^/^. \ 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 rei,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°, 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, 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 = 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 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 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->"."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, '? 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» = 31'5SH 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. i i-o-o.-^ w-i-oc oi-c«o <7i = -t-- i-o = -cr. oi s — d — CO 00 - I- « 1^ cr. o c o = - m o o 1^ c - c o m I- t^ -- i^ c^ 1- cr. g=::i;£= - = o2- ooo 00 O - O I- O n O - I- 0-. l^ »^ C CI -- - c C C - c^ O I- 0-. O l^ 00 0-, rr:ai=:^5 OCOJ^O Ot^O 3-. t^rOMO t^OO-r- Or-t-t^O oir^oi^o ooot^co^ 1? c^CL^-o moct^o CC-. OOl^ OOl-r-O o-ooo- CO ^ °°'^S 2°g2^ '"°® oiocot^ c-ci^o moo-o t-mcn-t- o>o-. oi^r^o CO to t» « « - .* CT) CO g Ot-r^c^O Ot^OOJCO «oo § t^O-t^.-O Ot^t^OO Oit^r^-.o 00 0-. i^o.«o-oot^ s OOOOt^ ©"I^COO O"^ 3 ot^ooio -oocoo t^o-t-o ooocoo r^OMOt-o CO o CD r^oor-" c^050t-.o r^ooroo co^omo os oi^ooi- CO cit^-«o 00-00 t-t>.o| 1^ ocoooio i^02i^-o ooi^o- mt^coo; CO c-ico.^o- ^ 1 S — ot^oo o-.^oo oomot^ COrot^OCO OOO — I^^^ g 1 ooox^r, oooo- r^gg OOOOt^ Ot^Mt^^ cot-oo>o t^cooot>. -O-OOlO sr-^" -°°° °='^ 00 o-t^OO — t^^ot^ i^oir-coeo mo-o-. ot^oioi-.o 5 coot^oi>. octooo — CO coor^-.c^ -ot^-— 000 ■* « -, CO — Tj. CI^O — o o-ooo> 00--l^0 OOl-OO t^-OOl-C ^ - - - " ^ - CO Ti< o 5°«-° °°°°j; -°°-S S?'-:°::a ^S'^*^'- CO o ooooio ooooir^ -.c^ro •-. CO « -^ ^ COOMOO t^OOlOO ^OI--C0 OI^OIOO oc^mt^oo ?:; ot»«ot^ — ot^o r-oo> -. ^ <£! CO « CO CO oot^o— oooot^ cooot^r^ OOt^OO .--.r^CO:^ ^ 0>-0t^0 CO'-COr^O t^O)« o — oc^t^ oocor^O oi:^«o« t^oot^.^ r-ooo-cr. ?3 t^c-, c-:o-i ooco — ro ocor* Ot^"Cr>0 Oi-O ^ t^oiocno -oi^-o ot^o—- ofoot^o ot^coot^- c5 -oooooo..g,o^-; CI 1-1 Ol-l-OO c«;0000 l^-^OlOt^ OC0I^-h05 oooooo ^looo^c, ooo^- t^2^ _. i-oo-t^ omot^o ocoocoo t-.r-ooc^ o — o«oo 0-«^ »-.oco ot^ot^o —oc CO o t^oooo -t>.t-.oo — t^ooo ooot^o COOOOt^O OM — Ot^. OOI^OO 0>OC o o ooooo ooooo ooot^o OOOOO I^«S00l-0 s ooooo t^ — ooc^ 00- |00-^r-.^ ^(M-MCCW TT.TrTJd.Ui »(iCCOr-l> t-l:-0CQ0Ol--i s§5§2 2ss5;;s? ^s^J 1-9000 O r^ I- •o — O C C 1- ?• « r: o 1- = i~ - 00 ?.■ 5 '- Jo- :: O Ol O O I- 050.-. ^ " ~ ~ -r " :: - i- i: - CO — ^ -^sz^a O CO o o CO >- « r* O Ol t^ r» ooog2 — r- — t~ CO CO r» ri i* CO e« — o t^ o o t^ t^ O O O O CO o» r^ c o « -=°2S 2j^r.05 CO - t- CO t^ t- O O CO 0> O - CO t- o> — CO — t^ CO t^ - O — Ol — o o t^ •* t^ '- O O O t- CO o l^ O CO -■ o t~ O O ^ - -i o-gjoo t^ r; o> '^ ;;;;2' = = O O t^ 1^ t^ -0 o c^ t^ ^ CO U3 t^ tj- - .'-. CI O CO I- — 00 -. — ^ o o — - CO f eo t^ 1- o CO «^ — (Jl r~ 1- - 1- a t^ — o 5-S°=° CO o r- - 1- - t^ — ■>!• T CO 00 to O O 05 - O 1- o O I- - ij- t- CO '- CO c* — ?: 1- - — . 2 CO -°;;5S° o o - -ss=»= °2 °5";^ s I- CO C- l^ o . :z I- n _ m CO t^ t^ CJ .. _ =; = = °5r - O t^ — O I^ CO CO CO -S;:z- = s o »-. o o o -2 = 5'^ ?. i^ .-: ^ « -■ o o o o — o o r» o O t- CO - O 1- ~ « S 5 w O O I- 1^ o c^ =1 -i t^ S J= I.- '- 3 = S = S i '^ — r» o (O r^ o m o o t^ . CO — SS'^?^ r>. o o -i t^ O O C CO t; o r^ c 2 => g ooogo r^ t^ t- j; '^ = s 2 '-i t- — «0 CO — coo O O CO I- O O „ CO O CO t- 1^ I— o 1- i^ r: o OOJOO.^ - 1- t~ 1^ --.:-: r- o ai o - ° r '-- = 5 ro - O CO o ro ^ l^ C - I- o °25::S 0002 = t^ => - 1- '- Tf ^ « t^ t;^ - o o o - i^ i^ CO o o ^-OOJ^O tJ< ,.o.. t^ -- - t^ CO - CO c - * CO t •« -O '- CO - CO (M — « O r- o o t- O O) t- t- o o m 00 CO O r: 1- - O CO I- — goooc. t^ CO t^ O t^ O COOOg,^ r. — o t^ CO o is - — O O t^ ro CO -- « t^ ro c^ = I- ■- c 2 — CO m •* CI o> o o o o> - °°<=S5- CO o o ?: o o — r> CO - t- 1- c CO t- 01 , t» o o O t^ t- Oi CO c CO O .^ - C r^ s? r°-°s 1^ a-, r^ CO a t- c^ CO - c. — - CI 1- 000 CO - 1- r- - S2«°'^S n°s i^ — CO CO ^ oogjoo- 00 r^ o — o o r~ o o o o S'-S 000 t^ t^ o> CO I^ '-I c t^ 00 — °sn CO c« t* 1^ i^ CO o> — t^ CO t^ '-i 000 t^ ^ m t- l^ 000 t- CO — 0000 oot-00 coi"— or» cj>o O O O l~ o o t>.oor~— cocot^oo o OOCOO t^t^r: 00 OOt^OOO Ot^OOCO I^OOCJIO cocot^o— O. OOOV O O O O t^ o COOCll^O o — t^o ooi^ot^ o — .nt^o coo 00 — oo> ot~o — t^ — ^ CO O O CO Ci o t^ o o 000 Ott^COOO 00 — coo OOOt»Cl3i ^ _j f, -- CO t^ 31 CO 3> ^ r» -^ CO t^ a> -* ic I.-5 tc -^r 'X ;r I- t- re y> ^ n -■ • o 53 "'-irsrs « o f^ ^o-.go t>. O t^ r- O 2-2 = :: o - o t- CO S;: = °°S o r. t» t- o o» to — — 00 CO S^°i:£ - O O O CO -- O t^ t^ Ol (O « - o t^ c» — o •* ^ooor.^ § — CO — S t:-- '^ S = 2 = 5S t^ -. O CO o Ol O O CO = — U5 .O -!■ g — — CO H*^- i^ oi ?: - = I- c: O C CO 0-. I- t-. = c o to i;- = 5 = - - -"^ = - CO CD " " - - = o cr. a-. 1^ o i:°S 05 o o t^ o -- O O t^ CJ> o — 00 CI t- 1^ ~ c- O ;-. t^ O O CO ci - o> o t-. § o CO o t^ t- CO i^ — t^ o 00 CO - - t^ eo -i •«• to s o c 1^ CO t^ O t^ O 1- Cl O O O O r-. t^ o o o t^ - o o 0-. CO - to C4 - - E^ '^ 5 — oi « -H m o o o o o r. CO - C - O t- o t^ o o t^ CO t^ CO O CO . o> o t^ t» o eo « eo O O g l^ ro t^ O O CO CO t^ o - C^ CO ^ CO t- - -^ o — ^ o -. o CO 1^ o c t^ o o> o - — i-« CO s o^-ogt^ eo O i^ O 1^ -1 n o o o> ^ - t^ t^ o o to ^ « o — o o O CO o t^ o O Cl o o - '^ = = 5 '^ ° ^ m CO CO ai o r^ o o o t^ t~. -H O ^ O O O t^ ?5 O O l^ o o t^ « o o t^ -1 Oi 2 = ^5? « O t^ O CO t^ J5 -°s-s C CO CO — o o t^ o 5J Ol O M - (3> O 2 " " i^ oi — CO 1^ W t^ " o t^ o o ^'^^s* O -- CO t^ CO o - l^ O CO — -« -to o> o — r^ CO eo 0> o 00 CO t^ O t^ 05 -- CO r- ■* 2S»«o O O) — -- t- o to - °Z^z° .-■ c» « CO S°2=° ^ CO c<5 a> o t^ - O CO t^ o -c 00 r- 0> O) t- O '^ t^ 5 '^ o t^ t- t^ o o eo - o CO CO — to i^ CO O CO = CO o CO CO t^ CO CO i^ — ^: O O r^ o — ■ i^ o = ^2 = S o Oi o :o CO 00 e» ^ — I> I- t- o o CO CO t^ CO t^ o o o» t^ o o t^ o o -< CD t^ O O 1^ o CO OOt^I^t^ or^^go o CO o r^ o o> o o -- - - o CO O O O O t^ 5: S '^ S ° O eo o o t- o o - O CO CO t^ o « I> r- o I- o r- o o - CO o en t^ -. - o t^ o CO ^ 1> 2°s;j:° i^ o - o t^ I^ o o CO CM o - CO t^ oi CO - o t^ o o « CO t^ o o CO - U5 -< t-. r- CO CTi O r- -J. 00 00 o o r» CO o t^ ^ o o> o> o o -. o t^ - o p- —CO or.« O C5 - = °55 '- :: S S ° o o t- o t-. - 1^ - o o CO - 2 '^ 5 '^ "o = ?3 o t^ o o o o t^^t^^ S5° !>• o^r.«o °2f;; = :: S°i:«'^ r2-S5 O O O O CO I- 00 1-H o -- c t^ o CO o o o o I^ O CO ■^ -0 — o ^ t^ « - o OJ o - t^ o t^ O O t~. CO t- CO o o t^ CO M O O o o -°i;;::g o - o |o a> S Z y-t 1^ o - o o - O O CO o o o t^ CO t^ oi o « to — T-t d _ - _ O -i * t^ o o O '- s 2 :: S -H o> r. o o r»t^ o o» o O O O t- M crj - o o o t^ § — O CO o t^ — t^ t^ t^ CO «) to - ~ °°;; or^gjoco r^ t~ O O - CO CO CO t^ — °s° = s r^ - o. o t- o CO o og, = go t^ CO o o t^ — o ^ -3 - CT> O 1- 1- ^ C r^ CO 31 t- o o t^ o CO o - o o o CO eo a> I- o - t- ■r - t CO o 1==.== t- o .-o o o O I- o ° ° - ?. :; t^ o o o a> °5'-<=S? p-5 t^ O CO O i^ - £ '^ S> 2 -§ CO I- o c «^ CO - 1- t^ CD o o — CO o eo t^ o o> oooo« ^^z^'- 2 ° "^ 2 - o o eo o -- o s 2«::'^:: «"=;:!;« t^ o o CO - - o t^ t- O O C7> t^ O O O CO O t^ t^ -< 1^ o t^ °^z:;°^ ^ '- » S S =" o o — o o O O t^ o o o o o - o> t^ O — 1^ So z O CO O i^ CO S '^ « 5 « = oJ Ol O (^ O M -...ogo «2;: 1 5 'o ::i 2 i: 2?:^?i;?^ ^ nt^T. o-oo-i~ « ui ^o0|-o»^ oot-t»- M'^ss® °;::®5" oon.-«- ?2» ^Sw'-SS ^S^**^" CO — ot^oo i^ocoot^ o--or»c^ og- — ojo e-ooooi^- i^ — — ts.oo» — — o oowono g t-«or^o oooo— oot^ot^ 1^22"° 2 — "^S®" OOO OOt^OOM t^->000)0 M t^ M M U3 5 ot^t^coPO t»cnoo>o> ^^'^Z^S °®5m2 Si2°o'*c on— ooMOt^— ot^t^^o^' t I--.000 -oi-oi^ -ooot^ a>--cot^o oioc — oi- — t^O OOOOOt* MOOt^Ot-- s °S°^'^ 5®5-° °£«?Jo ::'"2S'" z°°°'^'^. os>t» o — ocoo— noot>-ot>. -« — — CO ^ — s o o t- o CO <=> "^ - ^ f^ '^ 2 « S o °S'^S° o t^ o o 5 ;2 r»on oi^o — on coi^t-oo-- s t^eooc3»— o>ocot~— ot^O;rt^ °?°2!;r oo — t^-- »§S =2=;'-°5 j^o-ot^o s -t^l-O'- eoot^ot^ 00» — 1^— ooooo t-t-oo— C (NtO(0— o> I^O~ —^ - = „ r^googo noono.^. 05 '-i:::'-? =-°§° ^ = °2= ^^2°-- ;:®'^2z2 OOO oor-.r^rO'" t^CiOO'-.*: ^ t^t-w I-O I^OOOO -r-I^OO "r-^^n S^Slo*"" ooen nt^— o>»>-o> oi^-ot^o lo—x — — o w — r» CO o--go 00.^-2 °^n;;'- £2°*^= fiS^'S'^ 2'-^: S°°5g'- ::^--«° 5 — OOl^t^ I^COOO— 0>— OnO t^t^—Or- OOO"0t^r; o-t^ oino-. O — O OOOOOt^ ?:; -cc-o ot^o-- i-ciot^o oot^r-n oi^t-onc t~n3> -t^OO.-Or^ O — t^oioo ^ „-ooo -gj2'-S 0..0-0 5 = = 2= 2;i; = '^s = 0-- t-o«c^t^— nricoot^o fo—co t^ «-.oc» — CO oot^mo — ooinc^ t^oot^a> oo«oo "^o^^^^- --0 t^or-nnn ooio — t^o 00 r^ont^o =5 = -- ° ?:- = :! -2L-SS -2-« = = goo j^-r^rjoo ..oonogj ot-ooo> t^o — coo oot^og 2^02° 5°*°**-^ oa. o oon«t^o> Mt-nClO- CM o=^c?>- t^ot-coo or-oot- eooot^o ®~~'='2'" i:'^;: S = = ::s'^ g'-'-*;: CJ> oor^i-t^ o = 5>co oni-o- oi^oor^ oroooi^r: OOt- __-)n — o t-oi^oot^ — CO ■* e» t^ M aioo o»or»«-'0 CO Oi CO O O O n 1^ O r~ t^ - C~ O O -; «r - O 2 - 2 O « t^ O O or^o t^oi— t^oo o>oa»ot^i^ ■^ Ol — — t^ 0» O a © O t- O t^ t^ O O CO o o — or^o> nnt-oo o«o — t~ ooot^o oi^-gjo — ot^- coeoooot^ ooor~i- — g or-oot- --000 jonogo o.--ot- ^oo-jlii:,' — cot* ooor^oo 3>eoo--ot>. g ^SJ^S- 2-2«« -°<=:;2 =?:-^5 = — :i: o ?.• t»oo oi^o>noi^ oot^ot^o rv. °°s2® S*::'^® 0.--2* ?S2f:° ®«S'"°- C5 -1 r^ -• T~ t^ ^ r5 r> -- h- -< ~5 r^ r> — •■; in <^ -^ -j: ^ I- t- cc X n r. ^. :> |5?-'^- «t7?}«2 =!&::??:?•? ???.?r:r: x?^???;- ^lor^cro c^oooo ::.^-^j:;- °-22;: °5°^i;i^ CO 1 or^-^r^o mooit^o i-ot^ ^ - = -5= S = 2 = ° J^;;:^^'- °:;'-SS -°."°S° s or-Mt^o j^^^;j|2 '"■°;::; g 1 '-°i:°:: i^"'-" "-"^ _^ o-^;«— ot-.:«:Mt^ or^ooM o> — o-o mr^-oi-o _. OlOl^mO OI>— OO t-oo ^ '^ss;;^ :;°^-° ^'-SS- 5°s?^S <=°°°s^:: CD OOOlOt^ -Ot^OO -OO — t^t^oi— O'^cno MOoi^o no — i^t^ t^t~«r-.t^-- CO C O C O 1- t^ c ^. ?;^ l^ 2 "^ ° ^ t^OCnt^O OOWI^'-i t^OM-O rHOlOOCO OOt^OI^t- m(N OiO-H — o o> — o — CO — — U3C^ « — o CO C-, a>t^-0 c^coociO ^^i,c.;t^ Oi^r^oo oioi-oio-. en en oo>ot^ i^o-or- Or-omoo _, r-oooo oorcoo o-t^ ^ CD ^oi^ oi mt^o- oo« jOc-^- c*.* o CO — .|mt^^o>o rtOi-HOt^ MOOit^M ooior-o i^Mr-oi>o CO ^ C5 °°s°S'"::°2^2'::;: CO - S -oot^o OiO>t^r^— t^^^-coo OOMOO) ot^t^t^«i> .n ^ (^ en '-> t in-t- -oo mTfpH _^ or-o-r- MO-rct^ or-O ^|0"0t-- t^^coo- oot^oo i-.i^ooo r-r^r^t^a>c-. CO t-r^cocco oc-t^co ooo „, la. ^^i-l-:o t^^o-^i- Mcoooit- ocot^oo ooo-Cr- r-omno r^ooo— ooi^ «o Ot^OOO Mt-r^t^t^ t^MOOa> «0 1-M0 Mt^OCOCn« 0--i>i^0 t^r^o — « coc^ ,_|r-r-.«oo ooi^r^oi ot^ot-^ ccr-r^c^- oooc«700 i 1 lot^ooo ooi^ocji co-Hcri^r- ooooco i-^i-.o^ot^i- oi^o-M i^ot^-t^ en o a o2" --0O ^- ,^«„ O ,. Mooter-. t>.mrtOt^ oot^ — M ,-ir^r-^o ooot^— (>^ ^(N en ^ «coo ,-,r^,i.ci-H>o MO s -"°°- "°""° — ° oot^oo ^oo-r^ —ooot^ oot^coo ot^o?:--o „ OO — t^t- o en en s o r-i-- ^o>|--j:5« °::°2;5 °^- s ocg«o -l^OOt^ oo-mo 0-l-OC»1 Ot^OCl-O ,„ I- O t^ O — r-^ I^ 05 1- O n - ro 05 s t^OOO— ' OI^I^OO — I^OIOCO O — OI^O l^fO^O — p-; s Ot^coo«> OO. t^coo — .-rr: oi rj. en — to 00 CJ s -t^OOO Ol^M — Ol CCC-.Ot-.-H Or^M-O t^O-«^l-^ O — WOO l^tCCOOt^ OOI- OS tj °S25°«S5°-5"* to -> ooooM i^MOfo— omt^oio «t^ooo> t^o — r^cro S ot^o— o Pjcoor^o MOO 1^ •♦ — — O -s t^-.t^OM r^r.«{r5l-. OOOOt^ OOt^t^^ 0--M0010 m53 coMCj>t^o> ooooo r^— o 5'5:d2i: Sg^?J;5?5 q!2i^5f3 S^&ri.^ [z?t^^P.'?. ??!^?^ 2---C0 »^«^ 18000-27000 O - 1- O C l~ 1- 1- O O 1- o o 2 U -,-- = = :: = -S- 5°°«- °-i''- = = ^H"' ^..o 03-20.. SZl^-^iJ C3 M^-?*^ S*n®- ®®S«2 2*"®$" oooMOc oot, ooog«- -gj^oo.- CO S°£®® z '^ 2 !:^ ® «* w 2 ® 5 5: •* ° 2 5 ** ® z ° ~ 1^0— eot^j^oo-. IT o'Kt.— 00 _. O'-oioo wr«.or»M or»oo-« — i^. — oai — oci^t^c n- «5 - „ 2 '^ - « w j^ioi. t^oi- ow — ot>. oO'-t'W oowoo r.oit^t>.oc 0-0 l^foono— r-rrot-c - — pi •<)•— U1 (CO — i-OOi>.- Oi-Ci Cio WO-OO t^OOOW wr-r^-r-. - t^mo — 1^—0— r»o — woo ^ w.,oc.o r^ow-w - = -0- ^ooor; wogo..^ OWO — oot^t^— ot^mot^o ft^owwo w-i^o5a> ooomt. owwt^o 0. s. — — i- «J--*- 0--0 e< p«- 12^^-, ._|<--Ol~Ot. l-OOOW —00 WW — l^COt- I^WWI^OI- ^oe^ «-«" ^«.„t^ ^-e««o- ^j|ow---w ot^o-o t^ooCT. t^ t^oii^w- 01--. r-r^- --« ^r^cjooo -or^owo. ^pooooj --or^j. r-r^wj^o «;;;::«« °g--«o — oooi^-r- or-— ot^oj OOt^ — C<5 oooot^ i-mt^t^— r. oopo t. — ooor t-om t^erwot^o --0000 It^-Ot^O WO-OO f^t^Cl-O t^— oor- wr.r~o>o>c £2- a> ^ _„„„ ^ o>o«D-* CO I - - ES= °o..go- '--S^Sgl o|°'"'=='=' *^°°S° 02*^°° ®®C:*5 ^^^n*^*! -00 t^-c, or^o t^t^o — CO 1> CI wowoo — 01- 1-0 CWO-l^ -MOt^O -OCWWI- 1^— W CI c<°o w — ooor. ^|w-ot-o oi^owo f^oo- — not^ot-. wi^wo — — N§ 1 - - " - ^ - - - - * « .. t^W— Ct^ — WW— OOt^r^OO ^g.|-°5 = n ::°g'-° S"-*"?; «°s::° ^ri^'-'u Zn" ^ ^ 'S'^ ° ~ ~ ~ - Z " ~ " " '" - mot-, i^— «ooo oot^ — w-< it^r^ot^o w — — 00 — — r^o t^ooot. oot^— t^w ^ 0(N «M-„ ""S '^"S «-l- 01-0 w2''Sr'=£ t^ocioo- go 1 ° *" 2 w - '"»2S° £5*^-0 ° 2 ® 2 5 S 5 - 2 "^ ^ o>oo or-o> — t^i. or.— oww -. WO«« ^ Ui " g OggOOOg... W0 0r;W0.^£0-..«J^0000.~ *="£ =;:;;;°i:'- ^s-'-s* ^CM §;;;««'' «-«°2 ^^°«« 2'-° = '- s?°« = ?.- 00 - '- " §-- S????§f2ff SS^c?^^^ 15?=:!^;;:; r^s?????? ::c:r;Sfe J5S5Sf: SSSsSSg r;-„ -t^-(NCO — OCO — f 00 CO — - 00-001 t^co-.ot^ ooco joot-i^i^ 00--0 r^cooo— o>OCT.or5 cot-.t^cooi-» ^j -o ..- o. -U,C. ^ -^;oci-- -0000 ot^o 10 n " ^ -oocoo ^.-025 02-00 or^-oj^ g^^-25 _. Ot^l-I^CO -00£-- I^-CO S -2CI ^ - 22- ,_|rOOt^— r^ Or-Cr>-- OOOOt^ - — t-CcO CCOCCr. COO 00- COCO o>c);o- — t^ •»• '""<^' ^ - = S"° °?Snx '^'S ^ =-oot^ oci-t^o co = a>oo t^t-.co-t^ -- = oco> t^oocoo coji — oo> o>— t^ U5 — CO CO — — ^po.o = o 2-2=- S = SS= 00050 ^^o;^3 = ^00.,-= 0..0-2 o«- _.|t^— — CO r-Oi^oo or^cot^oi Cioot^eo --r^-oio 0.^ -- CC-- c ^1— t^— — 1^ ocot^oo — 00 O-.-ococ^ - 2*" n Ot^OiOO OOOCOO Ot^Oi^ro t^— -^— t-Oroo>c-- — - — r»o>tco — m -loco -__ t^Ot^COO t^OOlO>t>. — — CO "" oooi>.o -t^oot^ t^oooo oro-eo:o OOt^COOI- ^ 1^0 — 00 -OC3>0 — t^ Clt>.CO d- d-OtOdUS ^ -:<^-Cl- t^OOr-O 0>«t^00 MI^COOO -OiOI^OCO 1-1 CO a>t>.ocoo — t^t^i^ ri — — t^_ - „u,--CO- ^ coor-oo ococo-a> o-. t-— ot^ t^-r^oco 0000-0 u) TfOi--— dco— — 00 — r- — ,_ oocot^— t^cot-00 t^oo 00 .„ - cod- lo^root^ t^ooi>co 00 — t-t^ t^-ooit- i-o-cooc •^e< — 10 t^co— — — ■*— >o C^OOOirO -t^-OO OCOt^ S^ i-t-^^t^o ot-oi-t^ oot^eo- oo>a-. coo cot^ooo- TfOtC- -- --- d-UJ d C5 coot^— — t^cot^o OC0 — eo — — tc — 00 --2S° ^::-^2 - — °°2 °2'-''^* 2-°'^'^2 0> — S — '^inm CO— — — t^t^ooo C001 — 0— ooot^oi -i-o-o t^coo-i>.ai — • 00-00 t^t^OOO Or^OO— O-t^l-COl- (M^-c»- - U3 'i' -eo.r';or^ omot-t-o> 1-1 S-S-° °&2-= 5»S 1 _,^t^.^i^ t^rocTiOO eoociot^ coot^oi^ o--oi^o cOt-(N— — to-o — — CI CO — on. - - - j^l-oor-o co-ooro r^o» — __|t^ot^or-. oocot^eo ooio- t^ t^t^o- c^ t^t^nooco COo— — CI ««CJ 01— CO — - 1 - _„|c^ocoi^t^ C-. cnooco t^ 1 IC; t^OOcO -I^OOii^ ^co — - O-v^r-Ol- -. t-t^OOO „0-co -- --;nn. co--t^o>eoo t^ — Ol-^t^CO . - — CO o-i-i-n ot-oot^ co-o 1 .tcoo -d-d| ,^ t-ClOCOO — e^C^OOl — t^O?--© t-OCOt^O fOOOiOOO to - U) t^ CI CI CO •♦ - CO t^ -00- t^ — — oeoco — t^ coo«^— 000 1 ^ -0- «- - 1 -Ot^— — 01 r^CJlO- Ot^COt^l^ 00 — — t^— — t^O CO CI CO-U5-CD « -COCI -0 OCO '-2'° g: -gooc^r^^-ooc^ogj^j pofoot-o ---oo> t^ot^- CO 000-0 ocot^mot^ -0000 t^t^orseo — t^ol 1 -H _- — 1 — CI -M CO c<5 -* -1 ^ -H .0 10 CO ti i^ r^ 1^ 1^ ac QO =5 r-- 0000— — iMciroro n n -^ 27000 — 36000 -o« ..o^o^o Mj^2::°° en i^ (0-C.X f. — OO t^3>ort«-.o ot^oent^- O P5 r- - « * - tOt^OO— 00«^0>0 0(0(00 1^ oioeor^t^ O — — COOl- ra«— « 00 « — c« — _— ^ — « ci^n oo« t^ ooi-t^m- g t^coi-oic^ ooco-ot^ i^ — 00— 0«^0i^«^ oocoooc — — uj- e«>oeo«JO — — — — oot^ — t>.ot^oo oomont- ^ |o — (oa>o (ot>.ot^o r^oo>t^o 0(0 — t^weo eoo>o5o:t~M t^eoc^ma> ooo — t^ ootot^o ot^t^o— t^or~a> — - ^tO(N01< — — c^'*— „t^tO — ooo c^oiooct^ -cooi-ori 00 CO i^ooi^- oooMto (o — or-o t^o — oa> t^-t^oooi CO^ (N--*-0> -_-co CO-O 00 — - o — 2 ~ 2 Ot^fO — O t^cOOO— t^Ot-COtO I-00040 o— oot^c JO CT- — — — — 0»?5 rf> — CO OO- OMt-.— t^O Or^OOCTlO »>. - - ^ — 00 o — ocnoi ooii^i^n coo — — t^ j>cyir-t^i^ cor: oocoi^ CJ — OCO ^(O-*-— •♦— t^c^- OCJOO - CO „ „ « - S'^- ::5ci°r:'^ ^ = ^^'^'3 _, coooor^ — o>:ooo or-r:-o nt^oot^ tooo-oo -«.. OOgooo -OM-rjr- ,_ oooeoo r^t<.r»ot^ t^omot^ -CTij^t^- -t^ot-ooi CO — — «jc< o— — ooeo'*- o> — t^oo o«^-mci(o oitor^tooo •* c* m " — 00 •♦ CO °°^=°"='-^°^=°=°°°"°-^-l 5S° j;°n'^«2 ®g°2'- = oot- — - — oocot^ oicoor-o oooeoo r^o — or^O O o— coot^t^ — -«o o-o t^oo — o>o too- — OlO t, .-«;^2'"° ®°°2^ °S°2- "^S^^S Sn'^S*" «<-o -Clt-:^i-0 r^oooor- •^ -t^-Cico t-.-.-^r-oi cccot--co rorio- omcoot^co m ox- „u^-^ci_ «-x.o oocooor» t-t^o o>o — o>or^ ot^-i^fnoi ^|oo-ot^ 2g = ::° °g = °S ^-^^'^ =££'- = - ='S'- S5°°S?? -5°°°^^ « :: = 2 = ?5 ;;-°2° '--Pj^^ =2-2° S'^RH^S o — — «§ oi^o-i^ oi-^t^- ot^|o°^ ocoooeo t";;:;'^^:? oio?: coooi-oo o--i-t-3> nS -- "(Mr-(0 t-.«OC^ -CJ-CO 2 2 jogj- r.^000- 00^3^.0 CO as t^CTlOt^O -COOOI^ OCOOl.-OI^ f-OOOO) I-.Ot^-r-t- C< «^— J> t^OOlOO (OOt^Oi— — oco.-r — ooot^o ^0 CO — ^co— -l-c^c*- ®2S "-SS*-" R'-SSS^ § MOt^OO 00t>.0— CO — — t^t^ 00 — c^O nrjr^Cj^t- oi^o ooe>o(ot>. Motot^c- O U1 (N — - |ooocor» ofoooo — t^ooo ot^toor* t^r^- oo — oci- CT>M:r>-r-eo — o — or^t- S 0001- I- nt^ooo i^oco. 00 1^ 1---00 ooi- oco--*- t^ — «oco t*00 — t^tooi~«co oot^cooo ^|S°°m5 ^-S'^S 5'-°^pj g?°i:-2 ®°j:'^::S OO— t^OOt^OO eor^OOt^O Tf — — CO ^ 00.^05 oor-2- =»*-g 2-S55 '-°S£°5 ° S ® *" 2 " '^ S ® = ° Er "^ - 5? o> C« — — 00 to — (OO t^o- t^co t^oo-o otooo>r« t^(oo»<^— °®'"55;^ r^oto r^cot^- oio r^— o — ocn — — — «« OP« (0 »^ m — r^oi^io t^o»(ocoo — t>.oi^ or~-oo oococTir^o 5:j3f5 3S&§r3pi 5^S5?;? |S5::;^;J ?5Si?5?§g =:ii::?srs SSSf::?: SSSSSSSS ^ o .-r o - .- 1- 1^ o - 1- - ~ :. y. c: ^ i^ pi o c o i- o> o r^ o «| .- = - = ? °'^s^s 2° = "^^ ioi"" ^m'""!oco 2'"' 2°° c^rlui"^^ S^' 1^ o-oi^i^ c-t^oo o-o T^ o i^ c - 1- o o o o cr. 1 I- ■- o m a> o CO r^ o i- o i^ ^- S ° = ° = ='-°s = -" = 5 00 CO O-r^l-m Oim-mCl l^l-m?^0 -COOOiM t--t^OOl- S — 5ci— eoS^Si lo CO «c-. «mo i>— a>^m oot^-t^ mt^O)--o oo-Ht^oo CO CO =v i^ooo co — ot^o — t-=i 00 OOt^C M — OOI^O -MOj^t^ ^Ol-OOi COMOmO-, t^cj— CO— -ooio«5 ro 2"— ^ *^' CO CO c\ y. ~ t^ ^ O C 1- C i^ t^ Oi °> Ot--.?^t^ t^OOt^O COCt-CiO l-OOl-I- ^CroOcOCT. _^ t^OC -o o — t^r-co Otot>. CO cot-ococ3i i-t-t^c:>t^ oo — oco ot^o-o-. oi-oo — c SOl-O-H ot^oco— OCIO CO CJ i-ooii»- -ot^oo cot-t-o— o-fot^o -0-0500 OlOOOl^ COO-. t-.-o o-o ^ o ««, «, CO o t^l^COOO coco- I^ t^OOil^— t^OCOOO l^CT. oor^ — 0>Tl< — ^(M«^r^CO(N — t^t^o CO -a. --2.,-- - T}« oii^ooro i>.cot^eo-H oiot^^o — i^ocoi^ oiocot^or^ - ^ 00 « -. ^ « e» CO - -to - « i3 i^t>.ocoo oo-t^r- eooo ^ .- M ■* I^ ^ PI — i^oo eot^-o.^ oocoot^ oiot^n- oooot-co S oocnt^o t^OOOO t^-o CO o ocoi^oi>. oooi^t^ o — ooo r^050coi>. — oo — coo -co cn^ 001 — CO t^— o-o coot^ s -ot^-- oc^oot^ -ococoi^ ot^cocico oi>oi-oo CO o -ojr-or^ -t^o-c3> ooi^ — — « lo •* t^ o *g I- - 31 C - 1^ CI l^ O O O t- 1^ - O t^ CO I^ l^ I- O t- O O CO a. g 2--5" SS'^SS ::2° "S Ol^t^t^Ol — COOCOO — C-COt^O -Olt^O- t^ooot-co 0o«i^co oocoom t^35--05 oi-oot^ -HOt>.coo-. t^ COS; OIOOIO- O-OI-O Ot^« (N O 12 - «5 t- ^ S2 05 OCOOJOO t^OI^t^Ol OOt^(3iCO Ol^OOO COfOOt^-O ^ -r^t-O- OOlClt^t^ COOO g comi^co- OO-COCO OOOI>.t^ Ol-l^Ol- C0O-O0-5O rf— -— C0C4O — CO-HODrCIN-i- OS -oot^o OCK- o>o t^oeo 00 X O O O I>. ^ O O l^ O a. t^ O O — t^ ro o l^ l^ t^ CO o o> O — 00 00 t^O-i-t^ — O-COO -Ot^ in o CO - Tf - t^ s -Si:S5 g'^S"^ :i^°°° °°;:i:S S'^g-sS ^|°S-^2 — ss- s::- s t>M?0Ot^ Or^t-r^- cot-r^oo -.-ot^o ocooo-o S o-cor. i^ ot-t^oco oor^ — — — CO — to 00 cot^ooco oc; — ot^ oicoor^o r^ooio— c t^ocoot^o o -loo— ot^— ^ — rt ^ 05 oin'^S'co *^— — S'^ 2-2 s OOOl^O 1^1-OCOO t>.— OOr- (OOOOO coot-oot^ ^ 2;; = ;:g -'-^^^ -'^^ CO 03ii~ — o r^oc^i-co t^ot^coo o>i>.cocoo Or-coi^oco (Nc^'- ui cc c^ to a> —00- — c. oot- t^ot^ o 2 -^^ «>-« ^ .- O - 0-. I^ - m O I- o - o m .^ O 1- ^ 1- o ., o m o c. o o ^ -J:- = 0^oa;.-g--0- CO CO — O — oco or»r^r>.t^ 000>coo coco — «-0 — t^OO-r~ m ^ c^ «■* ,-o- <3> O^. t^ CO CO O l~ CO I- O CO CO - n§ a> ^ f) e* '- ri n o ^oio« nS -^t^i-IWt-^ 0>C<505i-Hl^ t-ltOt>.35CO CJ— 'h-f-IOS t^3iMJ5— '!> O O r-< ^ -H - ^M -M c-5 M ^ -# Tf. i. to WtC CO t- l^ l^ I- X X J5 .-^ 58SS^ 2=55^;;?? sssg': 36000 — 45000 - = ^ « t^ O - M O {; ° '^ = u 2 5 f. O 1- c a> 1- C. I- - " - " = ■" ~ b. '' * CO ~ " " - - o m r^ n^^'-S^ goj^s'-S s 2 ;; - ° ' — CO 3 l^ t^ «- o> Cl — to i" rr — - O t^ O I- o o o o o »^2° = ;;| 3| l^ C, CO 1- CO B"-'i — - t- »- n 2 ° S 1^ CO c ^-°z O O 1~ O O M ^ ° ° ?:• = S o c;c.-00 t^ - 1- 1- t^ C CO «s^° = i> CO 1- - o o a O 1- 0^ O J- - - •- O CO p g s -.3000 " « '^ - - c< I- 00 - Z w ® •" « s - - « C 1- -'^£- !- z " " " r-. r-. o i^ c a s = 20-.. CO C 1- CO - g c. ^ -'-L-f;'- " :_• •* O O I- «^ O o to cr> -J «c ^ 5 I^ 3| — CO t^ t- o» C« — i~ I^ CO — -0 " "2 C 1- I- CO C. 1- C» C I- CO e« — ^ — t^ o — — t'. o 2 o — o> O t^ O O 1^ s sns^* — CO t^ t^ CO «^ °°S5S lo22-S* — o o O O C t- C I- o CO o o - t- o s c. t- I- - " - - ° " c. .- t- i~ - — — CO " Ji; S ° 5 :; 2S = i^ - o> o — o o o — o o o to r-i s I- n t^ - I- SS-3° '" — 5 ^ £ t^ CO c C O Oi - C t- C. M o I, o 2 o o - s i^ c t>. ^ i~ CO c. t- c- C = 5 = 22 C C - J. 1- c C r^ O C-. o> t^ — t^ O -, CO 2 '- CO r- « o o t^ to 00000 S = - 1: 2 gj .. «5 = -r. CO c I- - I- CO i^ c — C 3> O CI t^ "SS'^S ° CO CO - 01 t>. C t^ — — t^ I^ CO — CO «o CO CO o> r^ - - »^ ogoo^o O — t^ C « - n o n t- 1^ CJ O - t^ o - t^ — o - r: t^ ="^ £ :-o^ ° t^ c - 2-^-^ = «^ — 1^ CO CO '' — o K. ^ ^ £■ ob — I- ro t^ C O O s CO I" C» CO 2 ! " " ° - t^ CI CO CO CD la ci - CO r- t^ 3^ -- 5°« O — 1^ t^ t-. Cl 00 M ^ »»■ — O CO - t* — « « CO t^ S ^ I^ "^ 2 3 t^ CO — « '^ S « 5 = ° •- i^ c o» CI — — e» '-i •♦ r^ O r: i^ - CO O O t- o - o ^ ■* 2 *3 -2 = -3 t^ — t^ o> CO t^ t^ S = ''°»5 c i~- n C* 00 O t^ l^ o — o -H CO i- - o ^: t- o c;g 5 '" 5 5 ° t^oo -0 C C I- - S = = 2« c C r: t- I- s° = = 25S'^:: CO t^ I- !0 t- O — " '^ 2 OS CO = 5^22 o-oj^.. CO t~ CJ> ^SS--*^ -- f^ o C O r^ rr « t^ c c. o t- o o I- m t- - = = 00 CO I- 1- " '^ 2 ^ " =i-2"= o ^ c - o t^ o r- — ,f « O I- O O I- l> lO CO — c> 05 00 = .^ = .. ^ c. -i:- = 5 t- t- c - " " H " - 1- o o C 1^ O O l^ M — I^ t^ C3 CO — to CO CO - CO t- .^ .. OJ t^ CO „ _ — t^ "t — sss o 2 - - = S O O - 31 t^ O s t^ CI o> t^ S;;«'^ = m CO r^ r-. — CO CO 31 CO . 31 — CO o -« t^ °£=S=* O O O O — t^ s CO - — S'-S*^ r^ CO t^ (j> — t^ t^ o o Ci t^ o o o o 2::^^°°:: «o n ^ S " 000 t^2 l^ — C7> — CO CO ^oor^go ^°:: O 3> O r^ — — (MO <0 - 1- o o t^ t^ :o COS -c^ogj -- t- 01 r^ O^Or,2 -0020 — CO — • $r?is ^ « 1^ O « 3i cc -o tr w 1- t^ SS^S5?;5 |S5^:::?5 ???i?J!?^ -* ■<* ^ lO o ??05^r: SS55?.?: . y CO ...-o = ;;-«- £ 5 [^ uj " °K-t:° — C S» C CO — — ^ - CO c-j t^ c I- Ci <= CO -• — --5 CO o o c - t- o - CO t^ c c CO o> - o " s B - " s ° ?i £ ? ;s CO »^ " =J g 2 « 1 !: " CO '^ ^ O P^ CO o o — t^ C CO l>- -2»«S ..OOiOO - 1~ CO CO C 1" " ')• ?? ro t^ to t^ 0-. t^ t- er. !>■ CO CO 00 CM c^ o - c o CO - t- CO o (O 1^ CO - l^ «3 ■*•<}■ ^ ° « s- ^ s r- C- CO CI OS CM CO CM — t- - CO °%" O I- C I- = S S = 2 2 O O — I^ o CO 0» t^ CO i^ t^ — r^ n°5 CM = - s •- = c.ogo* t>. Oi CO o o -H t^ CO -. CO i^ C 1- «3 CI CO = s X- •- ° (O - CO 31 2"- CD j:*^^^ " H ° " 2 O t^ t^ o — 1 ^ .^ <= ^S^'^*:: § ? " S ° " ^^^^^S -=- CD — O l>. o — r-. ^ I- O O CO CO o CO t^ eo-^o «°°::«i: CO 01 en t^ P^ 1^ '- i^ — o> CO o> o CO CO i^ Ol t^ O t- |-^ O O O O Oi 1^ — t^ t» CO « CO t- CO c^^ogo 2 001^ O ° = ?:• § :: - l^ - O t- CO o o -. en CO — 2 '- gi i,^ 2 S ;3 '2!:-°S l^ t^ CI r- CI — c> 1- o '^ " - - - t-- O I- 1- CO O t^ -^ Cl o CO « I- C t^ ?^ C P^ CTJ § CO S ci i^ '^ -. 1- - c cl' ~ - o CO t^ o -- o S ° " S CO. °r^z'^^ Ci - t^ •I' rH r^ r- a. 01 t^ c « ^ Cl S — t^ CO t^ CI CO i^ s = = ^3 ° g = '^ s ^ooo- 1^ ^ o -. o ■a- r-. *2""5 01 l> r- -^ !■» «>g 5:5°s? cooocg n ^ ^s c o o o - o :o t^ O CO O C5 r- O l^ O^ O " '^^ ° s CO 00 '^ 5- ^g g'-irSS t- CO 000 s oi - r~ i» CO o CO o -H CO o o o o t^ CO o o t^ o — 01 t^ m en — CO <3> r^ — — ■* t^ « CO c:5 CO = CO t- CI - c - CJ - CO x.= _ 03 CD Oi CO C O O CI I- S = 2^3 ^ CO " l~ o rj- C-1 -. ^ i^ Oi i-^ OOCO.-C CO Oi 5 S c ^ ^ s ~ 5 = 55: = CO r^ CO t^ w r-. — O Ol t^ o - o R::S2° I^ t^ ^ CO C C I- C ^: i 1-. - c= c C-. CO CO t- CO C CO _-. c o I- r- r- '2 - ° ° :: coo^jr^o 0-. « CO ^ 00 00 c< -H 00 ;^ ^ 5 s '^ ^ CO .. = —.,-.?» CO 00 - l- C C Ci i-"-° CO t- 1- t- - CO I, ^ — i^ nl ^ gg-ooo t^ ^ Ci C5 i^ = s§ o CO CO - -. t^ o IM '^ •* — O C CO — I- 1- CO Oi - CO ^ CO - t^ O) = 1> s C-; - CO c « c ^ 1~ -i ° = t: = ^ j;^« c - o c o 1- CO — 0> O —. r-i U3 I- CO i^ CO 1^ 1^ CO r^ to CO (N « c - i» g CO t- I^ t^ CO t>. - 0-0 t- O t^ l~ o o •- C t- CO C CO ° - i: S ^^ ° « '^ ?;? s ^ o> g CO s c ■- t^ en - = "B° l^ 0-- l- O - 1^ O C O I^ o I- - !>. C -i CO l^ •* J2::S°5S CM 1> l^ CO C I^ — CO C P^ CO Cl c ■- t^ CO o r< o o CO o c r^ o o r^ — a> c» rH CO CO i>. •-1 i^ — o> 1^ CO CO n s ® C:'"E5 c^^^o — CO CO CO " ° ° £■ 2 -. O I- Cj CO :: '^ g ° :2 2 c .. 1- CO 1- C C I- CO CO ° ?? z ^ '- O.^COO 000 1- 1- -- 1 CO c .^ o 2 o 2 CO '^ S 1- CO '^ = « ;: '^ ^^ CO CO = 2 = ° S t^ - 1- t^ ~ 01 -H '=: c< CO '" " 1- J. CO M — •* — r- (N c^ 1^ — o> CO — CO t^ — I- g r^ o> CO en — U, l^ -- CO « i^ r- ^s « en « en t^ — s - C. l^ o — CO o -- -- 1- CO CO O t^ CO 1^ - ~ 1* t^ — 1^ c s; 5 '^ ~ 5 'r^ - CO - i>. en °5? CO =1 C I- CO o 00 r- c^o-.o 1- - t- CO 1^ I- 01 1- '-. C CO CO O) - C« — '- — :^ 2 c r~ t- = r> = s « ^OgJ ^s — C Ol O l^ 2 = S^S i: « '0 2 » 1^ — CO ^ — — 1- CO «5 ^ « « i«3 1^ f? CO o> t^ e» •«. 1- 2 -• Oct- |s'^:=St: 2?l?^;^s :;^!j?s? CI — «^^ c^ ira (C «£ I- t^ l^ t^. OC 00 Oi J5 3SS82 25:3^?5e? ????^ 45000 — 54000 Ol-O 0- — mi^t- ooot^oo ?s C» O 1^ O O Ol - O I- n o - -, = l~ -, .- 1- :-. 1 - C -, i - °°- 2??i:°-5 =:;»='°- ?§ -oooo. oi^ooo .^-noo wnr^-w ot^i-noo — e< — — c40o>— rso — — -, ^ « ^ ^ CO CO «»S = ^ »«i:'-= ;:'';;»S «::S2° ° = S''g.:: 200 -00..0- 0-2^"^° o CO ° = -°5j -225- S«»-- s^^ri* -g^-^^- — CO— - n " ^ " '^ '2 ^l..=.» ,=.=„,=,,. .,=,»,=.,,, — M- OOl-Or^O I^OC - U;— — S— '"o2°^ Sii'" — r.t^Mo> — ;^t-i~''"lr «oo a — oot-o> ot^MK"*" CM CSI--C0O t^oi^i-t- t-)0-ot^ noMi^o oonocii- on— •»• CO— c«— -f— e« — s OCOi-Ot- OlOOI-O l-0>0>0!0 1-=0t^ O-; — l?iO = oo> 0 t^OlO — — r^OI'.MO Ot^OiX oo r~oo — i^oior:— mt^t^ooo CI °;:s°- s-**"" ®-So° 2°£-? 2 = ®-z5 o-o -oot^ori t--ooi^r^ - 2- " -:2 C5 o criot^on or^o — r« — cooi^o > ooj. i-i-i-= = -^ o lo- e o o t^ o> t^ ci CO = — c = 1^ ". .-) — — — (N oor-i^ot- i^oooMJi ^^ -t-OI^= I-CIOO- O — I-mi^ OO-t^O r-. OO-l^- O - g = ° 5:i:gS'^3 o'-°°i: = «§ -cor^-o ooii^o- coooot^ «coot^o . oco^i- „ ^ „ „ — — to -— .t^ nooot^- o^nooi-i~ i- = = on -i- = 02 '^"•^'^2 ~°'~o~ °'"1^Z~~ t^ — o -t^;r. o-t^ o-t~>--. I- Oi oo^oo =2i:'-° ^-52° zz°'^° -|oo- = = ojoo o — ot^i-o oot^ot^o _ l-Ot^OO 1--I-r- no-t-O O-OOr. t-OJ. OC-. J- 00 — „ — c«r^ i-oi^moo ooo — o- — -!• - — «^ - 00 1- c. c t^ - o = ro o o o .o t^ CO cn t^ o i~ = - co o i- = = ;o t^co— — Tr-*ot»co— ot^cooco oroa. r:i^i ^„— eJ-2 " '^'- — — o '"2'' Ss= z^^^'-n °-5? = s or^OO^ t^-r~ = -OOCvr^ 0>00r-0 or-i~i^— r- '--2 1 = 2 = 2- ogor^-o S -Or-I-I-- 0I-0C7-0 -0j'-=0 OI^^"^- '^ Z ^ ^^ ^ r. = t^ r-— — O-O nooooj^ CO oo — oo. i^i~o> — o t^2®°3 2° — «— ®'"®°5;i g -oicooco C4 — — t^t^ ot^o — o oo> — eoi->Of~cco -CU3--5.-OCO- - --oi-c.o> — d „ — — = -:: g^a'^-' oooj,.^^ 0rrt^-0> OCMt-t- noot^o o> ■^ O O O) »^ — 2jIi^o «0 --- cc--r -- °°« '^."isss ® = ?3:::2 = 3 ., _,ot-- ni-=.^o ooo-o t-j^-oo co-i-o2 = oo— occi^no- r^t-oo- o — »^ c* — — en S Ot^cOi-r> t^r:cooo> OOt^O— -crr^o— OOO — t-O — o— cjiioc* oeo- 2*° " — oo i^oo«^t^o I^I^O — — « - -oo ooi^oco 0^2° " Z;°^'"2 --®o'' — 5*^0 2 - 2 5 S *" <= 2 ° *^ ° ?i s Of-— eot^ oco — oco ot^— oo c. r-ooi- c»oiooi--c PSCTt^ 00>— t<.0>0 Or^O.-rO«>. -«o -w-2 t^-* •3 t-eooeo— — t- — oo* t^ot-eot- OOt-i-3> 0«~— OOt-- — io. o c?> o o> o CO -°^5=' 1^ CO - ^ 1^ I- 1- = c^ 00 CM = 2^£?5 - to 01 s'- T*4 o 1- 05 r: o r^ o « C7> a» 2 "* " ^^'^ss r^ 2 C ^^ t^ - S 1^ r^ — 5 S S '^ = S = 2 n ?o t^ o> — 05000 = g 2 - IZ Z ~ T-^ en - - CO S i~ t^ '^s^ 00 « C r^ O t^ - 1- I- 2 = 55:: t^ CO t^ to CTi ci cr, - •* ~ Oi 'n ^ °5 "2^°° — i^ c5 ^ m - O O 1- 1^ " 2 ?5 r- to i^ t^ ^ CO 2 = 0-0 °« = S5£ ci i^ — r- - CO |J = t^OO CO i^ - c^ m - t^ = °°-S — o> 1^ CO =1 t^ CO - t^ »g t> CI — to - 5 - - " 5 2'- ^ 2° = 2° 1^ t^ to t>. t^ giS^S t^ — « - — t> t^ — 1^ s CO 1> — — t^ — CO r- 00 m - o t^ CO ^5SZ° r^ t^ - - t-- ■- CO in CO a. t- - to "^ s to 01 — - to t^ — CO CO — r^ (N I- t^ - CO to CO e: t^ ci to C r- CO 01 — - 1^ CO 1>- = ;:s;;2 - I- 1^ i:-° = S n- 01 to r). — - l^ CT, ^ r^ i^ C — Cl s cr, 1^ t» to ° - ° S 5 CO Ji i^ = z2 = ;; = j:: to — s O I- O O CO 2^-°° CO Ol 1- Oi « - — 1^ 01 t^ CO CO CO CI to 1- TT t^ r- t^ - CO ^ o ?: O) t^ o ^ t^ to « « t^ r» i- o> 2S°S° r^ t-- t>. s r--'-" C31 CO t^ CO c m -- O I- i^ t>- to 55-»| 01 r~ to 00 — -=2-£° s 5 - = = ij - - in 00 - o t^ o o 000-0 Oi — t^ CO CO t^ Ol — t» o> •-• |=»S^= cji 01 t^ to CO o 1^ O r^ 1- C-, « i^ 1^ to « "^ — X 5 '^ 0-. — CO CO g t^ t^ = o> 00-00 go.^ O O CO o o C I- - .- S = 5°- to 1^ 3-. — I- - - CO 1^ I- Ol to - CO 10 t- Cl t^ CI -..„ 00 000 5 I^ o to =i o — t^ to t^ «-5°S i^ — t^ 2°«-S5 10 °2£°- CO — — 5rS2 CO o^j^co° gooo- - to l^ - -O CO — 1- CO t^ 01 1- l^ CO 5; r^oo|t^ (N — — CO ^3 - O I- 0> 00-0^ - ;; ° 1: 2 o> — 2 - 1^ g - ^ ° = !;?2S ^5 o to O O Ol ^ S S i: « 2 = - « 2 °'-2^S t^ — t^ t- «o5J 1^ — — r) «0 — UJ CO eo r. ■* c» 1 -.^-2';: C5 M -j r^ r-^ nt^Sicn kft CC CD t- 1^ r- n M ?! "-"t^ t- I- X X =^ -. 5?5S^ 2^t5;3?? 5;S?S? 54000 -ei^OOO 0..00. •-zsS = ;; g O to m 1- o to 12 2 ~ " = 2 2.0-^ .". " ' - ~ '~ T: o — — M O 0> M t^ — O to O e<; O 0> J5 O to 2 U ° 2 - to {^2 «"- ® 2 2 '^ «; 0> to — — 10 — t^ rs ca » t« to « o> w w >* ° = O^to rjg to — "-H — to to to to — — « 5 ?i O - 1^ °z:«SS° O O O to n to o o o> oog n to — to to — 00 0> (O to (O to to — * (0 «^ o o t- to o n o to 2 00 to O O to to - _ ro; = s ■0 » - to to to gto to 2 = - fi •? °;^z-s?j «2 = rr = ^ to r; o ■* 2 "■ O to - -, I, rj o c to O to s - '-^ ^ " .-r 2 '" " - a CO - = = - = i- - o o to lo a> -^ to -^ C« to to .n o - 2 » 1 ^- = T-l to M .too 2 — » 12 " to to n — M s::'^5 = 2 Mao O O to to O O •^SS°::5 § :;- °5« •^uSSS — rl '* 2 2 - CJ JO ^^OOtoO «5° Si2Z^'-S «"s°°2 s to O z^° g .0 _ -2 — Oi to e» r^ « ** — s ;; ; = 2 '- c^5 o to o a o to — Cl r^ to to ro e§ — 2 O to o> M a, o> — tn to -H to CO CO o> to S = '' - = - o> - o r? o ri - 3> to IN " - «g ^! E-° - to n - n to — in to 2 1: 5 -^ = u: ?i fi - " .^l^o O to o — — M to -t — O — to r> -. to S 2 — o o o gOOtojj o» to - ro c = .0 to jO ^ t^ = = ss= o> O M to o - o - - r: to c^ 2;:- - CO o "H '^S^ . to to to = 5=2 = to r-. - = = £2 to O = 3^ t_o O 25 = :: = ;: g to O O to O = £ = 2 ; to to = r-. to c a> - .0 JO = ^ r«° o - to r; n o to o m o o o — to 00 = - ^ CO a — a — — ~ 2 •1" ~ ^o^ = i: = 2 « =1 = = to r-. 2S2 I- = O = to o J! to o o r; -i to o -• S o o - to to _ PJ to CO t_o J, t, = S = ci 5 '^ " = t- = O - = — O to - O — to fO o 5 s = Ogto 2^ = 51; :r 00 — Cl to to 5SS=== r^c.^ Z^5?=2 O O O O O to 00 o o 2 iir io ^■^ = 5 = to = lo O I, = 2--S =-S2Z= -ia = to to o - r^ CJ to c to o - to ^ — — s = - to rr to — — o» M to n Ct to ~ '" 2 « = = -S = 2 -.Otoo. = £5 ^^ O - to c O - n = = to to CI -s = ^ X ^ '^ cr. 10 - CO ~ to ~ — i to ci n - 10 d — .r ** O — C< ci _2 = ooto s to o O to o = 2 = ^S - to ^- = = 55: Oi to s.s n ?: to o — ^> to - O t-- = o to — s hi '" — 5 t- to — — CO to — CO 00-00 = = = £'-;; c.S ^OOOto^ S'^x^-" CO to n — O CI to — M to « S5 s; = '^ to to CO ri to to o .- - 5=^ ^ ° S 5 *- toO.toOCJ s " O « to to « — ?: 10 -- to * M — CO o> S '^ — •♦ '^ to — Z2- a o o c. = n " !r ~ — CO ?! CO — ^ e« to » — Ci t- to go^ togo-OO g = = SS = 5 "^ 5 3'-2 rj o> o> to (M 10 « 2 - -s = s = — to — to n to CO — — «-2 — to to O O to — o — |2 togjgoOO ^ O to 000 to no ^ n c* to to CJ» to - — to to B°. ? — *^ — O to to o — n «)^ £ in 5 = 5 = = "2" — to to to g-Otoo JOOOMO.O 2r?r: ;^ « to crj CO Si ;^5^5^??^? 5? = l::?1 ?1 V^ ^ !? =: =:t :^f2r; ?? •? -3 ?^ P: S55??^? rH 1- = O O Ol O t- t^ - « O t^ CO 1- - — in -< CO C CO »- CO O 3. CO O I- - 00 1-H o> Cl t^ :: = '- g S a< -^ - 1- o O r- o-o-o a> o I- c^ o CJ> M oi CO — o o t^ O O -H t- o lO t^ « CO CO - -. tN. - 00 o O i^ ^ ^' •* « r- O =5 t-» r-~ o O «« t- CO - t^ 22-S^ t^ CO 05 t^ t- — o o CO — 22"-° O t^ — t^ ro - M C» CO en - o CO ^ CO t^ t- t^ O O O 1-. o s = = 2:;::; t- — t^ S° = O CD o - c t^ m o.oor^ t^ CO — a> o> M O O t- O £? CO '^ ui - ^ 5? CO cr. 7. - "" -i ■" rC uo ^- = CO ^ ,, o o - t^ Cl O M O CO o t- o t^ (71 I^ o o t^ o o> — t^ o o> O — UJ g 5^-52° - t- r- -- ^ s t^ S MM MM — — © a> o o o i^ t- CO I- (O O o - o o o - 5 ^S^'^M 00 — « O O t^ « — 0> '^ 5 O M O M t^ CD C* -> t^ O « O t^ Oi O) -. r- C^ t^ t^ IN ■* -< — -t CO t^ - t^ — - — . M — CO c~ 00 = o 2 ;^ ^ o £^ o ;J2 t^ Ol O M Ol o = S = £2 CO t^ O C-, = n o CJ -r r^ t^ a-, n «^= = o - B 1^ - C - l- - o I^ o o i: *- ° ;? 5 o ^: o t^ — O O m O t^ CO IN g = S ° - - CI r^ C t^ - c^ l- l- o o o O .-'. CI o - 2 :: = - g CO O 1- o o o CO '" X S ^ '~ K CO CO t^ - CO t^ CO CO — t^ I- g a-, r: .r i> — ^ (N -< <£) O O t^ C CO t^ O O O M O cr-, t- CO O '" io - S to - O t- I- o — M — s t^ - a> I. c- t^ CO 10 ,. -• — n CO CO O - I- C^ l^ ZI ci 2 2 -. O O CO t^ O O 1- - o CO °5°S2£ CO CO — i^ -22"" t^ Cl CO § 2 5; z: « '^ t^ CO o l^ o o o O - O I- M t^ - O O I- = 2 ° ° S I2 CO t^ C = i- CO — X -^ -r 2 '^ £5 o o n o o o i>. o o I^ CO Oi - 1^ >- 2 ° 2 S - °-|2° = " s 01 CO t^ — CO t- CO 000 CO l^ O c:i O — O O I-. C-. o J- .. o o g ^ = - " s — — CO 3) g - M 2 "^ t^ <3> CO 000 O t^ O c. n °°°-s O CO O t^ — CO t^ o o <= i^ o» f^ — s CO r i ~ 2 ** ., CO 2 S - s = s - m o I- o S in ^ - CI t^ — o o o It _ 0> .^ — O CD 1- - I^ 05 ° 5 ;; 2 S ooog- J- - in O o M ~ ^ '" 2 " I;: :: CO CI t^ t^ o --°2S t^ r^ CO ^ I- 10 o> i^ §m5 ^ O - I- 1^ o o o — o o to to 00 -li* I- Oi " w f- CO — r~ r« CO CO CO CO = o c - o 2 '" ^ U '" SS"=S O CO o o - i~ i^ .-0 CO - c; t^ CO ° '" ~ 2 "^ CO ..gooo MM^OO g.^0^0 c: « « '- ^ — — ^ |= = S = .- r^ I- o) - ^?. ^-25S - 2 ® ® 2 :^ S S*^ iJ 00,00,0 r» o» M tN. o> — OJ? s« = s° .^t^OO.- O) CO to " — c« SS^Sl:: e?,?,^'^ i-i CO t>. ri M -* -1" Tf -* lO O -> t^ ^ '^0 »-:, CT — ono o o t^ '- m m »»o-.OMr~ |C2»^^°"'«^0*00 OOt^O— OOWCl^ — frOO»*C OOO --IlCOOi^O Ol^OMOO o 2';:°55 °z'^o° 25'-^;" °f!J°T; °mS°-" ^ C- O C. O C I- Ol i- CO O fO I- - (7. cr ,, .-^ o 1- O - O I- ^ o . O C - 1- i. - c I- o C 1- - ^cix — tc- ^ ^-i- CO-- -n ~ o - — («oco oi^o — Ota ooot^— O •«0> CO o o o o 2 t^o ^ m -- o t^ cs t^OOt^Ot^ o — oooo O (O O O O t^ t-o — t^O oOeor»«^o »" — ■♦ CO — t^ e< * -" •♦ « t^ CO o o o l~ - I- O i I- o - o o oi^t-oo t»t~-i-oo r»ot'-coo ooeocoeo «^ !0 o o t^ o t^ c 1^ ?i r» o O t^ O t» O l^ CO oot^o— t^or^ai CO (^ X lO — — d ^ c -r — O Ol CO l~ o o t~ — O t^ o O O £0 O Ol o o ty> CO o C4 — t^ .O t~-— O CO(OCOO)t^O criOl^ Or^Ol^O OO ooci~ — ct^oco t^ — cnoio oni^ooo o =. o o — — ooinc^ro — •• O I^ CN <* — Ol^— — O I-OIO — — t^ o — — t^OO 0>l-OC0— OMOO o -> o o t^ o c to — i-» o> r^ o 1^ Ol o l^ O Ci l» O O O «-. — O 1^ CO t^ o — o o — c CI — l» o o t>. i^ o — i~ i^ o :? o> — OO e^OCOl^COO OO .0 O l- Tl- 00t»0 — COi^I^O O O — Ol IN — O C O O — f- O 'Oicoi^ oioooir-^oi Ol .-0 I- CV OO t-co — Oeo O — t^COO O O r^ i^ O i- — •» o 1~ O Ol o — o 1^ O O I- O CO O 1^ t- o — o O CO O 1^ o t- CO O O O CO Ol opooir»o ot-»>^oo t^OO OiOt>.Oc-» CO — — o eooioieooo c:; •-: c; c; r- I- ^ t. o t. t> z> o «^ " — o r- o rt ~ — ' I ^ 00 i>.000— Ore-OO 3-. -i^ s I- = - I- CI 1^ i^ o « i^ I- - o c c. - I- t- c o rt i^ n o o -■ g =oS i^ o = ~ ?^ CO OS i^ 1- 1^ o i^ o = o -- oi o i^ c o i^ 000 000l-0> OOOt^t-. C»300--0 *^"^°'^g r-S ooo'-t^ t^ot^i^t^ — 00 i-r:r:i^- ---t^oo r»i^tooo) oo~t>.e<: r^ot^ccr, i- CO C35 oj:--oo goooo ,^..^:e C5 cc^-r<:t-. t^t-oot^ ooi~.ci-. oi---t^o -csi-i-c. CO CD ~'";:;'^[2 ^z^ci'^^ '^"« 00 ooii^«« oMot^re oo-Mt^ M-t^t^— oomo--i^ " ,^ c< >^ « — 05 cco) — t^— — onot^ I- CO CO 00 r;omt^t^ cjo-i^n oio-o© t^-t--t^ oio-noo S CO 00 OOO-O t^t^O~t^ Ot-Ot^- I-nOCS- «t^!^r2t^'- 00 o?:t^o^ai — t^ci-— Oct- o 00 l^O-COC --l^C« Ot^l-OO; O'--.!^- rjom-rto tc- tC" 1^ —00 lo«o«c«--.c^ 53 '^"" 2 "^^ t^ t^ :<:t-oore oooio- «^cor-m ~ocir-o t^nor-.t^c 00 r-C*—! 40C^-. I^« ^ t^^-CT-r-O oo)-.o/t-.coo-m o-no- oot^oot- ^ C-. t-ooo n '~ n t-r-o r- r- I- r« eo « rl t^ g2 oi-co — o i--.o>t-n 000 § = £-:::: 0.^000 ^^o-t^ 22"°° S"°s::« § -coot-o ocomcit-- t-r^o ic^ooCTii^ OCT. cct^o oo-H-t^ i^oco-t^ nomo»-c g CO - c 1- = -, - I- - c I- - C-. c c c .-c « CTj c^ i^ c :^ n r-Ol^t^C^ I^t^I^t^O ni^O 1 t^CO-OCl -r-. t-CO Ot-^CO- CCi-Ol^I^ -ooooc § OCI-cr. t- C-. -I-C=i coo- 1 CO in C-. r-OMo or-oc»-. c: oi"---!^© coi^ — fo t^ooot^~ S5 t-OOO- r-O-OI- c. f. ol CO i-=i~i^- C200--0 i~oo-o a, -^occoi ocot^ooi- _u ooomx re 0-0- Ct^t^l 10 - .oom OMOt^— OMOt^cno c> n <^ o> "•*l< '^ w eo «M„ ^ J^S«::'-Sg-::5:'o = »- o ;;'^2r° S? = °c;o 2o°'^= l: = 2*S -- = = - = CO _. .„. _ c. r.-S-:: -j;2S° '^^S:::: =55::iP =t:-S^- CO — C<30> t^ooot>. wot^oo — omoi^cr. IJ oeoot-O) o~ci-o i-c- -^ CO i-oi^ot^ n — — t^o OM'- — i^ooeoi^ «^oo — -c^ „«o "COio i^^i-. U5-" in 0-- — ^ t-i-— O— oiocoore r-i-t--. t^s C. r- i^ « '^ i^ i~ 1^ « a CO -- o> 1^ t^ — 1- X«OC«" '^ «o „„^0) « r- t-S °°'^°i;? ^'^^irs 2 = 2 - ^•£-2t^ 2?5f^;?*c;; q:?^?'?? g5&f::t^ [z?^^^^'^. S8S82 ^^^^n J^gg : 72000-81000 n — a> — — CO e^trc* w — i-OOoa> Ol^OOOi i--.r^c- J. oi--.-. 1-1-3. c-1- 0" " o o>— 000 eoot^oii- COOI^ MO — t-OO MOr»P3l^Ci eo — o«^o— i^eoo- t* oooir^i- t-oo04— t^— oo>e^- ^-" """s*" =**i2°° COS i~o>a. i-o coai-no a: — oi^co t^« — oo coot^ncor-. — CI •Wl-.3>«5 — — — CI CI— t^O 00- 1, — - C< d - P-. — o i^o>r»o — t^ t^ooooo t^OT '^'^2~'2 r.-co-- coi-i-roo t^-o-o co-pon.-a, — t^— d «o — r»— d— I- vco -r — _cii^ o-. c.-io -J.OO-. t^o nr^coooc — — c po M i^ i^ X i — i^ — i.. .-0 i» .;. — — a. 1- d — d — pj — t» — no M po — CO s oo>o — i^ — ^<<500 ojooon ot^i^-i^ 0100 — a. d " ■zz'^ ::S2gS« ooo-o.^ 00 00 °°S2d °'^-M° "^nS^® n-'"-° -«-OOOl> •-2S -^^^1%° 2«-S°ci 00 OOOOCO c^o> — i-o o«^— 01^ co-ooo cr. 03>l-0r-. s — - - -0 ^ « I--0 I^OOl-OO t^-OOt^- 00 ot^t^t^— snt^oor^ ocot^c^A — rooio t^^o — o>o»co ci^M r>oo-.-^i^ 01--0-0 CI- _^^„t^„ U3 g t^wt-c^r- a, OiCOOi - Oi i^ — - « - «^ C t^ — ^d— — t^ coo — 1^ CO Ol-rl OOOOI^O Ot^OI--!^ o cj o CO — - ?2 SS;2°£ °S'^J:° ::°°2- o-o.^o cooc^^s- — — d — g 0!0 — o«^ or^o — — oo.-o^. — COt^Ot^t- OOlI-l-O- - -^ -d^ g CO -Od - oot^ « — oeooo rH«n — ot^ 5 -o> — ot^ cor^t^oro i^-:ooo o»a>t»oo ot^— »^— - -£>.•♦— t^r*- — d U3d cr> — T|i«) S CCT>cc-o oeo.-ot^— ot^oi«- t^ot^o- — Ot^Ol^ ort— „0> COCOClt-r^O O-CI^t^O — = = '=» CO r^— r^t-o — — joo— 0-. eo — 0— t^i^oi^co eooi^ — o> — ^^ "2" 2 - '^ 2 " 2 " " " - d t- — t^— CTir«i^eo — o t^a>M — t^co S ot^ocoo t-Ol-OO O-t^OO -0-. COOO — O-Olt^l- - 2S " ojjr^ 0-0 = ^2 °^!; = ?i'g 10 ^o = = 2 = = -s= ^-:;- = - ;:j^ = *-= s;:s = ::'^ 0J..O 052°°- 500..= - "^SS'- ^i^j^'g °?s"^ °'^S2- s;s::«2'c; t^Ot^ OCOO-Ol- (OO — t^O£^ ^ — 000— Ot^t^r^— t^t-.0)O— (Or»t>.0>(0 i^t>.oaic»!0 (O dt^ """2 — — "— O>00U5— — — tN.-0 -C^««I-0 OlOI^OO- CO t^OlOOCO — 0>t>.0> l^l^0»00 MO — — OOcOt^— CO 0t-.c0^, - — «„„_(OdCO -t^— 00 ~2~ "2 S -c-c ^ -i^mco t--e- = to t-Oi- oeo — 00 i-or5t>it-i- D-5 °°;;;S-° S-«2£2 53 5d — — 00 '^ — oi — o>*^-fd— *^2° — 2 S*^ 00 t^ — i^ot^ oot^eoto coi^eo — — oi^OOt» — 0!0003> .* - - - «5 1- - dO»<0-* CO d JO — d d 25*- OMOog- 0^-00.. ^ — c. 1^1^ — r~3iCt^ i^OOcoi^ oo>«>.— — t^O) — o»a, eo«ci^-.d- -co *" m*^-"^- «-.(-:s> o«^oooo> o>r^i^— cot^ oio 00— eo— d — r-SJ COCOOI-O OOOf^C3> cot^ooai — t^ooo — — o»-eot- o ,5SS; S?2&§?2?J SS?SS5:i — 1 -< ?■> >Ti ri -M CD rs -t -* lo u5 tc cs t^ t^ 00 00 OD X :7> 5i _ 1 - en CO :r CO I- CO CO - Cl CO t- t^ - O O I- O - I- I- O O 1^ M c 05 j CJ - ^^-t^ :oo-roo l-OO — O -l^OCO-O 00 en O t- 1- - = 1- I- CO M O a> 31 — to— " Z " -" _ t-t^—o- «^or-M=i r-t^orot^ o>- — r^^J -.->-.ooc-. ^O t^OCO-O OOI-t^- oo-co- t^cocoot-- (Mo __,^x,-. ,^„^ toot 1-- ~ 00 - - CO 00 Ol-ro:OI- r-ooot- ooco Ol ro — t — e* c.|::-— -s°? '-;^^5 - = 2° ^l^;--- o|--| =S2i;. ='^.^ CO t^cOCJJO— l^OOO— OOt^OCO -.t^OlOM cooot^oo I: o I- a-, e. 3> = o 1- I- =1 n o o 1 " = - " "- I CO r^ — t-oo oo — no i^Oicocor* ooit^'^-' moo — — c^ ^ -00.-2 2 = °^° --<= 1 O 1;^ o^r^oot^ cor-=-, t^o o--or; t-eocoot^ o-ocot^-. t^ o o - .-o o o - - O J. .-o t>. 5 t, ^o-t- ot^oeot- r^ococoo -cooooi t-t-ca. o^. „ „^_ ^ C4C*-. OU5 —O O^'O § --,,._,o ----^ ^'-^ to '-j:2°s S^'^SS °^gn° ^"^'^^ °^S2°io- CO o n =^ o i^ --t-oo -<=5J Oit^OOO --COOO —Ot-t-— 0)03i0t^ t-OOOlt-O CO OCO — OM t^— COCOt- COOO. CO 3»W-t^O OOIOOO t^OOCOO CO^Til-COCT) — t^t^OC0I>. -.It-O-Or^ --000 C3>t--l _|cot^cOOO t>.0 — OCO CO— 't>.OM t^r^— 'OCO OrOCOt- — o jn„co- «--2 "-^- --- ^Q o t- o o - o o o t- 2 g°° W3 oa. l-O- -OOOlO COOOCOt^ O-t^-3. :or--l-CJ>0 ro — eo o — ■ to — r^toioo — — — ■ CO t-OCOt-O CO — — OO t-— — C5 IN - t — t^310l-. ococot^i- OCOOt^— I^OO-t^ OOt-t-O — >ncj>— tc-T>c. o-r^gjo ot-go- 022 CO t^OCTiOO ot^t^co— oi^ooi^ —oimt^o ocncoooi^ ^ ^-^i:*^ g = '-°S °5:° l^t^OOO COCO-t^O) -OOt^O OOOC0-- t^cot-.Ot-CO r--- — CJ> i-ooot^ o-o ^ CO CO 2 t - - |o«g..,o 0..3-0 -;:2s.5 2°::2° =g'^S = '^ CO 1 - c. - CO 1 - - ; OOCTl-O t^OOlOC -Ot^Ol^ Ol^CSOM O-.-Hl^05° C-D — I- CO r- CO 1- t- — 1- « o Si Oil : -COa t^c^ — — CO- -co CJcntiNi- lOOOt-eo OOOOO r-MO " CO 1 -" - - ^ CO 03 — i^oot^ — cocot^o — i^orom t^ — coo>r-. oi^ooocTi it-o — i-o CO — cot^<34 i-ot-i: S| -- g2--S. 2 _^ s i-o-cjiO) Oit^oot^ oii-coo- oooo- -1-0. ^-o- (M (M t-0Jit3>0 OOt-OO Ol^-^ico COOOl-3> 2°'~2- — _-|ocoocor- co-t-t-— i-or^: CO — o oi~ — CO— ot^ooo t^O(35r-o — — ooo r-co — cot-i~ IMOOOO t^O — Ol^ OOM- tH — - 5^ °°°o5: -^°:2S? ="-5 = ot^j^ot^ 2='S'^r ;;®g ^ oo>i^oo oo«o— oooot* —cot^oi^ oo>o--mo o CP C£ l^ »- 1^ l^ 00 CO 03 r. ?gs§« 2;i?:j;??? ?;??5 ^ m 000 -90000 o ,-o p" n «~- 1^ o — 1^ c n a. t. 1^ o CO ^ rr r- ,, r~, - _ ,^ - ,," "- - - _ , 05 "o t^ "o — ~rC "r^ CI "o cT ~o -. O "n o.r. m o o o c „ I^ o» „ o t» o o CO o er o „ t^ o o ^ o „ PO c "* m •" " ^ CI _ ^ I, M O) — o> o n CO o ^ w o o> (O r» o t~ » «» t^ o" ~rC ~n o o> CI "a ~a CO en n CI lO c« r» m r, r, " ^ o o o o o ^ M „ o „ ^ „ f^ m fe " t^ " « '^ " X "" " CO ei v: •^ -^ !>. _ r^ ,^ ,^ C5 _ rr ,^ ,^ o a> ^ i^ _ O 1^ Oi o o _ „ CO ,^ c o o ^ n o ,^ "77 n c •r (O n c •r o o " ••* "" -■ - _ m _ ^ r^ CO Ol t>. t^ o M t>. m o » ?» „ ri on CI m ei (O n or. " - r^ o r« «T> ^ O ^ !>. „ „ !>. ^ o o o» CO c r^ o t^ to ^ M I^ o r^ eo t^ o» o (^ CO „ o r» o ej PO o fT, "0 ^ t^ >n - '■' ~ ~ - 1^ „ 1^ f^ ^ O o o t^ » r: I, „ ,^ fr. _ _ ■n (C - - " " ^' o n r. m „ ^ „ m o n r> -r> f. „ !>. C3 Cl f^ ~ fo o c 1^ c-. CO CO ^ CO o o o „ „ ^ ^ o o ^ _ _ o o to C7> - r. n o n r~ „ n - -^ r^ „ ,, „ fr- t^ O r; — „ -r; f^ o o o r- = ,, r^ o — , CO o Cl n m _ n r" 1^ r« r- _ n. X r- (C CI " "•' CI r- „ „ ^ ^ f^ _ _ CO _ 1^ 1^ I, a> Ci „ f _ lO r" _ ,., ^ _ c< (-) o •^ t^ n - „ f^ _ „ M CO «^ 1^ o t^ o CO » CO r^ o Ci o o i>. r^ r^ _ - c» «^ '" -_ _ ^ ^ 1^ _ r^ fO .T, ^ r^ „ o ^ V o p t^ t^ o o CO t^ o o i^ 1^ C7-. CO r^ „ ,^ „ ^ ^ f. r. o - _ (T, ^ -I- •* - '" "" _ ,^ „ „ f^ ■^ £^ o r-, o o r> — ~. I^ r; cy. 3-. 31 (^ „ - -T. „ ~ _ (^ "* ~ " ~ — _ r» 1^ fTl — , ~ — ^ f. o en t~ rr^ o o 1^ o to O ■^ ^ ■^ _ - I, — o 1^ - -1 „ t>- ^ _ ^. o ,-> c. o ^ t^ o a ■^ r^ o CO [^ „ o a> ,^ c _ r^ _ c ^ ,^ .,. 71 . .... ^ 1^ r^ -» _ " C-. z " C? " es " " CO " •n o r^ _ ^ - ■^ ,^ f^ — „ cr; ~ „ ^ „ 1^ C" ,^ m >» P- •» ,^ - _ ,, ., — r^ 'l' . „ ,^ ^ ,^ ,, •r If: " " ~ CO " S a. ~ _ c — _ ,, _ O o O o ^ ^ f^ ^ ... „ o o o 1- ^ p' o f-l es r^ o c :r (N •r o Cl X (O - - - o o _ o _ „ r- O „ r> o ,^ f-> ,^ I^ ^. — ,, — O „ o _ ^ ,^ -, ,-. _ ,» o p. o c - ,, ^ = o _ >-. <; " c» r: " " •* e: ~ CI t^ " U n " - CI -»• " (C "" „ l» _ ^ „ t^ r^ _ 1^ — C^ r^ _ — cr 53 "" *" " " o o> ^ o „ i~ o o fO r>> 1^ m pn f^ o 1 ^ ^ o 1^ ^ r- _ a. o CO o ,. o CO c. _ o => „ r» o n ^ ,^ c c* " " «o iii = c< •♦ "" d CI o r^ ■* (0 "" " 3-. 1» „ ,^ ^ ^ - ,, „ _ 1^ f _ o. t. •r C?l c — = - "^ c< o o I-. o (T. _ ,, o •oS o ,, ^ _ f» o eo o> o o ^ CO „ CO „ CO ^ „ ^ r» o c ~ eo a c ^ ri ec ■-T tr 1^ oc :r ■y> ?> - ^ "" •M "' "' ■' r; " '" o ■ - " tt '" - s o> I^ o o O t^ o o o CO o> — r» r^ o " " "^ — CO CO CO o «> - 00 ^ r^ o CO CO t^ — i^ e^ CO § o o> o o - ^^SS'^ oco. 1 OO a> o o t^ — °"°sB t^ O I^ CO o o -- o o> o o> O t^ o o t^ if5 00 2^°SS O CO — r^ O 00 - CO - C- o CJ 00 °-:^5S t^ o o - CO t^ o t^ — — CO t^ Cl t^ 05 O CO t^ 05 «s«°;: t- t- o CO ^ o-o«r. CO o r^ t^ « CO CO o O o O CJ t^ o CO o t^ O O O CO CO t^ C^ -H — s t^ — o o o — — CO r- a> co^g5^c« ss 2-;: = £ — t^ ~ O t^ C31 — O I^ ^ 0> CO •* CO t^ -^ 'ss = - Ol 1^ O -- t^ Ol ?^ O O t^ — CO CO 1-. 05 -< CO t^ o — CO i^ Oi t^ o O t- CO t^ o i;j°s°S° g O O O - l^ CO o t~ " O) 5 = 2 CO CO O 1^ t^ - 05 " ci -- t^ ? - O l^ t^ o> s — '^ CO O O O I^ 2 ?^ -::§;:-» s = °2S^ t^ - C^ O t- oog CO CO e-. CO 05 i^ CO -H o o o o to ^-25::° t^ CO o> - t^ to t^ (N -i ^ o — i^ CO — t^ s °^-i^ o»-o„ rt r- O s 02JJC0 "Ss-H = °-s° °'-^SS oi o CO r- t^ o s '^ '^ " (N t^ o> I- t- — r- — Oi o O 1- ai £; o o t>. o o coogoo 05 !>• O CO t>. r- t^ I> CO O -H c» O O O CO s o o c. r- o O - CS — t- c- CI rt :^ O CO M O t* «S«-2 O O « -- CO t^ O O O t^ S°«D=g t^ CO o c:i - 5S°°S CO c c- s O O O t- -H CO t^ « O t- a> o CO t^ o CO — ai o> CO O t^ « O O CO — O t^ Oi o O I- -H O o rt o CO CO (N CO t- CO o (J CO O O t^ — o o t- o o « O) CO 05 t^ o t» en Ol o - - — t- CO CO — ^ CO o o t- t^ ^ o — CO t^ c;> o — CI m c:i CO o i5 - t^ OJ o t^ - c< ■* — -1 o o t^ CO « -H oooc^« O O -- t^ O t^ — O — l^ o CO CO O ~. 1^ o r- o CO CO t- o - CO oo«..^ o c:i o i^ — t^ CO 05 t- o « O t^ — O o CO CO Ol t^ CO o t^ CO 2 = °°« 1- o o o t- in rt Ol t^ o c» ^ o -< o o o> C.O-.OJO °Sj-<=q: t^ JC* O O r- CO O O t* CO o rt O) 5 S-^^q: CJ> C- t>. l>. o Tt to — t^ rt o ^ O O t* CO OJ t^ - o m o o> ^ i^ -H r» !>. 1^ CO CO CO o OJ CO CO ^ »°s-« rt rt n o t- C CO CO CO CO o CO - t^ O O O t^ o ^ oj t^ t^ CO ■ c^ "^ S '^ ..gcoor. «=»^ss CO i^ CO o =■. a> O CO O CO CO O rt CO CO - i^ ^ °-i'5 t^ t^ — -1 t>. CO c:^ o -. o ° S 2 " 5 o t^ o o t^ o CO S ° '^ '- ° 2 .- gj = o rt«l:l Ol I^ O CO O I^ t^ CO o c CO — CO t^ t^ '^ rt o rt ^ CO f» rt (M - o o t^ — CO •n o t^ r^ m o> t~ O — CO O t^ CO c^ — o U5 J> — t^ CO CO OS 1-1 oi i^ CO o 1^ CJ t^ o t- o rt t- a> rt rt Tf goo lO CO 2 "^ ~ — o l^ o CO O CO o o t>. CO -- 1^ c^ ^';;°z°^ CO r-t f^ O t^ O 1^ CO C3> — O rt «^- rt O 1^ — CO 1^ O 04 rt « - c< O t^ 1- O 05 o s t- — t^ O CO c< S23 CD O t- - C3» - O O CO t^ O CO -« CO o.ocog f5 t>. rt l-» -1 oogjgo s CO o- or- CO 1- t- O - 00 o ^ c» «ZS CO o CO l^ CO O t^ S5SS2 - - 1^ l^ o "" "" 2 oggoo r^ o oi CO t^ o s O O O — CO t^ Ol rt rt 1- O O 3> c^§ o o o r> o oo^og *^5S^'' o 2 o o o en -H t^ O CO t* «s r- O O CO CO e«3 »- o> . (N — - ,^,^_ — oint^on ot^oi — «^g O — O t^O — OCTlO t>.o-c^oo t^-.-.c^o 001^-01 coooMO t-«i-ro- a. i~i^no- noo ooit^OMO t^oi^ooco g-og »co-c.- S-*-^. - " S t-.P50 — — 1^0— i-cioi — t^ 000 I'- ?^ — MOOr^ ,^|CT>-.oo»>. onoMt^ 1^0^-0 oi-o>ot>. -nsoa r- s CO n5i°°gl 2°2'-° S'^Siirs =:;°S^ :: = i:- = 2 CI ='°'-2° £° = S'" sj: = '^2 5 = 2 = ;; ^ = = »«5 10 — - ocn t^—t^ — t^M cnOMO — — 22*-'^*' ^ S'^ ^ t^orji-- OOC-; i^s-ci.-o t^-ri^os =r;t-oor- 0— ^^— — ^ — — a. a> — •♦ ioe-»3> «« CO — — ^ r-t^J. no t^OCiOO OOt~3>- ClOl'MO l-C:?. -t^M <£„^0 X c„o— - 0-(0<0 icjoooo — -t-.-cn t^-oot- «i-ot^- c-. Ci t~ '• ■-^ t^ ao CO Oioont> 1^0-nt^ — noi-t^ t^c^i^oi^ o> n n c - = -- cj- ^ « ^c; r-r:t^ o»o -= ooooor^ OM — t^— ot-o — w4 r»a-. t^oo cct^t^— ot^« — n — " - -« 2- 2- -=^^- t-on ot^— ocoo — or^— no — c» — •* — CO CO -ni--n nnOl^O 01-000 OiO-S>r^ OOOt~.C-. 31 — O OOOf»—— MSlOlOl^O g — nt^— noc>o>t>. o-ot~o on-jci t^ — cncn 0-. = t^-.OOr:r^ 0-.M03>re CO 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 -r — — c<— — r» — - - -00 0-00.. 2 S'^£=='S co'^iot^-nc^ — t^ -1^0 j:: OnO-t- r-3-. -OCl nn-OO 3. l~MOI^ Ol-non- «-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« :\: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 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 + ... + //-') 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 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+ ^/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 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. ) ...\\.'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•)• 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" (••■ -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, (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)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« .' { Pn = ftn pn 1 " '>„ />« .' •/« = + ^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„_! 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:.\'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 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 '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 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) + {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 -;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), '. ^ 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'' 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 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>)' 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 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) > ■<'• (^''^^^ 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- 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 // -■

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) = 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-) , / ('), and if

pr(uinnfti 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 (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 {.,•) = 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) ... 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 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, (cv) = .v'' + (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 *' 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^. I3' 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 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 oHang-(-(»,: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+/ ( 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 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 = 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 = 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'" ^ •" (^*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 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 — ■) = 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 \) + .^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 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, 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 trianC, 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-" ; 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^ ' 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 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 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) ((/, &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|^ 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 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, (lOS" 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. ' /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. 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 /"(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 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 (.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 * ... 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 (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

(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 (■/', //) = ;'' + 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) = 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 = ( {") ; 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 «|)'(^;, "{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, ) 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— 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'{,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 -' (.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^ = (//) //, ; 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^- 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 (y) = sin (y + a). (j>{z) in (1552) becomes {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

' (a) \P{u + l,) 4.{;) + l4''('(a) ..,,,. _ {0 (a)Y ;^>) m, .

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 l{.r,/::) ^ ^ ^.n/z) ' ruuoF.— By (1737), V'„a,4-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, 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 >/ + (hX\ + \//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)) 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/- ({fI,,-(L.)rl^{.v,i/) and rp{d,,-(l.,)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 + (.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 ..z... 1701 : (!>^^ = 'k,,.vl+--i4'...'ir~.+u^:+J.n,) = „,/,,. + cf>_,.. 1704 ■,/) = (^,,/y:.+2(^,„,//,.+(^,,.+(^,//,... 1705 K (,'//) = (^,,//^+;?(^,,...//:.+i?<^,.,.//.+<^3x. 1706 In these formnlff! tlie notation ^ 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 ..„ j>.,. (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' :A.^ Proof.— Bj (1708), ^, = 0. Therefore (1704) and (J70o) give these values of t/o, and y^,. 201 BIFFEUENTIAL CALCULUS. 1713 If 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{.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,„t-'i'^M * + iK'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 r (^y^) = <^. »/.,.+(^,.v.,.+(^.r... iMi-rjrrr functions. 205 1721 .r ('///-) = ■:,. »/;+(^,,//;+<^,.4 + Cf),, U.,, + „ //,., + are found by solving the last three of the same equa- tions, by (582). IMPLICIT FUNCTIONS OF TWO INDEPENDENT VARIABLES. 1725 If the equation

.,.,+ (^,,.//.-+(^,.-//.,.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,=:) = {(..,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 y = '"> ^s = « ; 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 =

, 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.— .r,. Substitute this, and iluiioto (n + hj-)"i/„, by U,, thcreforo //)> wliere x, u aro connected witli ^, ij by tlic c(jnations ?/ = 0, r = 0. It is reciuired to cliaiige the independent variables 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 — 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 ^-\-. ., d^ ) d- = (\ {d^, dy, d,) u = xj/ {d^, d^, d-) u. 1808 But if n 1)0 a quantic, then, as shown in (lOlM), >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, {.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 (./•) 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). 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

(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 / 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), 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 (x, y) Is a maximum or minimnm irlirii (p^ find y must ranish ; ., ^o,. — (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 ^ (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 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^, (^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- >■!.■ ..;, <#>,,-- 5 <^-2.r ^.-7/ <^.r.- <^.r,c. • K. 4',, f ,-- <^,ya- <^:;.v <^//-- ^ino !-,-, i.,, 4'.. zx 4>zu ^tz .'W w.c i>wy 4*tcz 4>>iv 1858 The theorem can be extended in a similar manner to («, y, ;<,.= ax + hy + fjz^p = (1) ^ a /' .'/ Ih; = Lc + hy+p+q = (2) I h b J \,a, = 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 (,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, 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

2^% 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 (./■) 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 = ) — (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 {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 = (,/', //) (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 ^ (.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.('^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 =

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 ;,,,., ^, ^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 { (^ 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 [ 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 •) = —a sin. r + Z> cos ar. Assume .1 Cvis x-t- JJ .sin .J -f- C = \ (.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 /(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 (»—/)-) 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 Vj<^ 111 ////.' 111-n. Add 1 to p, and siihfnict n from ra. Thrri integrate by Parts, fx"'ilx; and also hi/ ])lrif/ 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-''')-' 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 |(«=+..-^)'« • = 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\/(,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 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+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±$. 2106 f ^^iL__= l_sin-'l±^. [By (2181). Method.— Substitute (.i; + /0"\ as in (2101). Observe the cases in which A=l. '^^"' J (.r+/ov(«+^>''+^'0 ~ ^' V u+7i//+(y)' with tlie same values for A, B, C, and y as in (2104). The integral is reduced by (2097). 91 OR r ( ^'■+»') '^■'' INTEGRATION BY RATIONALIZATION. 341 :Mktiiod.— Substitute 1) liy puttint,' .>■ = p tan (0 + y), and (lotcrrninc tlio constant y by equating to x-cru the coenicicnt of sin 2« in the denominator. The rcsu .. . , ,. „,, f. f // cos « -f iV sin , Iting mtcgi-al is of the (orm . ,, — tT"''"- Separate this into two terms, and integrate by substituting sin in tlie first anil COS y in the second. 2109 C__±u)jrr wlicrc «/)(,'') mid F{,r) arc rational alo:('l)raic functions of .t, the former being of tlie lowest dimensions. Method. — Resolve ^, '^ into partial fractions. The resulting integrals are either of the form (2107), or oIfs they arise from a pair of imaginary roots of F(x) = 0, and are of the type f .-V^'-'l // '^"l! 7 _l, n ' Substitute a- — o in this, and the integral (21U8) is obtained. INTEGRATION BY RATIONALIZATION. In the following articles, F denotes a rational algebraic function. In each case, an integral involving an irrational function of x is, by substitution, made to take the form ]' F (z) d::. This latter integral can always be found by the method of Partial Fractions (1915). 2110 f;.'|„..,(_^:f,(5+|i;)Uc.( '^1..->(5^)M5$^)-.^^-('^'- 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 ( = 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), E{k,), 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 = <>, 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 ;/ 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 \/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) = (■'■) ^/'^' = 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 ^/('•) infinite, the value of [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(b)—f{a). Therefore % = 0' (&)=/(&) and n„ = —' (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 ('!'(") =^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 * + -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 '^ {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 ' « 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

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 = {(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-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), ! and l + /3=~ and by (2405) (1 -ft) + -^ (/'O = 2 (//3)^ - -j^ 7r=,

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 <^ («)+ 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/,'-^)-;^, 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 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'->''-) .(( ^\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) -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),

/, 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^)''''=;: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 _ + " ;; 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-- 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-)(t-2) _^ _^ , 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^ >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^, 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»», ^'-, 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{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 (0') + 2 -^, 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). 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«) (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 {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 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

('t^) cos fu cos iLV (In d.v = — f* f = — (() (x) sin tu sin uxdu dx (ii.) From (i ) and (ii.), by addition and subtraction, we get 2740 i (x) COS tu COS ^lxdudx = irlfp (t)+(l> ( — t)], 2741 ^ (•'^) sin tu sin ux du dx = tt [^ (J)— (~0]' Whence, by addition, 2742 \ f " 4> (ciO cos u {t-x) dudx = 27r (/), 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)-log8^1og(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

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:)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 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), (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 ?/, = j (z, x), ?/,, = ^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 + T" + &c. > 1 . Now, integrate for >/, ^ + &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

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'cos3-2e.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

(x) giving lij> ( — 1) when x = l. Thus the total value of the integral is Jf (l) + i(: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 —(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) 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 (^') be not a continuous function between x = and r/' = /, let the function be

(«) +^(«)} 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 ^ = 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

\^^!^^,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,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 : 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'-^\,)^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+\,)-(P+\ct>X-\-...}Si/ + [{N'-\-\,)-{P'^\,)^+...} 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'+\,-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,= —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 + ^^ = m^cpj.^, ((>2^=: 7i^^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, {^^\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; ; 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), 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^, 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.,{^), ... ,(^),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 {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 =

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^ = fj,, + 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 (.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^^(^^,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 {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 c,{'>',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 (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

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., = «^., + , y,, + 1>z -.v Substitute the value of ^.,, and eliminate - by means of y^, =

''> «? ^^)- 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 (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+ ... +{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

-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 = {x,y,a,f{). 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^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 {^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, ...) {D) ^{D-r) (t>{D-2r) ... {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 Catu + 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 « (.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. 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 = (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 =

(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 (.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^ (D) e'u+a, (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-\-{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) {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 + (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 !/„.,. + -^)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 -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-) ) 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^,

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^^^[ (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"^ = (-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 = (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., = {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 {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 {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 {f) = {D)e'-^ 3760 =^{0)+{(QO.t+{(h)0\^-^&G. PiiooF.— (log eO = ^ (log i') e°' (3757) = 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 (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) {x), f being an algebraic function. Proof. — Let a = e'", then the left = f{E)e'''^{x)=f{e")e'^^^ C^O (3475) ^a^f{aE) (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>< 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 + «^^•+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 )// -|- 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 , y = h siu —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 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 {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^ 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 {'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= ^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/» = 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 smm;>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 IXlC 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

(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) = {.ci'+,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 ("> 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) =

(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

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^-^^=^ (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, , m(i= ny sm' is 4764 loi cos ^'-\-mfi sin = 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. 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', ^" ■> ^"' | = &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

(V, im\ y') (X, II, v) = ((^^V+<^,,x'+ {\, 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- — 40 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 - + 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'-\-:' -h'^, 4980 C" = ab'-\-a'b-2hh', H"=f 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±{...v"+,!/"+..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 ' 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+/),^ = >•>',,. 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 = {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^ = '{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 (^) = 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 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;, ^2Syyl + ^2y.ryl-^H,^.I/.r-^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),

,, = 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 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 =■

(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 / 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 '. 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, 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 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; ), ^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

, 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 =

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 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 =

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 aib > 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 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, ^ 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

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 ^+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^ + my., (/)-/ 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^.+!/„ + ~'l>. or ^-^-^ -,, (5782,5549) the second form being the value of jj when the equation of the surface is 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^,, ,^,, .,„ (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 ^ d d• 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 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,,-^yj, = 1 ; and from these the invariants, 5863 4r.+.A^-y-ij,,, <<<^,+//<^.,. 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 {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 + , = 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'> 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 d0d(l>, Wf = l((r' siii^ 6 sin 4>ded, Vz = i nV* sin 6 cos ed0d, where 7=1 ((r' sin eded. 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 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;?/, '^ 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'^ 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 . 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) 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.. + ... + 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+

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. P and (.'/)} 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. ^ / = 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..« = H-/c: C.99. (p .fx = F. (px, to find

y = 0)

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

'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,. 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.

r, ijix when

$) when. 3 eqs. connect ^••,y,^,l: 1723. * of two independent variables : y,-: when (p {x, y,z) = : 1728. * 'P.i'i'yz), (P'iT, 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^{, 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 {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;) (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^(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\ =

: 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. vpj. 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;\ 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. M 4. ut u 1 1 >'' ,^0 I f X A 2. r)\ \g — V— -oY, ,,--' ^^ < 1 6- 1 i J' 6 r H J) 10. A -^V^ /^ X V f (7 M N X 4^ 16 30: 24. _:^ T M ¥ ^ARy nisivEHSiTs: Of cAi.1^2' **^ 16 303 24. / I T M N' 30. I L- 23. ( ) z N / ^1 T. •-■X L ( i4 ^ 31-453' >^ ^ 31-455. ^^ 1 '' -' -f -Y " •:> I t 1 t M / _J^_ f- ^ OF -iTB.^ _.-,. \ / 52--62)3' 53. 52 62): ' M 63 75^*' y 63 11 ' K\ Plate 7. 78. 76-86^ nateV. 76-8©. 83. 85. :o; /^w;, "i Hf-^liy [>, te 8. 87 -bb^. H > PJo lO Pic ¥ 89. T q p' p r -^ V P / / ^ h 93. riate u. I S 10 ^'iBte u. 112. X -^^^ iw TL JY' !34 95. 37. 39. XJb (c X o~^^ZI^\ ^ ^ ^ ^^~ ^ 101 102. I04-. 105. 108. L II o'.r 116-123-1 4 "O B ■ i tl ooU 116-1231 Hate 13 13 {-^"^ \<5ZJ 130 piatPis '3 ik:^-i3Zf 130 116 n F 117 ^ n K V (/ J^ W^ "--^^ " ^^ c pi"ei4 133—141 ^ Ha^w 133— 141 142-152^ V I tTK / 142-152 V ;r / V lOO lun, 1 s^KJ 3 \J T iiy IDD ~ 10/ IDO— ID/ li 168-17? li 168-177 H- 62 ^«l 1/b-lb2 TbR^ 178-182 H 82 170 168 i^- 3 169 / / ^ / M C . © 177 176 O \ M -jr X ^ j> 183 193 -jr 183 193 fe 'It I / K / 178 / ^ / r? ^ { \ / ---< \ / / O^x/ \/ / / X -DA 1 /\ ■^-__^ f-v \ 1 \ / \ / — V4 \ ^^ m "-^ i RETURN TO the circulation desk of any University of California Library or to the NORTHERN REGIONAL LIBRARY FACILITY BIdg. 400, Richmond Field Station University of California Richmond, CA 94804-4698 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS • 2-month loans may be renewed by calling (510)642-6753 • 1-year loans may be recharged by bringing books to NRLF • Renewals and recharges may be made 4 days prior to due date. DUE AS STAMPED BELOW MAY 2 fi 2001 (UL ' Z0Q3 OCT 3 2007 12,000(11/95) YD 0734