A Treatise on Electricity and Magnetism/Part I/Chapter X

[ page ]146.] NEARLY SPHERICAL CONDUCTOR. 179

where F is a function of the direction of r, and is a numerical quantity the square of which may be neglected.

Let the potential due to the external electrified system be ex pressed, as before, in a series of solid harmonics of positive degree, and let the potential U be a series of solid harmonics of negative degree. Then the potential at the surface of the conductor is obtained by substituting the value of r from equation (74) in these series.

Hence, if C is the value of the potential of the conductor and .Z? the charge upon it, C= 4,

��..-(j+l)B,arWFYr (75)

Since F is very small compared with unity, we have first a set of equations of the form (72), with the additional equation

= - Q -F + 3A 1 aFY 1 + 8tc. + (i+l)A i a i FY i

+ 2(.# / 0-U +1 >7,)-2 ((j+VSja-U+VFYj). (76)

To solve this equation we must expand F, FY 1 . . . FY i in terms of spherical harmonics. If F can be expanded in terms of spherical harmonics of degrees lower than Jc } then FY i can be expanded in spherical harmonics of degrees lower than i + k.

Let therefore

B Q - F- 3A 1 aFY 1 - ...-(2i+l)A i W<= 2 (Bj a-U+DJ}), (77)

d

then the coefficients Bj will each of them be small compared with the coefficients B Q ... B i on account of the smallness of F, and therefore the last term of equation (76), consisting of terms in BjF, may be neglected.

Hence the coefficients of the form Bj may be found by expanding equation (76) in spherical harmonics.

For example, let the body have a charge _Z? , and be acted on by no external force.

Let F be expanded in a series of the form

F = S 1 Y l + &c. + S t Y lk . (78)

Then S l Y l + &c. + 8 1g Y t = 2(S J a-V+VY j ), (79)

N 2

�� � [ page ]180 SPHEEICAL HAKMONICS.

or the potential at any point outside the body is

��(80)

��and if o- is the surface-density at any point

dU

4-770- = -- >

dr

��or 47700- = (l+fl 2 r a +...+ (-1)^7,). (81)

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 degree /, the ratio of the difference of the superficial densities at any two points to their sum will be k I times the ratio of the difference of the radii of the same two points to their sum.

�� � [ page ]CHAPTER X.

��CONTOCAL QUADRIC SURFACES*.

��147.] Let the general equation of a confocal system be

��~ 2

��where X is a variable parameter, which we shall distinguish by the suffix A 1 for the hyperboloids of two sheets, A, 2 for the hyperboloids of one sheet, and A 3 for the ellipsoids. The quantities

0, A 15 b, \. 2 , c, A 3

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 A 15 A 2 , A 3 , we find, by elimination between their equations, that the value of x 2 at their point of intersection satisfies the equation

X*(6 Z -a*)(C*-a*) = (A 1 2 - 2 )(A 2 2 - 2 )(A 3 2 - 2 ). (2)

The values of f and z 2 may be found by transposing a, b, c symmetrically.

Differentiating this equation with respect to \ ly we find

dx Aj / 3 x

~~T = r~9 - 9 * ^

d\ Aj 2 a 2

If ds^ is the length of the intercept of the curve of intersection of A 2 and A 3 cut off between the surfaces A x and Aj + ^A^ then

-^J 2 jb 2 ^ ~di\* ^ ~di\ 2 __ A 1 2 (A 2 2 -A 1 2 )(A 3 2 -A 1 2 )

3^1 = ^XT h ^| h ^r! B w-^w-^xv-^)-

• This investigation is chiefly borrowed from a very interesting work, Lemons sur

les Fonctions Inverses des Transcendantes et les Surfaces Isotherme*. Par G. Paris, 1857.

�� � [ page ]182 CONFOCAL QUADRIC SURFACES.

The denominator of this fraction is the product of the squares of the semi-axes of the surface A x . If we put

7)2 _ \ 2 A 2 7)2 _ A 2 \ 2 <) n A 7) 2 _ \ 2 _ \ 2 /K\

• • 1 A 3 A 2 ^2 A 3 A l J anQ -^3 A 2 A l > \P)

and if we make a = 0, then

d _ D 2 D 3 (

��It is easy to see that Z^ 2 and D 3 are the semi-axes of the central section of A x which is conjugate to the diameter passing 1 through the given point, and that D 2 is parallel to ds 2 , and D 3 to ds 3 .

If we also substitute for the three parameters \ lt A 2 , A 3 their values in terms of three functions a, (3, y, denned by the equations

da c .

-j = , . > A., = when a = 0,

��/ 2 /2 /^f =1 > A 2 = * When = > ( 7 )

VA 2 2 b 2 Vc 2 A 2 2

/?

A 3 = c when y = ;

��then ^ = -D 2 D 3 da, ds 2 = -D 3 D 1 dp, ds 3 -D^D^ dy. (8)

C

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

��__ _ L _ dV c 1- ds[" Jad Sl ~ "Jal)^!^

Since ^, ds 2 , and ^ 3 are at right angles to each other, the surface-integral over the element of area ds 2 ds 3 is

- dV c DD D,D

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

We have found the surface-integral for the element of surface intercepted from the surface a by the surfaces (3 and p + dfi, y and

�� � [ page ]I49-] TRANSFORMATION OF POISSON s EQUATION. 183

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

��da c

since D^ is independent of a. The surface-integral for the two opposite faces of the element of volume, taken with respect to the interior of that volume, will be the difference of these quantities, or

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

��. and

c dy 2 c

These six faces enclose an element whose volume is

727 2 7 2

��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 dadfidy,

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

If p = 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, 0, y, we may put them in the form of ordinary elliptic functions by introducing the auxiliary angles 0, $, and \//-, where

A x = sin0,

A 2 = V c 2 sin 2 $ + b 2 cos^), (13)

��sm\//

If we put 5 = h, and F + /2 - 1, we may call k and It the two complementary moduli of the confocal system, and we find

�� � [ page ]184 CONFOCAL QUADKIC SURFACES. [l 50.

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

��13 =

��, 2^

1 / 2 cos 2 </> where FJc is the complete function for modulus k ,

��y * 7 o * o "

V 1 k* sm 2 \lr

Here a is represented as a function of the angle 0, which is a function of the parameter A 15 /3 as a function of </> and thence of A 2 , and y as a function of \j/ and thence of A 3 .

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 Aj and A 3 are 4 F(k) and that of A 2 is 2 F(Jc ).

Particular Solutions.

150.] If V 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 x in the sheet under consideration. We shall thus be able to study one of these sheets at a time.

Let a x , a 2 be the values of a corresponding to two single sheets, whether of different hyperboloids or of the same one, and let F 15 F 2 be the potentials at which they are maintained. Then, if we make

-a 2 r i + a(7 7 i Tg) fio\ , (18)

��the conditions will be satisfied at the two surfaces and throughout the space between them. If we make V constant and. equal to V in the space beyond the surface a l5 and constant and equal to F 2

�� � [ page ]150.] DISTRIBUTION OF ELECTRICITY. 185

in the space beyond the surface a 2 , we shall have obtained the complete solution of this particular case.

The resultant force at any point of either sheet is R _ _dF_ _dFda ds l ~ da ds

or ^ = r i~ r 2 C . (20)

If pi be the perpendicular from the centre on the tangent plane at any point, and P l the product of the semi-axes of the surface, then p l D 2 D. 3 = P 1 .

Hence we find ^1^2 C P\ ^oi\

1 = a ~p~

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 a- may be found from the equation

4-770- = ^. (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

--iV (23)

��2 a l -a 2

The quantity on the whole infinite sheet is therefore infinite. The limiting forms of the surface are :

(1) When a = F^ the surface is the part of the plane of xz on the positive side of the positive branch of the hyperbola whose equation is #2 z z

To o := 1 \ /

b 2 c 2

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

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

The Hyperloloids of One Sheet.

By making /3 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.

�� � [ page ]186 CONFOCAL QUADRIC SURFACES.

Limiting Forms.

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

��The Ellipsoids.

For any given ellipsoid y is constant. If two ellipsoids, y : and y 2 , be maintained at potentials V^ and V^ then, for any point y in the space between them, we have

^) (26)

��71 72 The surface-density at any point is

��where p 3 is the perpendicular from the centre on the tangent plane, and P 3 is the product of the semi-axes.

The whole charge of electricity on either surface is

��a finite quantity.

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

If we make V 2 = and y 2 = F(k), we find for the quantity of electricity on an ellipsoid maintained at potential V in an infinitely extended field, V , .

^ c WTr\ v /

F(k)*-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 the elliptic plate whose equation is (25), and whose eccentricity is $, is

o- = - V __ X , (30)

��/

V

��and its charge is _ V

�� �� � [ page ]151.] SURFACES OF REVOLUTION. 187

Particular Cases.

151.] If 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 6, and the equation of the system of meridional planes to which the first system is reduced is

1-2 ,,,2

_? ?. = o. (32)

(sin a) 2 (cos a) 2

The quantity /3 is reduced to

^=/ s - = lo ^ tan t (33)

��whence we find

2 smd) = -5 - 5> cosc> =

���e* -f e~

If we call the exponential quantity \(e^ + er^) the hyperbolic cosine of /3, or more concisely the hypocosine of /3, or cos h ft, and if we call i (e^ e-P} the hyposine of ft, or sin^ ft, and if by the same analogy we call

the hyposecant of ft, or sec h ft,

��cos h ft

1

sin^/3 sin hf$ cost ft

��the hypocosecant of ft, or cosec Ji ft, the hypotangent of ft, or tan h ft,

��and COS 1 P the hypoeotang-ent of ft, or cot Ji ft ; sm/fc/3

then A 2 = c sec h ft, and the equation of the system of hyperboloids of one sheet is

= C 2 (35)

"

��(seek ft) 2

The quantity y is reduced to \ff, so that A 3 = c cosec y, and the equation of the system of ellipsoids is

O O 9

• + y + * = C 2. (36)

(secy) 2 (tany) 2

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

�� � [ page ]188 CONFOCAL QUADRIC SURFACES.

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

gastf-JL-, (37)

��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

V a = -- . (38)

W *,v/e 2 -r a Q = c~ (39)

2

152.] to0^ Ciw*. Let b c, then = 1 and = 0,

_ 9$ a = log tan - , whence A x = c tan ^ a, (40)

and the equation of the hyperboloids of revolution of two sheets becomes #2 ^2

��(sec/*a) 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 becomes log tan -- 5 whence A 3 = c cot k y, and the equation of the family of ellipsoids is

>2 .,2 I 2

��(cosec/5y) 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 a potential V in an infinite field is

Q = c (44)

If the polar radius is A = c cot h y, and the equatorial radius is B = c cosec Ji y,

��- /AK .

= log -- --- (45)

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