A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 14 of 28

(<0 of the expansion vanish except that containing Ac.

n

Expressing the operator on ^ in terms of differentiators with respect to the real axes, we obtain the equation

#Z_(_ (—)d'- d . y

dz*~* W ' 1.2 da?-* df '

_£(. + ,)!(.-„)!

2°-^!

from which we can determine the coefficient of any harmonic of the series in terms of the differential coefficients of ^ with respect to tv, y> z at the origin.

143.] It appears from equation (50) that it is always possible to express a harmonic as the sum of a system of zonal harmonics of the same order, having their poles distributed over the surface of the sphere. The simplification of this system, however, does not appear easy. I have, however, for the sake of exhibiting to the eye some of the features of spherical harmonies, calculated the zonal harmonics of the third and fourth orders, and drawn, by the method already described for the addition of functions, the equi- potential lines on the sphere for harmonics which are the sums of two zonal harmonics. See Figures VI to IX at the end of this volume.

Fig. VI represents the difference of two zonal harmonics of the third order whose axes are inclined 120° in the plane of the paper, and this difference is the harmonic of the second type in which o- = 1 , the axis being perpendicular to the paper.

In Fig. VII the harmonic is also of the third order, but the axes of the zonal harmonics of which it is the sum are inclined 90°, and the result is not of any type of the symmetrical system. One of the nodal lines is a great circle^ but the other two which are intersected by it are not circles.

Fig. VIII represents the difference of two zonal harmonics of

144 &•] DIAGRAMS OF SPHERICAL HARMONICS. 201

the fourth order whose axes are at right angles. The result is a tesseral harmonic for which n == 4, a- — 2.

Fig. IX represents the sum of the same zonal harmonics. The result gives some notion of one type of the more general har- monic of the fourth order. In this type the nodal line on the sphere consists of six ovals not intersecting each other. Within these ovals the harmonic is positive, and ia the sextuply connected part of the spherical surface which lies outside the ovals, the har- monic is negative.

All these figures are orthogonal projections of the spherical surface.

I have also drawn in Fig. V a plane section through the axis of a sphere, to shew the equipotential surfaces and lines of force due to a spherical surface electrified according to the values of a spherical harmonic of the first order.

Within the sphere the equipotential surfaces are equidistant planes, and the lines of force are straight lines parallel to the axis, their distances from the axis being as the square roots of the natural numbers. The lines outside the sphere may be taken as a representation of those which would be due to the earth's magnetism if it were distributed according to the most simple type.

144 #.] We are now able to determine the distribution of electricity on a spherical conductor under the action of electric forces whose potential is given.

By the methods- already given we expand *, the potential due to the given forces, in a series of solid harmonics of positive degree having their origin at the centre of the sphere.

Let Anrn Yn be one of these, then since within the conducting sphere the potential is uniform, there must be a term — AnrnYn arising from the distribution of electricity on the surface of the sphere, and therefore in the expansion of 477(7 there must be a term

In this way we can determine the coefficients of the harmonics of all orders except zero in the expression for the surface density. The coefficient corresponding to order zero depends on the charge, 0, of the sphere, and is given by 47ro-0 = a~2e.

The potential of the sphere is

144 £.] Let us next suppose that the sphere is placed in the neighbourhood of conductors connected with the earth, and that

202 SPHERICAL HARMONICS.

Green's Function, G, has been determined in terms of a?, y, z and #', y, /, the coordinates of any two points in the region in which the sphere is placed.

If the surface density on the sphere is expressed in a series of spherical harmonics, then the electrical phenomena outside the sphere, arising from this charge on the sphere, are identical with those arising from an imaginary series of singular points all at the centre of the sphere, the first of which is a single point having a charge equal to that of the sphere and the others are multiple points of different orders corresponding to the harmonics which express the surface density.

Let Green's function be denoted by Gpp>, where p indicates the point whose coordinates are #, y, z> and p' the point whose co- ordinates are #', y', z'.

If a charge A0 is placed at the point p' , then, considering x', y\ z' as constants, Gppf becomes a function of so, y} z and the potential arising from the electricity induced on surrounding bodies by An is ty •=. A. G » ( 1 )

0 pp ' \ /

If, instead of placing the charge AQ at the point jo7, it were distributed uniformly over a sphere of radius a having its centre at p') the value of ¥ at points outside the sphere would be the same.

If the charge on the sphere is not uniformly distributed, let its surface density be expressed, as it always can, in a series of spherical harmonics, thus

47r<22<r = ^0 + 3^1Z1 + &c. +-(2#-f l)AnYn. (2)

The potential arising from any term of this distribution, say

4ira2(Tn = (2n+ 1) AnYn) (3)

rn a>n

will be n+1 AnYn for points inside the sphere, and tM+1 ^w^n f°r

points outside the sphere.

Now the latter expression, by equations (13), (14), Art. 129, is equal to , , n . an dn I

or the potential outside the sphere, due to the charge on the surface of the sphere, is equivalent to that due to a certain multiple point whose axes are h^.»lin and whose moment is Ana\

Hence the distribution of electricity on the surrounding con- ductors and the potential due to this distribution is the same as that which would be due to such a multiple point..

144 &•] GREEN'S FUNCTION. 203

The potential, therefore, at the paint /?, or (a, y, 2), due to the induced electrification of surrounding bodies, is

where the accent over the d's indicates that the differentiations are to be performed with respect to x', y ', /. These coordinates are afterwards to be made equal to those of the centre of the sphere.

It is convenient to suppose Tn broken up into its 2n+\ con- stituents of the symmetrical system . Let A£ Y^ be one of these,

then d'" - J*> (s)

Th~¥Yn- v"

It is unnecessary here to supply the affix 8 or c, which indicates whether sino-^ or cosa-0 occurs in the harmonic We may now write the complete expression for *£,

But within the sphere the potential is constant, or

• • 1 4, + SS [ J£L f* J<?>] = constant. (7)

Now perform on this expression the operation Dr**, where the differentiations are to be with respect to x y> z, and the values of % and Oj are independent of those of n and or. All the terms of (7) will disappear except that in 7, and we find

_ o

We thus obtain a set of equations, the first member of each of which contains one of the coefficients which we wish to determine. The first term of the second member contains A0, the charge of the sphere, and we may regard this as the principal term.

Neglecting, for the present, the other terms, we obtain as a first approximation

W _ 1 22*V W g ()

A" - 2 "

If the shortest distance from the centre of the sphere to the nearest of the surrounding conductors is denoted by #,

204 SPHERICAL HARMONICS. [145 a.

If, therefore, b is large compared with #, the radius of the sphere, the coefficients of the other spherical harmonics are very small compared with A0 . The terms after the first on the right-hand side of equation (8) will therefore be of an order of magnitude

. /«

similar to -r

We may therefore neglect them in a first approximation, and in a second approximation we may insert in these terms the values of the coefficients obtained by the first approximation, and so on till we arrive at the degree of approximation required.

Distribution of electricity on a nearly spherical conductor.

145 a.~\ Let the equation of the surface of the conductor be

r. = a(l+F), (1)

where F is a function of the direction of r, that is to say of 6 and $, and is a quantity the square of which may be neglected in this investigation.

Let F be expanded in the form of a series of surface harmonics

y=/o+/1r1+/2r2+&e. +/„!„. (2)

Of these terms, the first depends on the excess of the mean radius above a. If therefore we assume that a is the mean radius, that is to say, approximately the radius of a sphere whose volume is equal to that of the given conductor, the coefficient fQ will disappear.

The second term, that in /i , depends on the distance of the centre of mass of the conductor, supposed of uniform density, from the origin. If therefore we take that centre for origin, the coefficient f^ will also disappear.

We shall begin by supposing that the conductor has a charge J0, and that no external electrical force acts on it. The potential outside the conductor must therefore be of the form

where the surface harmonics are not assumed to be of the same types as in the expansion of F.

At the surface of the conductor the potential is that of the conductor, namely, the constant quantity a.

Hence, expanding the powers of r in terms of a and F, and neglecting the square and higher powers of F, we have

145 a-] NEARLY SPHERICAL CONDUCTORS. 205

(4)

Since the coefficients Alf &c. are evidently small compared with A0, we may begin by neglecting products of these coefficients into F.

If we then write for F in its first term its expansion in spherical harmonics, and 'equate to zero the terms involving harmonics of the same order, we find

a =4,1, (5)

A.Y,' = A^f.Y, = 0, (6)

It follows from these equations that the Y"s must be of the same type as the Y's, and therefore identical with them, and that Al = 0 and An = A0anfn.

To determine the density at any point of the surface, we have the equation &Y dV

477(7= -- Y- = — -j- COS € j (8)

dv dr

where v is the normal and e is the angle which the normal makes with the radius. Since in this investigation we suppose F and its first differential coefficients with respect to 6 and 0 to be small, we may put cos e = 1 , so that

(9)

Expanding the powers of r in terms of a and F, and neglecting products of F into An, we find

A0~(l-2F) + &c. + (n + l)An^Yn. (10)

Expanding F in spherical harmonics and giving An its value as already found, we obtain

477(7 = ^0^[i+/ara+2/8r8+&c.+(»-i)/llrj. (n)

Hence, if the surface differs from that of a sphere by a thin stratum whose depth varies according to the values of a spherical harmonic of order n, the ratio of the difference of the surface densities at any two points to their sum will be n—l times,

206 SPHERICAL HARMONICS. [145 "b.

the ratio of the difference of the radii at the same two points to their sum.

1453.] If a nearly spherical conductor is acted on by external electric forces, let the potential, U, arising from these forces be expanded in a series of spherical harmonics of positive degree, having their origin. at the centre of volume of the conductor

U= B0+3irT^+B^Yt'+&0.+B.S>T.', (12)

where the accent over T indicates that this harmonic is not necessarily of the same type as the harmonic of the same order in the expansion of F.

If the conductor had been accurately spherical, the potential arising from its surface charge at a point outside the conductor would have been

Let the actual potential arising from the surface charge be Tf-i- W, where

CJ^+.^ (H)

the harmonics with a double accent being different from those oecurring either in F or in 77, and the coefficients C being small because F is small.

The condition to be fulfilled is that, when r = a (1 + F),

+W= constant = A0

tt

the potential of the conductor.

Expanding the powers of r in terms of a and F, and retaining the first power of F when it is multiplied by A or B, but neglecting it when it is multiplied by the small quantity C, we find

• (^ir+to. + C±jIK'=0. (15)

To determine the coefficients C, we must perform the multipli- cation indicated in the first term, and express the result in a series of spherical harmonics. This series, with the signs reversed, will be the series for W at the surface of the conductor.

The product of two spherical harmonics of orders n and m, is a rational function of degree n + m in sc/r9 y/r, and z/r, and can therefore be expanded in a series of spherical harmonics of orders not exceeding m+n. If, therefore, F can be expanded in spherical

145 c-] NEARLY SPHERICAL VESSEL. 207

harmonics of orders not exceeding m, and if the potential due to external forces can be expanded in spherical harmonics of orders not exceeding nt the potential arising from the surface charge will involve spherical harmonies of orders not exceeding m -f n.

This surface density can then be found from the potential by the equation ,

)=0. (16)

145 c.~\ A nearly spherical conductor enclosed in a nearly spherical and nearly concentric vessel.

Let the equation of the surface of the conductor be

r = a(l+F\ (17)

where F =/i Y1 + &c. +J™ Y*\ (is)

Let the equation of the inner surface of the vesselybe

r = 6(l + G), (49)

where G = ffl Tl + &c. +^> T&, (26)

the f 's and ^'s being small compared with unity, and Y^ being the surface harmonic of order n and type o-.

Let the potential of the conductor be a, and that of the vessel /3. Let the potential at any point between the conductor and the vessel be expanded in spherical harmonics, thus

l?T, (21)

then we have to determine the constants of the forms h and k so that when r = a (1 + F), ^ = a, and when r = I (l + G), * = /8.

It is manifest, from our former investigation, that all the ^'s and /fc's except h$ and kQ will be small quantities, the products of which into F may be neglected. We may, therefore, write

a =

(1 _J>+ &c. + *"«• + ) 7 , (22)

ft = M- *„ (1 - G) + &c. + ( b" + jj Y (23) We have therefore a = A<> + ^, (24)

208 SPHERICAL HARMONICS. [146.

l> (26)

whence we find for the charge of the inner conductor

*o = («-/3)^V (28)

and for the coefficients of the harmonics of order n

"n /QA\

_a2n+l » (30)

where we must remember that the coefficients/^, gn, hn, kn are those belonging to the same type as well as order.

The surface density on the inner conductor is given by the equation

_n -n

' n~ 2n + l_2n+l

146.] As an example of the application of zonal harmonics, let us investigate the equilibrium of electricity on two spherical conductors.

Let a and b be the radii of the spheres, and c the distance between their centres. We shall also, for the sake of brevity, write a — ex, and 6 = cy> so that x and y are numerical quantities less than unity.

Let the line joining the centres of the spheres be taken as the axis of the zonal harmonics, and let the pole of the zonal harmonics belonging to either sphere be the point of that sphere nearest to the other.

Let r be the distance of any point from the centre of the first sphere, and s the distance of the same point from that of the second sphere.

Let the surface density, o-lf of the first sphere be given by the equation

47T(r1a2 = ^-M1P1-f 34>P2 + &c. + (2m+l)JmPw, (1)

so that A is the total charge of the sphere, and Alt &c. are the coefficients of the zonal harmonics P1} &c.

146.] TWO SPHERICAL CONDUCTORS. 209

The potential due to this distribution of charge may be repre- sented by

for points inside the sphere, and by

(3)

for points outside.

Similarly, if the surface density on the second sphere is given by the equation

47r<72£2 = J + -B1PJ + &c. + (2» + l)J5llPw, (4)

the potential inside and outside this sphere may be represented by equations of the form

(5)

(6)

where the general harmonics are related to the second sphere.

The charges of the sphere are A and B respectively.

The potential at every point within the first sphere is constant and equal to a, the potential of that sphere, so that within the first sphere U'+F=a. (?)

Similarly, if the potential of the second sphere is /3, for points within that sphere, U+ 7'= ft. (8)

For points outside both spheres the potential is V, where

Z7+r=*. (9)

On the axis, between the centres of the spheres,

r+s=c. (10)

Hence, differentiating with respect to r, and after differentiation making r = 0, and remembering that at the pole each of the zonal harmonics is unity, we find

2!

a '"^

where, after differentiation, s is to be made equal to c. VOL. i. P

210 SPHERICAL HARMONICS. [146.

If we perform the differentiations, and write a/c — x and 6/c = y, these equations become

0 =

0 = A^ + Bx* + 3^ x*y + 6 -#2#3/ + &c. + J (n + l) («+ 2) B

0 =

(12)

By the corresponding operations for the second sphere we find, 0 =

(13)

To determine the potentials, a and /3, of the two spheres we have the equations (7) and (8), which we may now write

:m. (15)

y

If, therefore, we confine our attention to the coefficients A1 to Am and J5X to Bn, we have ^ + ^ equations from which to determine these quantities in terms of A and JB, the charges of the two spheres, and by inserting the values of these coefficients in (14) and (15) we may express the potentials of the spheres in terms of their charges.

These operations may be expressed in the form of determinants, but for purposes of calculation it is more convenient to proceed as follows.

Inserting in equations (12) the values of Bl...Bn from equa- tions (13), we find

[2 . 2 + 3 . 3^2 + 4 . 4^4 + 5 .

TWO SPHERICAL CONDUCTORS. 211

+6.1/4- 10. 1/4- 15. 1/

4. (17)

.&»44- A a?4/ [4 . 1 4- 10 . 1/ + 20 . 1/ . 2 4-10.3/ .3. (18)

fl/[5.2. (19)

By substituting- in the second members of these equations the approximate values of Al &c., and repeating the process for further approximations, we may cany the approximation to the coefficient to any extent in ascending powers and products of x and y. If we write _ -

we find

[2 + 3/+

• 30/+ 75/+154/ + 280/ + &C.

• 90/ + 288/ + 735 [32 + 200/ + 780/ + &c.

[144 + &C.

(20)

25/+ 36/-f 49/0 + 64y12+ &c. -f «7/ [6 +18^+ 40/+ 75/4-126/4- 196/° + &c. -f«9/[8 4-30/+ 80/4-175/ + 336/4-&C. 4-«n/[l04-45/4-140/4-350/4-&c. 4-^13/[l24-63/-f 224/4-&C. 4-«15/[144-84/4-&c.

V 2

212 SPHERICAL HARMONICS. [146.

• afiy*[ 16 + 72/ + 209/+488/ + &C.
• #10/[ 60+ 342/+1222/ + &C.
• a?12/ [150+1050/ + &C.

64 + &C. (21)

It will be more convenient in subsequent operations to write these coefficients in terms of a, b, and c, and to arrange the terms according to their dimensions in c. This will make it easier to differentiate with respect to c. We thus find

19. (22) 
    

(23)

(24)

-f 40«6£7k-13

(25)

146.] TWO SPHERICAL CONDUCTORS. 213

(26)

-f 5250909 + 336a7£n)c-18. (27)

14 (28)

:i7. (29) (30)

(31) (32)

15. (33)

(34) (35) (36) (37)

The values of the r's and ^'s may be written down by exchanging1 a and b in the ^'s and jo's respectively.

If we now calculate the potentials of the two spheres in terms of these coefficients in the form

(38) (39)

then £, m, n are the coefficients of potential (Art. 87), and of these m = cl +jt?i acz +p2a? c~* + &c., (40)

n = b-i-frac-t-qzatc-*— &c., (41)

214 SPHERICAL HARMONICS.

or, expanding in terms of a, d, c,

(42)

c-22. (43)

The value of I can be obtained from that of n by exchanging a and b.

The potential energy of the system is, by Art. 87,

jr=:%lA2+mAJ]+%nE2, (44)

and the repulsion between the two spheres is, by Art. 93 a, dW I0 dl

The surface density at any point of either sphere is given by equations (1) and (4) in terms of the coefficients An and J3n.

CHAPTEK X.

CONFOCAL QUADKIC SURFACES*.

147.] LET the general equation of a confocal system be

#2 f z2

A23^2 + Xtl^ + X*^2 - *i I1)

where A is a variable parameter, which we shall distinguish by a suffix for the species of quadric, viz. we shall take A1 for the hyper- boloids of two sheets, A2 for the hyperboloids of one sheet, and A3 for the ellipsoids. The quantities

a, A15 b, A2, c, A3

are in ascending order of magnitude. The quantity a is introduced for the sake of symmetry, but in our results we shall always suppose a = 0.

If we consider the three surfaces whose parameters are A15 A2, A3, we find, by elimination between their equations, that the value of tf2 at their point of intersection satisfies the equation

The values of y2 and z2 may be found by transposing a, b, c symmetrically.

Differentiating this equation with respect to A1? we find

dx Aj . .

d^-l^^X'

If ds1 is the length of the intercept of the curve of intersection of A2 and A3 cut off between the surfaces Ax and A1 + ^A1, then

dx

-f

dy

&Z /vl V"2 ~"1 ) V<V3 ~"1 /

^A ~ /\ 2 ~2\ /x 2 7,2\ /\ 2 ^2* I*/

• This investigation is chiefly borrowed from a very interesting work, — Lemons sur les Fonctions Inverses des Transcendantes et les Surfaces Jsothermes. Par G. Paris, 1857.

216 CONFOCAL QUADRIC SURFACES. [148.

The denominator of this fraction is the product of the squares of the semi-axes of the surface \t.

If we put

^2 = A32-A22, J022 = A32-V, and £<* = A,2-^2, (5) and if we make a = 0, then

It is easy to see that D2 and D3 are the semi-axes of the central section of Aj which is conjugate to the diameter passing through the given point, and that D2 is parallel to ds^ and D3 to ds3. 6 1 If we also substitute for the three parameters A1} A2, A3 their values in terms of three functions a, j3, y, defined by the equations

ct =

(7)

then da1 = -D2D^da, ds^ = -D3D1d^, dss = -D-^L^dy. (8)

c c c

148.] Now let F be the potential at any point a, /3, y, then the resultant force in the direction of ds^ is

dV_ d7cla dV c l~ " d§l '' dadSl= da -Z>2#j'

Since dsl3 ds2, and dsz are at right angles to each other, the surface-integral over the element of area ds2 ds3 is

Now consider the element of volume intercepted between the surfaces a, /3, y, and a + ^a, fi + dft, y + dy. There will be eight such elements, one in each octant of space.

We have found the surface-integral of the normal component of the force (measured inwards) for the element of surface intercepted from the surface a by the surfaces j3 and /3 -f d{3, y and

I49-] TRANSFORMATION OP POISSON's EQUATION. 217

The surface-integral for the corresponding element of the surface a + da will be

since D^ is independent of a. The surface-integral for the two opposite faces of the element of volume will be the sum of these quantities, or

Similarly the surface-integrals for the other two pairs of faces will be

d27D2 d2FD2

-T-Z — =- da dp dy and -j-^ — *- da d(B dy.

dp2 c dy* c

These six faces enclose an element whose volume is

7. 2 7) 2 7) 2 , , JL/-1 -LJn -L/n i-it

dst ds2 ds3 = — 2 — | — - da dp dy,

and if p is the volume-density within that element, we find by Art. 77 that the total surface-integral of the element, together with the quantity of electricity within it, multiplied by 4 TT is zero, or, dividing by da dp dy,

which is the form of Poisson's extension of Laplace's equation re- ferred to ellipsoidal coordinates.

If p = 0 the fourth term vanishes, and the equation is equivalent to that of Laplace.

For the general discussion of this equation the reader is referred to the work of Lame already mentioned.

149.] To determine the quantities a, p, y, we may put them in the form of ordinary elliptic integrals by introducing the auxiliary angles 6, </>, and ^, where

A1 = ^sin^, (12)

A2= v/<?2sin20 + 62cos24>, (13)

A3 = csec\lf, (14)

If we put b = kc, and k2 + V2 = 1, we may call Jc and k' the two complementary moduli of the confocal system, and we find

a =

218 CONFOCAL QUADRIC SURFACES. [l5O.

an elliptic integral of the first kind, which we may write according to the usual notation F(k,6}. In the same way we find

• (ie)

where F(k') is the complete function for modulus k',

, .,

1 _ ft* COS2 {f

Here a is represented as a function of the angle 0, which is ac- cordingly a function of the parameter \lt ft as a function of <£ and thence of A2, and y as a function of \j/ and thence of A3.

But these angles and parameters may be considered as functions of a, /3, y. The properties of such inverse functions, and of those connected with them, are explained in the treatise of M. Lame on that subject.

It is easy to see that since the parameters are periodic functions of the auxiliary angles, they will be periodic functions of the quantities a, /3, y : the periods of Ax and A3 are ±F(k), and that of A2

Particular Solutions.

150.] If Y is a linear function of a, /3, or y, the equation is satisfied. Hence we may deduce from the equation the distribution of electricity on any two confocal surfaces of the same family maintained at given potentials, and the potential at any point between them.

The Hyperboloids of Two Sheets.

When a is constant the corresponding surface is a hyperboloid of two sheets. Let us make the sign of a the same as that of SB in the sheet under consideration. We shall thus be able to study one of these sheets at a time.

Let al9 a2 be the values of a corresponding to two single sheets, whether of different hyperboloids or of the same one, and let 7[, 7£ be the potentials at which they are maintained. Then, if we make

the conditions will be satisfied at the two surfaces and throughout the space between them. If we make V constant and equal to ?i in the space beyond the surface a1} and constant and equal to T^

15°-] DISTRIBUTION OF ELECTRICITY. 219

in the space beyond the surface a2, we shall have obtained the complete solution of this particular case. The resultant force at any point of either sheet is

-n cW dV da

or

-i "?2^A- (20)

If PI be the perpendicular from the centre on the tangent plane at any point, and Pl the product of the semi-axes of the surface, then

Hence we find ^ = ^-^ Wi y (2l)

al~~a2 -^1

or the force at any point of the surface is proportional to the per- pendicular from the centre on the tangent plane.

The surface-density <r may be found from the equation

477(7 = 7?!. (22)

The total quantity of electricity on a segment cut off by a plane

whose equation is x — a from one sheet of the hyperboloid is

J *

= c

The quantity on the whole infinite sheet is therefore infinite. The limiting forms of the surface are :— *v<fyXI' =-^\

(l) When a = F(k] the surface is the part of the plane of x'z on^'" the positive side of the positive branch of the hyperbola whose equation is #2 ^2 <*•< j£T4r 4? '-'' *'"

(2) When a = 0 the surface is the plane of yz.

(3) When a = —F(k) the surface is the part of the plane of xz on the negative side of the negative branch of the same hyperbola.

The Hyperboloid of One Sheet.

By making ft constant we obtain the equation of the hyperboloid of one sheet. The two surfaces which form the boundaries of the electric field must therefore belong to two different hyperboloids. The investigation will in other respects be the same as for the hyperboloids of two sheets, and when the difference of potentials is given the density at any point of the surface will be proportional to the perpendicular from the centre on the tangent plane, and the whole quantity on the infinite sheet will be infinite.

220 CONFOCAL QUADRIC SURFACES. [150.

Limiting Forms.

(1) When ft = 0 the surface is the part of the plane of xz between the two branches of the hyperbola whose equation is written above, (24).

(2) When (3=F(k') the surface is the part of the plane of xy which is on the outside of the focal ellipse whose equation is

(25)

The Ellipsoids.

For any given ellipsoid y is constant. If two ellipsoids, yx and y2, be maintained at potentials V^ and 7£, then, for any point y in the space between them, we have

The surface-density at any point is

'=- ~Mf> (27)

4*yi-y2 P3

where p3 is the perpendicular from the centre on the tangent plane, and P3 is the product of the semi-axes.

The whole charge of electricity on either surface is given by

and is finite.

When y — F(k) the surface of the ellipsoid is at an infinite distance in all directions.

If we make P2 = 0 and y2 = F(Jc}> we find for the quantity of electricity on an ellipsoid maintained at potential V in an infinitely extended field, y

The limiting form of the ellipsoids occurs when y = 0, in which case the surface is the part of the plane of xy within the focal ellipse, whose equation is written above, (25).

The surface-density on either side of the elliptic plate whose equation is (25), and whose eccentricity is Jc, is

= 47rx/^T2/pj" /ri^?r

V ~?""^r^

y

and its charge is Q = c ™TX • (31)

I 5 !•] SURFACES OF REVOLUTION. 221

Particular Cases.

151.] If c remains finite, while 6 and therefore k is diminished till it becomes ultimately zero, the system of surfaces becomes transformed in the following manner : —

The real axis and one of the imaginary axes of each of the hyperboloids of two sheets are indefinitely diminished, and the surface ultimately coincides with two planes intersecting in the axis of z.

The quantity a becomes identical with 0, and the equation of the system of meridional planes to which the first system is reduced is

(\9 "" / sin a)2 (cos a

As regards the quantity /3, if we take the definition given in page 216 (7) we shall be led to an infinite value of the integral at the lower limit. In order to avoid this we define /3 in this particular case as the value of the integral

If we now put A2 = c sin <£, /3 becomes

i.e. log cot |0.

e?-e-t

£

Whence cos d> =

and therefore sin <£ =

If we call the exponential quantity i (eP -f- e~?) the hyperbolic cosine of /3, or more concisely the hypocosine of fi, or cosh £, and if we call \ (e&—e~P) the hyposine of /3, or sinh /3, and if in the same way we employ functions of a similar character analogous to the other simple trigonometrical ratios, then A2 = c sech /3, and the equation of the system of hyperboloids of one sheet is

(sech/3)2 (tanh/3)2 The quantity y is reduced to /r, so that A3 = c cosec y, and the equation of the system of ellipsoids is

Ellipsoids of this kind, which are figures of revolution about their conjugate axes, are called planetary ellipsoids.

222 CONFOCAL QTJADRIC SURFACES. [152.

The quantity of electricity on a planetary ellipsoid maintained at potential V in an infinite field, is

where c sec y is the equatorial radius, and c tan y is the polar radius. If y = 0, the figure is a circular disk of radius c, and

<r=- (38)

(39) 152.] Second Case. Let I = c, then Jc = 1 and lc' = 0,

o = log tan — - — , whence Aj = c tanha, (40)

and the equation of the hyperboloids of revolution of two sheets becomes •

(tanha)2 (secha)2 The quantity (3 becomes reduced to $, and each of the hyper- boloids of one sheet is reduced to a pair of planes intersecting in the axis of x whose equation is

(sin/3)2 (cos/3)2 " This is a system of meridional planes in which (3 is the longitude.

The quantity y as defined in page 216, (?) becomes in this case infinite at the lower limit. To avoid this let us define it as the

i PA.I. • A i value of the integral

A3 ^3 ~~

If we then put A3 = c sec |r, we find y — I — — r , whence

J^, sin y

A3 = c coth y, and the equation of the family of ellipsoids is

(cothy)2 " (cosechy)2 "

These ellipsoids, in which the transverse axis is the axis of revo- lution, are called ovary ellipsoids.

The quantity of electricity on an ovary ellipsoid maintained at potential Fin an infinite field becomes in this case, by (29),

(44)

sin where c sec |/0 is the polar radius.

If we denote the polar radius by A and the equatorial by ^, the result just found becomes

1 54-] CYLINDERS AND PARABOLOIDS. 223

(45)

' — -D~

~B

If the equatorial radius is very small compared to the polar radius, as in a wire with rounded ends,

AV

~ log 2^ — log^

When both b and c become zero, their ratio remaining finite, the system of surfaces becomes two systems of confocal cones, and a system of spherical surfaces of which the radius is inversely proportional to y.

If the ratio of b to c is zero or unity, the system of surfaces becomes one system of meridian planes, one system of right cones having a common axis, and a system of concentric spherical surfaces of which the radius is inversely proportional to y. This is the ordinary system of spherical polar coordinates.

Cylindric Surfaces.

153.] When c is infinite the surfaces are eylindric, the generating lines being parallel to the axes of z. One system of cylinders is hyperbolic, viz. that into which the hyperboloids of two sheets degenerate. Since, when c is infinite, k is zero, and therefore 6 = a, it follows that the equation of this system is

/g2 4/2 " __ ^2 (47)

sin2 a cos2 a The other system is elliptic, and since when k = 0, /3 becomes

2 -, or A2 = b cosh/3,

A/V^J*'

the equation of this system is

(cosh/3)2 (sinh/3)2 These two systems are represented in Fig. X at the end of this

volume.

Confocal Paraboloids.