, c respectively; hence, since rt'A= B' C — AA\ we obtain
by this substitution
a' 4) (//) -^ [ah + A) [{h - h) \h -c)- a'] - [h'h + B') [c'h -f C) ;
, ., .^ , , A' B' C
and, it we write X, //, v tor r , rr , r ?
' a h c
4> [h) = {h - X) {{h - h] [h - c) - a'-^} - '' Jf ' [h -f^){h-y)- (1 )
therefore [h] = [h - a) [h - h) [h -c)- h-' [h - h) - c" [h - c) ;
tlicrefore (co ), > (i), ^ (c), ^ (— cc ) are -f — | — supposing
J > c, and the roots will be separated by h and r.
Cauchy's method of separating the roots is given in
Todhunter's Theory of Equations^ in the chapter on Cubics.
373. To find the conditions that the discriminating cubic ma>/
have equal roots.
In the case of two equal roots, suppose yS = 7, thou n, can be
derived by transformation from
a./- + /S (/ -}- .r) or [a - /9) x' f /3 {.>■' + / + .:-, ;
UENEIJAL E(iUATION OF THE SECOND DEGUEE. 24U
.-. f/._. - (a - /S) [Ix + my + nz)' + /3 (u;' + 1/ + z') ;
.-. a = (a - ;S) /"^ + /9, a' = (a - -S) »»*,
/> = (a-/3)m-^ + ^, U ^[a.-iB)nJ,
a
., //o' , cV/' a'/)'
.'. p = a r = , r = c r •
a u c
These arc obviously tlie coijJitioua that the coijlcolJ may be
one of revolution.
If all three roots be e(iual, u.^ must have been a{x^ + y^ + z'^)
before transformation ; therefore a = b = c and a =b' = g' = 0.
o74. i\nothcj' form of the eonditions for two equal roots may
be obtained ; for {jS - a) {^~c) = b"' and (/3 - a) {^-b)= c"' ;
... ^/3-a){b-c)=b'-'-c''',
J/-' _ r" , c"- a'- . a" - b'
a
.-. ^ = a + =6+ =c+ , ,
a — b
and we may observe that, if a' = 0, b' or c' = 0, ai)d if a, d be
the two whieh vanish, /3 = //.
375. To find the equations of the coordinate axes lohicli inahe
the terms in u,^ invoicing t/z, zx^ xy disappear.
When u,^ has been reduced by transformation to ax^+^y'^ + yz^
one of the new axes is the intersection of the two plaijes whoso
equation is, referred to the original axes,
u^ - h [x^ + y'^ + z'^) = 0, where /< = a, /S or 7 ;
therefore, by Art. 89, the equations of the axes are found by
writing a, /i, 7 successively for h in
x [b'c — [a - h) a] =y [c'a - [b - h] b'] = z [nU — (c — h) c\.
These equations do not give the position of the axes directly
if two of the three quantities a', b\ c vanish, but, if a', b' be the
two which vanish, it is obvious, from the original equation, that
the axis of ~ will be in the direction of one of the axes.
Iv K
2oO GliNERAL EQUATION OF THE SECOND DEGREE.
.376. The direction-cosines of the axes can be symmetrically
expressed in terms of the roots of the cubic (/> [h) = 0.
For [a'a + A'] {{a-h){a- c)-a"'] = {b'a + B'] (c'a+ C) (Art. 372) ;
therefore [a'a + A'f [(a - l) [a — c) — a"^] is a symmetrical function
of the coefficients, hence, if /, ?«, n be the direction-cosines of
[p. - 1>) [a - c) - a"' {aL-c)[oi-a)-b"' [a - a) {a. - h) - c"'
1 1
(j>\a) (a-/9)(a-7)'
AVe give also below the method of determining the directions of
the axes by means of the definition of a principal plane.
,377. Tojind the equation of the locus of middle jwints of a
si/stem of parallel chords of a conicoid determined by the general
equation.
Let the equation of the conicoid be f{-r^ ?/, z) = 0, and let
(X,, /u., v) be the direction of the chords to be bisected, (^, ?;, f)
the middle point of any chord.
Then the equation /(| + Xr, t? +yur, ^-\-vr) = must have
its roots equal and of opposite signs.
This gives the condition
, df df df ,
d^ drj dX
or (r/f + c't) + b'K) ^ + [c^ + I'V + «T) y^ -f {l>'^ + (ii) + cX) V = 0,
■which is the equation of the diametral plane.
.378. To determine the j^rineipal planes of any conicoid.
A principal ])lane being perpendicular to the chords wliicli
it bisects, we shall have the direction-cosines given by the thi'ce
equations
(i\ + (•' jX + b'v = .S'X,
r'A. + A/i + «V = .s/i , (1)
b'\ + «'/x 4- cv = sv^
where .s is a constant given by the cubic
(.• - a) {s - b) (,v - c)-V'= (>,
J
GKNF.RAL Kca'ATION OF THK SF.COND DKC 1;F,1:. 251
tlic discriminating cubic wliicli has been already discussed.
Since to each of tlie three vahies of s there corresponds one
system of values of X : /i : v, there are, in general, three and
only three principal planes.
If, as in the case of a surface of revolution, there are an
infinite number of values of \ : /x : v, we obtain from equations (1)
a- s c h' , c' h — s a
- , = , = - , and Y'= . = — ;
e — s a b a c — s
h'c J c'n ah'
.-. s = a 7=0— J , = c - , , as in Art. 3 1 3.
a b c
379. To shew tltat the three lyrincipal 2y^(tnes of any conicoid
are mutiiaJhj at rujht angles.
Let .!, ., -}- Z>V, = .'?|X^,
c'X, + hfi^ -f aV, = s,/i,, (1)
Z^'Xj + (/'/X| 4- CV| = s^v^.
^Multiplying by X^, /x.^, v^ and adding, we obtain
(«X,^ -I- c>.^ + t'vj X, + {c'\ + hfi^ + aVJ /x, + {h'\ + «>.^ + cv,) v^
= 5, (X^X,^ + ^,^^+v,vJ;
.-. s\^ . X, + s,^^Jb^ . fi^ + s,,v,^ . V, = .?, (XjX^ + ^l^f^^ + v,vj,
whence (./^ + yz^ -{■ 8 = 0, in which a, /3, 7 are
the roots of the discriminating cubic, for, on ti-ansformation,
the coefficient of x" will be
aX,' + hfi^' + cv^' + 2a'fj,^v^ + -^'»',^, + -t"'^,/*,
and similarly, /3 = s , y^s^.
252
GENERAL EQUATION OF THE SECOND DEGREE.
380. To distinguish the surfaces represented hy an equation
for which the roots of the discriminating cubic are finite.
In this case there is a centre at a finite distance, to which
if the origin be transferred, the direction of a new system of
axes can be clioscn (Art. 3tO), such that
u ^ ax"" + hf -t- cz" + dw'
+ 2a' yz + 21' zx + 2c xy -|- 2a" xw + 2l" yio + 2c"zio = 0,
will become by transformation ax^ -\- ^y^ -\- 72;"''+ 8m?''' = 0, lo being
written for the unit.
The transformation will be effected by substituting
Ix + my + nz + ^w for x^ and similar expressions for y and 2,
w being unchanged ; the discriminants, being invariants, are
therefore equal,* since the modulus of transformation
L m, n, I
n\
n\
0, 1
.'.^[^c) =
h\ a"
a', h"
c, c"
rt", h",
= a/37S
a, 0, 0,
0, ^, 0,
0, 0, 7,
0, 0, 0, 8
a^7 *
Hence, we have the following table for the case in which
0/37 or A is finite, and a > /3 > 7, by which it may be seen how
the loci are distinguished :
ax" + ^y' + 72' =
a
^
7
//(«)
+
+
+
+
+
+
Ellipsoid
Hyperboloid, one slieet
+
+
~
-
-
Hyperboloid, two sheets |
-
Cone, real i
+
+
+
Cone, imaginary, or jwint |
+
+
+
+
Imaginary locus |
♦ Salmon's Higher Alf/cbra, Arts. 118 and 23,
GENEKAL EQUATION OF THE SECOND DEGKEE. 253
In order that o£, /3, .iiul 7 may be all positive, a + h + c and
A must be positive. If the locus be a point, or rather an in-
definitely small ellipsoid, the section of i(^ = by each coordinate
plane must be a point-ellipse ; therefore each of the quantities
hc^a'^j ca — />'•, and oh — c"^ must be positive.
The conditions for surfaces of revolution are obtained in
Art. 373.
' 381. To distinguish the surfaces represented hy an equation^
for lohich one of the roots of the cubic vanishes, and the centre is
single and at an infinite distance.
The conditions that the centre may be at an infinite distance
arc that A = 0, and that one or more of the three quantities
bch)\v shall be finite,
a" A +Z'"C"4 c"B'^
a"C'+h"B -t-c"^',
a"B' + h"A' + c"C.
The surfaces will be the elliptic or hyperbolic paraboloid,
according as the roots of Ji^ - {a-\- b + c) h+ A + B+ C= 0, have
the same or opposite signs, i.e. iis A -\- B -]■ C is + or - ; but, by
Art. 363, A, 7?, C have the same sign, hence
A, B, and C + gives an elliptic paraboloid,
A, B, and C— „ hyperbolic paraboloid.
382. To distinguish the surfaces represented by the general
equation when there is a line of centres at a finite distance.
The conditions that there may be a line of centres are A =
and II{u] = - [a" ^f[A) + b" ^/[B) H- c" V(C)}' = 0; or, if A\ B\ C
a" b" c
be finite -p + 77- + Tt =^- The equations of the line of centres
arc A'^ — a a" = B'-ij — b'b" = C'^— c'c" = p suppose ; therefore, if
\ic transfer the origin to any point in the line of centres de-
fined by some value ofp, the equation of the surface will become
ti., + a"^ + b"r) + c"^+ fZ= 0,
an"' h'b"' c\r , ^
since the cocfiicicnt of p vanishes.
254 GENERAL EQUATION OF THE SECOND DEGREE.
ir a', h',c' he all finite, A', B', C
for instance, vanish, the other two being finite, yl' will be finite,
and, by Art. 363, if B' = 0, then A = 0, and C' = 0, and h and c
vanish ; hence, recnrring to the original equations for determining
1 -1 1 • 1 • 2a" b" ab"' , ^
the centre, wc easily obtain the equation u^ , H ^ +«= 0,
and the condition b'b" = c'c".
If two roots of the discriminating cubic be finite, since ii^ is
reducible to the form ^7/ + jz\ the surfaces represented by the
equation will be in the general case in which
a'a"'^ b'b"'^ c'c"'^ 7 • ,- •
— .r -f -w + ,v + " '^ tiuitey
A, B, and (7-f , an elliptic cylinder,
Aj Bj and (7-, a hyperbolic cylinder;
when ., +...= 0,
A, B, and (7 + , a line cylinder,
A^ B^ and C— , two intersecting planes.
If only one root be finite, 11,^ is reducible to yz'^j but in tJiI«
case, since A + B + C=^0, A, 7?, C, being all of the same sign,
must vanish separately, from which it follows that A\ B\ 6"
also vanish, and there cannot be a line of centi-es at a finite
distance.
383. To di'sti'nfjuish flic surfaces wJien there is a line of
centres at an infinite distance.
In this case A' ^B'= C = ; therefore A = B=C=0\ two
of the roots of the discriminating cubic must therefore vanish ;
also a'a!\ b'b", c'c" must not be all equal.
Since aa = b'cj &c., u^ can be put into the form
y z "
+ ;? + ?
and the only surface represented Is a parabolic cylinder.
384. To distinijnish (Tie surfaces for icJtich there is a plane
of centres.
GENERAL Ec/L'ATION OF THE SECOND DEGl^EE. 25.3
In this case, as in tlie last, the minors all vanish, and wc
have in addition a a" = h'b" = c'c" ; the equation Jnay therefore
be written
„, , fX 7/ Z\'^ , ,, fx 11 Z\
ahc[-+ 4 + - +2a'«" - + •/-, + -) + (Z=0.
\rt be) \a c J
The surface repi-escnted consists of two parallel planes unless
aoca = a'a ", or d= - = , = -~ . \n whieii case they are
' a b c ^ ^
coincident.
One of the planes will be at an infinite distance if a, b^ r,
a', b\ c all vanish while one at least of the other quantities
remains finite.
385. The results In the general case may be tabulated as
-\- (L and v for
' "2 T 't'li
r u -r 1 • ^ a a bo
follows, if V be written for — yr + —frr +
A B
(a'a" - b'b"f + [b'b" - c'c")'\ where / denotes ' finite' and a, /9, 7
are the roots of -the discriminating cubic, a > /3 > 7.
y
A
a
/3 y
i/M^^^'i ^
"■1
+
+
+
+ +
- i
1 Ellipsoid
+
+
+
+ '
Hj-perboloid, one sheet
- -
-
1
Hyperboloid, two sheets
+
+
+
^1:
1
Cone, real
+
+
+
+ 1
Cone, imaginary, or point-ellipsoid
+
If-]
f
4- 1
Paraboloid, elHptic
+ ' -
Paraboloid, hyi>erb<^)Hc
1
+ +
1 + ' /
Cylinder, elliptic
1
+
+ ' 1 +
1 Cylinder, line
+
- !
-
f 1 Cylinder, hj-perbolic
+
— i
-
Planes, intersecting
+ i
■ / i Cylinder, parabolic
«
+ ; 0,
, Planes, parallel
a"'
//'* c"'
Fur coincident i)lanes d= = ,
a b
256 GENEllAL EQUATION OF THE SECOND DEGUEE.
For two planes, one at an infinite distance, a, J, c, a', h\ c' = 0,
one at least of a", h'\ c" finite.
For two planes at an infinite distance, d alone finite.
38G. Pi'OCf'Sses for Jindinej the locus of any given equation.
When a particular equation of the second degree is presented
to us, in order to discover what species of surface it represents,
we would recommend the student first to form the discriminating
cubic, and it will then be seen whether the last term A vanishes
or not.
I. If A be different from zero, we must find the centre,
transfer the origin to it, and by changing the directions of the
axes reduce the equation to the form ax^ + ^y'' + 72"^ + 6 = 0,
where a, /S, 7 are the roots of the discriminating cubic, which can
always be found approximately, at all events their signs can
be determined by Des Cartes' rule ; and 5 has been shewn to be
— -^- , or in particular cases may be found more easily without
the use of the determinants.
II. If A = 0, and A-\- B-\- C be not zero, In which case the
two roots /?, 7 will be finite, it will be best to determine the
directions of the axes which correspond to the three roots 0, /9, 7,
And to suppose the origin so chosen that the equation becomes
leithqr
^/ + 7.t;-^+2a"./; = 0, (1)
or ;3/ + 7-"'+S = 0. (2)
If we do not require the position of the vertex, the value of
a" in (1) can be found by equating the discriminants, by which
we obtain ^ja"^ = — j (a"A +h"C' + c"B')^. and if we find
that a" vanishes, we take case (2).
]>ut if in case (1) the coordinates |, ?;, ^ of tlic vertex be
required, we think that the best method is to transfer to this
})oint, and observe that the result must be
GliNKKAL LQUATION UF Till: SiaoND DKiiUKI-:. L'-x
if (/, 7?/, «) be tlic direction of the new axis of x ; so that
Avc obtain the equations for detenninln<^ ^, ?;, ^ and a",
''^ +077 + h'i^ + a" = la.",
c'^ + hi] + rt'^ + h" = ))ioi",
h'^ + at] + c^ + o" = »a",
«"| + J"7; + c"^+ c/ + (/| + m-n + w^) a" = 0.
From the first three equations
a"A + h"C + c"B' = (LI + mC" + 7«S') a".,
which gives a" without ambiguity, and the fourth equation with
two of the former determin<3s f, ?;, and f.
If a" = 0, we shall have in case (2) three equations, equivalent
to two Independent equations, which will determine the axis of
the surface, and we can obtain 8 from a fourth equation com-
bined with two of the former; thus eliminating ^ and t] from
the last three,
h [a'c — hh') = l>" {b'b" — a' a") + c" [ba" - c'b") + d [a'c - bb'),
since the coefficient of ^= a" A + b"C' -f c"B' = 0, when a" = 0.
Tims the position of the locus Is determined completely in
both cases.
III. If zl=0 and A-\-B+ (7=0, the equation is reducllle
to one of the forms yz'^ + 2a"x = (1), 2 + 18 = 0.
The discriminating cubic is
(A -32; (A- 1/-9;A-32)-128(^-1)-6.G4 = U,
the last terra of which is and the roots 0, 36, — 2.
The equations of the axes are known from
[64 + 3 (A - 32;; X = [- 24 - 8 (/* - 1)) 3/ = [- 24 - 8 (A - 1;| z,
which give 1x = y = z\ x = — iy = - -iz ; x = 0, y = z\
and fhe direction-cosines arc
U, ?, \U (§V2, -Jv2, -iv'2), and 0, i , 2, - J n^2).
OlINERAL EQUATION OF THE SECOND DECSREE. 2')d
The equation being reduced to 36/ - 2z' + 2a".r = 0, by the
equation of invariants
-i(3.84G.lG + G.lG)' = -36.2a"'; .'. a" = ± 0.
If (^, v^ ^) be the vertex referred to the original axes, when
we transfer to this point the equation must reduce to
».^ + 2a"(i.c + iy+^^)=0;
... ?,'2^-Hr)-S^-3 = ^0L"
- y|+ 7; + 3r-G = §a"
- 8| + 3»7+ ^"-G = ?a";
.-. - 3-2.6-2.6 = 3a"; .-. ct" = - 9,
and since the constant terra vanishes
K(l + 2'7 + 2?)-3|-G;;-GC+lS = 0,
or | + 2»; + 2^=3;
389. To find the surface ichose equation is
x' - 2if + 2z^ -\-3zx-xij-2x-\-ly-bz-Z = {).
The discriminating cubic is
(A-l)(A^-4)-f(/. + 2)-i(/.-2) = 0, roots 0, '^-''^^ ,
H [u) = _ i [Aa + C'b" + B'cy = 0,
and the equation is reducible to
3V(3)-H ,_ iV(3V-_l ,, , g^^,
2 ^ 2 "
The equations determining any point (|, tj, ^) of the axis of the
surface are
- ^-l'? +7 = 0, (1)
3^ +1^-5 = 0,
which give the straight line
^=-lr; + 7 = i(-ir+.>,; (2)
2Gi) GENERAL EQUATION OF THE SECOND DEGKEE.
tlierelbrc multiplying the equations (1) by |, ■»?, t, and adding
Tlio equation therefore represents two intcrseeting planes whose
line of intersection is given by (2).
390. To find the axes of the conical envelope of a central
couicoid.
The equation of tlie cone referred to its vertex as origin is
o- [ax^ + hjf + cz') = [afx + hgjj + chz)\
a being written for af'^ + hif + cJi^—lj the discriminating cubic
in this case is
[s - ca f af'] [s - ah + h\f) [s - ac + c'^h^)
- b'^/h'' (s - <7a + df) - c'aVif {s - ah + hy)
- a'hfY {s-ac + cT) + 2a'hVfyh'' = 0,
or writing s^, .9^, s,. for s — aa^ s — ah, s — ac
a'r h' direction-cosines of axes in the ratios
{1, ± V('i) - 1. 1). (1. 0. - J)- 4. Hyperbolic cylinder. 5. Hyperboloid
of one sheet, centre (J, ", - "/). 6. Cone, direction-cosines of axes
{0, i v'(-). - ^ V(2);, {+ i \/(2), i, h]. 7. Ellipsoid, point or impossible
according as t/ < = > 55, 8. Hyperbolic cylinder.
(2) The equation Ix* + 6y* + 4s' - 7yz - llsj - 7x^ = a* represents an
hyperboloid of one sheet whose greater real axis makes with the axis of s
an angle tan"' ^,'('2).
(3) The equation
ax' 4 4^' 4 Uz^ t 12/yr i- 6zx 4 Ix// f 2'j".r ^ 2h'y 4 2c"z t =
will in general r<.pie^ent an cUip'ic i)arabuloid, a parabolic cylinder, or a
262 PKUBLEMS.
hyperbolic paraboloid, according as a > = < 1. What surfaces will it re-
present in the following cases :
i. 36" = 2c", a> = y) sin 0, with
similar clianges for y and s.
Hence, defining an axis of a conicoid as a diameter, such that by revo-
lution about it through two right angles every point of the surface returns
to the surface again, deduce the ordinary cubic equation for the determina-
tion of the axes.
CIIAPTP]R XVII.
.DEGREES AND CLASSES OF SURFACES. DEGREES OF CURVES
AND TORSES. COMPLETE AND PARTUL
INTERSECTIONS OF SURFACES.
391. Having already fully investigated the nature of the
surfaces represented by the general equation of the second degree,
we will proceed to the loci of equations of higher degrees, which
we may consider as equations either in three-plane or four-plane
coordinates : in the latter case we may suppose the equations
homogeneous, without loss of generality.
392. Surfaces which are represented by rational and integral
algebraical equations are arranged according to the degrees of
these equations when plane coordinates are used, and according
to classes when tangential or point coordinates are used.
A surface is of the ?«'" degree when the equation of which
it is the locus is of the n^ degree in the coordinates of any
point of the locus; the geometrical equivalent being that a
surface is of the w" degree when an arbitrary straight line
intersects it in 7i points, real or imaginary.
A surface is of the n"^ class when n tangent planes, real or
imaginary, can be drawn to it through an arbitrary straight
line.
If p\ (?', /, s and ^/', q\ ?•", s" be the point coordinates
of two planes, the coordinates of any plane passing through
their line of intersection will be Ijj -\-m2)", Iq -\- mq"... [Art. 128],
I : m being an arbiti'ary ratio, and the particular planes which
touch a surface whose tangential equation is ^(p, i?, »*, s) =0,
supposed a homogeneous algebraical equation of the ri"> degree,
will be determined by the values of I : m which satisfy the.
equation i^(/// + 7/y/', ...) = 0; the number of values of the
ratio will be 72, and this will therefore be the class of the
surface.
M M
2GG COMPLETE AND PARTIAL
393. Curves and Torses arc arranged according to their
degrees.
A curve of the n^^ degree is one wlilch intersects an
arbitrary plane in n points, real or imaginary.
A torse of the n^^ degree is one to which 7i tangent planes,
real or imaginary, can be drawn through an arbitrary point.
Other classifications of curves and torses will be explained
hereafter.
394. Among the various methods of treating of curves
which have been proposed, one is to consider them as the inter-
section of surfaces whose equations are given. In this method
the difficulty arises, to which allusion has been made (Art. 13),
viz. that extraneous curves may be introduced which are not
the subjects of investigation.
If any curve be supposed to be given in space, it is impossi-
ble generally to determine two surfaces which shall contain no
other points but points which lie on the proposed curve ; but
among all the surfaces which may be drawn through a curve, it
is desirable to obtain the simplest forms of surfaces of which the
curve shall be the partial intersection.
395. The number of points in which three surfaces inter-
sect, which are of the m^\ n*'>, and p^^ degrees respectively, is
vin2)j unless they intersect in a common curve, in which case
it is infinite.
For the proof of this proposition, the student is referred
to Salmon's Treatise, on Higher Algebra^ Lesson VI ir., on the
number of solutions of three equations in three unknown
quantities.
The student may be able to satisfy himself of the truth of tlic
proposition, by considering that the number of points in which
tlie surfaces intersect will, by the law of continuity, be unaltered,
if we substitute particular instead of the general forms of the
surfaces. If the surfaces respectively consist of 7», ??, ^) arbi-
trary planes, it is obvious that the number of their common
points of intersection will be ?»??;>, each point being the inter-
section of three planes, taken one from each system.
iNTKKsix'TioNs OF si:ia\u'ic><. 2G7
390. Tlte comphte intersection of two surfaces of l/ie 7/i^'' and
n^^ degrees respect i veil/, is a curve of the tnn^^ degree.
Let a plane intersect the surfaces, the number of points of
intersection of the pLine with the surfaces is mn, this is there-
fore the number of points in which the phuie cuts the curve,
and the curve is of the mn^^ degree.
397. To find the number of conditions ivhich a surface of
the n^^ degree may he made to satisfy.
The number of constants in the general equation of the »^''
degree is evidently the number of homogeneous products of four
things of n dimensions, and is therefore
^ 4.5...(4 + n-l) ^ (/i + l)(n + 2)(» + 3) ^
1.2. ..n ~ 1.2.3 '
but in estimating the number of constants with reference to the
number of conditions which the locus can be made to satisfy, we
must diminish this number by one, since the generality of the
eciuation is unaltered if we divide by any one of the constants.
The number of disposable constants, so obtained, is
( ;^ + l)(n + 2)(n + 3) _ (n^ + 6n +ll) ^ ,
1.2.3 ~ ' 6 -^^ ''
Thus <^(2)= 9, <^(3) = 19, )(4)=3-l,.
4> (5) = bb^ (f> (G) = 83, and so on.
Since, when a point is given, we may substitute its coordi-
nates in the general equation of a given degree, and thus obtain
a linear equation of condition between the constants, a surface
of the third degree may be made to pass through 19 arbitrarily
chosen points, and one of the fourth through 31, &c., and cj) [n]
arbitrarily chosen points will completely determine the position
and dimensions of a surface of the 7i^^ degree.
A surface of the n^^ degree is also determined by (f) [n] inde-
pendent linear equations of any kind between its coefficients.
398. All surfaces of the n^^ degree lohich pass through
[n] — 1 given p>oints have a common curve of intersection.
If ?f = 0, V = be the equation of two surfaces passing througli
the given points, \u-\- fiv = Q will be the equation of another
268 COMPLETE AND PARTIAL
surface of the ?i''' degree which passes through the ^(z^ — 1 given
points I and since, by giving proper values to the ratio A,: yu,
this surface may be made to pass through any additional point
which is not common to the two surfaces u = 0, u = 0, this equa-
tion will be the general equation of all surfaces which contain
the ^ («) — 1 given points. But this equation is also satisfied
by the coordinates of all points which lie on the curve of in-
faces containing the [n] — 1 points be given, it
will be possible to eliminate from the general equation of the
surface of the n*^ degree all the constants but one, which will
enter iflto the resulting equation in the first power only. This
equation will then be of the form xi + X?; = 0, where m, v are of
the ?i'" degree, and A, an undetermined constant. All surfaces
represented by this equation will pass through the curve given
by the equations
which curve is therefore completely determined. For example,
eight points determine a curve which is the complete intersection
of two conicoids.
In the case of complete intersections of surfaces the nature of
INTERSECTIONS OF SURFACES. 269
the curve is not given when the degree is given, except in the
case of prime numbei's, when it must be a phinc curve.
For example, a curve of the twelftli degree might be the com-
plete intei-sectiou of pairs of surfaces of the degrees (1, 12), (2, 6),
(3, 4), and these different species, belonging to the same degree,
would require a different number of given points to determine
completely the surfaces.
The following proposition serves to obtain the number of
given points sufficient to determine a surface of the «'" degree
which, bj its complete intersection with a surface of a lower
degree, gives a curve of the yj/^'" degree : this is given by
Pllicker, but may also be proved directly by a theorem given
by Cayley.*
402. All surfaces of the n^ chgree lohich pass tlxroufjh
given points of a surface of the [n)-cf>{p)-h
lie on a surface of the rj^^ degree, where n =p -f q^ whose equation
is My = 0, then u^u^ = will be one of the surfaces which contain
the ^ (h) — 1 points, and may be obtained by giving a certain
value to the ratio X : fi'm the equation \u + /it' = 0, so that
\u + fivE. ?,?/,.
The curve of intersection of all the surfaces of the n^^ degree
containing these points lies on the surfaces Uf, = and ", = 0.
Hence if (f) {n) - {n — q) — 1 points be taken on any fixed
surface u^ = 0, all surfaces of the 7i"* degree, which pass through
these points, will intersect the surface of the q^^^ degree in the
same curve.
Thus, if«^ = l, the proposition is reduced to the following:.
All surfaces of the n^^ degree which pass through !« (« + 3)
♦ Xouvclles Annaks, xii., p. 3DC.
270 CoMPLKTt: AM) PARTIAL
given points in a plane determine a curve of the 7i^'' degree
in that plane.
If 2' = 2, the proposition becomes :
All surfaces of the ?«'^ degree which pass through n [n + 2)
points on a conicoid intersect the conicoid in the same curve.
403. When it is said that a curve or surface is determined by
a certain number of points, tliese points must be supposed
arbitrarily taken, for it is possible so to select the points that this
number would not be sufficient. Thus, a plane cubic is generally
determined by 9 points, but, if those be the nine points of
intersection of two of such curves, an infinite number may be
drawn through them. A curve of the fourth degree of one
species can be determined completely by 8 arbitrary points,
but If these given points be the intersections of three conicoids
which have not a common curve of intersection, taking these
surfaces two and two, we may obtain three curves of that
species passing through the same eight points.
404. If two surfaces of the n'^* degree pass through a curve
of the nr^^ degree situated on a surface of the r^^ degree, they
will also intersect in a curve of the n [n — rY^ degree, situated
on a surface of the (n — r)^^ degree, because one of the surfaces
which passes through the intersection of the two w*'*^ surfaces
will be the complex surface formed of two of the degrees r
and n — r respectively.
Thus, If a conicoid intersect a cubic surface In three conies,
the planes of each of these will Intersect the cubic In a straight
line, making the complete Intersection ; and since the three plaucs
form a cubic surface, part of whose curve of intersection with
the original cubic surface lies on a conicoid, the three straight
lines will He in one plane.
405. The theory of imrtial intersections of surfaces was
first discussed by Salmon.* AVithout an examination of such
partial Intersections it Is not possible to analyze different species
of curves of the same degree. If we considered only complete
* Qiuirtcrli/ Journal, vol. v.
INTERSECTIONS OF SURFACES. 271
intersections of siu'ftices, curves of the third dcgTce couhl only
be considered as phanc curves, whereas it will be seen that
thej may also be imrtial intersections of conicoids.
406. Tojiml the surfaces oj xcliicli a given curve is the partial
intersection.
In order to find the surfaces which may contain a curve of
the m^^ degree, it is observed that through ijc) points a surface
of the U^ degree can be made to pass. Now, the total number
of points which are common to a proper curve of the m^"^ degree
and such a surface, supposing the curve not to lie entirely
on the surface, is mlc^ since this is the number of points in
which h planes intersect the curve ; and the law of continuity
makes the statement general.
If [h] = ink + 1 , one such surface can be drawn containing
the curve ; if mk+ 1, two surfaces of the k^^ degree can
be drawn, and therefore an infinite number. Thus, for a curve
of the tliird degree, if Z- = 2, ^ (1-) = 9 > 3.2 -f 1, hence an infinite
number of conicoids may be drawn containing any curve of the
third degree.
When ^ (Jc) —7nJc + 1^ one surface of the A;'^ degree contains
the curve, and another of the k-\- If must also contain it, for
(f)[k+l)- <}) [Jc]^ = -I [k -i 2) {k + 3), therefore
,(/.•+!)- [m (/.• + 1 ) + 1] = i- {Jc + 2) [Jc + 3) - m,
Avhich is always positive.
Modifications are required when the surfaces are not proper
surfaces. Salmon gives as examples of this modification a plane
curve of the third degree through which it is possible to de-
scribe an infinite number of conicoids, but since each conicold
must necessarily consist of the plane of the curve and an arbitrary
plane, the intersection of the plane and conicold will not deter-
mine the curve. Again, if a curve of the fifth degree, which,
according to the above laws, ought necessarily to be determined
by surfaces of the third degree, lie entirely on a conlcoid, every
surfiice of the third degree which contains the curve may be a
compound of the conicold and a plane, and we must advance to
surfaces of the fourth degree to determine the curve.
272 POINTS OF INTERSECTION OF SURFACES.
If a curve be given of the m^^ degree, and k, I be the lowest
degrees of surfaces upon which it can lie, any surface of the /c^"*
degree constructed to pass through mJc + 1 points will contain
the curve, and similarly for the other surface.
If ml+l points known to lie on the curve be given, and
I > ^•, all the rest can be found.
407. The number of arbitrary points through which a curve
of the m^^ degree can be drawn cannot exceed a certain superior
limit which is easily determined, for suppose k arbitrary points to
be given, and a cone to be constructed containing the curve, and
having its vertex in one of the assumed points, the degree of this
cone will be wi — 1, since any plane through the vertex must
contain m— 1 points of the curve besides the vertex, and therefore
7)1 — 1 generating lines of the cone, and the number of its gene-
rating lines sufficient for its complete determination is the same
as that of the number of points necessary to determine a plane
c xi 71th 1 . wi (m + 1 ) .
curve or the m—l\ degree, viz. — ^— — 1.
The greatest value of h for which such a cone can be con-
structed is — ^-- — ; this is therefore a superior limit, although
lower limits to the number h may be obtained in general from
other considerations.
Thus, a curve of the third degree cannot be made to pass
through more than six arbitrarily chosen points.
408. Jf {n) - 2 given planes,
they will touch n'' - (f> («) + 2 additional fixed planes.
Similarly for other theorems.
In illustration of the points which have been considered In
this chapter, relating to the intersection of surfaces, we give
here some elementary properties of cubic and quartic curves.
Cubic Curves.
413. If two conic.oids have a common generating line, any
plane which does not contain this generating line will intersect
the two conicoids in two conies which have four points in
common, one of which will be in the generating line ; hence the
curve which with the generating line forms the complete inter-
section of the conicoids, being met by an arbitrary plane in
three points, is a curve of the third degree; such a curve is
called a cubic curve.
Conversely, if we take any seven points upon a given cubic
curve and an eighth on any chord of the curve, we can make
an infinite number of conicoids pass through these eight points,
which will have for their common curve of intersection the
cubic curve and the chord, for each conicoid meets the curve
in seven points and the chord in three, and therefore contains
both entirely.
414. A cubic curve^ ichieh is the intersection of ttco conicoids
having a common generating line^ intersects all the generating
lines of the same system as the common line in tu-o iioints^ and
those of the opposite system in one point only.
Call the two conicoids A and B^ and the common line L.
Any generating line of A intersects B in two points, neither
of which will lie on Z, if it be of the same system as Z, but
one will lie on Z, if it be of the system opposite to that of
L] but the points which do not lie on L must lie on the
cubic curve, which proves the proposition.
276 QUARTIC CUKVES.
415. The common generatinfj line of two conicoiJs ivJtich
determine a cubic curve is twice crossed by the curve.
A plane which contains the common generating line intersects
each of the conicoids in a generating line of the opposite system,
and these two lines intersect in one point only ; but the plane
contains three points of the curve ; hence two of the three
points must lie on the common generating line.
416. When tico cubic curves lie on a given conicoid^ to fad
the number of points in which they intersect.
Each of the cubic curves is the partial intersection of the
given conicoid with another which has a common generating
line with it.
Call the three conicoids ^1, Z?, and B\ and the curves C, C\
and let the complete intersection of A and B be the curve G
and the line Z, and let that of A and B' be the curve C" and
the line U .
The eight points which are common to A^ i?, and B' must
be the intersections of the complex curves CL and C'L ; and
two distinct cases arise according as Z, U are of the same or
of opposite systems.
If they be of the same system, L will meet B' in two
points both of which will be on C" ; U and G will intersect in
two points, therefore G and G' will Intersect in four points.
If they be of opposite systems the two points in which
h intersects B' will lie one on JJ and the other on G' ; hence
Z, Z' ; Z, C" ; and Z', G will intersect in three points, and
therefore G and C" in the five remaining points.
Quartic Gui'ves.
All. The intersection of two conicoids is a quartic curvcj
since a plane must meet the two conicoids in two conies which
intersect in four points ; but this is a particular kind of quartic
curve. An arbltraiy quartic curve will intersect an arbitrary
conicoid in eight points, and only one conicoid can be con-
structed which will contain nine points of the curve, and
therefore th» entire curve.
QUARTIC ClIUVKS. 277
The general (juartic curve may tlicrefore be considered as
the partial intersection of a conicoid and a cubic surface drawn
througli thirteen points of tlie curve, and the remaining portion
of the complete intersection must be either (i) two straight lines
which do not intersect, or (ii) a conic which may be two inter-
secting straight lines.
i. In the first case a generating line of the conicoid which
is of the same system as the two straight lines common to the
two surfaces, meets the cubic surface in three points which
must be on the quartic curve, while one of the opposite system
meets the cubic surface in one point only, besides the points
in which it cuts the two common lines, and therefore intersects
the quartic curve once.
ii. In the second case every generator of the conicoid meets
the common conic in one point, therefore two of the three
points in which it intersects the cubic surface lie on the quartic
curve.
If i<2 = be the equation of a conicoid containing the quartic
curve, If, = that of the plane of the common conic, the equation
of the cubic surface must be of the form i\u.^-\- u^v^ = 0, and the
quartic curve, in this case, must be of the particular kind which
is the base of a cluster of conicoids, viz. that determined by
the equation \i(.^ + fi\\ = for all arbitrary values of \ and fi.
418. To find the number of jyoints of intersection of two
quartic curves which both lie on the same cubic surface.
Let the surface be denoted by S^ and the conicoids which
contain the two curves by S,^ and S.\ and suppose the remain-
ing parts of the complete intersections to be two non-inter-
secting lines, so that the complete intersection of S,^ and >S, is
C„ L, J/, and that of S; and S^ is (7/, L\ M'.
The three surfaces S^^ S.^, and ^S^, intersect in 12 points,
and since L intersects S^ in 2 points, and similarly for the
other lines, 8 of the 12 points lie on the extraneous lines, and
the two curves (7,, C'/ intersect in four points.
This supposes that the lines X, L' do not intersect, but if
they intersect, the modification is easily made ; for example, if
the four lines form a skew quadrilateral, the numltcr of points.
278 PKUBLEMS.
belonging to these lines will be reduced to four, and C'^, C/
■will intersect in eight points.
By similar reasoning, if the remainder of the intersection of
S,^ and >§.,, or of S,^ and S^ be a conic, it can be shewn that
6'^ and C^ will generally intersect in four points ; and if the
three surfaces all contain the same conic, by Art. 411 this \\'ill
count as eight points of intersection, and therefore C,, C/ will
intersect in the four remaining points.
XVII.
(1) Every cone containing a curve of the third degree, in which the
vertex lies, is of the second degree.
(2) Prove that an infinite number of curves of tlie third degree can be
drawn through five points arbitrarily chosen in space, but that six deter-
mine the curve : what limitations are necessary that such a curve shall
pass through the points?
(3) Through a curve of the third degree, and a straight line meeting
the curve in one point only, a conicoid can be drawn, of which the
generating lines, which do not inteisect the given line, meet tiie curve
each in two i)()ints.
(4) Tlirough any point in space a straight line can be drawn which
meets a curve of the third degree, not a plane curve, in two points.
(5) If P, Q be two points on a cubic curve, all the conicoids which
contain the curve and the chord PQ have common tangent planes at
P and Q.
(6) A conicoid can be drawn tlirough a given chord of a cubic curve
containing the curve and touching a given plane through the chord at a
given point of the chord.
(7) The projection upon any plane of a curve of the third degree, not
plane, by straight lines drawn from a given point, is a curve of the third
degree having a double point.
(8) No straight line can cut a curve of the h'^ degree, not plane, in-
more than M - I points.
(9) The locus of the centres of a cluster of conicoids is a cubic curve.
(10) 'i'hree conicoids which have a common generating line meet only
in four points besides the generating line.
(11) Through five points of a conicoid, wc can draw two curves of the
third drgrce lying entirely in the conicoid.
PROBLEMS. 27y
(12) A quartic curve is the intersection of two conicoids, prove that
a cubic surface can be construcled which contains the curve and two given
conies, one on each conicoid, if these conies do not lie in the same plane.
(IJ) Shew that, if normals be drawn to a conicoid from every point of
a straight line, their feet will lie on a quartic curve.
(H) Among the conicoids forming a cluster there are four cones, real
or imaginary ; each of these cones has four of its sides tangents to the
curve which is the base of the cluster, and the four points of contact are in
one plane.
(15) Through a chord AB of a quartic curve, which is the base of a
cluster of conicoids, a plane is drawn determining a second chord ab; shew
that, as the plane turns round AB, the chord ab generates a conicoid;
shew also that a plane which passes through ah and a fixed point £ of the
curve also passes through a fixed chord EF.
(16) The projection of the base of a cluster of conicoids on a plane is a
curve of the fourth degree having two double points, real or imaginary.
(17) Two quartic curves which lie on the same hyperboloid, each in-
tersect one system of generating lines in three points, j»rove that the
curves intersect in six points if the generating lines be of the same system
for both curves, and in ten points if the systems be opposite.
(18) Through a given straight line planes are drawn touching the
sections of a conicoid made by a plane passing through a second straight
line; shew that the locus of the points of contact is a quartic curve passing
through the four points in which the two straight lines intersect tlie
conicoid, and four of whose tangents intersect both of these straight lines.
(19) If three straight lines be the complete intersection of a culic
surface with a plane, three planes through these lines will intersect the
surface in three conies ; prove that one conicoid can be drawn containing
the three conies.
(20) Find the number of points in which two curves of the fifth degree
on the same surface of the third degree, and on two conicoids, intersect.
(21) The eight points given by the equations lx* = mif = nz* = rid', are so
related that any conicoid passing through seven of them will pass through
the eighth.
(22) Three cones of the same degree have their vertices on a straight
line, and two of their three curvcjs ot intersection are plane curves; shew
that the third curve is also plane, and that the planes of the three curves
intersect in a straight line.
pr
CHAPTER XVIII.
TANGENT LINES AND PLANES. NORMALS. SINGULAR POINTS.
SINGULAR TANGENT PLANES. POLAR EQUATION OF
TANGENT PLANE. ASYMPTOTES.
419. In this chapter we shall, for reasons given in the
eface, confine ourselves to the consideration of surfaces whose
equations are given in Cartesian coordinates, and in discussing
singularities of contact we shall only consider those of a simpler
kind, reserving for a later portion of the work those which
are interesting merely as subjects of pure geometry.
420. It will be convenient to state here, that we shall often
employ the following notation : when the function F[x, y, 0),
for which we shall write F, is used, U, F, W will be written for
dF dF dF , , , ,, d'F d'F d'F d'F
dzd^' d^d^' ^^"^ "^^'^^ ^=/(-^' y)^lh 1^ '•> h < will be written
„ dz dz d'^z d'^z d'^z
dx^ dy'' dx^ ' dxdy ' dij^ '
421. To find the relation between the direction-cosmes of a
tangent to a surface at a cjiven ordinary point of a surface.
Let the equation of the surface be F= F{^, rj, ^) = 0, f , t), {"
being the current coordinates of a point, and let [x, y, z) be the •
point Pat which the line is a tangent.
The equations of a line through P, whose direction-cosines
are X, /x, v, arc
L-'^ = n^=i_-_^=^ (1)
At the points where this liiu* moots the surface, the values of r
tan(;i:nt links and ri.AM-.H. /^ . /■ 281^/-'
' '/^ '^Z' '/^
arc given by the equatloij i'(.c + Xr, y -\- iir^ ^ + ^n'y^\ ^^^^\y v^
since F[x^ y, ^) = 0, this equation may be written ' '^ /'
'■(^ciia/ji
another point (} will become coincident with 7', and the line
will be a tangent line.
• dF dF dF- ^ „ . , ,
At an ordmary pomt -r- , -7-, , do not all vanish, but
•' ' dx^ dy ^ dz '
there may exist points on a smface for which this docs happen ;
such points are called singular points : we shall presently considiu*
this peculiarity.
422. To find the equations of the tantjent plane and the.
normal to a surface at an ordinary point.
The equation of the locus of all the tangent lines which
can be drawn through an ordinaiy point is found by eliminoting
\, /i, V between the equations (1) and (3\ which gives
,^ ,dF , ,dF ,^ ,dF ^
shewing that the tangent lines all lie in a plane, whic Ii is
called the tanrjent plane.
The normal is perpendicular to this plane, and its equations
arc
fz "^ ^ - .V _ ^- ^ r
dF dF dF /{fdFv 7dF7' TTTT^^
dF /\l''J\ (dF\' /dFy
dz \/\[dx) "^ \dy) "^ [dzj
dx dy dz \/ [\dxj \dyj \dz J )
the equation (3) represents that the normal is perpendicular
to every tangent line.
00
282 NUI.'.MALS AND TANCiKNT PLANKS.
42iJ. To Jiiul the niunher of normals which can he drawn
from a given point to a surface of the n'" degree.
Let F[^^ 7]^ ?)=0 be the surface. The number of normals
Avill be the same from whatever point they be drawn, the
number may therefore be found by Investigating the number of
normals which can be drawn from a point at an infinite distance,
which we may assume in Ox produced.
The number will therefore be equal to the number of nor-
mals parallel to 6>.r, together with the number of normals to
the section by a plane at an infinite distance.
Jf (.f , _?/, z) be the foot of a normal parallel to Ox, F' [y) = 0,
F'[z) = 0, which combined with the equation F {x, ?/, ^) =
give n [n - \Y solutions.
Again, any plane section of the surface will be of the n^^
degree, and the number of normals drawn to any curvey (a-, ?/) =
of the n^"^ degree is, in like manner, the number of normals
parallel to Ox, together with the normals which can be drawn
to points at an infinite distance, the mimber of the latter is n,
and the number of normals parallel to Ox is given by the num-
ber of solutions of /' [y) = 0, and f'[x, y) = 0, which Is n[n — 1) ;
hence, the number of normals to the plane section at an infinite
distance is 7i^
Therefore, the number of normals which can be drawn to the
surface from any point
= n {n - ])'' + n^ = ii^ - n' -f n.
424. To obtain the form of the equation of the tangent iilane
irhen F{^, t], ^) is represented as the sum of a series of homo'
(jcneous functiu/is.
Let F{x, y, z)^F^-\- F„_^ +...+ F^ + c
where /'', denotes a homogeneous function of the .s'" degree
in .r, y/, z ; then, by a known property of homogeneous functions,
/ d d d\ ^ „
therefore the equation of the tangent plane may be written
^dF dF ^dF „ , , -,
TANGi::vr lim:> and I'Lani;s. 283
01', since l'\ 4 7^,,_, +...+ <; = 0,
^'1 + "',/y + f',/f -^ ^■■- + =^- +•■■+ (" - » -'•'. + '" = "■
425. To find the equation of a tanrjent plane to a surface^
when the direction ofthej^lane is given.
Let {Ijtn^ n) be the given direction, and 1^A-mT]->rnt=p tlie
equation of a tangent plane to the surface jF'(^, ?/, ^ = *^ 5 *''<^"
if (a-, ;/, z) be the point of contact, since this erpjation must
be identical -with
^dF dF ^dF dF dF dF
dx dij dz dx ^ du dz '
•vve have
I dx m dij. 11 dz ^; V dx "^ dy dz )
\
and these equations, with that of the surface,, give the coordl-
ivates of the points of contact of any tangent plane in the given
direction, and also determine a relation between /, ?», n and ^7,
sueh as was found in Art. 253 in the case of a conicoid ; this
relation is the tangential equation with the JBoothian coordinates
I m n
P' P' P'
426. To find the locus of the jyoiats of contact of tiufjent planes
drawn to a giccn surface from a given point.
Let F=F{^, 7], ^) = be the equation of tlie given surface
of the n^^ degree, and let (/, //, h) be the given point. ]f
(x, 1/j z) be one of the points of contact, the tangent plane
to the surface at (a:, ?/, z) must pass through (/, y, //).
This gives the condition
•which, combined with the equation of the surface, determines
the required locus, or the curve of contact.
It has been shewn, (Art. 124), that .r , +V , + .- , mav
dx ^ (III dz
by means of the equation of the surface be reduced to an ex-
2^4 SINGULAR POINTS.
presslon of the [n- 1)*^ degree in u", ?/, z ; the equation (4) so
reduced gives a surface of the (« — l)'** degree, called the frst
polar^ Avhose intersection with the given surface is the curve of
contact.
The curve of contact for any conicoid is therefore a conic,
the first polar being in this case a plane.
The general theory of polars will be considered hereafter.
/Satgalar Points.
427. To find the relation between the direction-cosines of a
tangent line at a singular point.
oince at a singular point i , -- , - and -r- separately
vanish, the coefficient of r in equation (2) vanishes for all
values of \, yti, v, which shews that the line (1) meets the
surface in two coincident points, in whatever direction it be
drawn through P; in this case we find the direction of any
tangent line by taking a point Q near the double point P, and
moving it up to P until a third value of r vanishes, the direction
of PQ will then be that of a tangent line, and the relation
•between its direction-cosines will be
^ d d d
ax dg dz
or. written in full,
«X' -^ vix" + wv" 4 2u'fiv + 2v'v\ + 2w'Xfj, = 0. (5)
If all the partial differential coefficients as far as those of
the {»— 1)'" order vanish, the equation (3) will have 5 roots equal
to zero, and the point will be a multiple point of the 5*" degree ;
it is easily seen that the directlon-coslncs of any tangent line
win satisfy the equation
^ d d dy ,,
which shct\-s that the tangent lines all He In a cone whose
tertcx Is the point I*. This cone is called a tangent cone.
SINGULAR I'uINT.S. 285
428. To find the equation of the tangent cone at a laidtiph
2)0 Int.
If tlic multiple point be of the s^ degree, the direction-
cosines will satisfy the equation
and the equation of the tangent cone is found by eliminating
X, /i, V between this equation and the equations (1), we thus
obtain
{(i-)^+(.-i/),:J,+(r-^);^.fi'-=o,
where it must be remembered that in the performance of the
operation indicated ^-cr, ii—y and ^—z must be treated as
constant, in other words, the symbol of operation must be ex-
panded before the differentiations are performed.
429. To find the equation of the normal cone at a douhle point.
The equation of the tangent cone at a double point Is, by (5),
and that of the tangent plane at any point of a generating line
of this cone whose coordinates are x + Xr, y + /^r, z + vr Is
[u\ + tc'iM 4- v'v) (I - a-) + [io'\ + i"/i + u'v) [v-y)
-f- (i-'X4 u'fi^icv)[i;-z) = (\]
lience, If , , = ' , = ~ , be the equation of the normal
' X /i V
to this plane at (a-, y, 2),
«X + 10 fi + v'v xo'\ 4- VIM + \iv v'\ A- u'fJL + in
X' /a' v
.'. mX -f ic'fi + vV — X'p = 0,
io'\ + Vfi + u'v — fi'p = 0,
r'\ -f n'fi + irv - v'p = 0,
X'\ -|- yti'/i 4 v'v — ;
= P
283
srxciUi-Ai: points.
M, v.- , V
\' i
v'
(^)
X', fi'
and tlic equation of the locus of the norraals to all the tangent
planes to the tangent cone is
where p = vw — u"\ ^/ = v'w — ini\ &c.
430. The condition that the tangent cone shall degenerate
into two tangent planes is
u\ 10
')) becomes
0, Art. 88,
and in this case, the equation
- {pX' + r'fju' + q'v")
0, Art. 3G4
so that the normal cone degenerates into two coincident planes ;
this may be accounted for geometrically in the following
manner: the generating lines of the normal cone are each
perpendicular to the plane containing two of the generating
lines of the tangent cone taken indefinitely near to one another ;
if then the tangent cone become two planes, we can take the
two generating lines on one plane, which gives a normal to
that plane ; or we may take one on each close to the line of
intersection of the planes, which will give a normal in any
direction we please in the plane perpendicular to the line of
intersection, and a double plane will be formed, because these
two generators may be on either side of the double point.
The equations of the line of intersection of the two tangent
planes will, by Art. 89, be
[v'lc'- uii) (I — x) =[ic'u'— vv') (77 - 1/) = [n'v- wio') (^- z)^
or ]>'{^-x) = q{rj-ij)==r'{^-z),
unsymmi:ti:ical equation. 2S7
ami tliat of one of the coincident planes In which the normals lie
p{^-^) + r'{v-?/) + q'{^~^) = o,
and this plane is perpendicular to the line of intersection, since
J>p' = >•' (Art. 3(33).
h' T=0, X=0 be the respective conditions that the tangent
and normal cones may become two planes, and I\ (J^ li,
i", fj\ R be tke minors of A', X= Pp + R'r -^-Q'q, but
2' = qr-2)"' = uT, Art. 363,
F ' = q'r' — pp' — u 2] &c. ;
.-. N=T{up-i-lc'r' + v'q')=T\
431. To find the equation of tlie tangent plane and normal at
arty jioint of (he surf tee (jlven hi/ the equation ^=f[^-, v)'
Let a Hue be drawn through .r, ?/, sr, whose equations arc
^-.t'=m{^-z), rj-ii^n{^-z),
the points In which this line meets the surface are those for
which ^ is given by the equation
r - z =f[x + m (^- z), y + n (?- z)\ -f[x, y)
= [mp + nq) (^- z) + | [nn^ + Ismn + tn') (^- z)' +. ... ,
and If the line be a tangent line two values of ^- ^ are zero ;
therefore 1 = mp + nq^ and eliminating m and n by means of the
equations of the line, we obtain the locus of the tangent lines
^- z =p) (^ — yl(), PAq at the multiple point.
If the pLane move past V to the position ir, the curve of
intersection Avill gradually assume an oval form, which will
degenerate into a conjugate point at a.
It is clear also that a plane may meet the ring in the circle
GIIKL, in which case it is a tangent plane at every point of^
the curve in which it meets the surface ; this curve is composed
of two coincident circles, as may be seen by moving the plane
inwards parallel to itself.
It will be shewn also, that a tangent plane, drawn through
a line COD perpendicular to Oz^ intersects the ring in two
circles.
OF SURFACE AND TANGENT PLANE. 291
436. To find the equations of the tangent line at oni/ puint of
the curve of intersection of a surface loith its tangent plane.
Let the equation of the surface be F[^^ 77, ^)=0; that of
the tang-ent ph\ne at [x^ y, z) will be
,^ , clF , , dF ,^ , dF ^
or i^-cc)F'ix) + [v-!/)F'{^) + {^-z)F'{z) = 0. (1)
Let the equation, of the tangent line at any point {x\ ?/', z') of
the curve of intersection be
L^' -V-!/^_ ?:
z
since this line lies in (1),
\F'{x)^tiF'{y) + vF'[z)=i), (2)
and since it meets the surface in two coincident points at
(-^'j y'l ^'),
\F' {x') + fiF' if) + vF' [z') = ; (3)
these two equations determine X : /j, : v when (a;', y', z') is an
ordinary point on the curve and the surface.
437. To find the singular points of the curve of intersection
xoith the tangent plane at any point.
If the point be a singular point on the curve of intersection,
any line drawn through this point will have two points coin-
cident at the point considered ; hence, the two equations
obtained in the preceding article will be satisfied by an infinite
number of values of X. : /* : v; this will happen in any of the
following three cases :
(i) When F' [x].^ F' {y) and F' [z) vanish simultaneously,
which occurs when there is a singular point at (.r, y^ z). in
which case there arc an infinite number of tangent planes.
(ii) When F' {x'\ F' [y) and F' [z] vanish simultaneously,
in which case [x\ y\ z) is a singular point on the surface.
H ^vi,on^;-g)=£:g;)=^;;j,i„ „.,..,. ...e .1,0
tangent plane at (.r, y, z) is a tangent plane at (.<•', //', z') also.
292 KULED SURFACEi?,
In case (1) one of the tangent planes to the tangent cone
touching it along a generating line (V, /u,', v') must be the
l)lane considered, and the equation (2) must be replaced by
[uX + iv'/jk' + v'p') \ + {w'X' + Vfi' + u'v) IX
+ {v'X + u'[x' + icv) V = 0, (Art. 4t^9),
thus the ratio \ '. [x : v will be determined, except in cases where
{x\ y\ z) is a singular point on the surface, or where the
tangent plane considered is also a tangent plane to the surface
at(a;',2/', s').^^
In case (ii) a third point at least must be coincident with
(a;', y\ z\ and the equation (3) must be replaced by
where s is 2, 3, ... according to the degree of multiplicity of
the singular point {x\ y\ z').
In ease (iii), if neither [x, y, z) nor (a:', y'^ z) be singular
points of the surface, the equations which determine \\ [iw
will be Xi^' {x) + iiF' [y] + vF' [z] = 0,
whether (cc', y'j z.) be coincident with (a;, ?/, z) or not.
This ease includes the singular tangent plane, a portion of
whose curve of intersection consists of two coincident curve
lines, which will be considered immediately.
riuUd Surfaces.
438. The student is already familiar with certain surfaces
which are capable of being generated by straight lines, or
through eveiy point of which some straight line may be drawn
which will coincide, throughout its length, with the surface.
For example, — a plane, a cone, a cylinder, an hy[)crboloid
of one sheet, an hyperbolic paraboloid.
He is aware that any portion of two of these, the cone and
the cylinder, may, if supposed perfectly flexible, be developed
into a plane without tearing or rumpling.
DEVELOPABLE SURFACES. 293
We shall now give sonic account of the general character
of surfaces which have this property, distinguishing them
from those which, although capable of being generated by
the motion of a straight line, arc incapable of development
into a plane.
439. Def. a liuJed Sioface Is a surfiice which can be
generated by the motion of a straight line; or a surface
through every point of which a straight line can be drawn,
which will lie entirely In the surface.
A ruled surface, on which each generating line intersects
that which Is next consecutive, Is called a Devehimhle Surface^
or Torse.
A ruled surface, on which consecutive generating lines do
not Intersect, is called a Skew Surface, or Scroll.
Developable Surfaces.
440. Explanation of the developunent of developable surfaces
into a plane.
Let Aa, Bb, Cc, ... be a series of straight lines taken in
order, according to any proposed law, so as to satisfy the
condition that each Intersects the preceding, viz. In the points
a, b, c, ... .
Since Aa, Bb Intersect In a, they He in the same plane,
similarly the successive pairs of lines Bh and Cr, Cc and Dd^ &c.
lie In one plane ; thus, a polygonal surface Is formed by the
successive plane elements AaB, BbC, &c.
This surface may be developed Into one plane by turning
the face AaB about Bb, until It forms a continuation of the
plane BbC, and again turning the two, now forming one face,
about Cc until the three AaB, BbC, CcD are in one plane,
and so on ; the whole surface may, therefore, be developed into
one plane without tearing or rumpling; the same being true,
however near the lines Aa, Bh, ... are taken, will be true in
the limit, when the surfixce will become what we have dcfnied
as a developable surface, this name being derived from the
property just proved.
294
DEVELOPABLE SURFACES.
Edge of Begresslon.
441. The polygon ahcd^ ... whose sides arc in the direction
of the lines Bh^ Cc^ ... becomes in the limit Ji curve, generally
of double curvature, which is called the Eihje of Rcgressioii,
from the fact that the surface bends back at this curve so as to
be of a cuspidal form. Every generating line of the system
is a tangent to the edge of regression, which is therefore the
envelope of all the generating lines.
DEVELOrAHLE SURFACES. 295
In the case of a cyliuder, the edge of regression is at an
Infinite- distance.
For a practical construction of a developable surface having
a given edge of regression, see Thompson and Tait, Nat. FltiL^
Art. 149.
442. To find the general nature of the intersection of a tan-
gent plane to a developable stafiace with the surface.
The plane containing the element DdE of the surface repre-
sented by the figure evidently becomes in the limit a tangent
plane to the developable surface at any point Z> in the genera-
ting line Dd, since it contains two tangent lines, viz. Dd and the
limiting position of a line joining such points as D and E, which
ultimately coincide ; and again, supposing I)dE in the plane of
the paper, Ff meets this plane in e, Ggf meets it in some
point /', Ilhg in g'j &c., and similarly for Cc, Bb, . . . on the
other side.
The complete intersection of the surface and tangent plane
is therefore the double line formed by the coincidence of l)d^ Ee^
and the limit of the polygon ah' clef g ... which is a curve
touching the double line Dd at the edge of regression.
Cor. To find the nature of the contact of the edge of re-
gression and the tangent pilane .
The plane containing the generating lines A/, Ee contains
the three angular points c, <:/, e of the polygon in the limit,
therefore the tangent plane contains two consecutive elements
of the edge of regression, and is, as will be seen later on, what
is called the osculating plane at that point.
443. The shortest line which Joins tico j^oints on a develop-
able surface is the curve., the osculating plane at every point of
which contains the normal to the surface at that point.
If the surface be developed into a plane, the shortest line
must be developed into the straight line joining the two points.
If on the polygonal surface in the figure on page 294, ADCD...K
be the polygon which in the limit becomes the shortest line
joining A and iv", since on development this becomes a straight
line, two consecutive sides EF^ EG must be iuclincd at equal
206 SKEW SURFACES.
angles to line Ff. Ilcnce a straight line, drawn through F
perpendicular to the line Ff in the plane bisecting the angle
between the planes EFf\ GFf^ will evidently lie in the plane
EFG^ and bisect the angle EFG. This line will be in the
limit the normal to the surface, and the plane EFG will be
the osculating plane of the curve ABCD ... at the point F.
Therefore the shortest line is the curve, the osculating plane
at every point of which contains the normal to the surface at
that point.
Such a line Is called a geodesic line of the surface, and it
will be hereafter shewn, that the property enunciated for de-
velopable surfaces is true for geodesic lines on all surfaces.
If the geodesic line, joining two given points, be drawn on
a right circular cone, the equation of the projection upon the
base can be shewn to be
- sin (7 sin a) = j sin [0 sin a) + - sin [(7 — ^) sin a},
a, h being the distances of the given points from the axis,
7 the angle between these distances, and a the semi-vertical
angle of the cone.
Skew Surfaces and Curves of greatest densitij.
444. Let AA\ BB\ CC\ DD\ &e. be straight lines drawn
according to some fixed law, such that none intersects the next
consecutive; let aa\ hh\ cc\ dd\ ... be the shortest distances.
Suppose now that we take two of the generating lines as CG\
1)D\ and imagine DD' twisted about c so as to be parallel
to CC", and united with It by means of an uniform clastic
membrane : if now DD' be returned to its original position,
the portion of the membrane near cc being unstretched,
will be denser than any other portion. If the same process
be adopted for every line, the series of membranes will gene-
rate a surface which will ultimately, as the lines approach
nearer to one another, become a skew or twisted surface.
The curve which is the limit of the polygon formed by
joining «, h^ c, fZ, ... at which the Imagined membranes would
have the greatest density, is called the curve of greatest density ;
It is also called the lino of strict ion.
SKEW SURFACES.
297
It may be observed, that the shortest distances between the
conscciitive generating lines of a scroll are not generally
elements of the line of striction.
445. To explain the nature of the contact of a tangent plane
to a shew surface at any point.
Let P be any point of a skew surface, AA' the generating-
line passing through P, suppose a plane to be drawn tlirough 1*
containing BD' the next consecutive position of the generating
line, this plane will intersect the third line CC in some point 7/,
QQ
298 SINGULAR TANGENT PLANE.
and, if PR be joined, it will meet BB' in Q ; PR will therefore
be a tangent line at P having a contact of the second order at
least, so that, if the surface were of the second order, it would lie
entirely in the surface. The tangent plane at P is the plane
containing A A' and PR\ R will change its position for any
change of position of P, thus the tangent plane at any point
in AA' will always contain AA'^ but it Avill move about AA
through all positions, as the point of contact moves along AA'.
The tangent plane, therefore, at any point of a skew surface
contains the generating line and some other curve which must
be a straight line in the ease of a surface of the second degree.
446. To shew that the equation of the tangent plane to a
developahle surface contains only one 'parameter.
Since the general equations of a straight line involve four
arbitrary constants, we must, in order to generate any ruled
surface, have three relations connecting the constants, so that
it may be possible, between these equations and the two equa-
tions of the generating line, to eliminate the four constants, and
thus obtain the equation of the surface which is the locus of all
the straight lines. In developable surfaces the generating
straight lines are such that any two consecutive ones inter-
sect, and the plane containing them is ultimately a tangent
plane to the surface. The equation of this plane will then
involve the four parameters, and by means of the three relations
we may eliminate three, so that the general equation of the
tangent plane to a developable surface will involve only one
parameter, and we may write it in the form
a being the parameter, and 0(a), "^[o) functions of that para-
meter, given b any particular case.
Singular Tangent Plane.
447. Def. a singular tangent j)la)ie is a plane which, instead
of touching a surface in any finite number of points, touches
along the whole of a curve line.
SINGULAR TANGENT PLANE. 299
If the curve of intersection of any plane with the surface be
composed, in part at least, of two or more coincident lines, the
other part being made up of simple curves, either the plane will
be a tangent plane to the surface at every point of such a mul-
tiple curve, or it will contain a multiple line of the surface, such
as would be generated by the rotation of a cross round any
fixed Hue not passing through the angle of the cross.
Conversely, if a tangent plane touch along a curve line on
the surface, this curve line will be a multiple line on the tangent
plane.
Thus, in the case of the anchor ring (Art. 435), the plane
which touches the ring along a curve has for its curve of inter-
section the two circles coincident in LKII] also the taiigent
plane to a cone contains two generating lines which ultimately
coincide, and is therefore a tangent plane at every point of the
generating line which it contains; any more general develo])-
able surface is an example of the case of a tangent plane which
contains a double line, at every point of which it is a tangent,
combined, as shewn in Art. 442, with another simple curve.
A surface of the fourth degree admits of the case of a double
conic, as in the example of the anchor ring, or of a quadru])le
straight line, as when it is made up of two cones touching
along a generating line.
A surface of the fifth degree might be composed of one of
the third degree and one of the second, in which case a tangent
plane might meet the former in a triple and the latter in a
double straight line.
448. To find tlie condition that a tangent plane may he sin-
ffidar.
Since a line, drawn in any direction in the tangent plane
through any point of the double curve in which the tangent
plane touches the surface, will contain two coincident points, but
if it be drawn in the direction of the two coincident tangents to
the curve of contact it will contain four coincident points, we
have to express that at every point of the double curve there
are two coincident tangents, and that a line in their direction
contains four coincident points ; and we may observe that
300
SINGULAR TANGENT PLANE,
= 0,
these tangents arc what have been called inflexional tangents
(Art. 432).
Since the two inflexional tangents coincide, their direction-
is given by the equations
u\ + to'fi + v'v _ w'\ + Vyw + tt'v _ v'X + u'fi + wv
Ur - V ~ W ''
and \U+fxV+vW=0.
Tlic condition that these equations shall hold i»
w, w', v\ U
w\ Vj u'j V
v', m', w, W
U, V, IF,
and the condition that a fourth point may become coincident is
that for the values of \ : /^ : v given by the above equations
the further conditions wlien the curve has a higher degree of
of multiplicity may be easily obtained.
449. The conditions of the" existence of a singular tangent
plane may also be found, by considering that the point of con-
tact, determined by the equations of Art. 425, may be any point
of a curve line, and its coordinates are therefore indeterminate.
4.")0. For a surface given by the unsymmetrical equation
^ =/(!■, 7j), the equation of a tangent plane at any point (a:, ?/, z)
is
a tangent line whose equations are
meets the surface in points for which p is given by
vp = (p\ + ^At) p + y {rX"" + 2sX/M + tfjC')
ANCnOK RINO. 301
If the tangent plane be singular, for all the points in wliicli
it meets the surface, v=pA, + 2/A, and for all the points of the
double curve four values of p are zero, and the two inflexional
tangents coincide ;
/. 7-V + 2s\^l + tiji' = and (x ^ + fi -^ J z = 0,
and the former has equal roots, therefore rt = 5^, and either
?-X + *f/x = or s\+ f/j. = 0.
451. Every tangent plane to a developable surface is a sin-
gular tangent plane, since it contains two consecutive generating
lines, hence its curve of intersection with the surface consists of
two coincident straight lines, and, as shewn in Art. 442, a single
curve line. The analytical conditions of singularity are satisfied,
since, if \, yu., v be the direction-cosines of the double line, which
lies entirely in the surface, the coefficients of all the powers of
p will vanish, and rt = s^ in consequence of the coincidence of
the two lines.
At any point of the single curve, the values of p, q being
y, q, the direction cosines X., /i, v of the tangent are given by
v = \p-\- fiq and v = \p' + mq', which are independent equations,
since //, p and q, q are generally unequal.
452. We have selected the following illustrations of the
points which have been considered in this chapter, and we
call attention especially to those relating to cubic surfaces and
the wave surface as of intrinsic importance.
453. Tangent plane to an anchor ring.
Let the plane containing the centres of the generating circles
be taken for the plane of a-?/, and the axis of rotation for the
axis of z ; and let r be the distance of any point (a?, y, z) from
the axis, c that of the centre of the generating circle, a its
radius; then r^ = x^+f/^ and z^ -^^ {r - cf = a'' ; the equation of
the anchor ring is
ie + V' + r + c' - ci'f - ic' (r 4- rf) = ;
302
IIELICOID.
that of tlic tangent plane at a point (a?, y, s) is
x{r-c)[^-x)Ary [r - c) [rj -7j) + zr{^- z) = 0,
or (r - c) {x^ + 1/7]) + rz^= ?•' (r - c) + rz' = r {a" + c{r-c]].
To find the curve of intersection of the surface with a tangent
plane which j^asses through the centre.
Suppose that it passes through axis of ?/, and is inclined at
angle a to that of a^, so that a — c sin a; and at any point of the
curve of intersection r = c— a cos^, e = a sin ^, x = z cota ;
.'. y^ = ?•'■' — x'^ — c^ — 2ac cos^ + a^ cos""*^ — a^ cot"''a sln^fl
= c'"' — 2ac cos 6 + a" cos^ 6 — (c' - a^) sin'
= (c cos^-a)";
•'• {y ± cf = c^ cos'"' 6 J
and x' + z'' = c^ sin"' ^ ;
.-. x' + [ij± af + z' = c' ;
hence the curve Is two circles which intersect in the points of
contact, forming two double points.
To find the form of the curve EAF in the figure of the ring.
The equation of the tangent plane is ^ = c — a^ and the form
of the curve of intersection is given by the equation
[ri' + r + 2c (c - a)Y = Ac' {v' +{c- a)'],
or iv" + D' - 4«C77'^ + 4c (c - a) ^' = 0.
When c = 2a, the curve is the lemniscate of Bernoulli.
454. Tangent plane and normal to a Helicoid.
Def. The Helicoid is a scroll generated by the motion of
a straight line which intersects at right angles a fixed axis,
about which it twists with an angular velocity which varies
as the velocity of the point of intersection with the axis.
If the axis be taken for the axis of z^ and that of x be one
position of the generating line, the equation of the surface
generated will be
r=rtan->|,
and the tangent plane at a point (.r, ?/, z) will bo
EXAMPLE OF SINGULARITIES. 303
or (a;" + y^) (^- s;) = c {xrj - yf ) ;
at the point (a;, 0, 0), the equation becomes x^^ct], hence the
tangent of the angle which the tangent plane at any point
makes with the axis varies as the distance of the point from
the axis.
The equations of the normal at (a?, ?/, z) are
^- X ^ v-y ^ c (g"- z) ^
y -X x'^if '
and for the normal at (.r, 0, 0), ^ = a;, xr]-\ c^= 0, hence the
locus of the normals at points taken along a generating line is
an hyperbolic paraboloid ; which is true for any scroll.
455. Tojind the singularities of the surface lohose equation is
{z' + 2x' + 2/)' - [■^' + f) [x' + y'+\Y = 0.
Wc consider this surface as represented by the given equation,
in order to illustrate the general methods given in Arts. -4:27
and 448, for discussing singular points and planes; but the
student will see clearly the results to which we shall be led,
if he first trace the plane curve whose equation is ?/^= a? (1 — a;)^,*
and then imagine the form of the surface which would bo
generated by its revolution round the axis of ?/, which it is
easily seen is the surface proposed.
To find a singular point we have, writing r"^ for x'' + y\
U =2x [Iz' -iJ^^r"- 3/) = 0,
V =2y [Az' _ 1 + 4r' - 3/) = 0,
TF= Az [z' + 2r') = 0.
The systems of values of x, ?/, z which simultaneously satisfy
these equations, and that of the surface are ^ = 0, and either
(i) cc = 0, ?/ = 0, or (iij r'' = 1 ; (i) shews that the origin is a
singular point — it will be found that the tangent cone of
Art. 428 becomes an infinitely slender cylinder or cone,
given by V +/*'' = ; (ii) gives a circle of singular points — the
conical-tangent at any point (a?, y^ 0) of this circle becomes the
two tangent planes [x^ + yTj—}y=z ^\
* Frost's Curve Tracing, Plate II., Fig. 8.
304 WAVE SURFACE.
To find a singular tangent plane vvc have, by Art. 448, the
equations
2 [Xx + fxy) [Az" - 1 + 4r'' - 3/) + 4s {z' + 2r') v = 0,
and 2 [X' + /x*) (4s' - 1 + 4r'' - 3r*) + 4.v^ [Zz' + 2r')
+ 32s [\x + /x?/) v + ^[\x + firjY (2 - 3r') = ;
there will be two coincident tangents if
j/ = and 4s''-l+4r'-3r* = 0,
also by the equation of the surface [z^ + 2ry — r'^ (1 + r")' = 0,
the only solutions of these equations are s"'' = 0, r''=l, and
z^ = -^j^ r^^^f the first solution gives no tangent plane, but
two cones intersectingN^ a circle, any generating line of either
of which is a tangent line; the second solution gives two
tangent planes s = + § \/3, each of which is a singular tangent
plane touching along a circle ic'* + ?/^ = ^, the direction of the
tangent to which is given by \x-\- fiy = and v = 0, the re-
maining part of the curve of intersection is a single circle of
radius |.
In this case the condition of four points being coincident is
X-f -^ H- -r] F=Oj which becomes
[3 V dx "^ d7j)
2,0 (^xx + fxy) {\' + /.^) - 8 [\x + li^yY = 0,
it is therefore satisfied by Xx + fiy = 0.
Wave Surface.
456. The equation of the Wave Surface may be written in
either of the forms
y' z'
+ -/^, + -i 5 = 1,
r' — a^ r'^ — ¥ r^ — 6^
aV V'f d'z' ^
or -s i + -r,-^, + -5 Q = 0,
r -a r — b r ~c
where r' = rc^ + ^^ + z* ; and we shall suppose a>h> c.
The existence of singular tangent planes to this surface is of
great importance in explaining a peculiarity in the transmission
of light through a biaxial crystal.
In order to shew that such planes exist, we shall employ the
method of Art. 449.
WAVE SURFACE. Wi)
457. To jind the point of cojitact of a tangent plane whose
equation is Ix -f my -{ nz =pj and the relation between ?, ?>?, n
and p.
Tlic equations for cleterminlng the point of contact arc
V V W xU-\-yV+sW
-y = - = — = ' = -2cr suppose,
I m n p 117
Op
where U =-^i^^-..-2jl\
r - a'
r' — c
p 0^ v" ^^
.-. X Z7+ yV-\-z W= 2 - 2/-'P= - 2 - c
Also by squaring and adding, and observing (2),
^=^^f^^'-f^-fT 1(7^. ^ (^73^3 + ^TZTri+Zi
(3)
The equations (1) and (3) give the values of x, y, z at the
point of contact, and (2) is the required relation between the
constants.
R It
306 WAVE SURFACE.
If a, y9, 7 be the Bootbian coordinates of tbc tangent pUino,
VIZ. — , — , — , and a. o . c be written lor - , 7- , - , tbe tan-
gcntial equation of tbe surface will be
aV h'^^-^ eV
p — a p —0 p —c
wberc p*"* = a'^ -f ^'^ + 7^^, an equation of tbe same form as tbe
Cartesian.
458. To find a singular tangent idane of the xoave surface.
Tbe point of contact in tbis case being any point in a
curve, tbc coordinates must be indeterminate ; now y Avill be of
tbe form — if ^; = b and m = 0, and tbese values also make r'
indeterminate, and tberefore x and z.
And since, by (2) and (3),
r^-f-P \[f-aj [f-6y\'
T li' 1
we baVC -^ jr, = y^j -„ = -^ :, .
a —b b —c a —c
Tbe curve of intersection is a circle given by tbc })lanc
lx + nz= p, and eitber of tbc spberes
, d'-b""
. b'-c'
or r' - — — z = c\
nb '
p = a, l — O and p = c^ n = give imaginary planes, bonce tbcrc
are four real singular tangent planes.
459. To find tlie singular 2^oints and the corresponding normal
Tbc singular points may be founxl by investigating for wliat
definite points of contact /, ?/?, n can be indeterminate, wo
sball tbus obtain
ax c z a c
,j^O,r = !,, ana ^.~^, = ,.,-, = -^. , (4)
wbicb determine four singular points.
CUBIC SURFACES. 307
By equations (1) of Art 157,
a" — b' , d^ 1 h* — c' c'
— l = p and ?? = w ;
X P ^ V
a^_J-^ y^-c"- a'-c'
.-. 1+ n = -;
X z p
.'. [a^xl + c^zn) [Ix + nz) = aV, by (4),
or (iVf" + c'zhf + {a' + c') xzln = d^c\
... (,r - h') r + {h' - c'-') n' + ^^^^ V(«' - 1^') \/{h' - c') In
which gives the eqiuation of the normal cone at the singular
point.
Cubic Surfaces.
460. On every surface of the third degree there are 27 straight
lines and 45 triple tangent jylanes^ real or imaginary.
This theorem was iirst discovered by Cayley.*
An arbitrary straight line intersects a cubic surface in three
points, given by an equation of the form
u^ Du.r-^\D-u. r' + 1 Dhi . r' = 0.
Now the four constants in the equations of a line may be
chosen so us to satisfy the equations m = 0, Du = 0, D''u = Oj
l)'u = 0, and, since the above equation will then be satisfied
by all values of r, all straight lines having such constants will
lie entirely in the surface ; and the number of such straiglit
lines will clearly be limited, speaking generally, although in
particuKir cases, as in that of a cylindrical surface, it may be
infinite.
If a plane be drawn in any direction through such a straight
line, its line of intersection with the surface will be composed
of that straight line and a conic forming a group of the third
degree; and the two double points in which the straight line
♦ Camhriil'je and Dnhlin Matkernnticnl Journal, vol. iv.
308 CUBIC SURFACES.
intersects tlic conic arc two points of the surface at which the
plane is a tangent plane to the cubic (Art. 434).
JSow, there will be five positions of the plane for which the
conic will become two straight lines.
For, if the axis of a? be a line which lies entirely in the
surface, the equation of the surface will be of the form
where m.^, v^ are quadric functions ; and if the surface be cut by
a plane whose equation is ^ = - = r, the conic, which is part of
the line of intersection, will have for its equation /u,u'„ + vv\ = 0,
where u\^ v',^ have each the form
a/' -f h^x' + c^ + 2clpc + 2(?/ + 2/ra = 0,
in which a^, e^, f^ are homogeneous functions of /i and v of
the degrees denoted by the suffixes.
Hence the equation of thd conic will be
cuf + /9,a;'' + 7, + '2h^x + 2 a./ + 2^/a! = ;
it will therefore become two straight lines if
^A% + 2a,£,r, - o^K' - ^.<'iX: = 0,
which gives five values of the ratio A, : /i.
In each of the five particular positions of the plane the com-
plete intersection is three straight lines, which give three double
points, and the plane is a triple tangent plane touching the
surface at each of these double points.
Through each of the three straight lines In a triple tangent
plane four other triple' tangent planes besides the one considered
can be drawn, giving rise to 12 new triple tangent planes and
24 new straight lines, making in all 27 ; and the surface cannot
contain any but these 27 lines, for the point in which any line
on the surface meets a triple tangent plane ABC must lie on
one of the three lines AB, BC^ CA, which form the complete
intersection of ABC with the surface, and the plane which
passes tlu-ough the new line and AB, supposing this to be the
line which it cuts, must contain a third line, and, therefore, must
be one of the five triple tangent planes drawn through AB;
the line considered must therefore be one of the 27 lines.
LINE OF STRICTION. 309
Five triple tangent planes can be drawn through each of the
27 Ihies, which would make 5 x 27 planes in all ; but since each
plane contains three of the lines, we have in obtaining this
number reckoned each three times, hence the number of triple
tangent planes is 45.
Line of Striction.
461. To fnd the line of strict ion of a scroll.
Let the equation of a generating line be
^ = ;..! + a, r=*'H/3, (1)
where the constants are functions of one parameter 6] the
equations of a consecutive generator corresponding to a value
6 + (16 of the parameter arc
77=(m + f7»0l+a + f^«, ^={n + dn)^-^^^d^. (2)
Let P be a point in the line of striction ; PQ the shortest
distance between (1) and (2) ; x^ y^ z and x-\-hx^ 2/ + %? ^ + ^^
the coordinates of P and Q.
Since FQ is perpendicular to both generators,
Zx + m'^y + n^z = 0,
and Ix + [m + dm) By+{n-\- dn) Bz = 0]
.'. dni8y + dnSz = 0.
Also, by the equations (1) and (2),
8y — mSx = xd7n + d(x,
hz — nBx = xdn + d^ ]
hy Bz _ Bx
' ' dn — dm ndm — mdn
_ By — mBx _ Bz- nBx
(1 + m") dn — mndm — (1 + li^) dm + mndn '
/. {xdni + da) {(1 + n^) dm — mndn]
+ [xdn -I- /3) {(1 -I- m') dn - mndm] = 0.
If the parameter 6 be eliminated between this equation and the
equations
y = mx-\-oij z = nx-i-/3,
310 POLAR EQUATION
we shall obtain two equations which will be those of the line
of strictlon.
462. Line of strict ion of an hyperholoid of one sheet.
For the hyperbolold y^+-2 = -5+l, the equations of a
generator being
i- = - cos 6 4 sin 6, - = - sin ^ — cos 0,
b a c a
xdm -^da=l{-- sin 6 + coii9]d0= '- dd,
xdni + dlS = c( - cos ^ + sin e\ dd = ^| dO,
( 1 + n') dm - mndn = ( -^ -f -^ ) sin ^ dd.
'dO:
(1 + m"^) an - mndm = — ( ^t, + — , )
^ ^ a \b' ay
/I l\?/.2 X /z' y" x' \\ ,
\c h J be a \c a a J
the intersection of this surface with the hyperboloid gives the
line of strictlon for one set of generators.
Polar Eqnatioii.
463. To find the j^olar equation of the tancjcnt i)lane to a
surface at a given point.
Let the equation of the surface be — = m'=/(^', 0'), and
let ft, 6^ ({) be the coordinates of the point of contact of the
tangent plane.
OF TA\(iKNT ri.ANE. ^JU
The equation of the tangent plane is of the form
jpu =cosa cos^'-f sin a sin^' cos(^' — yS), (Art. 7G),
anil the constants^, a, and /3 are to be determined from the con-
sideration that the tangent plane contains not only the point
of contact but adjacent points which have moved up to and
ultimately coincided Avith that point.
Hence the values of -rr^ and -rj fit the point of contact are
da dcf)
the same for both tangent plane and surface, let r, iv be those
values ;
.'. ^)u = cosa cos^ -t- sin a sin 6^ cos((/) - /3),
jyv = — cosa sin^ + sina cos^ cos(^ — /3),
jno = — sin a sin 6 sin {(f) — 13) ;
.•. ]) {u sin^ + V cos^) =siua cos(<^-/3),
2) {u cos 6 - V sin 0) = cos a ;
the last three equations give readily the values of the constants ;
and the equation of the tangent plane becomes
u = {u cos d — V sin 6) cos 6'
+ {u sin^+ V cos6) cos(0' — 0) sin^'
+ to cosec 6 sin (<^' — >) sin 6'.
Tiiis equation can also be written in the form
r^ d
— = -—[r {sin 6 cos d' - cos 6 sin 6' cos (0' - ^)]]
r- cosec ^ sin^' s'm icb' — 6) .
dq)
464. To find the perpendicular distance from the pole xqion
the tanr/ent 2)lane.
This may be obtained from the first three equations of the
last article by squaring and adding, whence
p' (m" + v' H- w" cosec* ^) = 1 ,
1 , (duV {duV .a
312
POLAR EQUATION.
405. We may arrive at the al30ve result hy the l\)llo\vlng
process, wliicli serves to shew the geometrical signification of
the partial differential coefficients, and will be useful as an
exercise.
Let P be the point of contact, Fit a tangent line passing
through OZj and FQ a tangent line in the plane through OF
perpendicular to the plane FOZ; take E and Q points very
near to P, and in OQ^ OF. take Op, Oj) each equal to OP;
then F/) = r &m 6d(fi and Fp'^rcW ultimately, and Qp^—Fjy
are respectively the values of dr due to changes of 6 and >,
considering the other constant,
dr
j:=-eotOPP,
and
= -?!'=-
coiOFQ.
r sin 6d(f> Fp
Draw OF perpendicular to the tangent plane QFR^ and on a
sphere, whose centre is P, let a8/3 be a spherical triangle with its
angular points in FQ, FO, PP, join 8y, 7 being the intersection
of FY and a/3, then By is perpendicular to a/3, and a8^ is a
right angle, llencc
cota8 = cotS7 cosa^Y, and cotyS5 = cot67 siuaS7;
ASYMPTOTES. .".l.'>
.•. cot'aS + cot'/SS = cot" 87 = ./ ;
V
1 1 1 fd)-\' 1 {rh-\^ 2^
466. Def. An asymptote to a surface is a straight line
which meets the surface in two points, at least, at an infinite
distance, while the line itself remains at a finite distance.
An asi/mjytofic plane is a tangent plane whose point of ,con-
tact is at an infinite distance, the plane Itself being at a finite
distance.
An asymptotic surface is a surfcicc which is enveloped by all
the asymptotic planes to the surface.
467. General considerations on ayym2)totes.
If we imagine any tangent plane to a surface, and consider
the result of supposing its point of contact to be at an infinite
distance, we shall be led to the following conclusions :
Smce the plane at infinity intersects the surfiice in a curve,
real or imaginary, there are generally an infinite number of
directions in which a point of contact may be supposed to move
off to Infinity ; to each of these directions will correspond an
asymptotic plane.
Each asymptotic plane is the locus of all the corresponding
asymptotes, and these asymptotes will all be parallel, since they
pass through the same point at infinity at which they are
tangents.
►Since there are two tangents In every tangent plane at an
ordinary point which pass through three consecutive points, viz.
the tangents to the curve of Intersection at the point of contact,
there are in each asymptotic plane two corresponding in-
flexional asymptotes which pass through three points at an
infinite distance.
Since any plane which passes through an inflexional tangent
intersects the surface in a curve which has a point of inflexion
at the point of contact of such a tangent, the curve of Intor-
.ss
314 ASYMPTOTES.
section of the surface and any plane drawn through an in-
flexional asymptote has a point of inflexion at an infinite
distance.
4G8. The peculiarities which arise in the case of singular
points at an Infinite distance can be examined without much
difficulty by a comparison with what takes place at a finite
distance.
If, for example, there be a double point at infinity, in the
place of the conical tangent at a finite distance, there will be
a cylinder of the second degree formed by the asymptotes
which correspond to the direction in which the double point
lies.
Of the generating lines of this asymptotic cylinder there
are six which meet the surface in four points at infinity.
The curve of Intersection with any plane parallel to these
generating lines has a double point at infinity.
The curve of intersection with any tangent plane to the
cylindrical asymptote has a cusp at infinity.
4G9. To find the asymptotes to a given surface.
Let F=F{^j ?;, ^) = be the equation of the given surface,
(ic, y, z) any pomt ni an asymptote, --- = =- — =r
its equations; and let F[\,fx^v) be arranged in a series of
homogeneous functions of the degrees n, n- ], ... , so that
F{X,fi, v)ee(I>,,-\- <^„_,-f...+ (^, + c.
The points in which the asymptote meets the surface are
given by the equation
F{x + \r, y + fir, z + v7-) = 0,
.,, T. 1 1 • (^ d d
or, il JJ denote the operation a- ,— -Vy—j — V z — ^
>•>„ + r'-' [D^h,^ + «/,„.,) + r"-' {\D'4>^^ + i},, ., + <^„...) +...= 0.
Kow for a sim])le asymptote two roots are infinite ;
.-. 0„^() (1)
and />>)„ -f (^,,., = 0. (2)
ASYMl'TOTCS. y,\')
The first equation shews that all asymptotes arc parallel to
generating lines of the cone
F„{^,V,^) = 0, (3)
where I\ consists of the terms of the /<"' degree in F.
The second equation
d(b (Id) dd) ,
shews that all the asymptotes parallel to any generating line
of the cone (3) lie in one plane, which is the asymptotic plane
parallel to the tangent plane touching the cone along the
generating line.
Again, corresponding to inflexional tangents in tangent
planes at points at a finite distance, there are generally two
asymptotes in each asymptotic plane which meet the surface
in three points at an infinite distance, the condition of this is
^Z)V„4i^^„-. + <^„-. = 0, (4)
and the two inflexional asymptotes are the lines of intersection
of the conicoid (4) with the plane (2).
It can be shewn that the conicoid and plane Intersect in two
parallel or coincident lines by proving that, if {x, y, z) be any
point in which they intersect, a line drawn through this point
In the direction (\, /i, v) lies entirely in both surfaces.
Write a; + \/* for Xj &c., and ^ for the operation
. d d d
^di+^d,.-^\lv^
X, /A v being considered constant in the difterentlations.
(2) becomes {D + rA) (f)^ + 0„_, = ?*-A(^„ = ?*«0^^,
(4) becomes J {D + rA)' (^,. + [D + ;-A) >„_, + ,,_^
= r {n - 1) {D,^ + >„_,) + ^r'n {n - l) <^,. = Ir'/i [n - 1) <^,, ;
therefore, since ^,, = 0, (2) and (4) arc satisfied for all values
of r.
470. Should the student be Interested in the discrimination of
the various singularities which may occur, he will find a guide
31G ASYMPTOTIC SUIfFACES.
ill two articles by rainvin,* who has nearly adopted our method
of treatment, and has carefully followed out the consequences
of supposing the conicold (4) to have the various forms of which
it is capiible.
471. A singular asymptotic plane is one which touches
the surface along a line at infinity, if considered as the limit
of a tangent ])lane ; and if considered as the locus of asymp-
totic lines, it is a plane such that lines drawn in any direction
in it meet the surface in two points at an infinite distance.
The analytical conditions are obtained by considering that
the equation D(^^^ 4- >„_, = must be independent of the values
of X, yu,, V.
Asyinjjtotic Surfaces.
472. To find the asymptotic surface of a given surface.
The asymptotic surface being the surface enveloped by the
asymptotic planes, which are tangent planes whose points of
contact are at an infinite distance, is a developable surface cir-
cumscribing the surface along the curve of intersection with the
plane at infinity.
The equation of an asymptotic plane is
\ /i, V being connected by the equationa
<^,^ = and \'' + /a"'+ v'=l.
We shall write w, u'... for ,^ , t-^' , ..., and t/_, Z7_ ... for
~dx' ~dx' ••• •
Considering a consecutive position of the asymptotic plane,
we have the equations
dP ,, dP , dP ,
dx'^^-^d/^'^^d'.'^'^''^
U_, dX+ K, dfx,+ Wdv = 0,
\ dX+ fi d/ji+ V Jv = ;
* Cl-elles JnnrnnK vol. «5.
AS V MITOTIC SUHFACES. 31^
tlicrefure, by arbitrary imiltipliers,
-''/' + -lII'„ + 77>' = 0,
and, imiltiplying by X, /i, v,
(«-l)P+^l»(^,. + ^ = 0, .: B=0,'
:
7//+U^„-,
dv
f, /i, ?;i
c?//
now the degrees o^ — ^vio-u"^ and U , arc 2(?t-2) and
da ""' ^ '
?i — 2; the degree of tlie equation is therefore 3?i — 5, and
the number ot values of X, /a, v which satisfy this equation,
and ^^_ = Is n (3/2 - 5), which Is the degree of the asymp-
totic surface.
474. Or we may proceed thus :
The asymptotic surface contains 3n [n - 2) lines in the plane
at Infinity which are the Intersections of the planes of Inflexion
of the cone ^„ = 0, and contains, moreover, the curve of the
rj'^ degree. In which the plane at infinity Intersects the cone;
hence, the number of points In which the asymptotic surface
is met by an arbitrary line In the plane at Infinity
= M [n - 2) + n = w (3» - 5).
For limitations of the number arising from the existence of
singular points, see Palnvlu's second article.*
tl, vol. G5.
Mirnion of ArruoxiMATiux. 319
Mdhod of Approximation.
. 475. Although it is necessary to know general methods of
hantlling the equations of surfaces, yet in order to find the shape
at particular points or at an infinite distance, it is most in-
structive for the student to employ peculiar methods to suit
peculiar cases.
The method of approximation by transferring the origin
to the particular point in question, and rejecting all terms
■which can be shewn to be small compared with those retained,
gives immediately conical tangents or any other form which
nearly coincides with a surface in the neighbourhood of a
singular point.
The form of a sui-ftice at an infinite distance may be found
by a careful consideration of the relative magnitude of the
coordinates in the same manner as the author has treated the
subject in his Treatise on Curve Tracing. The kind of con-
sideration required may be seen by the following example.
476. To find the plane and parahoUc asymptotes of the surface
whose equation is
ck' -h ?/' -I- 2' - Zxyz - 3a {yz + zx + xy) = 0.
The equation may be written
^lv - a {li' — v) = 0,
where u = x + y + Zj v = cc'-i y' + z' - yz - zx- xy.
If xl' and v be of the same order of magnitude when a?, y, z
are very great, we have for a first approximation m = 0, and
for a second t/a = 0, the plane asymptote touching along a
circle at infinity.
If w"^ be large compared with v, the first approximation
gives v = aif, and the next gives v = a[u-a\ which is a para-
boloid of revolution.
The same results m.ny be obtained by making the line
x-y = z one of the axes of coordinates, so that the equation
becomes
320 PROBLEMS.
in which, if x, ?/, z be of the same order of magnitude,
\/(3) x-\- a = 0- and if x be large compared with 7/ and z^
The conical tangent at the origin is if -^^ z^ — 2x\
XVIII.
(1) Prove that the tangent plane to the surface xyz = a^ forms with the
coordinate planes a tetrahedron of constant volume.
(2) Find the equation of the tangent plane at any point of the surface
xyz + 2abc = box + cay + ahz, and find the conical tangent at (a, b, c).
(3) If tangent planes be drawn at every point of the curve of inter-
section of the surface a [ijz + ex + xy) = xyz, with a sphere whose centre is
at the origin, shew that the sum of the three intercepts on the axes will be
the same for all.
(4) A surface is given by the elimination of « between the equations
F{x, y, z, a) = and f[x, y, z, a) = 0; shew that the direction-cosines of
the normal at a point {x, y, z) are in the ratio
F'{x)f\a) -fU) na) : F'{y)fia) -f{y) F'{a) : F'{z)f («) -/'(=) F'i")-
(5) The points on a conicoid, the normals at which intersect the normal
at a fixed point, lie on a cone of the second degree, having its vertex at
the fixed point,
(6) Prove that the projections on the plane of .ry of the normals to the
ellipsoid '- -^ ^.+ —^= I, at points whose distance from that plane is c coso,
* a" 6' c*
touch the curve (ox)^ + (%)■' = {a' - i'j* sin^«.
(7) Given {x* + y' + z' + c* - a')' = 4c' (x* + »/'), find the points the normals
at whLch make angles a, ft, 7 with the axes, and the loci of points for which
(i) 7 is constant, (ii) a is equal to ft.
(8) A chord of a conicoid is intersected by the normal at a given point
of the surface, the product of the tangents of the angles subtended at the
point by the two segments of the chords being invariable.
Prove that, O being the given point, and P, P the intersections of the
normal with two such chords in perpendicular planes containing the normal,
the sum of the reciprocals of OP, OP' is invariable.
(9) Find the tangent cone at the origin to the surface
(x' 4 J/' + ax)* - (c' - a*) (x' + z') = 0;
and shew that as n diminishes and ultimately vanishes, the tangent cone
I'KOBLEMS. .'Jlil
contracts, and ultimately becomes a straight line, and as a increases up to c,
it expands, and finally becomes a plane.
(10) Shew that the 27 lines in a general cubic surface intersect in 135
points.
(U) Apply the method of Art. 448 to find the singular tangent planes
of the wave surliice.
(12) Shew that the normals to any scroll along a generating line lie
on an hyperbolic paraboloid.
(13) If tangent planes at two points on a generating line of a scroll be
at right angles, prove that the rectangle under the distances of the points
of contact from the line of striction measured along that generating line
will be constant.
(1-1) If a series of straight lines, generating a surface, be described
according to a law such that the shortest distance between two consecutive
lines is of a degree superior to the first, it will be at least of the third.
(15) Shew that the lines of striction of an hyperbolic paraboloid
V- = a: are its intersections with the planes ,-. -, = 0.
be "■ b^ c^
(16) A straight line intersects at right angles the arc of a fixed circle,
and turns about the tangent with half the angular velocity of the point of
contact round tiie circle.
Trove that the surface so generated intersects itself on a straight line,
and find the tangent planes at any point of this line.
Shew that the line of striction is a plane curve, whose plane is inclined
to tiie plane of the circle at an angle tan'' 2.
(17) Find the asymptotic planes and the asymptotic surface of the
conicoid ax* + by* + cz* = 2x.
(18) Shew that the coordinate planes are the three singular asymptotic
planes of the surface rijz = «'.
(19) From difTcrcnt points of the straight line - = ^ , z = 0, asymptotic
x" ? ' z^
straight lines are drawn to the hvpcrboloid - + "^^ - -, = 1 ; shew that they
o ^^ a* b* c*
will all lie in the planes — ^ = ± - \'2.
a b c
(20) Shew that the asymptotic planes to the surface
z (x* + t/) - ax' - by* = 0,
are parallel to the plane xy, and that the locus of straight lines in these
planes having contact of the second order at infinity is s = a, or 2 = i;
and that the axis of r is an evanescent asymptotic cylinder.
322 PUOBLEMS.
(21) If the cone of asymptotic directions have a double side, shew that the
surface will generally touch the plane at infinity, and that the section by
this plane will have its inflexional tangents in the intersection with the
tangent planes at the double side of the cone.
(22) Shew that the conicoid which determines the inflexional asymptotes
of the surface, whose equation is x* - j/V - 2a^i/z = 0, is an hyperboloid
of one or two sheets, the latter giving imaginary asymptotes.
(23) Discuss the form of the surface s (x + j/)* - a (^' - t/*) + i'z = at
an infinite distance.
(24) Shew that the asymptotic surface of z (a; + yf - oa* + tj:* = is a
parabolic cylinder.
(25) Shew that there is a conjugate line in the surface
a' {2 {xf + s') - x'f = {xf + c») (2/' + 2« - ay.
(25) Shew that the surface
{x" - 2') [x^ + 3!/' - 8* + 9a')« = {6a (a;* + ?/* - 2*j + AaJ
has a conjugate hyperbolic line in the plane of sr.
CHAPTER XIX.
VOLUMES, AREAS OF SURFACES, &C.
477. To fnd the differential coefficients of the solid contained
between a surface^ given in rectangular coordinates^ the coordinate
planes^ and jjlanes parallel to them drawn through any point of
the surface.
Let ic, ?/, z and x + Aa?, y + Ay, s + As be the coordinates
of two points P and Q upon the surface.
Draw planes through P and Q parallel to the planes of
yz^ zx^ and let V be the volume CRPSOM cut oft' by these
planes from the given solid. If ^V be the increment of T,
324 VOLUMES, AREAS OF SUKFACES, &C.
when X is, changed to a; + Aar, -while y remains constant, and a
simUar interpretation be given to the operation A^^, the volume
Pyjl/=A^F; also the volume PQNM^ which is the increment
of A_^ F when ?/ changes to 2^ + Ay, =A|^(A^r), which is easily
seen to be the same as A^(AyF).
Let 2;,, 2^ be the least and greatest values of z within the
portion of the surface PQ^ therefore PQNM lies between z^AxAt/
and z^AxAy,
A F\ . /A..F>
"K Ax J '\ Ay J
lies between z, and z„.
Ay Ax ' ^
Proceeding to the limit, in which z^ = z,^ = 2, we obtain
cPV ^ cTF ^
dydx dxdy
We may observe that, since the volume PrM is ultimately
equal to the area RM x Aa;, the partial differential coefficient
dV dV
-^ represents the area RM^ and similarly -, the area SM.
478. The differential coefficient of the volume of a wedge
of the solid contained between the planes of zx^ xy^ a plane
through the axis of 0, and a plane parallel to yOz may be
obtained as follows.
Let Fbe the volume included between the planes zOx^ ^Oy^
the surface, the plane whose equation is ?/ = tx^ and a plane
parallel to yOz through any point (a*, y, s), then A,Fis the in-
crement of F when t changes to < + A^, x remaining constant,
and is the volume which stands on a base whose area is \xAt.x)
A^(A,F) is the increment of A, F when x changes to x + Ax^
and is the volume which stands on a base whose area is
\ [x + Axf At - ix'Af. = {x + 1 Ax) Ax Af ;
hence, as before, -^ ' is between z^ (a^ + ^Aa:) and z^[x+lAx),
d''V
and, proceeding to the limit, , -, = zx.
VOLUMES, AliEAS OF SURFACES.ll&cT ' , /:325 ' ,
X ' // • /' ■ ^
479. To find the differential coefficient ofthej>oiti6^of\/.
surface given in rectangular coordinates, cut off by the coordi/iate j
l)lanes, and plaiies parallel to them draion through any point of .
the surface. ''. •
Let P, Q be the points {x, ?/, z) and [x + Aa-, ?/ + Ay, z + Az), •
S the surface PROS, cut off by the planes through P. A^, which is the increment of A^/S
when y is changed to ?/ + A^/, and is evidently the same
as A^ iA,^)-
Let 7,, 7^ ^c the greatest and least inclinations of the tangent
plane to the plane of xy for any point within the surface PQ.
Therefore PQ is intermediate between AicA^/ sec7, and
Ax^y 86072-
Hence — — ^— or is intermediate between scc7,
Ay Ax-
and sec72, which are, in the limit, each equal to sec 7.
T'"='''='''"-'=' 1^ "■• from the plane z O.r, and
let rbe the volume of the wedge of a cone contained between
the planes zOx and zOP, and the given surface, the axis of the
cone being Oz, and 6 the scmi-vcrtical angle.
OPRrS is the increase of the volume when 6 increases by
A^, remaining constant, therefore OPRrS = A..V.
326
VOLUMES, AHEAS OF SURFACES, &C.
OPSQT'is the increase of A, F when ^ becomes 4 A^, and
therefore = A4. (A^ V), and shnilarly = Ag (A^ V).
If OP, OS, OQ, OT intersect a sphere, whose centre is
and radius OP, in P, s, q, t the volumes of OPSQT and OPsqt
will be ultimately equal, and Ps = r/\d, Pt = r sin ^. A^, therefore
A^ (AjF) is ultimately equal to -Jr" sin^ A0A^;
482. To /?«(/ f7//•,, and 1 : cos-v/r,, each of which becomes
ultimately r : ^), where p is the perpendicular from on the
tangent at P;
.-. A* (A^.S') = - sin ei {x, ?/J - (f> [x, ?/,)}
JXy
484. The student will have to determine in every particular
case the best order in which to make the summation of the
elements ; in some cases it will be advisable to take elementary
slices of the surface, instead of the elementary parallelepipeds,
as when the area of a plane section is known.
Thus, in the case of an ellipsoid, the area of a section BPQ
is irQN, RN, and a slice of the thickness dz = ~f- (c'-s')c/^,
, , , . irah
whence the volume is
- .,- 1 [c' - z") dz = -^ irahc
VOLUMES, AREAS OF SURFACES, &C
329
485. He must also judge wlietlier it is advisable to use
other coordinates than those in -which the equation of the
surface is given.
Thus, the equation of an anchor-ring being
{x' + f + z' + 6' - a^f - 4c' (ic' + y') = 0,
if we make a? -^^ y" =■ r"^ ^ z^ = a' — [r — c)\ we can sum the
elements which have their projections on the circular ring
2'iTrdr^ and the volume is
[ ^irrdr V[a*-(r-cy-'] = [ 47r (r' + c) dr s/ {a' - r") = 27rc.7ra\
J c-a •' -a
486. To find the volume contained bctiveen the surface ichose
equation is [x-\- yY = 4:az^ the tangent plaiie at a given j^oint, and
the planes of zx and yz.
Let the given point be (/, //, /<), the equation of the tan-
gent plane is ^ -^ y = a/ [j] {- + h) ] the volume required is
/ (h\ ^x-\- y]'
jjjdxdydz^ the limits being from ^ = a / (~) {•^+!/) - ^' to ^~T '
then from 2/ = to y = 2 \J{ah) — a*, since the tangent plane
meets the surface where {x + y)'^ — 4 \/[ah) (a; + y) + 4a/i = 0,
lastly from a; = to a; = 2 sj{ah). The volume is
//:
— \x^y-2,J{ttl,)Ydydx
l-Jx.,^l,akW,uJl^-^f=yU-
u u
330
t'OLUI^IES, AREAS OF SURFACES, &C.
This result may be verified thus. Let A OB be the surface,
ACB the tangent plane along the line AB^ ADB a plane
parallel to xO>/, adh any section of the surface parallel to xOy.
Then area adh : area ADB :: ad'' : AD' :: Od : OD ;
therefore volume A OBD = I 2ah . j dz = alt' ;
also volume A CBD = ^2ah . 2h = ^ali' ;
hence the volume required Is -— .
y' z'
487. To find the volume of the elliptic paraholoid ; + ~ = 2.r,
cut off hy the plane Ix + my + nz =^?.
Perform the integration in the order a*, y, 0,
_ y' z' _p — my — nz
^'" 26"^2"c' ^'^" ~~l •
For a given value of 2;, the values of y at the curve of intersec-
tion are given by the equation x^ = ir.^,
, 21)111 h ., 2h . . ^ ,,,
or y -i ^y+-z --j{p-nz) = 0, (1)
VOLUMES, AUKAS <>F SUKrACES, &C. 331
of which y,, ?/., arc the roots, aiul z must be taken between the
limits which correspond to y, = y„, that is 2;,, z,^ are the roots of
the equation
-j^fJ"j^{l/-I/.)(i/.-y)dy^^^ by (1),
= ^ /^^ £■' ((2/ - 2/J fy. - Z/.) - (y - !/X] ^^Udz
therefore the volume
= f ^ j (7' - w')' <^«, where 27 = z,^ - 2„ u = s - ^ (.2, + ^J
4 • ci 2* '
cos^dtW^ putting ?< = 7 sin I
and i(^^_.~^)^= .^-+ I + -^-;
.'. volume = - ^/{bc) ^-^ j^ .
4 6
The student may verify this result by the summation of
elementary slices bounded by planes parallel to the given plane.
488. Tojind the volume contained hetwcen surfaces
cos a COS0
V(l -cos'^'a sin*^)
d(f>
= — [C-sin '{cos a sln(C-y8)} - sin'' (cos a sln/3)]
o
since cos B = cos a sin ( C - /9) and cos A = cos a sin /3.
VOLUMES, AREAS OF SURFACES, &C. 333
Wc have given this as an example of the determination of
the limits in the case of polar coordinates, but the result is
obtained immediately from the area of the spherical triangle,
the volume required being the sura of an infinite number of
pyramids whose vertices are in the centre, the volume of any
one of which is ^adS, and the whole volume = ^a x area of the
spherical triangle.
490. To find the volume of a icedge of a sphere cut off hy
a right circular cylinder^ a diameter of xchose base is a radius of
the sphere.
Let the equation of the sphere be p^ + z'^ = a", and that of
the cylinder p = a cos(/).
ra ra cos (p
The volume is I I 2p \/[d' — p') dpd)d<^
= l«M«-ii;;(3sin)-sin3,)(/)}
= §a' {a - f (1 - cosa) + -/^ (1 - cos3a)}.
The surface =//y{/ + (;J)' , / (|)] ,/„/^,
between the same limits,
= d^ j' (1 — sin <^) d^ = a^ (a — 1 + cosa).
491. The following method of dividing a surface into
elements was employed by Gauss in treating of the curvature
of surfaces.
The coordinates of a point are considered as functions of
two parameters a, ;S, the elimination of which would lead to
the equation of the surface.
If a vary while /3 is constant the corresponding points on
the surface will lie on a curve, and a system of curves will
will be formed by giving /3 successive constant values.
334 VOLUiMES, AREAS OP SUKFACES, &C.
Another system of curves will be obtfilncd by making /3 vary
while a is- constant.
The element dS is the quadrilateral figure whose sides are
portions of the curves which correspond to constant values a,
a + da in one system and /3, /3 + 17/3 in the other.
Let If W2, n be the direction cosines of the normal to the
surface at a point in the element dSj IdS is the projection of
the element on the plane of yz^ let this be FQRS, the coordinates
of the angular points in order being 1/^2;
y+TJ"^
cla--^ "^iJ"'^
dy ^ dy Tr. dz ^ dz ,^
and y + -^d/3, ^"r—d^.
The equation of PQ is — r-^ = ""TT" ' '^"^ ^' ^^ easily seen
da da.
that PQ and SR arc ultimately parallel ; the pcq)cndicular from
S on PQ is
dy_ dz dz dy\
di3 da "^ d^ da) ^
K
also PQ = iyj;j -r^j^
and if we write AdadjS, Bdad^^ Cdad/3 for the projections on
the three coordinate planes,
dS={A' + B' + Cy-dadl3.
The surface-integral jj\hi + vw-\- 7m)dS, where m, v, to are
given functions of the position of dS may thus be expressed
in terms of the parameters a, ^, viz.
Jj\Au + no + Civ)dad^.
VOLUMES, AREAS OF SURFACES, &C. 335
402. To find the surface of an ellipsoid exp'cssed in elliptic
coordinates.
Let the equation of the ellipsoid be
x^ 1/ z' _
and let /i, v be primary semi-axes of the hypcrboloids, which
are the elliptic coordinates.
By Art. 283,
••y d^- / dz__dzdy\^ fiv{X'-^'){X'-ry')[fj:'-v')
•'• ^^ [dv df^ dv dfji) ~ I3y {y' - (3') '
,,^ 1 fdi/ dz . dz dy\ 1 ,
'''=iUd-^-d-.di)'^''
- ^ X f^-i^'-^')i^'-y")(f^"-'l dud.
_ X(m--v-)V{(X -- /3'-)(X---7^)1 . ,
y Vl(/^'^ - /3-^) (7-^ - /.^j {/:i-^ - O [i' - v')] ^
_ (/.'-OV{(V -A.-)(V--v T; ,7^,7, Art 286
The area of the surface cut off by four confocal hypcrboloids,
for which /i = yti, and fx.,, v = v^ and v,^ Is
j'^'fi'Mdfi X I"' Kdv-T'MdfM X [" v^NWv,
where J/= y/f^. 1^.^ (y .y^ , -^= ^ (^-^Z. Vj-f/^T) '
493. If the position of a point be given as the intersection
of three surfaces
F{x,ij,z) = a, G[x,y,z) = ^, and II{x, ij, z) = y,
the expression for a volume may be obtained similarly as
follows; when 7 is constant, the variation of a and /3 determines
33G
VOLU^FES, AREAS OF SURFACES, &C,
a surfiicc of which an elementary portion is {A' + B^ A-C'^)^ dad^^
and the equation of the tangent plane at this element is
the perpendicular on which plane from a point determined by
ot, ^, 7 + dy is
. dx „ dii ^dz\ ,
dy dy
dyj
dx
dy
dz
da
da
da
do.'
da'
da
dx'
dy'
dz
dx
dy
r//3'
dz
d/3
and J' =
d^
dx'
d^
dy'
d^
dz
dx
dy
dz
dy
dy
dy
dy'
dy'
dy
dx'
dy'
'dz
{A' + B'+cy-
hence the volume of the elementary parallelepiped, whose
opposite faces correspond to 7, 7 + dy constant, &c., is
and the volume = / / jJdad^dy = I j ~ dad/3 dy,
here =
494. To find the volume of a solid whose hounding surfaces
are given by tetrahedral coordinates.
Let ^, 17, C be coordinates referred to rectangular axes of a
point whose tetrahedral coordinates are x, y, z, w.
Since X, y, z are linear functions of ^, 77, t,
JMdvd^= CJJJdxdydz,
and if Fbe the volume of the tetrahedron of reference
jmdvd^=V;
but the limits for the tetrahedron arc, since x-\^ y + z + io = 'lj
z=-0 to 10 = or z = l —X — ?/,
y = to y = l — X,
x = to a^= 1,
and with these Wimts fffdxdydz = -}. • therefore GV=G.
PROBLEMS. 337
Hence, if F{x^ y, z, to) = be tlie equation of any closed
surface, the volume will be GV ffJJxdydzj the limits of the
integration being obtained from
F{x,7/,z, l-x-i/-z) = 0.
This method is due to Slesser.*
XIX.
(1) Find the volume of the surface xi/ i- ijz \ zx - a* = 0, cut off by
the plane x + y + z = c.
(2) State limits which can be used to find the volume of a closed
conicoid whose equation is ax* + by^ + cz* + 2a'yz + 2b'zx + 2c'xt/ = 1.
(3) Find the portion of the cylinder x' + ?/*- 2rx = 0, intercepted be-
tween the planes ax + by + cz = and a'x + 6?/ + cs = 0.
(4) State between what limits the summation of dxchjdz must be
taken in order to obtain the volume of the cone whose equation is
a:* + y* = (a - z)*, cut off by the planes a; = and x = z.
(5) Find the volume contained between the surfaces
J/* + z' = Aax, and x - z = a.
(6) Frove that the volume included between the surfaces r = a, z = 0,
= 0, z = mr cosO is ^wja^, r and being polar coordinates in the plane xy.
(7) Shew that the volume enclosed by the surfaces x- + y^ = az,
x^ \- y* = ax, and z = is — - , and draw a figure representing the progress
of summation.
(8) Prove that the volume included between a cylinder j/' = 2rx - x',
^ ' t/' / 5 3 \
a paraboloid — + T ^ •^'^ ^^^ ^^^ plane of xy is tt?-* f— + ri) •
(9) Prove that the volume cut off from the cone
ux* + vy* + tcz^ + 2/yz + 2gzx + 2hxy = 0,
X* v* z*
by the ellipsoid -; + r; + -. = 1 is *-ra6c (1 - k), the curves of intersection
' "^ a* b* c* ■> \ J
of tlie cone and ellipsoid being ellipses, and k given by the equation
ffh ^ hf ^ fff _l
gh - iif hf - vg fg - wh k* '
Quart, Jour, of Math., vol. ii.
XX
338 PKOBLEMS.
(10) Prove that the volume cut off by the i)lane rj = h from the surface
«»x2 + iV = 2 {ax + bz) f is -^^ .
(11) A cavity is just large enough to allow of the complete revolution
of a circular disk of radius c, whose centre describes a circle of the same
radius c, while the plane of the disk is constantly parallel to a fixed plane,
and jjcrpendicular to that of the circle in which the centre moves. Shew
that the volume of the cavity is ~ (3/7- + 8).
(12) Two cones have a common vertex in the centre of an ellipsoid, and
their bases are curves in which the surface is intersected by plHiies parallel
to the same principal plane, prove thai the volume of the ellipsoid con-
tained between the cones varies as the distance between the planes.
(13) Prove tliatthe volume contained between the plane z = {c - x) cot «
and the surface a-z' ^ {x - c) (j:* + »/') = is
— (3 cot « cosec a - 2 cos^ a - 3 log cot ^a).
(14) The volume contained between the surface
c' & \a hj c \a bj ab
and either of the planes yz or xz is „„- .
(15) Shew that the whole volume of the surface whose equation is
(x* + J/' + s*)' = cxyz is equal to ^-7- .
ytio
(16) Investigate the form of the surface whose equation is
and shew that its volume between values of tan'' - from to 27r is -,7r^a^.
X
(17) Shew that tlie volume of the closed portion of the surface whose
equation is 4a (?/' + z' - 40') + (x' - a") (3z + lOa) = is %\.\-rr {oaf.
(18) If ^5 be an element of the surface of an ellipsoid at any point, and
ui the area of a section by a jilane drawn through the centre, parallel to
the tangent plane at that point, prove that the limit of 2 — =4, the
summation being taken over the whole surface.
Find AS in terms of a, /3, if a: = a cos a, y = 6 sin a cos/i, and z = c sin a sin/3.
(19) If 5 be a closed surface, dS an element about P, at a distance r
from a fixed point O, the angle which the normal drawn inwards n?akes
I'KlMJLK.MS. 3.'}U
with OP, shew tliat llic volume contained by the surface = ij J r cosr/)d>S,
the summation being extended over the wliole surface.
O being the centre of an ellipsoid, apply the formula to find its volume,
interpreting geometrically the steps of the integration.
(20) Shew that 1/ extended over the surface of an ellipsoid is
equal to _ f 3 + p + -Jx volume of the ellipsoid.
(21) Prove that the area of a closed surface, no plane section of which
has singular points, may be expressed by the definite integral
sin dfpdO
u:"-
p
where p is the perpendicular from the origin upon the tangent plane.
(22) If each element of a closed surface be multiplied by -^ cos 0, where
r is the distance of the element from a point O, and is the angle between
the direction of r and the normal to the surface measured outwards, shew
that the sum of all such products is or 47r/t, according as O is without
or within the surface.
(23) If r be the distance from a point O of any element dS of a spherical
surface, determine the form of the function/ (r) when / / (/5', the sum-
mation being effected over the whole surface of the sphere, is constant for
all positions of O within the sphere.
(24) Shew that the shortest distances between generating lines of the
same system drawn at the extremities of diameters of the principal elliptic
section of the hviicrboloid, whose equation is — + — - ^ = 1, lie on the
■ ' a* h* c-
. Prove also that the volume
cj:?/ ahz
surfaces whose equations are , ' , = + - ,.
* X* 1 y* a -
included between these surfaces and the hyperboloid
ahc /a* - i' „ , a\
CHAPTER XX.
TORTUOUS CURVES. CURVATURE. TORTUOSITY.
495. We have already shewn that curves may be con-
sidered as the complete or partial intersection of surfaces, but
in the investigation of the equations of tangents, osculating
planes &c. we shall also look upon a curve as the locus of
points which satisfy more general laws, the algebraical state-
ment of which assumes the form of equations between the
coordinates of any point of the curve and variable parameters,
the number of equations being two more than the number
of parameters.
Instances of the latter mode of representation of a curve
occur in dynamical problems, in which the curve is defined
by equations between the coordinates of the position of a
particle and the time of its arrival at that position.
If the parameters were eliminated from the equations con-
necting the coordinates and parameters, the result would be
two final equations which would be the equations of two
surfaces whose complete or partial intersections would be the
curve in question.
49G. If the coordinates of any point on a curve can be
expressed as functions of a single parameter t, so that for
each value of t there is a single value of each coordinate,
the curve is called umcursal.
497. As an example of an unicursal curve, Ave may take
the Helix, which is generated by the uniform motion of a
point along a generating line of a right cylinder as the gene-
rating line revolves wltli uniform angular velocity about the
axis of the cylinder.
TOIiTL'OUS CURVES. 341
If \vc take the axis for the axis of z^ and tlie axis of x through
tlie generating point at any initial time, 6 the angle through
which the generating line has revolved when the point has moved
through a space z on the generating line, we have, for the co-
ordinates of the point, a being the radius of the cylinder,
x = a cos ^, y = a sin d^ z = nad ;
here 6 is the variable parameter, and the curve is the intersection
of the surfaces x^ + ?/ = a^. and y = x tan — .
498. In order to explain the terms employed in the ex-
amination of curves which are not plane, we shall consider such
curves as the limits of polygons whose sides are indefinitely
small ; and we observe that the plane which contains any two
consecutive sides of the polygon of which the curve is the
limit, does not generally contain the next side.
The term double curvature, as is remarked by Thomson and
Tait,* is not a proper expression, since there are not two
curvatures ; and the property, that the plane in which the
curvature is taking place at any point changes as the point
changes, would be better represented by calling the curve
tortuous and the measure of the corresponding property tor-
tuositij.
499. Osculating plane. The plane containing two sides of
the polygon of which a tortuous curve is the limit is in its
ultimate position an oaculating plane of the curve.
500. Normal Plane. Any side of the polygon in its limiting
position is a tangent to the curve, and a plane drawn per-
pendicular to the tangent though the point of contact is a
normal plane ^ being the locus of all the normals at the point.
501. Principal Xormal. The particular normal which lies
in the osculating plane is called the pfrincipal normal.
* Xatural Philo.iophi/. Art. 7.
342 TORTUOUS CURVES. CURVATURE.
502. Binormal* The normal wlilcli is perpendicular to
the osculating plane is called a binormal, being perpendicular
to two elements of the curve.
503. Polar Be.velojmble.-^ Let an equilateral polygon be
inscribed in a curve, of which consecutive sides are FQ, QR^
BS, ST, and let p, q, r, s be the middle points of these sides.
Let Aai), Bbq, Ccr be planes perpendicular to these sides,
forming the polygon ABCD by their intersections.
If the sides PQ, QR, ... be diminished indefinitely, their
directions are ultimately those of tangents to the curve, the
planes Aap, Bhq, ... are ultimately normal planes to the curve,
the planes PQR, QRS, ... are osculating planes, and the surface
generated by the plane elements Aah, Bhc, Ccd, ... is ulti-
mately the developable surface enveloped by the normal planes
of the curve, of which ABCD ... is ultimately the edge of
regression.
The developable enveloped by the normal planes is called
the Polar Developable.
504. Circle of Curvature. A circle can be described con-
taining the points P, Q^ R] when the sides are indefinitely
diminished, this circle lies in the osculating plane, and its
curvature may be taken as the measure of curvature of the curve
in the osculating plane. Let the plane PQR meet Aa in J/,
and let pU, qU be joined, then since PQ \s perpendicular to
the plane Apa, it is perpendicular to ^;Z7, similarly QR is per-
pendicular to q ?7, IT is therefore the centre of the circle through
PQR.
Therefore the centre of the circle of curvature Is the point of
intersection of two consecutive normal planes and the osculating
plane.
505. Polar Line. Draw pa, qa to any point in Aa, then,
since Pp = Qp, a Is equally distant from P and Q, and similarly
from Q and R, and therefore from every point in the circle of
St. Venant. t Mon<'c.
Tofac^p 3^2 ■
MunjJ/t k S^n, Uth
TOKTUOUS CURVES. CUKVATUKE. 343
curvature. The line of luterscctlon of two consecutive normal
planes is called by Monge the ^^olar Une.
50G. AtujU of Continfjence. The angle ^jL^*/, which is equal
to the angle between the two consecutive sides PQ^ QR of the
polygon, is ultimately equal to the angle between two consecu-
tive tangents, and is called the angle of contiiigence.
507. Sphere of Curvature. Any point in Aa is equally
distant from P, Q and i?; also any point in Bh is equally
distant from Q^ II, and S] therefore their point of intersection
is equally distant from the four points P, Q, P, S.
Hence, it follows that a sphere can be described whose
centre is P, and which contains the four points P, Q, P, JS, this
sphere is ultimately the sphere which has the closest possible
contact with the curve, since no sphere can be made to pass
through more than four arbitrary points, it is therefore called
the S2)here of curvature : the locus of its centre is the edge of
regression of the polar developable.
508. Evolutcs. It has been shewn, Art. iiS, that, if a be any
point in the intersection of the planes normal to PQ, QH, at their
middle points p, q, ap and aq will be equal and will make
equal angles with Aa. Produce qa to meet Bb in b] then a
string, placed in the position baj), would remain in that position
if subject to tension, since the tensions of the portions ab, ap
resolved parallel to Aa would be equal, and, if its extremity
were then moved from^> to q it would occupy the position baq.
Similarly, if rb be produced to c in Cc, and if sc be produced
to d in Dd.
If we proceed to the limit, it follows that a string may be
stretched upon the polar developable in such a manner that the
free end, starting from any point in the curve, would describe
the curve, if the string were unwrapped from the surface so that
the part in contact with the surface remained stationary. The
portion in contact lies on a curve called the evolute.
Also, since the position of the line pa is arbitrary, the curve
which is the limit of a,b,c,dy.. will change its position ac-
344 TORTUOUS CURVES. CURVATURE. TORTUOSITY.
cording to the position of «, hence the number of cvoUites \s
infinite.
All the e volutes of a curve are geodesic lines of the polar
developable.
509. Anyle of Torsion. The plane pUq perpendicular to
A Ua contains the sides PQ^ QR^ and the plane q IV perpen-
dicular to BVh contains the sides QB^ ES, and, since qU^ qV
are perpendicular to the line of intersection QR of the two
planes, the angle UqV is their angle of inclination.
This angle, which is ultimately the angle between consecutive
osculating planes, is called the angle of torsion.
Also, since a circle goes round BVUq^ the angles UqV and
ZTSKare equal, and the angle of torsion of the curve FQR^
is equal to the angle of contingence of the edge of regression
of the polar developable.
510. Locus of Centres of Circular Curvature not an Evolute.
Since q U will not, if produced, pass through F, because q U and
q V include an angle in the same normal plane, the locus of the
centres of circular curvature is not one of the evolutes.
511. Rectifi/ing Developable. If through every point of a
curve a plane be drawn perpendicular to the corresponding
principal normal, these planes will envelope a torse on which
the curve will be a geodesic line, since its osculating plane will
contain the normal to the surface at every point ; if therefore
the torse be developed into a plane, the curve will be developed
into a straight line. On account of this property the torse is
called the Rectifying Developable.
512. Rectifying Line. The line of intersection of two con-
secutive planes, enveloping the rectifying developable, is called
the rectifying line for any point of the curve, being the line
about which the curve must turn at that point in order to
become straight, when the torse is developed into a plane.
It may be observed that the rectifying line is not generally
coincident with the binormal, which is the normal perpendicular
to the osculating plane.
TORTUOUS CUUVLS. CUUrATUKE. TOKTUOSITV. 345
In the figure at p. 3-42 the surface whose edge of regression
is the limit of ABC... is the rectifying surface to the curve
which is the limit of abc An is the rectifying line at a,
and the binormal does not coincide with the rectifying line
unless ^)rt be perpendicular to Aa^ or a be the centre of curva-
ture of the involute of ahc...
513. If the polygon FQRS... were transformed into a
plane polygon by turning the portion QBST... through the
angle of torsion VqU about QB, and the portion BST... about
BS through the corresponding angle of torsion, the inclination
of any side ST in the new position in the plane of FQR would
be inclined to PQ at an angle equal to the sum of the inclina-
tions of the sides taken in order, and estimated in the same
direction.
Proceeding to the limit, we see that if, as a point moves
along a tortuous curve, at every position which the point
assumes the curve be turned about tiic tangent line through
the angle of torsion, the curve will be replaced by a plane curve,
such that the inclination of the tangents at the starting point
and any other point will be the sum of all the angles of con-
tingence ; if, therefore, s be taken for the angle between the
tangents in the plane curve, dz will be the angle of contlngcnce
corresponding to the extremity of the arc traversed by the
moving point.
514. Bate of Torsion. The rate per unit of length of arc
at which the osculating plane twists about the tangent line at
any point, called the rate of torsion^ is measm-ed by the limit of
the ratio of the angle of torsion to the arc at the extremities
of which the osculating planes are taken.
If, as we pass from PQ to QB^ see figure, p. 342, QB be
turned in the plane PQB so that PQB is a straight line, and
the plane QBS be then turned through the angle VqU^ the
process being repeated along the whule of a given arc, the
perimeter will become rectified, and the inclination of the last
to the first position of the plane containing two elements will
be the sura of all angles such as 176" between the extremities
of the arc so rectified.
Y Y
34G TANGENTS.
Proccedinj^ to the limit, it follows that, if osculating planes
be taken along the curve, and the elements of the arc be rectified
in each osculating plane in order, the angle between tlic first
and final positions of the osculating plane when the curve is so
rectified will be the sum of tiie angles of torsion throughout the arc.
If, therefore, t be this angle, ch Avill be the angle of tor.-^ion,
corresponding to the point at which the last osculating plane is
drawn.
515. Integral and Average Curvature:'^ The integral curva-
ture of any portion of a curve is the angle through which the
tangent will have turned as we pass from one extremity to the
other, the average curvature is its whole curvature divided by its
length.
Let a sphere of unit radius have its centre at a fixed point,
and let radii be drawn parallel to the tangents to the curve at
successive points, the length of the curve traced on the sphere
by the extremities of the radii measures the integral curvature
oi" the portion of the curve considered, and the average curvature
is the integral curve divided by the length of the curve.
51G. Integral and Average Tortuosltg. These are respectively
the angle through which the osculating ])lane has turned in
passing from end to end of any portion of a curve, and this angle
divided by the length of the arc considered.
On the sphere described in the last article let a curve be
described by the poles of the tangents to the curve which
measures the integral curvature, the length of this curve
measures the integral tortuosity, and this length divided by the
length of the arc of the tortuous curve the average tortuosity.
Tangents.
517. Tangent to a curve at a given imint.
Let s, s + A.s be the lengths measured along the arc of a
curve from a given point to the points P and Q^ whose coor-
dinates are x^ y, z and x + Aa-, y + A_y, z + A.:, and let c =
chord rq.
* Thomson and Tait. Not. Phil., Arts. 10-12.
TANGKNTS. o47
As Q approaclics to ami ultimately coincides ^vItll I\ the
cliord PQ and arc As become equal, FQ is the direction
of the tangent at P, and the direction cosines of PQ^ viz.
Ax Ai/ Az , 1.- . 1 (^-^ ^y (^z
-, ■- , become ultimatelv -7-, , , -?- .
c. c c • as ' (/s ' ds
The equations of the tangent arc therefore
dx dij dz
Also since c' = [Ax^ + [AyY + (A^)^
(1) Let the equations of the curve be given in terms of a
variable parameter ^, in the form
x^4>[e), y = ^{d), z = x{0),
then dx : d>/ : dz = f [6) : i|r' (^) : ;^' [0),
and the equations of the tangent at a point corresponding to 6 arc
^ — x_r) — y_^—z
(2) Let the equations be those of surfaces containing the
curve F{^, 77, ^ = 0, and O (f , 77, ^ = 0.
Then, at any point P of the curve,
F'{x)dx-^F'{y)dy^-F'{^dz=^0,
and (7' (.r) (/a; + (7' (3/) dy + 6-" (;r) c7^ = ;
■whence the equations of the tangent PQ may be written
iT' (.,) [^-x)^F' iy) Iv -y) + F' [z) {^-z] = 0,
and G' (x) [^-x)+ a [y] {ri-y)+ Cr {z) {^- z) = 0,
■which equations represent analytically the fact that the tangent
to the curve lies in the tangent plane to each surface at the
common point P.
(3) If the surfaces, the intersection of ■\vlilch gives the
curve, be cylindrical surfaces whose sides are parallel to the two
axes of z and ?/, and their equations be v=f[^\ ?={l)> ^''^
equations of the tangent will be
r-~~ = f(.r)fi-.r).
348 MULTIPLE ruINTS.
These equations are the analytical representation of the fact
that the projections of the tangent to the curve on the co-
ordinate planes of xy^ zx are the tangents to the respective
projections of the curve; which is obviously true, since the
projections of P and Q have their ultimate coincidence simul-
taneously with that of P and Q.
518. To find the directions of the branches of the curve of
intersection of two surfaces at a multiple j^oint of the curve.
The equations of the surfaces being
i^(?,^,r) = 0, and 6^(?,^,r) = 0,
and (x, i/y z) being a multiple point P on the curve, let
tz^ = '^~y = ^^ = r (I)
be the equations of a line through P; the points in which this
line meets the surfaces are given by the equations
F{x + Xr, y + /.u; .z + vr)=0)
2)
and G{x + \r, y ^ [xr^ 2 + j/r) = 0j'
there are an infinite number of directions which give two values
of r equal to zero, since the curve has a multiple point at P;
therefore the two equations
must be one or both identically satisfied, or else they must not be
independent equations,
i. If only one of the equations (;}) and (4) be Identically
satisfied, suppose this to be (3) ; then (.f, ?/, ~) will be a multiple
point on the surface P(f, Vi ?) = ^ j ^^^^t i^' t''"'s be a double point,
the line (1) must be one of the tangents whose directions arc
given by
MLi.Tii'Li: roiNTS. 349
and, since it lies In tlic tangent plane to G (^, 77, ^) = 0,
These eqnations give the directions of the two tangent lines,
which are the Intersections of the conical tangent to the first
snrface with the tangent plane to the second ; and, similarly,
for higher degrees of multiplicity.
il. If (3) and (4) be both Identically satisfied, the line (I)
■Nvlll be in any of the directions of common tangents to i^(|, 7;, ^)=0
and G (|, ?;, ?) = ; the directions are therefore given by
where s and t arc the degrees of multiplicity of the multiple
points of the two surfaces at (cc, y, z).
ill. If neither (3) nor (4) be identically satisfied, but the two
equations be identical so as to be satisfied by an Infinite number
of values of X : /i : v, there will be a surface AF-\- BG = 0^
which will pass through the intersection of F= and G = 0, for
which (^4; 4 /^ f + vj]{AF+BG) = win be identically
satisfied, if j be the value of the equal ratios -tttt \ i v ,- 1 i 1
' A ^ G [x] G [y)
In this case, therefore, \ : /z : v is determined by one of the
equations (3), (4), and
If in any of these cases two values of X : /i : v be equal,
there will be either a point of osculation or cusp on the curve.
519. As an example of case HI. in the last Article, suppose
we wish to find the directions of the tangents at the point
3,j0 koi:mal plane.
(a, 0, 0) In the curve of intersection of the liypcrboloiJ and
hyperbolic paraboloid, whose equations arc
?1 + -^ - - = 1
,.2 ^2
and J -y =2(^-o).
At this point the surfaces have a common tangent plane,
whose equation is x = a\ the third surface, on which (a, 0, 0)
is a multiple point, is in this case the cone
X V /I IN, /I 1
and the direction cosines of the tangents to the curve are given
by
520. Normal plane of a curve at a given point.
The normal plane being perpendicular to the tangent to the
curve, its equation is
521. To find the eage of regression of the polar developable
of a curve.
The edge of regression Is the locus of the Intersection of
three consecutive normal planes to the curve.
. The equation of the normal plane at (ic, y, z) is
{^-x)dx + {'n-y)dyV{l:-z)dz^O, (1)
that of the normal plane at a consecutive point Is found b}'-
writing in this equation x + dx for a*, &c., the line of intersection
of the two normal planes will lie in the plane
{^-x)d'x+{'n-y)d'y^-[^~z)d'z- [dxY-[dyf- {dzY=0.i2)
Again, writing x-\- dx for ic, &c., we obtain a plane In which
the line of Intersection of the second and third normal planes
lies,
(^-x)d'x^{v-y)d'y + {^-z)d'.z
- 3 {dxd'x + dyd'y + dz d'z) =■ 0, (3)
OSCULATING TLANE. 351
and the coordinates of the point of the edge of regression satisfy
these three equations. If we elaninate r, 3/, z from tlie equations
(1), (2\ (3) and the equations of the curve, Ave shall obtain the
two equations of the edge of regression.
The line of which (1) and (2) are the equations is ^Monge's
polar line, which is the axis of the osculating circle.
The point given by the three equations (1), (2), (3) is the
centre of spherical curvature corresponding to the point (ic, y, z)
of the curve.
b'1'2. To find the differential coefficient of the arc referred to
2)oIar coordinates.
Transforming to polar coordinates
x= r sin 6 cos 4> = P cos <^,
y = r sin B sin 4> — p sin cp^
z =r cos 6 J
p = r sin 6 J
(dz\' (dp\' (d.
10 J -\d6j-^''^'-'''''"\d6
The equation is easily obtained geometrically by observing
that ultimately
{AsY = [Iry + [rAd)' + (/• smO A(f)j\
Also, \f J) be the perpendicular from the pole upon the tan-
gent, and ^jr the angle between /• and the tangent, ^) = r sln^/r,
A.s , ,.. . , fds'
Ai
and = sec-vlr ultiniateh', .*. ( ,- = „ ..
\r ^ •" \drj r--p
Osculating Plane.
523. Equation of the osculatincj idane.
The osculating plane may be considered as the plane whicl
passes through three consecutive points, whose coordinates are
cr, y, 2! ; X + dx. . . . and x + 2dx 4 d^x. . . .
Ml
352 OSCULATING TLANE.
Let the equation of the osculating pLanc be
.-. AJx + BcIi/-\-Cch = 0,
and A {2clv + iVx) + B {2J>/ + d'y) + C [2dz + dh) = 0,
or Ad'x + Bd'y + Cd'z = 0,
hence the equation Is
^-x, 77-?/, ^-z I
dx^ dy^ dz 1=0.
d^x^ d^y^ d^z \
It may be noted that the equations of the tangent and
osculating plane are of the same form, whether the axes be
rectangular or oblique.
524. It should be obsei-ved with respect to the notation used
above that if x^ ?/, z be supposed given as functions of ^, and
we take points corresponding to values t^ < + t, ^ + 2t, which
is the same as making t the independent variable, the values
of X for t + T and t + 2t are
dx d^x
dt at 2
dx ^ d'x (2tY
and.+ ^2r+^^i^-f...;
and if the first be w^rltten x + Ax, the second will be
x + Ax + A{x + Ax) or x + 2Ax + A''x ;
hence if d be written for A, where t is indefinitely diminished,
dx = — , T and d'^x = -,^ r^ ultimately.
dt dr
525. As an exercise the student should find the equation
of the osculating plane, considered as given by any of the
following definitions :
i. As a plane containing a tangent and a point In.definltcly
near the point of contact.
ii. As a plane containing a tangent and parallel to a con-
secutive tangent.
OSCULATING PLANE. 353
lii. As a plane which has a closer contact with the curve
than any other plane.
In employhig the definition ii. he may shew that the shortest
distance between the tangents at the extremity of any arc (7s
is generally of the order of (7/.
526. Direction cosines of the hinormaJ.
The direction cosines of the blnormal, which is perpendicular
to the osculating plane, are in the ratio
dyd'z — dzd'y : dzd^x - dxd'z : dxd'y — dyd'x^
and the sum of the squares of these expressions
=[[dxy^{dyY-^{dzY][[(rxy-^{d:-yy->r{' [x, y, z) — be the given equations,
then, using the notation of Art. 420,
Udx +Vdy +Wdz =0,
U'dx + V'dy + W'dz = 0.
II U V W 11
Let Z), -E", i^ denote the determinants ' ' ^y, ;
dx dy dz ,
.-. -^ = -^ = -p = /c suppose,
whence d^x = IcdD + Ddk^
d^y = hdE + Edh^
d'z=kdF + FdJc',
.-. dyd'z-dzd'y = h'[EdF-FdE),
hence the equation of the osculating plane is
[EdF- FdE) [^-x)+...= 0.
530. Equation of the osculating plane in terms of the equations
of the tangent planes to the su?faces.
Employing the notation of the preceding article, we see that
I)U+EV+FW=0;
.-. UdD + VdE+WdF+DdU^ EdV^ FdW=0,
THE INTERSECTION OF TWO SURFACES. 355
, ,^, , dU J dU , dU
and dU= ax ~,~ + du -y- 4 «2 -7—
dx '' dy dz
~ \ dx dy dz] dx
if r denote the operation In the brackets, in the performance of
which Z>, E^ F are considered constant ;
.-. DdU+ EdV+ FdW= JcV (0) ,
hence UdD +VdE -^-WdF =-kr'{c{>),
simikrly U'dD + V'dE+W'dF= - JcV' ((/>') ;
.-. EDF- FdE=k[Ur-{4>']-UT' {c}>)],
and the equation of the osculating plane becomes
r{')[U{^-x)+v{v-y) + w{!:-z)} = r{ci>){u'{^-x)+...}.
531. To find the oscidating plane of the intersection of two
concentric and coaxial conicoids.
Let the equations be
ax'' + by' + cs"* = 1 ,
ax' + l3y' + yz' = l,
Z) = 4 (^7 - cl3) 7JZ = Ayz^ E= Bzx, F= Cxy,
EdF- FdE = E\l f Ci = BCz'x'd (y
(1)
.E) \z,
= BCx' [zdy - ydz) = hBCx' {Ez - Fy),
and by (1) Ez -Fy = 4:{(x- a) x]
.-. EdF- FdE= XkBC (a - a) x\
and tlie equation of the osculating plane is
"-- %' (f - .,) + ^ - -S' (, - 2,) + ^- " .' (r- ^) = 0,
which may be reduced to
BG ,. CA , AB ,Y, .
C^-Z') (7~o) ^^+(7^Kor:^j ^^ + (a-rO {0-b) " ^"^ '
356 PKINCIPAL NOUMAL.
532. Or, by the method of Art. 530, since
the equation may be written
{D'a + E'^ + F'y) [ax^ + hi/v +cz^-l)
- {B'a + K'b + F'c) (aa:| + ^i/rj + yz^- 1) = 0,
and the coefficient of
= BC [a- a) x^, as before.
533. To find the condition for a stationary osculating plane
of the curve of intersection of two surfaces.
The equation of an osculating plane is
{EdF-FdE)[^-x)+...=^%
the line of intersection of this plane with the next consecutive
osculating plane is in the plane
{Ed'F-Fd'E) {^-x) +...- {EdF-FdE) dx-...= 0',
the last three terms are identically zero, since dx = kD^ and in
order that the two osculating planes should coincide,
Ed'^F- Fd'E _ Fd'B - Bd'F _ Bd^E- Ed^D
EdF-FdE ~ FdB-BdF ~ BdE-EdB '
which are clearly equivalent to one distinct equation ; and each
c ^. c c ' ix d'B\Ed'F-Fd'E\+... ,
ot the tractions is equal to —fr^rrnrr^ — frrW i the nume-
^ d'B[Ed]^ -FdE)-\-... '
rator of which vanishes,
.-. d'B{EdF-FdE)+...= 0.
Principal Normal.
534. To find the equations of the principal normal at any
point of a curve.
The principal normal is perpendicular to the tangent line
and also the binomial, the direction cosines of which are pro-
portional to dx^ dy, dz^ and dyd'^z-dzd'y^ dzd'x-dxd'z^
dxd^y — dyd^x respectively.
Now we have identically
d'x[dyc['z - dzd'y) + d'y [dzd'x - dxd'z'j + cPz {dxcPy-dyd'x = ;
MEASUKE OF CURVATURE. ' 357 '
and if we make 5 the independent variable,
d:'xdx + d^y dy -\-d^zdz = d^s ds = 0.
These two equations shew that direction cosines of the principal
, . , d^x d'y d^z , . .
normal are proportional to -yj , -,% , --p^ , and with a general
Independent variable, its equations are
d ldx\ d fdy\ d fdz^
dd Us) dd \ds) J9 \dsj
535. If from any point in a curve equal distances he measured
along the curve and its tangent^ the limiting position of the line
joining the extremities of these distances is the principal normal.
From the point (re, ?/, z) let equal distances a be measured
along the curve and the tangent to the points Q^ T, Tiie co-
ordinates of (2 are a; + ^ o- + [-— 4 £ ) ^ , &c. and those of T
ds \ds J 2 '
dx ., .,..,,..
« + -,- 0", (xc, £ vanisliing in tlic limit.
The equations of the line QT are
f
— X-
dx
-ds""
V
-y
-P
?-^
dz
-ds""
d'x
ds'
+ £
d:y
ds'
+ £'
d'z
ds'
+ €"
therefore the limiting position of QT'is the principal normal.
Cauchy proposed, as a definition of the Principal Normal at
any point, the limiting position of the line joining the points on
the curve and tangent, whose distances from the point of con-
tact measured along the curve and tangent respectively are
equal, by which means the definition was made independent
of the osculating and normal planes.
Measure of Curvature.
536. To find the radius of curvature at any point of a
tortuous curve.
The reciprocal of the radius of curvature is the measure of
curvatiu-e, or the rate per unit of length at which the tangent
358 MEASURE OF CURVATURE.
to the curve clianges its direction. If p be tlie radius of
curvature at a point P, and ch be the angle of contingence
corresponding to the arc Js. - = -f .
p as
Draw Oj), Oq of unit length through the origin parallel to
the tangents at F and Q the extremities of the arc c7.9, join jJ^',
then, since the plane 2^^Q. '^^ parallel to the osculating plane,
p^-, which is ultimately perpendicular to 0^?, is parallel to the
principal normal.
The cosines of the angle made by Ojj and Oq with the axis
floe fi'JT Cl3(*
of X are -r- and -r- +(? ,- , and, if /, ?n, n be the direction
its els els
cosines ofjoq, projecting OjjqO on the axis of x, we have
dx , , /dx Tdx\ ,^
since j^q = dz ultimately.
, d^x 1 . ., 1 (^^V ^'^
.-. l = p j^, , and, sunilarly, vi = p j^^ ,n = p ^^, ;
I _ {^^ /^V I^S"
•*• p' ~ V/.V "^ UsV ^ \dsV '
If G be the centre of curvature at P, the projection of OFG
on Oic = a; + p/, hence the coordinates of the centre of curvature
are
537, The radius of curvature may also be found without
projections, as follows :
Let \, /A, V and A, + A\, p, + A/z, i/ + Av be the dij-cction
cosines of the tangents at the points Pand (), Avhose coordinates
arc cc, 2/, 2 and ic + Aa;, 3/ + Ay, 2; + A;? ; and let As be the
angle between the tangents
cos A£ = X. (X -I- AX,) + /i (/i + Ayx) + v (v + Av),
and (X + AX)' + (/i -t- A^)' + (v + Av)= - X'"' - ^a' _ v"' = ;
.-. 2 (XAX + /iA/i + vAi') + (AX)" + ( A,a)'-' + (Av)' = ;
MEASURE OF TORTUOSITY. 359
.-. 2 (1 - cos Ac) = (AX)"-' -f {AfjiY + (Av)"' ;
.". ultimately, when PQ is indefinitely dimiuislied,
{cky={cixy+{df.Y + {d.Y;
If s be not the independent variable,
7 dx dsd'^x — dxd's „
.-. since [dxY + {dijY + {dzf = {d>>f,
and dxd^x + dy d'^y + c?-3 d'^z = t/scf s ;
i^* = [d'xY + (t/'y)'' + UFzf - id'sY.
p
538. The student should, as an exercise, find the radius and
centre of curvature, when the latter is considered as the point
of intersection of two consecutive normal planes and the oscu-
lating plane.
Measure of Tortuosity.
539. To find the measure of tortuosity of a tortuous curve.
Let Z, W2, n and l+dl, m + dm^ n + dn be the direction cosines
of the binormals at two points P, Q^ whose distance along the
curve is ds. Draw unit lengths Ojy, Oq parallel to the two
binormals, and let X, //., v be the direction cosines of ])q ; the
angle q Op = dr is the angle between the osculating planes, and
-T- , the rate at which the osculating plane turns round the
ds
tangent line per unit of arc, is the measure required, which wc
shall call — .
a-
Projecting Oj^qO on the axis of .r,
l-\- dT.\-{I + dl)=0]
.'. dT.\ = dl and X = a--j-:
ds
•■• a' ~ \ds) "^ [lis) "*" [TTs
360 GEOMETRICAL INTERPRETATIONS.
which iDcay also be obtahicd as in Art. 537. a is sometimes
called the radius of torsion at P, but It Is better to look upon
- as the measure of tortuosity.
540. The measure of tortuosity may be expressed In another
form.
Since I '. m : n :: X : Y : Z, where X^dyd'z — dzd'y^ and
similar expressions for i', Z^
Idx -i- mdy + ndz =0,
Id^x + md:y + nd^z = ;
.*. dldx + dm dy + dn dz = 0,
and dl .1 + dm .m ■{■ dn .n = Q ]
dl _ dm _ dn
mdz — ndy ndx-ldz Idy — mdx
_ dld^x + dm d^y + dn d^z
Id'^x + md^y + n d'z _ Xd'x + Yd^'y + ZiF.
IX+mY-vnZ X'+Y' + Z'' '
and {m dz - n dy)'' +...
= (Z'^ 4. ,n^ + n') {{dx}' + {dyf + [dzf] - {Idx +...)' = ds' ;
1 _ Xd'x+Yd'y + Zcrz
" a- X'+ Y' + Z' '
If there be no tortuosity, or the curve be plane,
Xd'x+Yd'y-\-Zd'z =
.at every point of the curve, as in Art. 527.
Geometrical Intoyretations.
541. Saint Venant observes* that, if we take three con-
secutive points P, (?, R for which | = ic, x + dx, x + 2dx-\-d'x
respectively, the projections of P^, QR upon the axis of a; will
be dx, dx 4- d'x ; and if the parallelogram PQRM be completed,
by projecting the sides of the triangle PQM in order, since
♦ Journal de TEcok Pol., vol. 18.
GEOMETRICAL INTEia'KKTATIONS. 361
rM= Qli, wc sliall have dc ■+ projection of Q21 - {Jx + cT'j:) = ;
tlieretbrc d'x is the projection of QM.
542. In the general case, if a figure be drawn in which
(Ix, d'x, (I>/^ d'y are all positive, the projection of twice the
triangle PQM on the plane of xy will be easily seen to be
dxd^ij + dydx — dy {dx + d'x) = dxd^y — dyd'x.
Again, if Mm be drawn perpendicular to PQ^ PQ = dsj and
ultimately m Q=QR-PQ = d's ;
.-. M»r = QM' - Qnc = [d'xY + ((^y)' + [d'zy - [d'sf,
- 1 ... ,(MoxV {d^xy + id'yY+id'zy-id'sY
and - = limitof' ' _v y • -y \ 7 ^ — /_
If we make s the independent variable, this implies that
QB = PQ, in which ease QM wnll bisect the angle PQB and
be ultimately in the direction of the principal normal, the
direction cosines of which will be as d^x : cFy : d'z.
The radius of the circle circumscribing the triangle PQR
po-
will be ^\r', hence, if p be the radius of curvature,
1 _ ^^^-^V /"AY A^''^Y
p'~ [dsv "^ U'v ^Uv '
543. Oscidatinfj pJane^ hinormal and curvature of the luh'x.
In the case of the helix, Art. 497,
dx = — a sin 6 dd^ drx = — a cos 9 {dOy^
dy = a COS0 dd, d'y = — a sin 6 [dOy^
dz = 7m dd , d'z = ;
dy d'z - dz d'y = nci^ sin d {ddy,
dzd'x - dx d'z = - nd' cos e [doy,
dxcT'y-dyd'x^ a' [ddy.
hence the equation of the osculating plane is
(I - x) {n sin 6) -{■ {v - y) (- n cos ^) + ^- .^ = 0,
or n{^y-'nx)^a{i:-z) = 0.
.'>G2 RADIUS OF CURVATURE.
Tliis plane contains tlie point (0, 0, ;r), and tlicreforc tlie radius
of the cylinder uliicli passes through the point (a-, ?/, .-), and
this radius is the principal normal.
The direction cosines of the binomial will be sin a sin ^,
— sina cos^, cosa if a = tan"'/2 be the pitch of the screw.
The measures of curvature and tortuosity arc respectively
cos''' a , sin a cosa
and .
544. To find the radius of curvature of a curve wJnch is the
intersection of two surfaces whose equations are given; and to
express it in terms of the radii of curvature of the normal sections
of the two surfaces and the angle between thevi^ the i^lfine of each
section containing the tangent to the curve.
Employing the same notation as in Arts. 529 and 540,
X=d>jd'z - dzd'y = F {EdF- FdE) = /c" { UV' ((/>') - UV (>)},
and if p be the radius of curvature,
{dsY ^ ^^., _^ ^., ^ ^, ^ ^., ^^ _^ ^,, ^ ^^.,^ , ,^,. ,,
p
- 2 ( uu' -f vv + WW) r (6) r-'(<^') + ( w'+ v"+ w") [r (,)}'^],
and {dsY = I^'{D' + E' + F']-
1 _ {U' + v' + w') {ru') Y-:..
•'• p^- [D'-{E''+Fy
Let 0) be the angle between the tangent planes to the surfaces
at [x, ?/, z) and P^ ^U' + V'^ W\
... UU' + VV' -^WW' = rP' (Msco,
and D' + E' + F' ~ P'P" sin'^ w ;
. \_ P' [r-\')Y-2PP' co^a>.r {4>')-rw + F-^ {r\^)Y ,..
•• p'^ P'P"s\u'(o{D'-\-E' + F'f " ' ^ '
As an example of the use of the preceding formulae, we shall
obtain the radii of curvature r, r of the normal sections by
replacing the equation of the surface )' = by the equation
(})\ = Ix + my + nz — ^> =
of a nonual plane ; in which case, if P,, P",, P,, and V^ be the
OSCULATING CONE.
3G3
corresponding values of Z>, E^ F^ T, T," (0,') = 0, and, since the
normal plane contains the tangent to the curve,
D : E : F=dx: du : dz = I):E:F:
\ I 1 •J '
licncc, since &) = |7r, wc obtain from (1)
[r.»{<^)r
{^'m'
•■' ~ P' {d; + e; + F;y r [W + e' + Fy '
1 r[4>')Y
anu, sun
I'^'b'j ,:^- p-^D'->rE' + l^y'
I _ 1 / 1 1 2 cosci)
p'' sin'^o) V'*^ ?•■' »•'■'
which result will be obtained in the next chapter by ^Icunicr's
theorem.
545. To find the vertical angle of the oscidatuKj cone of a
curve.
L,Qt 2>Oo^ qPp^ rQ'i bo three consecutive planes which become
ultimately the osculating planes of a curve OPQR] these planes
intersect in P.
Take P as the vertex of a circular cone which touches each
of the phaups, this cone Is in the limit, the osculating cone of the
curve at P. Let PII be its axis
, op, 2n^ v
the sections
of the three planes made by a plane perpendicular to the axis,
and /, u the points of contact wlth^>(/, (ji-.
Draw ///, J/// perpendicular to the planes pP>i, 'lQ''\ ^''^"
tlie angle tl[u will be the angle of torsion, and pPi the angle
3G4
EADIUS OF SCREW,
of contlngence, and we shall have dr = ,| and ch = ,, ulti-
mately ; therefore, if 2\/r be the vertical angle of the cone,
, lit (h (7
tan V" = -rr = T" = - •
^ rt ih p
546. 77ie rectifying line is the axis of the osculating cone at
any poi»f of the curve.
For each of the planes through the tangent lines PQ^ QR
perpendicular to the osculating planes lyPq^ qQr ultimately
contains the axis PII of the cone,
547. To find the rate per unit of length at which the angle
between principal normals increases.
Let PQ^ rQR^ sRS be the directions of the sides of a polygon
■which are ultimately tangents to a curve.
In the planes PQr, rPs respectively draw QU, QV perpen-
dicular to rQR^ sRSj these arc ultimately in the direction of
consecutive principal normals.
Draw QU' in the plane QRs, perpendicular to QR, so that
UQU' is the angle between the consecutive osculating planes
= Jt, and U' Q V is the angle between consecutive tangents = dz.
Let dx be the angle between the lines QU, QV, and let
these lines and QU' meet a sphere of radius unity, whose centre
LOCUS OF CENTRES OF CURVATURE.
VjGo
is In Q, In U, V, U', the angle VU' U is a right angle ; there-
fore, we have ultimately
or {dxT = (dry + {ch)%
^X
the rate per unit of length of the increase of the angle
between principal normals is the reciprocal of what we shall
call the radius oi screw ; let k be this radius,
1 I 1
then ., = .. + - .
«" cr' p'
It may be noticed that (7t = ^7;^ cos \/r and (/3 = (7% sin-v/r,
which represents that the angular velocity round the rectifying
line, which is normal to the plane UQV^ is the resultant of
two angular motions round the tangent line and the binomial
which produce the rectification.
548. To find the anfjle of contingence^ and, the element of the
arc^ of the locus of the cenf)-es of curvature of a given curve.
Let ()7?, ES be sides of an equilateral polygon wliii-h arc
ultimately tangents to the given curve; let BV be the in-
4G6 RADIUS OF SPIIEKIAL CUKVATURE.
tcrsectlon of planes pcrpciullcular to Qli., RS through q^ r
their middle points; BV is therefore ultimately a tangent to
the edge of regression of the polar developable; let BU^ CW
be the tangents preceding and succeeding BV.
From q draw qU^ <^F perpendicular to BU an^ BV, join rl\
which will be perpendicular to BV^ and draw rW erpcndlcular
to CW.
If iri' be produced to 2V, UVw and UV will ultimately be
the angle of contingence dt and element ds required.
Let B be the radius of spherical curvature at q^ and <^ the
angle which it makes with the polar line BU. Since circles
can be circumscribed about the quadrilaterals BVlTq^ CioVr^
qVU=qBU=cjy, and VBU= VqU=dT, ultimately,
also r Vic = rC W= r C V + VC W= (/> + J) + dr ultimately.
With V as centre and radius unity describe a sphere, and
let q, r\ ?/, 10 be the projections of 5-, r, Z7, w upon it, and draw
ux perpendicular to rV?, then q'r = c/o, ux=dz cos^, xr =nq = 0,
and ion'' = ux^ + xid^ ultimately ;
... [dz] = {dz cos (/))■-' + {d(f> + dry.
By drawing a diameter through V to the circle BVUq^ it is
easily seen thjit VU=R smVBU, and therefore that dn'^Bdr.
549. To find the radius of spherical curvature.
If in the figure BVl/q^ UM be drawn perpendicular to QV^
VM= dp ultimately ;
.-. {ruiTY = ipdry-,{dpY;
also Bdr cos^ = VM= dp ;
therefore the distance between the centres of circular and
. 1 -n , '^P (^P
spherical curvature = ii cos(^ = -7- = tr- .
^ dr ds
550. To find expressions for the radius of curvature of the
edqe of repress ion of the polar developable of a curve.
COOKDINATLS IN TERMS OF ARC. 3<)7
These arc readily obtained by a method su^^gested by
Routli,* which can be exphiined by the last figure.
Considering the curve BCD^ U and V are the feet of the
perpendiculars from q on the tangents to the curve; and
substituting the corresponding letters in the known formnlEe
7; + -7 : , , ?* -r- 1 we obtain two expressions for the radius of
a-, Oy the principal normal, Oz
the binomial, the planes of yz, zx^xy being the normal, recti-
fying, and osculating planes, and let s be the distance of g.
point (a*, y, z) from (9, measured along the arc.
Then, at the origin,
^-1 ^^-0 ^'''-0
ds~^' ds'^' ds~^'
d'x ^ (Fy cFz
p ,2=0, Px^ = l, p 7. = 0,
'^ ds ' ds ' "^ ds
these quantities being the direction cosines of the tangent and
principal normal. If a be the angle made by the tangent at
dx
(.r, y, z) with the tangent Ox^ , = cosa ;
d'x [da.\' . (Pa
''•d? =-''''''■ \ds} —ds^
therefore, at 0, since doL is the angle of contingcnce, and a = 0,
ds'~ p''
♦ LliinrU'rli/ ,/ounifil, vol. Tit., p. 42.
368 COORDINATES IN TEKMS OF AKC.
Again, If jB be the angle made by the principal normal at
[x, y, z) with (9y,
d^y ^ dp d'y rfy . _^y3
da ds ds ds ds
tl)crefore, at 0,
1 dp d'y ^
p ds '^ ds '
and if 7 be the angle made by the binormal at (a*, ?/, z) with
the principal normal at the origin,
(dz d^x dx d^z\
dp fdz (Px dx d^z\ fdz d^x dx d^z\ _ . dy
''• 'tb \ds ds' ~'d's~div'^^\dsi?~'ds d?) "~ ^'"'^ ;^'
therefore, at 0, p -y^^ = sin y -r = - , since dy is the angle
ds ds a '
of torsion ;
therefore, by Maclaurin's theorem,
^ = ^-(J'
s' s' dp
y~2p Gp'ds'
s^
~ 6/30- ■
552. To find the avrjle hetween two consecutive 2^rinc>j)al
nortrnds.
The direction cosines of the principal normal at (.r, y^ 2),
.9 1 s
a point near the origin, are as - ^ : - : , and the secant
. P' P P^
of the angle between this normal and that at the origin is
/- I +.sM - + — ji ; therefore, the angle required is ulti-
mately.y(l-fi);
• 1 - 1 1
/c" p" O"' '
where k is the radius of screw, as in Art. 017.
COORDINATES IN TERMS OF ARC. 3G9
553. To find the shortest distance between consecutive irrincipal
normals and its position.
The equations of a principal normal at (a:, y, z) are ap-
proximately
- P [^ - x) = s {v - y) =
the plane xy being the plane of reference.
The differential equation is, in this case,
^^dx-^,dy = 0;
.'. a" logx -Z'Mog?/ = log(7;
.-. x*= Cxf.
The lines of greatest slope are the intersection of the cone
with the cylinders represented by this equation ; and it may
be observed that no generating line, except those in the prin-
cipal planes, is a line of greatest slope, unless the cone be a
right cone.
Line^ Surface^ and Volume-Integral.
660. We give here two theorems relating to line, surface,
and volume-integrals, which are of great importance in certain
problems in Electricity and Conduction of Heat, and which serve
as illustrations of the subjects of this and the preceding chapter.
Line-Integral and Surface- Jyitegral.
561. Def. If R be any quantity having direction, called
a vector quantity, and s be the angle between Its direction and
that of the tangent to a curve at any point (.r, y, z) taken
in a definite direction, the Integral jR cossds is called the line-
integral of Ji along the line s, s being measured from a fixed
„„ . . , , . ff dx dy dz\ ,
pomt. ihe nitegral may be written 11 " -, + ^ j^ "^ ^;rj ""')
w, V, v) being functions of ic, y, z.
If rj be the angle between the direction of R and a normal
to a surface at any point (a;, y, 3), the integral jjR costjdS is
called the surface-integral, the summation being taken over
SURFACE AND VOLUME-INTEGRAL. 373
the whole of a surface S. The Integral may be written
JJ{Ul+ Vm+ Wn)dS, U, F, IF behig functions of {x, y, z),
and /, 7«, 71 the direction-cosines of the normal to the surface
at [x, y, z) measured in a definite direction.
562. To shew that a line-integral tahen round a given closed
curve can he represented as a surface-integral of a surface bounded
hy the given curve.
Suppose the closed curve L to be filled up by any surface S^
and suppose S to be divided into an infinite number of small
elements, one of which is o- bounded by the line \. If we take
rals for two o
jtimated in the
portions of the sums taken over fi will be taken in opposite
directions, and being of the same magnitude will vanish ; those
lines X which abut upon L are the only portions which will not
be traversed twice ; hence the sum of all the line-integrals for
the elements \ will be that of the line L.
The proposition will, therefore, be proved if we shew that
it Is true for any elementary line \ and corresponding surface cr.
Let [x^ y, z) be any point on o-, and (^ + f , y + Vi ^ + H
any point on A,; the line-integral for X, u^ y, lo being given
at (^, 2/, ^), = IK" + -^ ^ + ^ ^ + ^' ^) ^^^ +•••} ultimately.
Since \ Is a closed curve, Jd^ = 0, /|^/^ = 0, and if we sup-
pose the summation taken in the direction from x to ?/,
hence the line-integral for X is -^ ( -r -j-] n +...[ a.
[\dx dgj J
The line-integral of L is, therefore, equal to the surface-
integral JJ{Ul + Vni + Wn) ds, when U= -. , , &c.
563. The surface-integral of a directed quantity or vector^
taken over a closed surface, may be expressed as a volume'
integral of a certain function.
We observe that if the theorem can be provcil for an
elementary portion of the volume enclosed within the surface,
374 SURFACE AND VOLUME-INTEGRAL.
■within which tlie directed quantity is supposed to be continuous,
the general theorem will follow, as well as Its modification, when
the enclosed volume Is intersected by surfaces across which
the directed quantity changes discoutlnuously.
For, if fj, u^ be two elementary volumes enclosed by the
surfaces o-^, o-.^ to which a portion a' is common, the normal
components along o-', which belong respectively to cr, and o-„,
being In opposite directions and of the same magnitude, will
disappear In the summation.
If, therefore, we sum for all elements within a volume F,
throughout which the value of the vector changes continuou^^ly,
the only points of the resolved vectors which are not destroyed
are those which belong to the points of the elements which
abut on the enclosing surface S.
If the vector change discontinuously In passing surfaces
2„ 2.^, &c., the theorem will hold for the portions F,, V^... into
which they divide F, and the volume-Integral over F would be
equal to the surface-integral over S^ together with the surface-
integrals over S,, 2,^; the differences of the vectors on opposite
sides of these surfaces replacing the vectors in the first Integral.
Let an elementary volume v be Inclosed by the surface o-^
(a:, ?/, z) being any point within u, and let (•« -f- ^, ?/ + »7, 2 + ^) be
any point upon o-, and ?<, v, w the components of the vector
at (a*, ?/, z) parallel to the axes.
The surface-integral for a is
//
|(„ + ;^«j+...),+(,,+J,+...)„,+..,jrf,,
hence
the surface-integral for o- = (-- -j- -. -f- . ) uultimatclv:
® \(/^ di/ dzj ' '
therefore, if u,, ?*,' be the values of ?/ on opposite sides of 2,,
//(,..™..„),;«=///(,t.|.|>r
+ " f
which represents the theorem in Its most gencfal form.
PROBLEMS. 375
XX.
(1) Tlie equations of the tangent to the curve of intersection of the
surfaces ax* 4 by* + c;' = 1 and bx* + ctj* + «2* = 1 are
ab - c' be - a* ac - b*
The tangent line at the point x = y = z lies in the plane
(a - 6) x + (6 - c) y + (c - fl) z = 0.
If ac = b*, the tangent lines trace on the plane of xy the two straight
lines whose equation is -, — r, = -,——r. •
^ r - 6' a' - 6^
('2) The equations of a sphere and cylinder being
X* + y* + z* = 4a' and x" + z' = 2ax,
prove that the equations of the tangent to the curve of intersection at the
point (a, ft, 7) are
(a - a) r + 73 = aa and (3y ^- ax = a (4a - «),
and that the equation of the normal plane is
^^(■-")(^l)•
(3) Find the tangent line of the intersection of the surfaces
2 (x + z) (x - a) = a' and z (y + z) {y - a) = a^
and ehew that it consists of plane curves.
(4) The paraboloid whose equation is ax* + by* ,= 4z has traced upon it
a curve, every point of which is the extremity of the latus rectum of the
parabolic section through the axis of z ; shew that the tangent to the curve
traces upon the plane of xy the curves whose equations are
r sin 20 = + (« - b).
(5) Shew that the equation of the normal plane at any point /, y, h on
X* y* z*
the curve defined by the equations -: + Vi + -^ = ^ i x' + v* + z' = tf' is
^ a' b* c'
a*{b*-c*) b*{c*-a*) c*{a*-b*) „
f 9 A
(6) A point moves on an ellipsoid so that its direction of motion always
passes through the perpendicular from the centre of the ellipsoid on the
tangent plane at the point; shew that the curve traced out by the point
is given by the intersection of the ellipsoid with the surface
x'"""2/""'2'"' = constant,
/, m, n being inversely proportional to the squares on the semi-axes of the
ellipsoid.
376 PROBLEMS.
(7) If the osculating plane at every point of a curve pass through a
fixed point, prove that the curve is plane. Hence prove that the curves of
intersection of the surfaces whose equations are
a;* + yl + s' = a' and x* + ?/* + 2* = Ja*
are circles of radius a.
(8) Prove that if every pair of consecutive principal normals to a curve
intersect, the curve must be a plane ; and find/ {0) so that the curve, whose
coordinates are given hy x = a cos6, y = b sinO, z=/ (0), may be plane.
(9) A curve is traced on a right cone so as always to cut the generating
line at the same angle ; shew that its projection on the plane of the base is
an equiangular spiral.
(10) If a string be unwound from a helix so that the straight portion
is a tangent to it, shew that any point on the string will describe the
involute of a circle.
(11) Prove that the locus of the centres of curvature of a helix is a
similar helix ; and find the condition that it shall be traced on the same
cylinder.
(12) When the radius of curvature is a maximum or minimum, the
tangent to the locus of the centres of curvature is perpendicular to the
radius of curvature.
(13 j Coordinates of any point in a curve are a: = 4a cos^0, y = 4a sin'6>,
z = '6c cob'O; find the equations of the normal and osculating planes; and
find the relation between c and a that the locus of the centre of the sphere
of curvature may be a curve similar to the original curve.
(14) A straight line is drawn on a plane, which is then wrapped in a
cone ; shew that the radius of curvature of the curve on the cone varies as
the cube of the distance from the vertex.
(15) If a tortuous curve be projected on a plane, the normal to which
is inclined at angles a, /3 to the tangent and binomial at any point, the
curvature of the projection will be to that of the curve as COS7 : siu'a.
(16) If ^ be the radius of curvature of a curve, then that of its projection
on a plane inclined at an angle a. to the osculating plane is p sec a if the
plane be parallel to the tangent, and p cos' a if it be parallel to the principal
normal.
(17) If the measures of curvature and tortuosity of a curve be constant
at every point of a curve, the curve will be a helix traced on a cylinder.
(18) If - , - be the measures of curvature and tortuosity at any point
p '{'-'-QT-
The equations of a cylinder and cone are
r sin = a and coiO = ^ (c^ - c"*).
If Ju A J, and A-^ be the areas of the cone reckoned from = to (f> = fi-a,
ft and ft ^ a respectively; then will yi, + A3 = (e" + e"*) At.
CHAPTER XXI.
CURVATURE OF SURFACES. NORMAL SECTIONS. IXDICATRIX.
DISTRIBUTION OF NORMALS. SURFACE OF CENTRES. INTEGRAL
CURVATLTIE. DIFFERENTIAL EQUATION OF LINES OF
CURVATURE. UMBIUCS.
5G4. In this chapter we proceed to examine the curvature
of surfaces, and to explain how the amount of curvature may
be estimated.
If the student will consider the simpler surfaces with which
he is familiar, such as a sphere, an ellipsoid, or an hjperboloid
of one sheet, he will have examples of the kind of flexure
which may take place at an ordinary point of any surface.
Any point of a sphere or a pole of a prolate or of an oblate
spheroid is an example of a point of a surface from which,
if we proceed along any section made by a plane containing
the normal ; the curvature is the same.
Any point of an ellipsoid is an example of a point on a
surface at which, if a tangent plane be drawn, the surface in
the neighbourhood of the point lies entirely on one side of the
tangent plane ; such surfaces are called Si/7iclastic.
If a tangent plane be drawn at any point of an hyperboloid
of one sheet, the surface will Intersect the tangent plane, and
bend from it partly on one side and partly on the other ; such
surfaces are called Antidastic.
5G5. Let two planes be drawn through the same tangent
line at any point of a sphere, one containing the normal and
the other inclined at an angle 6 to the normal, the sections
made by these planes will have their radii In the ratio 1 : cos^.
This simple relation between the radii of curvature of a
normal and oblique section, containing the same tangent line,
will be proved to be true for any surface at an ordinary point.
380 NORMAL SECTIONS.
The student may for an exercise prove it when the tangent
line is drawn through the extremity of a principal axis of
an ellipsoid parallel to another principal axis.
566. Consider next tlie curvature of the sections of an
ellipsoid made by planes passing through OA, the normal at Aj
an extremity of one of the principal axes; if AP be one of
these sections intersecting the principal section BC, perpen-
dicular to OAy in P; OP and OA will be its semiaxes, and its
radius of curvature at A = -p^—r ; also -p^-.- » -rr-r '^vill be the
UA (JA (J A
radii of curvature at A of the principal sections BA and CA^
and since OB^ OP, OC arc in order of magnitude for all
positions of OP, we see that of all the normal sections through
yJ, the two sections which have their curvature a maximum
and minimum have their planes perpendicular to each otlicr.
This property of- normal sections will be found to hold for
any ordinary point of all surfjvces.
rjQ7. If POB=^ ^^ _- + ^^^ = __ ; hence, if p, p' be the
radii of curvature of the sections AB, A C, and B that of the
.„ cos'^ sin'^ 1
section AP. i , - = t, •
' /? p L
This relation between the radii of curvature of the normal
sections of least and greatest curvature, called principal sections,
and that of any other normal section inclined at an angle 6 to
one of the principal sections will also be found to hold for
any surface.
5G8. These three properties which are true for all surfaces
will enable us to determine the radii of curvature of all plane
sections through any point, when those of any two sections,
not containing the same tangent, arc known.
AWmal Sections.
569. To find the relation between the radii of curvature of
sections made hy planes containing the normal at any point of a
surface.
NORMAL SECTIONS. 381
Let the surface be referred to the normal at the point as
the axis of ;:, and the tangent pLine at as the pLane of xt/.
(h , (h
— and -y-
dx ay
s, t be the vaUies of , ., , 7 , , , .. ,
dx dxc/i/ dy
z^\ {rx^ + Isxy + ty^] + «S:c.
Let tlic surface be cut by a plane passing through (9.r, and
inclined at an angle Q to the plane of zx\ at every point of
this plane x = u cos^, y^xi sin Q ;
.-. z = \ [r cos'^ + 2s cosfi* sin^ + t ain'^) x^ (1 + s)
where e vanishes in the limit.
If R be the radius of curvature of this section,
1 . . 2.r
-75 = limit —., = ;• cos"^ + 2s cos d sin d + t sin""'^.
li II'
The directions of the normal sections of -svliich the curvature
is a maximum or minimum are given by the equation
-{r-t) sin2^ + 2s eos2^ = 0.
If a be one solution, the rest will be included in the formula
|«7r + a, hence the sections of maximum and minimum curvature
are at right angles.
These sections are called the Fn'ncqyal Sections of the surface
at the point considered.
If the planes of the principal sections be taken for the planes
of zx and yz, a = 0, and therefore s = 0, and the expression for
the curvature of any section will become ^ = r cos'^^ + t sin'6 ]
let p, p' be the radii of curvature of the principal sectionSj
I 1 1
then - = ?• and — = t.
P P
1 _ cos^^ sin'^
•'• 7Z - "7" "^ 'p" '
also, if i?, W be the radii of curvature of any perpendicular
normal sections,
1111
11 li P P
These theorems arc due to Eulcr.
382 THE INDICATRIX.
The Indicatrix.
570. Euler's theorems and other theorems relating to the
curvature of plane sections of surfaces are deduced with great
facility by means of a curve called the indicatrix, employed
first by Dupin for this purpose.
Def. The indicatrix at any point P of a surface is the
section made by a plane parallel to the tangent plane at P and
at an infinitely small distance from it.
In cases in which,, as in antlclastic surfaces, the curve of
intersection extends to any finite distance, the name of indicatrix
only applies to the portions of the curve which arc infinitely
near to P.
571. At any ordinary point of a surface the indicatrix is
a conic.
Taking the axes as in Art. 5G9, the equation of the surface
is of the form z = ^ {rx^ + 2sxy + ty^) +&c.^ and by transfor-
mation of axes the term involving xy may be made to disappear,
so that z = ax^ + hy^ + terms of higher dimensions.
If the surface be cut by a plane parallel to the tangent plane
and very near to it, for which s = A, in the neighbourhood of
the point of contact h = ax' + hy'^ ; the indicatrix is therefore a
conic whose centre is in the normal.
572. Pendlebury has noticed^'- that the indicatrix may be,
at particular points of some surfaces, of any form, and the-
number of directions of principal curvature for such points may
be any number, in fact, equal to the number of apses in the
Indicatrix. He gives as an example a surface x' + y'^ = azj) f-j
generated by a parabola revolving round its axis, its latus rectum
increasing or decreasing with the angle through which its plane
has revolved ; such surface would look like a paraboloid with
ridges and furrows radiating from the vertex.
* Messenger oj Mathematics, vol. I., p. M8.
THE INDICATIilX. 383
573. The radius of curvature of a normal section of a surface
varies as the squaix of the corresponding central radius of the
indicatrix.
Let CP be the central radius of tlie indlcatrlx which lies in
any normal section whose radius of curvature at is i? ; tlien
2^ = limlt— — -, (see figure, p. 386); therefore, since OC is
constant for all normal sections It x CP'.
Ilcnce all theorems in central conies which can be expressed
by equations homogeneous in terms of the radii and axes, can
be replaced by corresponding theorems in curvature. Eulcr's
theorems follow at once, and if i?, R' be radii of curvature of
normal sections inclined to a principal section of a surface at
angles 9^ 6', such that tan tan 6' = -—,
P
then ii + ii' = p -f p',
and BR' sm'^d' - 6) = pp.
574. AVhcn the Indicatrix is an ellipse, the surface is syn-
clastic at the poiirt considered.
A point of a surface for which the indicatrix is a circle is
called an umhilic^ the curvatures of all sections made by planes
containing the normal at an umbilic being equal.
575. AYhen the form of the indicatrix Is hyperbolic, the
surface Is antlclastic at the point considered ; In this case the
radii of curvature of normal sections containing the asymptotes
arc Infinite, such sections pass through the Inflexional tan-
gents, and their directions arc given by tan''^= — , p, p being
the absolute values of the radii of curvature.
In order to deduce theorems from geometrical properties of
the hyperbola, It may be necessary to suppose two Indicati-Ices,
one on each side of the tangent plane at equal distances from It.
If p = p and R be the radius of curvature of a normal
section inclined at an angle 6 to a principal section
7? cos 2^ = p.
384 THE INDICATRIX.
576. When the section made by the plane parallel to the
tangent plane is a parabola, the part of the section which is
called the indicatrix is two parallel lines which become ulti-
mately, as in the case of a developable surface, two coincident
lines.
Such points are called FarahoJic iwints^ sometimes also
Cylindrical jjoints.
As an example of a parabolic point, take a point of the
'2 2 2
cone -^, -^ 'jr. = —, ^ at a distance I from the vertex In the
a c
generator - = -; transform the axes so that the normal at
this point is the axis of ^r, and the generator the axis of .r,
the resulting equation of the cone \?> h = —yr^y^ — zx — s^,
let z = h and a = ctana, then ?/ = ■ — hix -\- 1- h coi^y). the
section by a plane parallel to the tangent plane is therefore
a parabola, the distance of whose vertex from the normal at the
point considered hl — h cot2a, and since this remains finite, when
h is made indefinitely small, the degeneration into two nearly
coincident parallel lines in the neighbourhood of the point is
explained.
T2T
The finite principal radius of curvature Is — .
577. The intersection of tivo consecutive tangent ^ylanes and
the direction of the line joining the points of contact are p)(tralkl
to conjugate diameters of the indicatrix.
Let CP, CD be conjugate semi-diameters of the indicatrix
for the point of a surface ; since the tangent plane to the
surface at P contains the tangent to the indicatrix at P, its
intersection with the tangent plane at is parallel to CP,
and proceeding to the limit, when OC vanishes, the proposition
follows.
Def. Tangent lines at any point of a surfiice drawn parallel
to conjugate diameters of the indicatrix, are called Conjugate
Tangents.
OBLIQUE SECTIONS. 385
578. It follows from this property of consecutive tangent
planes, that if a torse envelope any surface the directions of the
generating lines at any point of the curve of contact aro
conjugate to the tangents to the curve.
579. To find the relation between the radii of curvature of a
normal and oblique section of a surface made by planes passing
through the same tangent line.
Let the tangent line through which the planes are taken be
the axis of x, anS, S' \ and let 7>'/, (ja
be normals to the surface 6', which, since they intersect, must
Intersect in the polar line alJA^ perpendicular to the osculating
plane pU(i\ similarly qah^ rb, normals at q and r, intersect in
the polar line bVio the consecutive osculating plane, and
brV=bqV=aqU-\- UqV.
Let a', b' be corresponding points for the surface S' ;
.-. b'rV^-a'qU-itUqV;
.'. b'rb = a'qa\
396 LINES OF CURVATUBE.
hence, normals to SS' at consecutive points ^, r are inclined at
the same angle, therefore the surfaces cut one another through-
out at a constant angle.
Reciprocally, if tioo surfaces cut one another at a constant
angle^ and their curve of intersection he a line of curvature on
one surface^ it will he a line of curvature on the other also.
Let it be a line of curvature on S^ and let the normals to S'
at 2 and r be qa!'h' and rh" meeting Ua in a" and Uh in h\ h" ;
then since the angle between the normals to S and S' at
2', r are equal, hrh" = bqb', and hrh" = hqh'\ therefore b' and b"
coincide, and the normals to S' at q^ r intersect; that is, the
curve is a line of curvature also on S'.
Cor. If a line of curvature be a plane curve, its plane will
cut the surface at a constant angle.
602. The analytical proof given by Bertrand is very simple.
Let P, Q be consecutive points on the curve of intersection
of surfaces >S', S' ; x, y, z and x + dx^ y + dy^ z + dz their co-
ordinates ; Z, 9n, 71 and Z', wi', n the direction-cosines of the
normals at P to S and S'.
If the curve be a line of curvature on S, the normals at P, Q
will intersect ;
.*. X — lp = x + dx— [l + dl) pj
dl _ dm _i^n _1 . .
' ' dx dy dz p' ^
Since PQ is perpendicular to both normals,
ldx + 9ndy-i-ndz = 0j ,^x
and I'dx + m'dy + n'dz = 0.
i. If the curve be a line of curvature on both surfaces,
ldr + mdm' + nd7i=0. , ,,. , ,,,x
.'. d [W -f mm' 4 nn) = 0,
or the cosine of the angle between the normals is constant.
ii. If the curve be a line of curvature on /9, and the surfaces
cut one another at a constant angle,
I'dl + m'dm + n'dn = 0,
dupin's theorem. 397
and d {W + mm + nn) = ;
/. Ml' + mdm' 4- ndn = 0,
also I'dl' -f in' dill + n'dii = ;
therefore, by (2), — = -—=-., tlic condition tliat the curve
shouhl be a line of curvature on S'.
603. When three series of surfaces cut one another orthogo-
naJly^ the curve of intersection of anij two of them is a line of
curvature on each.
Let the origin be a point of intersection of three surfaces, one
of each series, and the tangents to their lines of Intersection the
axes. The equations of the three surfaces may then be written
x^- o^f + lhyz-^cz' +...= % (1)
y + a'z' + 2h'zx + cV +. . .= 0, (2)
5; + a'V+25"a'_y + c"/+...= 0. (3)
At a consecutive point on the curve of intersection of (2) and
(3), we have 2/ = 0, z = 0^ x=x'^ and the equations of the tangent
planes are, ultimately,
X .2c' x -{■ y -\- z.2h'x' = 0^
X . 2a" x' + y . 2l}"x' + 2 = 0,
and since these also are at right angles,
Aa'c'x"' + 21)' X + 2yx = 0,
or, ultimately, I' + h" = 0; similarly, Z»" 4 ^ = 0, Z» -h h' = 0, wlilch
can only be satisfied by Z» = 0, Z»' = 0, i" = 0, and therefore the
axes are tangents to the lines of curvature on each surface.
Hence, the tangent lines, at any point of intersection of three
surfaces, to their curves of intersection, are tangents to the Hues
of curvature of the three surfaces through that point, and, con-
sequently, their curves of intersection must coincide with the
lines of curvature. This Is Dupin's tiieorem. A proof is given
by Cayley,* which puts in evidence the geometrical ground
on which the theorem rests.
Quurttrly Juumal, vol. XU., p. 186.
398 gauss' measure of curvature.
Measure of Curvature.
604. Gauss gives the following definition of Integral and
Total Curvature.
Def. The Integral Curvature of any given portion of a
curved surface is the area enclosed on a spherical surface of
unit radius by a cone whose vertex is the centre, and whose
generating lines are parallel to the normals to the surface at
every point of the boundary of the given portion.
Horograph. The curve traced out on the sphere as described
above is called the horograph of the given portion of the surface.
Average Curvature. The average curvature of any portion
of a curved surface is the integral curvature divided by the
area of the portion.
Specijic Curvature. The specific curvature of a curved surface
at any point is the average curvature of an infinitely small area
including the point. This is the measure of curvature which
was shewn by Gauss to be the reciprocal of the product of
the two principal radii of curvature at the point considered.
605. To shew that the reciprocal of the product of the
principal radii at any point of a surface is a proper measure
of the curvature.
Let an elementary area QRS be described including the
point P of a surface, and let a series of lines of curvature
divide this area into sub-elementary portions, such as p)qrs^ and
let /3„ p,' be the principal radii of curvature at p in the directions
p>1-)PS] the horograph for ^jjrs will bo a small rectangle whose
Bides are — and — , , and area = ^ -'— .
Px Pi PxPx
CURVATURE. 399
But, if p, p be the principal radii of curvature at /',
where s vanishes in the limit ; therefore the specific curvature
= lini.-££i:= \.
This expression is independent of the form of the clcmcntaiy
portion including P, and is analogous to the measure of cur-
vature in plane curves, the solid angle of the cone corresponding
to the angle between the normals to a plane curve at the
extremities of the small arc on which a point of the curve lies.
606. To determine the radius of curvature of the normal
section of a surface through a given tangent line at a given point
in terms of the coordinates.
Let the equation of the surface be /''(|, 77, ^)=0; and let
(a:, ?/, x) be the given point P, (X, fx, v) the direction of the
given tangent ; also let [x + dx, y + (7y, z + dz) be a consecutive
point Q taken in the normal section, so that ultimately
dx : dy : dz = \ : fi : v.
Then, if QIi be perpendicular to the tangent plane, P the
radius of curvature of the normal section will be the limit
°^ 2QB-
The equation of the tangent plane is
U{^-x) + V{v-2/)-\-W{^-z)^0',
.__ Udx + Vdy + Wdz
•'• ^^^- ±F '
where r' = U' + V'+U'\
]>ut, Q being a point on the surface,
Udx + Vdy + Wdz+l [m(^7x)» + ...+ 2u'di/dz+...]=0j
neglecting terms of degrees higher than the second ;
400 CURVATURE.
• ' - 2{Udx^Vdy-{-Wdz)
= +
Since we have the conditions
U\ + Vix + Wv = () and X''' + /i"-^ + v' = l,
the problem of finding the directions of the principal sections
and the magnitude of the principal radii of curvature is the
same as that of finding the direction and magnitude of the
principal axes of the section of the conicoid,
ux^-\-...+ 2uyz-\-...= \^
made by the plane Ux-\-Vy-\- Wz = 0.
607. To determine the ijrincipal normal sections^ and the
radii of principal curvature at any ])oint of a surface^ in terms of
the coordinates of the point.
The radius of a normal section containing the tangent whose
direction is (X, n,. v) is given by
p
uX' + V+ lov' + 2i(>v + 2 vVX + 'iw'\^l - -p (>'' + /*' + r) = 0, ( 1 )
where f/X -f T>-h TFv = 0, (2)
and when B. is given, the corresponding tangent lines are the
lines of intersection of the cone and plane represented by these
equations, X, /i, v being considered current coordinates. When
iZ is a maximum or minimum, these directions coincide, and the
plane is a tangent plane of the cone ; hence the direction of
the principal sections are given by
(m - cr) X + w yi. -f vv _ lo'X + {v — a) fjb + u'v
Ij - V
- w ' wncreo--^,
whence we obtain
X
U[[v-a-){io- o-)-ti'"'} H- V[i(!v'-io'[io-a)] i-W[w'u' - v'{v-a-)]
CURVATURE. 401
which, by the equation U\ + T> + Wv = 0, leads to
V [[v - a)[w - a) - ?<"-'}+...+ 2 FIF{yV-?/(?f-a)]+...= 0. (3)
This equation gives the values of the principal radii of curva-
ture, and the values of \ : /* : v, corresponding to each ruot,
arc given by the preceding system of equations.
Cor. The product of the roots of (3) is
IP {vw - It") +...+ 2 VW{v'to - vu') i-.. .
U' + V' + W
= [U^ -{V- + Tr") X measure of curvature.
Since the measure of curvature vanishes at every point of
a developable surtacc, the numerator equated to zero is the
condition that a surface should be developable.
608. We cannot help calling attention to another form of
the quadratic giving the principal radii, which was set in an
examination paper for Clare and Cains Colleges in 1873.
Since 2V]ViJ,v= U'-X'-V'fM'- W'^v'^ &c., the expression
P
for p can be put into the form AX' + Bfi^ + Cv\ where
A^u-k- -r/TTr ( ^^'^' ~" ^'^■' ~ ^^"^io") , &c.
Construct the conicoid A^- + Bif -\-CX'' = P, having its centre
at the point (.r, y, z) of the surface, the directions of the axes
of the section made by the plane f'^ + !'»; + ir^= are the
directions of the tangents to the principal sections of the surface,
and the corresponding values of It will be the squares of the
section.
Hence, by Art. 234,
AR-p^ nn-p'^ en -I' '
a quadratic giving p, p the principal radii of curvature.
Also the direction cosines of the tangents to the lines of
curvature are as J^_^: ^/_ ^ : ^,^._^„ where p., p.^ are to
be written for R.
¥ V F
402 UMBILTCS.
G09. To determine the conditions of an unibilic.
At an urabllic R retains a constant value for all directions
(X, yti, v), satisfying the two conditions (1) and (2). Ilcnce at
an unibilic the cone (1) must break up into two planes, one
of which is the tangent plane (2).
The left-hand member of equation (2) must therefore be a
factor of the left-hand member of (1), and the other factor
will therefore be
^^^{v-a)-\-^^[iv-a).
Multiplying the two, and equating coefficients,
V ,
-a)=2u\
T^/
^u^''-
-^)+^rb'^
-a) = 2v',
^(.-
a) = 2w,
which, on
eliminating a,
lead to the
two conditions
W'v + VSo - 2 VWu' Uho + W'u-2 WUv
r^-fir
•
W'-fU'
_ V'
i + ir'v-2UVio'
tf^+v " '
These two equations, together with the equation of the surface,
will, in general, determine a definite number of points, among
which are included all the unibilici. It may happen that a
common factor exists, so that the three equations are satisfied
by the coordinates of any point lying on a certain curve. Such
a curve is called a line of spherical curvature.
It should also be observed that fZ, F, W have been assumed
to ha finite in the foregoing investigation. Should one of them,
as C/", vanish, we must have, in the same manner, V^i-\Wv a
factor, and must therefore have
(?<-0-; V+...+ 2(/yUV+...
= ( T> + Wv) |/.-\ + (. - t dx is written
for X, &e,, and ^ = a,- + 6'j,, &c.,
cr a
0z=d>/ + -dv - r -, */o-,
dy, dV, F ! = 0,
(7^, dW, W '
which is the differential equation of the lines of curvature.
Expanding dUj dV^ dW^ and eliminating dx, dy, dz, and da,
It — cr, ^c'
The coefficient of U'^ is -
w, V- a, u, V
V, n\ ?/' — cr, W
V, V, ir
V— (J. u
= 0.
u , w — a-
, whence we obtain the quadratic given in
Art. G07.
614. The foregoing equations for determining the principal
curvatures undergo a considerable simplification, if the equation
of the surface be of the form
406
CURVATURE.
Wc shall then have ?/, v\ xo all zero ; the equation giving
the length of the radius of curvature of any normal section,
whose tangent line is (\, /a, v\ will be
P
the quadratic equation for the principal radii of curvature will be
Ru-P'^ Rv-P"" Rw-P~^'
the diflferential equation of the lines of curvature will be
U[v — w) dydz + V{io — u) dz dx + W[u — v) dxdy = 0.
The conditions for an umbilic in this case reduce to ic = v = io
when U, F, IV are finite, but since this is the exceptional case
mentioned in Art. 609, in which ?(', v\ and lo' vanish iden-
tically, there are other umbilics which are given by U=0 and
{v — u) W^ -^ [w — u)V^ — 0, and similar equations when V=
and W'—Q. The whole number of umbilics is therefore, as
before,
71 [{n - 2Y + 3 (n - I) (3n - 4)| ^w (lOn' - 25m + 16).
615. To obtain the differential equation of the lines of cur-
vature^ and to find the centres and radii of jirincipal curvature
when the equation of the surface gives one of the coordinates
explicitly in terms of the other two.
Let the equation of the surface be t=/(|, 77), and let P, Q
be consecutive points on a line of curvature whose coordinates
are a;, 3/, 2;, and x + dx^ y + dy^ z + dz^ then the normals at P, Q
intersect ; and if (|, 17, ^) be their point of intersection,
^-x^p[K-z)=^0, and 7;-7/ + ;2(e-^)=0, (1)
but f , 77, ^ remain the same when x + c?.r, y + <7y, z 4- dz are
written for ic, 3/, 2; ; therefore
d2){X-z) = dx^ 2)dz^ and dq {^-s)= dy + qdz ; (2 )
rdx + sdy _ dx + p{ jjdx + qdy) ^
sdx + tdy dy + q [pdx + qdy) '
.-. {(1 -f q') s -pqt] {dyY + 1(1 + ql r - (1 +/) t] dxdy
CURVATURE. 407
■wliicli is the dilToivntlal equation ot" the projection of the lines
of curvature on the pKinc of xy.
Let p be the radius of curvature of tlie principal section
through PQ^ hence by (1) p''= (1 +/ + -?')(::- TA therefore,
writing in (2). for. -r or ^^^^^._^^.^ ,
{rJx + sdy) a + dx +p [ikIx + qihj) = 0,
.-. [r'f) dx = 0]
... [ra- + 1 +/) (f(r + 1 + 2') " {^<^ + M)* = »,
or (r^ - s'-'} 0-- + { ( I + 5-) r - 2/) js + ( 1 + 2>') t ] a
+ 1 + ir + (f = 0. (4)
Cor. Gauss' measure of curvature Is
J_ _ 1 I _ _ rt-s" . . ^
pp ~ aa 1 -^f + q' " (1 -^f + iJ ^
which vanishes for a developable surface.
G16. To find the umlillcs of the surface z =/(j', ^).
Since the normals at points passing in any direction from
an umbilic intersect the normal at the nmbilic, neglecting
small quantities of the third order in dx and f/y, the equation
(3) must be true independently of the value of dy : dx^ and
this condition is satisfied by
1+;/ _V3_ ^ 1 + 7' .
r' ~ a t "*
these equations, with the equation of the surface, determine the
nmbilics.
Curvature of Com'coids.
G17. To find the radii of principal curvature at any point
of a ci ntral conicoid.
Let P be any point on the conicoid, supposed in the figure
to be an ellipsoid, POP' the diameter through /', CL the radius
parallel to the tangent at P to any normal section whoso radius
of curvature Is rcfiuircd, PQL the central section having the
408
CURVATURE OF
same tangent. Let a plane be drawn through a pohit Q near
F parallel to the tangent plane at P, meeting CP in F, and
let 2>j "^ be the perpendiculars on the tangent plane from
C and Q, so that -ct : ^; : : VF: CF. The radius of curvature
of the normal section is the limit of — — or -^ — , and
2-BT 2ot '
QV''.CUy.PV.VP''.CP''
:: -UT.VF' : 2}.CP=2'S} :p ultimately,
hence the radius of curvature = .
P
If a, /3 be the scml-axes of the central section parallel to the
tangent plane, — and — will be the principal radii of curvature,
which we shall call p and p.
618. To find the coordinates of the centres of curvature.
Let the equation of the conicoid be
?' + 2^' + - = 1 (1)
^8 + j. + ^. -I, u;
and let (/, ,7, h) be the point P, then f , 7;, ^, the coordinates of
the centre of curvature corresponding to /?, satisfy the equations
d' W c'
CONICOIDS. 409
and by Art. 28'), if the equations of oonfocal conicoiils through
Pbe
a* and /3' are respectively a* - a'"' and «" - (i"* ; tliercforc the
coordinates of the centres of curvature arc
fcr gV" lu-"^ fa" g}/' he'
a' ' />" ' c'" ' a' '' b' ' c" ■
CAd. If three confocal conicoids (^), (5), (C) intersect in P,
the centres of principal curvature of {A) at P are the poles with
respect to [B] and [C] of the tangent plane to [A] at P.
Let the three conicoids {A\ (P), and [C] be
a" y' z" , a:" v' ^^ , , ^* '/' -' ,
« 6 c a ^ c a b c
intersecting in (/, 17, //).
The coordinates of the centre of curvature of the normal
section containing the tangent to the intersection of {A) and {D)
are ., , ^z- ^ ^1 and Its pohir, with respect to [C], is
a b c
1 + t« ■«" ~5 = ^) *^^ tangent pLane to {A) at P.
a o c
SlmUarly for tlic other centre of principal curvature. This
proposition is due to Salmon.
620. The curve of intersection of tia confocal conicoils is
a line of curvature on each.
Let PT be a tangent at P to the curve of intersection
of two confocals S and .S", P^V, PN' normals at P to S and 5' ;
and suppose a central section of S made by a plane parallel
to the tangent plane .V'PP, and therefore to the indicatrix to
S at P. Now it is shewn (Art. 285) that PN' is parallel to
one axis of this section ; therefore PT is parallel to the other
axis • hence, the tangent to the curve of intersection of S and S'
at any point is parallel to an axis of the Indicatrix of cither
surface at that point, and the curve Is a line of curvature.
uou
410 LINES OF CURVATURE
621. At any point in a line of curvature of a conicoid^ the
rectangle C07itained hy the diameter parallel to the tangent at that
point and the perpendicular from the centre on the tangent plane
at the point is constant.
Let the line of curvature on the conlcoid S be the curve of
intersection with /S", and let FT be a tangent to it at any
point P; PN^ PN' normals to S and S' at P; then, if a, /3 be the
semi-axes of the central section parallel to N'PT^ the tangent
plane to /S, which are parallel respectively to PT and PN\ it
is shewn (Cor., Art. 285) that ^ is constant, and if p be the
perpendicular from the centre on the tangent plane, pa/3 is
constant, therefore pa. is constant.
622. The following proof is independent of the properties
of confocal surfaces.
Let P, Q be consecutive points on a line of curvature, the
corresponding centre of curvature, p the radius of curvature,
p the perpendicular on the tangent plane, a, /3 the semiaxes
of the section parallel to the tangent plane, a being parallel to
P^, and let CP=r, then by the triangle OOP
0C' = p' + r'-2pp,
and since, for a change from P to Q, 0, C are unchanged in
position and p is unaltered, rdr = pdp. But, by Art. 273,
(e + ^'' + r' = ci'^V'-^c% a^p = ahc]
.-. ada + (3d/3 + rdr = 0, and ^ + ^ + ^^' = 0,
multiplying the last equation by a'^ orpp, and subtracting the
preceding, we obtain (a'^ — ^'^) c?/3 = ; therefore /3 is constant,
unless a = /3, which is only true at an umbilic, therefore ^)a is
also constant.
623. To shew that the curves of intersection of a given
conicoid with all confocal conicoids ivhich intersect it satisfy
the differential equations of a line of curvature.
Let the equation of the surface be
a c ^ ^ '
OF CONICOIDS. 411
then the differential equations of the lines of curvature arc
^dx+^^d^+'-oints intersect.
If the axis of s be a normal at an umbilic, the equation
of the surface is of the form ^= a {^"^ + tj') + u^ (1 + e), where u^
is of the third degree in | and 77, and t vanishes in the limit ;
the equations of a normal at (ic, 3/, z) are
but if this normal meet that at the umbilic, the equations are
satisfied byj|^ = 0, 97 = ;
du du
dy ^ dx '
which gives three directions in which the point (x, y, z) must
be taken.
628. To find the three directions for which normals to a conicoid
intersect the normal at an umbilic.
Let a^"^ + br)^ +c^'^ — 1(1) be the conicoid, (a, 0, 7) the um-
bilic, (a + X?-, /Mr, y-{- vr) a point adjacent to it in the direction
(\, /A, v), the equations of the two normals are
^ — a — \r 7) — fxr ^—y—vr
« (a 4- \r) bfir c (7 + vr) '
aoL cy
one condition that they may intersect is /a = 0, one direction
is therefore that of the principal section containing the umbilic ;
for the other conditions
f - a = \r- -v (a-l- \>-) and ^- 7 = I'r— - (7 + vr) ;
.'. cy [b — a)\ = aa{h-c)v'j
or, since
b — a c — b
THROUGH AN IMnil.lC. 415
aoik + 07^ = ; (2)
and, by (1), n (a + X?f + bfi'r' + c[y + vr' = 1 ; (3)
.-. aX' + bfjL' + cv' = 0', (4)
(2) and (4) give the two other directions for wliich the normals
intersect ; and, since : 3) is satisfied for all values of r, they
are the directions of the imaginary generatrices through the
umbilic.
G29. ^Ye may observe also that since
«'-'a-'\-' = cV^'^ (i - «) X" = (c - J)" ;
.-. a\' -f bf^' + c'v' = b {X' + /i'-' + v') ;
.-. X' + fi' + v' = by (4),
which shews that these generatrices pass through the imaginary
circle at infinity.
(Since the argument of Art. G28 is independent of the mag-
nitude of r, it is true that all the normals at points along one
of the umbilic at generatrices intersect, and they have therefore
this character of lines of curvature, but Cayley has remarked
in a note on a paper upon " the direction of lines of curvature
in the neighbourhood of an umbilicus,'"* that they are the
envelopes of the lines of curvature, and belong to the singular
solution of the differential equation of these lines, as oppeai-s
from Art. 624.
630. In the note referred to above, Cayley remarks that
since, at an umbilic, -^ is determined by a cubic equation,
there are generally three directions of the line of curvature,
which may arise from three distinct curves, or from n curve
with a triple point at the umbilic ; and, referring to a paper
by Serret,t he states that the lines of curvature on the surface
xyz = 1 are its intersection with the series of surfaces
/* = [x" + wy" + wV)* + (a;" + o»y + ctz') «,
• FroBt, Quart. Joum. of Math., toI. x., p. 78, and Oaylej, ibid. p. III.
t Liout. Jaum., t. 12 (1847), pp. 241— -.'64.
416 LINES OF CURVATURE.
where to is an imaginary cube root of unity ; now at the umbilic
(1, ], 1), corresponding to which h = 0,
{x' + a>if + <^'zy = {x' -f wy + (ozy ;
.-. X^ + (Jiy^ + Oi'z^ = X' + Co'y + Oiz\
or = (u (a:"" + w'Y'' 4- (nz^)-, or = o)'^ {x^ + wy -f (w^'") ;
.'. 7f = z\ or x^ = y\ or 3^ = a;"'';
hence, through the umbilic (1, 1, 1) three distinct lines of cur-
vature pass, viz. the curves
y = z^ xy^ = 1 ; x=y^ zx^ =^\] and z = x^ yz^ = 1.
631. The differential equation of the line of curvature of
xyz = 1 is
xdydz [y^ - z') -f ydzdx [z^ - ^) + zdxdy [x'' — y') = 0.
Multiply by xyz^ and let x^ = j?, y' = q^ z' = /• ;
.*. p[g_- t) dqdr + q{r- p) drdp + ^{p — q) dpdq = 0. (1 )
Again, if h={p + a>q+ toV)^ + (/> + (o^q + (or/
[dp + codq + (o'^h-y [p + coq + co'^r)
- [dp + (o'^dq + codrY [p -f sin' \' + 2-^> ^ Cyp f ^> 4 -gg -t- C ^ J3 4 Cg* f 2^'g
r a " « *
(19) Shew that the projection, on the plane of xy, of the indicatrix at
any point of the surface : = (e" -f e ") cosj is a rectangular hyperbola.
(20) Shew that the indicatrix at any point of the surface y = x tan - is
the part of a rectangular hyperbola which lies near the point. Prove that
the section by the tangent plane near the point is the generating line and a
portion of a parabola.
(21) Deduce the conditions for an umbilicus from the equation giving
the radii of ciu'vature, by making the roots of the equation equal.
(22) Shew that a sphere whose centre is the origin, and the reciprocal
of whose radius is - + r + - touches the surface whose equation is
(23) Prove that the radius of curvature of the surface x" + j/"* + s"* = o"
at an umbilic is x 3 "" .
in - 1
(24) Prove that the measure of curvature at any point of an ellipsoid is
proportional to /)', p being the perpendicular from the centre on the
tangent plane.
(25) Prove that the measure of curvature at any point of the paraboloid
?L + 1 = X varies as (-) , p being the perpendicular from the origin on the
tangent plane.
(26) Prove that the measure of curvature at every point of the cllipUo
b
paraboloid 2: = - 4 '^ where it is cut by the cylinder ^, + |. =" ^ "'* *<1"*^
420 PROBLEMS.
(27) Shew that the specific curvature at any point on the surface xyz = abc,
varies as the fourth power of the perpendicular from the origin on the
tangent-plane at the point, and that at an umbilic it is I (^abc)'^.
(28) If a plane curve be given by the equations
- = cosO + log. tan 10, - = sinO,
the surface produced by the revolution of this curve about the axis of x
will have its measure of curvature constant.
(29) In a surface, generated as in (15), if = log taniO, the measure of
curvature will be the same at corresponding points on the fixed line
and on the circle.
(30) The integral curvatures of the portions of the ellipsoid — + |j -t- , = 1
x* v^ z'
cut off by the cone -r + ^r - -r = are in the ratio of -/S - 1 to V2 + 1«
^ a* b* c*
(31) Shew that the integral curvature of the whole surface
(32) Shew that the integral curvature of the portion of a surface of
revolution cut off by any plane perpendicular to the axis of revolution
is Att sinVo, where a is the angle which the normal to the surface at any
point on the curve of intersection of the plane and surface makes with
the axis.
(33) If any cylinder circumscribe an ellipsoid, it divides the ellipsoid
into portions whose Integral curvatures are equal. Hence, if three
cylinders circumscribe an ellipsoid, the integral curvature of the portion
of the ellipsoid cut off is tt - POQ - QOR - MOP, where O is the centre,
and OP, OQ, OR are the directions of the axes of the cylinders.
(34) Prove that the integral curvature of the portion of the surface of
3 ellipsoid -.+ ?-,+ -,=
hyperboloid of one sheet
X v z
the ellipsoid —. + ?;+ -^ = 1, bounded by its intersection with the confocal
'^ a* b' c*
+ ~-^, + -^^. = 1
. _ c» /r(a*-\«)(6«-X«)\
x' v^ 2'
(35) Find the umbilic of the surface — + ~ + — •= k^, and shew that, at
^ ^ abc
the umbilic, - = ^ = - , the directions of the three lines of curvature are
abc
, . ^. clx dtf ihi (h . (h dx ^. ,
given by the equations — = t" > t" = — > ^"^ — = — respectively.
PROBLEMS. 421
(3G) If the inclination of two surfaces at any point of their curre of
intersection be (', 5 the arc of the curve of intersection, p^, p, tlie principal
radii of one surface ; « the angle between the tangents to the curve and to
a principal section, and />,', i^.', a corresponding quantities for the other
^ do Bin 2a /I 1\ sin 2a' /I 1 \ ,, .
surface, prove that -r = — :z— I .,-—-—)• "ence, shew
as a \pi pj 2. \p^ Pi/
that if two surfaces intersect always at a constant angle, and the curre of
intersection be a line of curvature of one surface, it will also be a line
of curvature on the other surface.
(37) If one series of lines of curvature on a surface be plane curves,
lying in parallel planes, the other series will also be plane curves.
(38) The planes drawn through the centre of an ellipsoid, parallel to
the tangent planes at points along a line of curvature, envelope a cone
which intersects the ellipsoid in a sphero-conic.
(39) On an umbilical conicoid, the projections of the lines of curvature
on the planes of circular section, by lines parallel to an axis, form a series of
confocal conies, the foci of which are the projections of the umbilics.
(40) Find the differential equation of surfaces possessing the property,
that the projections, on a fixed plane, of their lines of curvature cross each
other everywhere at right angles. Prove that it is satisfied by surfaces of
revolution whose axes are perpendicular to the fixed plane ; and obtain the
general solution.
(41) Prove that the three surfaces yz= ax, VC** + y*) + -/(*' + «*) = *.
v'(x' + y*) - ;{x* + s*) = c, intersect each other always at right angles ; and
hence prove that, on a hyperbolic paraboloid, whose principal sections are
equal parabolas, the sum or the difference of the distances of any point on
a line of curvature from the two generators through the vertex is constant-
(42) In the helicoid, whose equation is t/ = z tan- , the lines of curva-
ture are the intersections of the helicoid with the surfaces represented by
the equation —^ — — ^ = ca"' f - «'" for different Talue* of c.
Also, prove that the principal radii of cur\ature are, at every pomt,
constant, equal in magnitude, but of opposite signs.
(43) Tangent planes are drawn to a scries of confocal conicoida- from a
fixed point on one of the axes, the locus of the points of contact it a
surface ; prove that three such surfaces corresponding to three poinU ooo
on each axis cut one another orthogonally.
422 PROBLEMS.
(44) Prove that the lines of curvature on the surface
a ax - O* + i"* aX - a + C
are two systems of circles, ^vho8e planes are parallel to the axes of y and 2
respectively, and pass each through one of two fixed points on the axis
of z.
THE END OF VOL. I.
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