. 5 :
L.45: M.13.
elements of arc : N.733.
generation of : C.94 : L.83 ; by two
pencils of corresponding right
lines, G.23.
with horaog. coordinates : Q.o.
the loci of similar osculating ellip-
soids ; d.e of same : C.78.
with loops: M.18.
with a max. or min. property: Mem. 13.
on a one-fold hyperboloid : 0.53.
with the same polar surface, d.e :
0.78^.
quadrics : J.20.
quartics : C..54., ; 1st species, J.93.
en. of two : C.53.
intersection of quadrics: C.54.
quintics : C. 543,58.
* radii of curvature and torsion : 5730 :
L.48: Mem.lS.
* and right line method : 5743.
sextics, classification of : 0.76.
■47:
(see also
A.37: 0.
L.44.
I *Tortuous curves — (continued) :
singularities of: 0.67 : J.42.
* spherical curvature : 5728, — 40
A.19.
triple, and their parallels : A.65.
*Tractrix: 5279; area, E.35.
*Trajectory: 5246: M0.8O.
of a displaced line ; oscul. plane, &c. :
0.70,76.
of 3 homofocal conies : An.64.
of meridian of surface of revolution :
L.46.
surface of points of an invariable
figure whose displacement is
subject to 4 conditions : 0.76,77.
of a tortuous curve : L.43.
Transcendental : arithmetic : J.29.
* curves : 5250 : An.76 : M.22.
equations: 0.5.59,72,94: G.6,10 : J.
22 : L.38 : N.55,56.
mechanical solution : 0P.4.
separation of roots by " compteurs
logarithmiques " : C.44.
without a root : J. 73.
*Transcendental functions
" Functions ") : 1401
86 : J.3,th9,20 : L.51.
of alg. differentials : J.23
arithmetical properties : M.22.
classification of : L.37.
connected with elliptic : 0.17.
decomposition itito factors by calcu-
lus of residues : 0.32.
and definite integrals : P. 57.
expansion of : J. 16.
integral : 0.94.95 : G.23 : J.98.
reduction of: Me.72.
squares of : Me.72.
theorem of Sturm : L.362.
whose derivatives are determined by
cubic eqs. ; summation of the
same : J.ll.
which result from the repeated integ-
ration of rational fractions : J. 30.
Transformation : bilinear : M.2.
birazionalc : G.7 ; of 6th deg. in 3
dimensions, LM.15.
contact : M.8.
* of coordinates : pl.4048 ; cb.5574 — 81 :
A.13: A.26: OM.l.,: J.2: JP.7 :
N.63.
in 3 dimensions : A.13: Q.2j J.8.
rectangular into elliptic : Mel. 4.
Cremona's : M.4.
of curves : L. 49,50.
of differential equations : Me.82.
of eciuations : A. 40 : N.64., ; 3 vari-
ables, G.5.
of a characteristic e({. by a discri-
minant : An. 56.
INDEX.
933
Transformation — (continued) :
quadric: CM.2 : N.75 : M.23.
simultaneous: Q.ll.
of figures in a plane : A.4 : N.64,774.
in space : C.94,95o,96 : N.79,802.
double reciprocal by normals to a
sphere : 0.56.
formulas : J.32.
of functions : An. 50 ; in an inf. series,
A.3.
by substitution : Mem. 31.
quadric : Q.2 ; quadric differential,
J. 70.
(ciy— 6,e)3 + (&2— ci/)3-f(c*— az)2 :
CM.1,2.
two variables : Mem.ll.
homographic plane : L.61.
in geometry : C.71 : J. 67 : JP.25.
descriptive: G.13 : Z.9.
* linear: 582,1794: CD.1,6 : CM.3,4 :
M.2: Mo.84.
groups of: M.12,16.
methods of : C.92 ; which preserve
an invariable relation between
derivatives of the same order,
C.82: L.76.
which preserve the lines of curva-
ture : C.92.
* modulus of : 1604.
* orthogonal: 584, 1799.
for equations of dynamics : C.67.
plane : M.5 ; and in space, G.16.
of powers into binomial coefficients :
No.75.
of product of n factors : An. 61.
quadratic, of an elliptic differential :
M.7.
in rational space : An. 73 : LM.3.
i-ectaugular : LM.14.
reciprocal: C.92: G.17: N.64,82,83.
of rectilineal space coordinates : J.63.
by series doubly infinite : An. 56.
space, for representation of alg. sur-
of surfaces : M.21 : N.69 : Q.12.
of symbolic functions into isotropic
means : C.43.
of tan"^ . / -—2 + symmet. y, z -.
CD.9.
Tschirnhausen's : An.58 : P.62,65 :
Pr.11,14.
ext. to quintics and higher : E.lo,4 :
J.582.
* unimodular : 1605.
of variables : G.5 : No.78.
Transitive function : of 24 quantities :
L.73.
doubly, of 6 or 6 variables : C.21,22.
Transitive function — (continued) :
reduction to intransitive : C.21.
*Transversals : 967—74: A.13,18,27,30,
56: CD.5: CM.l: J.84 : G.22 :
Me.t.c68o,75 : N.432,48 : TE.t.c24.
orthogonal : AJ.3.
of plane alg. curves : Z.19.
of two points : A. 66.
parallel : A.13,57.
of spherical triangle and quadrangle :
A.45.
Trees, analytical : A.J.4.
Triads of once-paired elements : Q.9.
*Triangle: 700: A.17,19,22,29,33,36,43,
cn46,61: G.21:J.50: M.geol7 :
Me.q.c62: N.42„,43.
* angles of : 677 : 738—45 : A.65 : P.28 :
division, A.51, 58: sum,
C.69,70 : difference, Q.8. '
* area : 707, 4036—41 : A.45,57 : 0M.2
Mem.l3.
* bisectors : of angles : 709, 742,
932, 4628,-30.
* of sides : 738, 922i, 951, 4631 :
Mem.prl0,13.
* central line : 957, 4644.
and circle, ths. : A.9,40,47,60 : Q.7,8,
10.
and 3 concentric circles : Me.85.
of 3 intersecting circles : Q.21.
circle and parabola : Q.15.
and conic : N.70.
* of constant species : 977.
* construction of : 960.
* equilateral, sum of sqs. of distances
of any point from its vertices :
923 : A.69 : gzl094.
formed by joining the feet of bisec-
tors of a triangle : A.64o.
Gauss's equations for a plane tri-
angle : A. 5.
* notation: 4629: E.44.
* orthocentre : 952, 4634.
pedal line of : Me.83.
* perpendicular bisectors of sides : 713,
4639.
* perpendiculars on sides : 952, 4633 :
Mem.l3.
and point : Q.5.
and polar s : circle, G.ll ; conic. Q.7.
of mid-points of sides, t.cQ.8.
quadrisection of : Mem. 9.
rational: A.51, 56.
* remarkable points of (see also"In-
ceutre,"&c.): 955 — 9 : A. four, 47 ;
two, 48 ; five, 52 ; 66,67 : E.28,
30,40 : LM. nine, 14„ : N.70,73,83 :
Z.11,15.
* of reference in t.c : 4006.
and right line : A.59.
934
mDEX.
Triangle — (continued) :
scalenity of: Me.66,68.
sides of: bisectors : A. 17.
division of : A.63 : N.83.
containing conjugate poles with
respect to four conies : An. 69.
cubic eq. for in terms of A, B, and
r: J.20.
similar: C.792; under 3 or 4 condi-
tions, C.78.,.
* solution of : 718, 859 : A.3,51 : J.44. :
TE.IO.
six-point circle : Q.4,6.
symmetrical properties : A. 57 : Pr.ll.
theorems : A.9,43,45,o5,57,60,6l3,63,
6S ; J.t)8,71,pr28 : Q.9,t.c7 and 8,
geoi,2, and 4.
relating to triangles of same peri-
meter under four other condi-
tions : C.84.
*Triangular : numbers : 287 : E.30 :
J.69: L.ths63.
prism : thN.42.
pyramid : At.l9 : No.pr32.
Tricircular and tetraspheric geometry :
An.77.
*Trigon. in t.c : 4006.
*Trigonometry : 600: A.lo,2,8,ll,13,30 :
G.13: extM.35.
formula; : A.27,65 : Me.81,gzl : 'N.77:
geoQ.5,10.
* formula : 627,700,823 : A.33 : K46,80.
functions : P. 1796 ; in factorials,
A.43.
in binomial factors : L.43.
in partial fi-actions and products :
Z.13.
functions closely allied to : A.27.
tables, en: A. 1,232, 25.
theorems : A.2, 50,51 : Q.pr7.
on product of 4 sines : Me.81.
on fl+ -
1 +
!)
Q.15,16.
cos3asin-i^-f 2sinacosa sin ^ cos(f)
+ sin2acos^^ = :
Trihedroii : about a parabola, locus
of vertex : N.63.
and quadric : N.71.
and tetrahedron : A.57.
Trilinear forms : C.92,93.
Trilinear relation of plane systems :
J.98.
*Triplicate-vatio circle : 4754b : E. 40,42,
435,44.
Twisted surface : see " Scrolls."
Twist of a bar : Me. 80.
*Umbilics : 5777, 5819—23 : J.65 : JP.
18 : M.9 : Q.3.
* of qua dries : 6603, 5834 : C.96.
Underdeterminants of a symm. deter-
minant {i.e., successive first
minors) : J.91.
Undeterminants, &c. : A.592.
Unicursal curves: A.60 : C. 78,94,96 :
LM.4.
cubics, ths. re inflexion : E.28.
quartics : LM.16 : N.84 : Z.28.
surfaces, transf. of: M.66.
Uniform functions : C.92,94;,95,963.
with an alg. relation : C.9I2.
of an anal, point xy : Ac.l : C.94.
two points : 0.96,97.
with " coupures " : C.96.
with discontinuities : C.94.
doubly periodic : C.94.
from linear substitutions : M.19,20.
with a line of singular points, decom-
position into factors : C.92.
monogenous : Ac. 4.
in the neighbourhood of a singular
point : C.89.
of two independent variables : C. 94,95.
*Units of elasticit}', electricity, and
heat : p. 2.
*Vanishing fractions : 1582: M.15 : Q.l.
Vanishing groups : CD.2,3,6,7,8.
ap. to quantics : CD.2,3,6.
*Variation: 76: C.17.
of arbitrary constants : L.38.
* calculus of: 3028—91: A.3,prs42 :
An.52o: C.16,50: CD.3 : CM.2 :
J.13,prl5,41,54,65,prs74,82 : JP.
17: L.41: M.22,15o: Me.prs72 :
M0.57 : N.83 : Q.prlO : and d.e,
L.38 and C.49.
history : N.51.
* immediate integrability : 3090.
and infinitesimal analysis : C. 17,40.
* relative max. and min. : 3069 : L.42.
of multiple integrals : J.15,56,f59 :
Mem.38.
ti'ansformations : J.55._;.
* two dependent variables : 3051.
* two independent variables : 3175.
integral, of functions : C.4O3.
* of parameters : th2714, 3243 : N.77.
* of second order : 3087 : A.4 : J.55 :
Q.14 : Z.23.
* of higher order : 3089 : A.27 : Z.26.
*Versiera or Witch of Agnesi : 5335.
♦Volumes : of solids : 5871—83 : A.
31,32,36: J.34: N.f57,80.
approximate : C.95.
of frustums : A.33 ; of conicoids,
J.44.
of right cylinders and cones in abso-
lute geometry : A. 59.
of surface loci of connected points :
LM.14.
INDEX.
935
Volumes — {continued) :
and surfaces by curvilinear coor-
dinates : C.16.
*Wallis's formula : 2456 : A.39,gz39 :
JP.28.
Waring's identity: N.49,62 : extMe.85.
Wave surface : O.13,47,78,geo82,92 :
CD.7,8 : CP.6 : L.46 : Me.66o,73,
76,78,79 : N.63,82 : Pr.32 : Q.2,33,
4,5,9,15„17.
asymptotes : C.97.
and cone: Q.23,26.
cubatureof: An. 61.
generation and en : C.90.2.
lines of curvature : An. 59 : C.97.
normals and centres of curvature :
geoO.64.
umbilics, geo : 0.880.
Wear of gold coins : E.43.
Web surfaces : see " Net surfaces.
Weierstrass's function
2tit"cosa"a;7r
G.18 : J.
with a >\ and fe < 1
63,90.
expansion in powers of the modulus :
C.82.,85,86 : L.79o.
*Wilson's theorem : 371 : A.48 : CD.9 :
J .8,19,20 : Me.83 : N.43.
generalisation : J.31 : Me.64 : Mel.2 :
N.45.
Wronski's methods : 0.92 : L.82,83.
formula of 1812 : N.74., : Q.thl2.
Zetafuchsian functions : Ac. 5.
Zonal conies of tetrazonal quartics :
Q.IO.
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