A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 16 of 28

each other orthogonally in D and .Z/, and let the straight line DI/ cut the line of centres in C. Then C is the image of A with respect to the sphere B, and also the image of B with respect to the sphere whose centre is A. If AD = a, BD = ft then AB= x/a2 + /32, and if we place at A, B, C quantities

of electricity equal to a, ft, and f respectively, then both

Fig. 12.

spheres will be equipotential surfaces whose potential is unity.

We may therefore determine from this system the distribution of electricity in the following cases :

1 68.] TWO SPHERES CUTTING ORTHOGONALLY. 243

(1) On the conductor PDQI? formed of the larger segments of both spheres. Its potential is 1, and its charge is

a + /3 -- -~ - =///)+ 7? 7) _/?/}

Va2+/32

This quantity therefore measures the capacity of such a figure when free from the inductive action of other bodies.

The density at any point P of the sphere whose centre is A, and the density at any point Q of the sphere whose centre is .B, are respectively

ikC'-t&f) and

At the points of intersection, D, D' , the density is zero.

If one of the spheres is very much larger than the other, the density at the vertex of the smaller sphere is ultimately three times that at the vertex of the larger sphere.

(2) The lens P/DQfJ/ formed by the two smaller segments of

the spheres, charged with a quantity of electricity =

and acted on by points A and B, charged with quantities a and /3, is also at potential unity, and the density at any point is expressed by the same formulae.

(3) The meniscus DPD'Q' formed by the difference of the segments charged with a quantity a, and acted on by points B

and C, charged respectively with quantities /3 and - , is also

in equilibrium at potential unity.

(4) The other meniscus QDP/J/ under the action of A and C.

We may also deduce the distribution of electricity on the following internal surfaces.

The hollow lens P'DQ'D under the influence of the internal electrified point C at the centre of the circle DJ/.

The hollow meniscus under the influence of a point at the centre of the concave surface.

The hollow formed of the two larger segments of both spheres under the influence of the three points A, B, C.

But, instead of working out the solutions of these cases, we shall apply the principle of electrical images to determine the density of the electricity induced at the point P of the external surface of the conductor PDQD' by the action of a point at 0 charged with unit of electricity.

R 2

244 ELECTRIC IMAGES. [l68.

Let OA = «, OB = b, OP = r, BP = p,

AD = a, BD = ft ^5 = x/a2+/32.

Invert the system with respect to a sphere of radius unity and centre 0.

The two spheres will remain spheres, cutting each other ortho- gonally, and having their centres in the same radii with A and B. If we indicate by accented letters the quantities corresponding to the inverted system,

-_

If, in the inverted system, the potential of the surface is unity, then the density at the point P7 is

If, in the original system, the density at P is o-, then

and the potential is -. By placing at 0 a negative charge of

electricity equal to unity, the potential will become zero over the surface, and the density at P will be

1 g2-a2x __ /33

" 3 V ' ** *

47T a

This gives the distribution of electricity on one of the spherical surfaces due to a charge placed at 0. The distribution on the other spherical surface may be found by exchanging a and b, a and ft and putting q or AQ instead of p.

To find the total charge induced on the conductor by the elec- trified point at 0, let us examine the inverted system.

In the inverted system we have a charge a at A', and /3' at .Z?',

and a negative charge — f at a point C' in the line A'B',

such that A'C' : C'B' : : a2 : /3'2.

If OA'= a', OB'= b', OC' = c't we find

c2 =

169.] FOUR SPHERES CUTTING ORTHOGONALLY.

Inverting this system the charges become

245

a _a

a a

and

a'/3'

7

a/3

Hence the whole charge on the conductor due to a unit of negative electricity at 0 is

a/3

£.£_

Distribution of Electricity on Three Spherical Surfaces which Intersect at Eight Angles.

169.] Let the radii of the spheres be a, /3, y, then

BC = x/jS^T?, C^ = Vy* + tf, AB =

Let PQR, Fig. 1 3, be the feet of the perpendiculars from ABC on the opposite sides of the tri- angle, and let 0 be the inter- section of perpendiculars.

Then P is the image of B in the sphere y, and also the image of C in the sphere /3. Also 0 is the image of P in the sphere a.

Let charges a, )3, and y be placed at A, B, and C.

Then the charge to be placed at Pis

Fig. 13.

V W + 7

Also ^P =

sidered as the image of P, is

so that the charge at 0, con-

r /

In the same way we may find the system of images which are

246 ELECTRIC IMAGES. [170.

electrically equivalent to four spherical surfaces at potential unity intersecting at right angles.

If the radius of the fourth sphere is b, and if we make the charge at the centre of this sphere = 8, then the charge at the intersection of the line of centres of any two spheres, say a and ft, with their

plane of intersection, is

1

1

The charge at the intersection of the plane of any three centres ABC with the perpendicular from D is

1

/T~ T ~T"

V 1? + /32 H" >2

and the charge at the intersection of the four perpendiculars is

1

VI 1 1 1 T + ^2 + -2 + ^2 a2 /32 v2 52

/3

System of Four Spheres Intersecting at Right Angles under the Action of an Electrified Point.

170.] Let the four spheres be A, B, C, D, and let the electrified point be 0. Draw four spheres A19 JS19 Clt J)19 of which any one, A19 passes through 0 and cuts three of the spheres, in this case B, C, and D, at right angles. Draw six spheres (ab), (ac), (ad), (be), (bd), (cd), of which each passes through 0 and through the circle of intersection of two of the original spheres.

The three spheres B1, C19 D1 will intersect in another point besides 0. Let this point be called J?9 and let B', C', and I/ be the intersections of Cit Dl9 A±, of Dif A19 BL, and of A19 JS19 C^ re- spectively. Any two of these spheres, A19 JBlt will intersect one of the six (cd) in a point (</£'). There will be six such points.

Any one of the spheres, A19 will intersect three of the six (ab), (ac), (ad) in a point a' . There will be four such points. Finally, the six spheres (ab], (ac), (ad), (cd), (db), (be), will intersect in one point S.

If we now invert the system with respect to a sphere of radius R and centre 0, the four spheres A, B, C, D will be inverted into spheres, and the other ten spheres will become planes. Of the points of intersection the first four A', B', C', D' will become the

1 7 1-] TWO SPHERES NOT INTERSECTING. 247

centres of the spheres, and the others will correspond to the other eleven points in the preceding article. These fifteen points form the image of 0 in the system of four spheres.

At the point A' , which is the image of 0 in the sphere A, we

must place a charge equal to the image of 0, that is, — - where a

a

is the radius of the sphere A, and a is the distance of its centre from 0. In the same way we must place the proper charges at JP, C\ If.

The charges for each of the other eleven points may be found from the expressions in the last article by substituting a', £', /, 6' for a, /3, y, b, and multiplying the result for each point by the distance of the point from 0, where

[The cases discussed in Arts. 169, 170 may be dealt with as follows : Taking three coordinate planes at right angles, let us

place at the system of eight points ( ± - — > + — - - , + - — ) charges

^ 2 a 2 p 2 y'

• e, the minus charges being at the points which have 1 or 3 negative coordinates. Then it is obvious the coordinate planes are at potential zero. Now let us invert with regard to any point and we have the case of three spheres cutting orthogonally under the influence of an electrified point. If we invert with regard to one of the electrified points, we find the solution for the case of a con- ductor in the form of three spheres of radii a, /3, y cutting ortho- gonally and freely charged.

If to the above system of electrified points we superadd their images in a sphere with its centre at the origin we see that, in addition to the three coordinate planes, the surface of the sphere forms also a part of the surface of zero potential.]

Two Spheres not Intersecting.

171.] When a space is bounded by two spherical surfaces which do not intersect, the successive images of an influencing point within this space form two infinite series, all of which lie beyond the spherical surfaces, and therefore fulfil the condition of the applicability of the method of electrical images.

Any two non-intersecting spheres may be inverted into two concentric spheres by assuming as the point of inversion either of the two common inverse points of the pair of spheres.

'248

ELECTKIC IMAGES.

We shall begin, therefore, with the case of two uninsulated concentric spherical surfaces, subject to the induction of an elec- trified point placed between them.

Let the radius of the first be b, and that of the second be^, and let the distance of the influencing point from the centre be r = be".

Then all the successive images will be on the same radius as the influencing point.

Let Q0, Fig. 14, be the image of P in the first sphere, P3 that of §o in the second sphere, Q1 that of Pl in the first sphere, and so on ; then

and OPs.Oq8.l also OQQ = be'*,

\ &c. Hence OP8 = be(u+2*™\

OQ8 = be-(u+28^,

If the charge of P is denoted by P, then

Fig. 14.

Next, let Qi be the image of P in the second sphere, P/ that of

/ in the first. &c.,

OP=

OQ;=

Of these images all the P's are positive, and all the Q's negative, all the P"s and Q's belong to the first sphere, and all the P's and Q"s to the second.

The images within the first sphere form a converging series, the sum of which is

-P

This therefore is the quantity of electricity on the first or interior sphere. The images outside the second sphere form a diverging series, but the surface-integral of each with respect to the spherical surface is zero. The charge of electricity on the exterior spherical surface is therefore

-iW— P

172.]

TWO SPHERES NOT INTERSECTING.

249

If we substitute for these expressions their values in terms of OA, OB, and OP, we find

^OA PS charge on ^ = -P__,

-T.OB AP

charge on £=-P — _ .

If we suppose the radii of the spheres to become infinite, the case becomes that of a point placed between two parallel planes A and B. In this case these expressions become

charge on A = — P

AP charge on B= — P-r^

172.] In order to pass from this case to that of any two spheres not intersecting each other, we begin by finding the two com- mon inverse points 0} (7 through which all circles pass that are orthogonal to both spheres. Then, if we invert the system with respect to either of these points, the spheres become concentric, as in the first case.

Fig 15

If we take the point 0 in Fig. 1 5 as centre of inversion, this point will be situated in Fig. 14 somewhere between the two spherical surfaces.

Now in Art. 1 7 1 we solved the case where an electrified point is placed between two concentric conductors at zero potential. By inversion of that case with regard to the point 0 we shall therefore deduce the distributions on two spherical conductors at potential zero, exterior to one another, induced by an electrified point in their neighbourhood. In Art. 173 it will be shewn how the results thus obtained may be employed in finding the distributions on two spherical charged conductors subject to their mutual influence only.

The radius OAPB in Fig. 1 4 on which the successive images lie becomes in Fig. 15 an arc of a circle through 0 and (7, and the ratio of O'P to OP is equal to Ceu where C is a numerical quantity.

250 ELECTEIC IMAGES. [172

, a A

Ifweput 0= = log^p> a==1°£oZ' ^= g~O then /3 — a = or, % + a = 0.

All the successive images of P will lie on the arc OAPBO'.

The position of the image of P in A is (J)0 where

That of Q0 in J? is Pl where

Similarly

6(Pt) = 0

In the same way if the successive images of P in B, A, £, &c. are Q0', P/, &', &c.,

6(Ps) = 0— 2st3-, 0(Q/) = 2)8— 0-f2st3-.

To find the charge of any image P8 we observe that in the inverted figure its charge is

P/V OF'

In the original figure we must multiply this by OPS. Hence the charge of P8 in the dipolar figure is

/OP..Orp.

^V OP.&P

If we make f = VOP.C/P, and call ^ the parameter of the point P, then we may write

P — At p

s — £ '

or the charge of any image is proportional to its parameter.

If we make use of the curvilinear coordinates 6 and (/>, such that

x + A/ — \y--k where 2k is the distance 00', then

Jc sinh 0 Jc sin $

cosh 0 — cos </> ' ~ cosh 6— cos 0 '

(x+k coth 0)2 +/ = k* cosech2 0,

73'] TWO SPHERES NOT INTERSECTING. 251

_

vcoshtf— • cos</>

Since the charge of each image is proportional to its parameter, £, and is to be taken positively or negatively according as it is of the form P or Q, we find

P - P Vcoshfl-

-t a ~~ ' • '

V cosh (0+ 25OT-) — cos Pvcoshtf — cost/)

V cosh (2 a — 0 — 2 sv?)— cos<£ p, P /cosh 0 — cos $

/cosh (0 — 2 <y t*r) — cos 0

/)/__ P / cosh 0— -cos <£.

v cosh (2 (3 — 0 + 2 SOT) — cos </>

We have now obtained the positions and charges of the two infinite series of images. We have next to determine the total charge on the sphere A by finding the sum of all the images within it which are of the form Q or P'. We may write this

P v cosh 6 — cos $ ^s=i

i //> Vcosh(0—

— Pvcoshfl —

,-o / . / ,, Vcosh(2a — 0 —

In the same way the total induced charge on B is

_ •*r+t = <X>

Pvcoshtf —

^* :1 /cosh (0 + 2 six] — cos </>

_ ^-#=00 1

— P vcosh 6 — cos(f> ^S=Q / w \

173.] We shall apply these results to the determination of the

• In these expressions we must remember that

2cosh0 = ee+ ee, 2sinh0 = ee-e6,

and the other functions of 9 are derived from these by the same definitions as the corresponding trigonometrical functions.

The method of applying dipolar coordinates to this case was given by Thomson in Liouville's Journal for 1847. See Thomson's reprint of Electrical Papers, § 211, 212. In the text I have made use of the investigation of Prof. Betti, Nuovo Cimento, vol. xx, for the analytical method, but I have retained the idea of electrical images as used by Thomson in his original investigation, Phil. Mag., 1853.

252 ELECTRIC IMAGES. [173.

coefficients of capacity and induction of two spheres whose radii are a and I, and the distance between whose centres is c.

Let the sphere A be at potential unity, and the sphere £ at potential zero.

Then the successive images of a charge a placed at the centre of the sphere A will be those of the actual distribution of electricity. All the images will lie on the axis between the poles and the centres of the spheres, and it will be observed that of the four systems of images determined in Art. 1 72, only the first and fourth exist in this case.

If we put

k =

2c

k ' k

then sinh a = > sinh 3 — T •

a o

The values of 6 and $ for the centre of the sphere A are 6 = 2 a, <£ = 0.

Hence in the equations we must substitute a or — k -j— = — for P,

sinh a

2 a for 6 and 0 for $, remembering that P itself forms part of the charge of A. We thus find for the coefficient of capacity of A

•#=oo 1

"^#=( = * 2.s=(

h(«?CT— a) for the coefficient of induction of A on B or of £ on A

=00 1

We may, in like manner, by supposing B at potential unity and A at potential zero, determine the value of qbb. We shall find, with our present notation,

To calculate these quantities in terms of a and #, the radii of the spheres, and of c the distance between their centres, we observe that if __

K= Va* + b*+c*—2b2c*-2(?a'2-2a2b2,

we may write

. , K . , £. . . K

smha = --- , smh/3 = —j- . Binhar = — — ,

2ac 2bc ' 2ab

cosh a = - > cosh 8=. - = - > cosh or = - -7 2ca 2cb 2ab

1 74-] TWO ELECTRIFIED SPHERES. 253

and make use of

sinh (a + /3) = sinh a cosh (3 + cosh a sinh /3, cosh (a + (3) == cosh a cosh ft -f sinh a sinh /3.

By this process or by the direct calculation of the successive images as shewn in Sir W. Thomson's paper, we find

=

ab

174.] We have then the following equations to determine the charges Ea and Eb of the two spheres when electrified to potentials Va and Tb respectively,

If we put qaa qbb—qab2 = £ = -

and paa = faff, pab = —faff, pbb = faff,

whence PaaPn—Pab = & ;

then the equations to determine the potentials in terms of the charges are Va = paa fia --pab Eb,

and^?aa, pab, andj% are the coefficients of potential. The total energy of the system is, by Art. 85,

The repulsion between the spheres is therefore, by Arts. 92, 93,

where c is the distance between the centres of the spheres.

Of these two expressions for the repulsion, the first, which expresses it in terms of the potentials of the spheres and the

254: ELECTRIC IMAGES. [174.

variations of the coefficients of capacity and induction, is the most convenient for calculation.

We have therefore to differentiate the ^'s with respect to c. These quantities are expressed as functions of k, a, /3, and <ar, and must be differentiated on the supposition that a and b are constant. From the equations

. , . , sinhasinh/3

k — — a smn a = o smh/3 =— c r-y— — >

smhtn-

dk cosh a cosh /3

we find

dc sinh w

da sinh a cosh

dc k sinh OT d{$ _ cosh a sinh (3 dc k sinh CT

</ar 1

whence we find

dqaa cosh a cosh # £aa ^u=oo (sc.+ b cosh ^) cosh (s vr — a)

sinh -07 ^ -^*-< c(sinh(^OT — a))2

cosh a cosh /3 ^T,

~dc' sinhw T

bb cosh a cosh^S qbb ^^=00 (*c — a cosh a) cosh (/3 + ~dc ' sinh «• T ""

Sir William Thomsom has calculated the force between two spheres of equal radius separated by any distance less than the diameter of one of them. For greater distances it is not necessary to use more than two or three of the successive images.

The series, for the differential coefficients of the ^'s with respect to c are easily obtained by direct differention.

dqaa _ 2 a2 be 2azt>2c(2c2-2b2-a2)

~'' ~ ~

dc " c2 c2(c2-a2-b2)

aH* {(5c2-a2-b2)(c2-a2-b2)-a2d2}

<?(c2-a2-62 + ady(c2-a2-b2-ad)2 d 2ab2c 2 _ I 75.] TWO SPHERES IN CONTACT. 255 Distribution of Electricity on Two Spheres in Contact. 175.] If we suppose the two spheres at potentitt unity and not influenced by any other point, then, if we invert the system with respect to the point of contact, we shall have two parallel planes, distant — and — from the point of inversion, and electrified by the action of a unit of electricity at that point. There will be a series of positive images, each equal to unity, at distances s (- + ~r) from the origin, where s may have any integer value from — cc to +00. There will also be a series of negative images each equal to — 1 , the distances of which from the origin, reckoned in the direction of When this system is inverted back again into the form of the two spheres in contact, we have a corresponding series of negative images, the distances of which from the point of contact are of the form — -— > where s is positive for the sphere A and negative for the sphere B. The charge of each image, when the potential of the spheres is unity, is numerically equal to its distance from the point of contact, and is always negative. There will also be a series of positive images whose distances from the point of contact measured in the direction of the centre of 0, are of the form — • When s is zero, or a positive integer, the image is in the sphere A. When s is a negative integer the image is in the sphere B. The charge of each image is measured by its distance from the origin and is always positive. The total charge of the sphere A is therefore o> 1 ab -=oo J. 256 ELECTRIC IMAGES. [175< Each of these series is infinite, but if we combine them in the form ™ ._ ^*=«> * 4&»1 g(a + 6 the series becomes converging. In the same way we find for the charge of the sphere B, oo cib ab *=-oo 1 The expression for Ea is obviously equal to in which form the result in this case was given by Poisson. It may also be shewn (Legendre Traite des Fonctions Elliptiqnes, ii, 438) that the above series for Ea is equal to b ab a + b' ) a H- b where y = -57712..., and V(x) = ^- logT(l +#). rf^z? The values of # have been tabulated by Gauss (Werke, Band iii, pp. 161-162.) If we denote for an instant b -=- (a + b) by a?, we find for the difference of the charges Ea and Eb, d . ab — -=- log sm TT^ x Tib cot tf + d a + b When the spheres are equal the charge of each for potential unity 18 ru=«> 1 = aloge2 = -693147180. When the sphere A is very small compared with the sphere the charge on A is a2 ^-*=o> 1 ^ = T 2*=i ~2~ approximately ; o $ or Ea = — ~ • I 7 7-] SPHERICAL BOWL. 257 The charge on B is nearly the same as if A were removed, or The mean density on each sphere is found by dividing the charge by the surface. In this way we get 47ra2 = 246' Hence, if a very small sphere is made to touch a very large one, the mean density on the small sphere is equal to that on the large o sphere multiplied by — , or 1.644936. Application of Electrical Inversion to the case of a Spherical Bowl. 176.] One of the most remarkable illustrations of the power of Sir W. Thomson's method of Electrical Images is furnished by his investigation of the distribution of electricity on a portion of a spherical surface bounded by a small circle. The results of this investigation, without proof, were communicated to M. Liouville and published in his Journal in 1847. The complete investigation is given in the reprint of Thomson's Electrical Papers, Article XV. I am not aware that a solution of the problem of the distribution of electricity on a finite portion of any curved surface has been given by any other mathematician. As I wish to explain the method rather than to verify the calculation, I shall not enter at length into either the geometry or the integration, but refer my readers to Thomson's work. Distribution of Electricity on an Ellipsoid. 177.] It is shewn by a well-known method* that the attraction of a shell bounded by two similar and similarly situated and concentric ellipsoids is such that there is no resultant attraction on any point within the shell. If we suppose the thickness of the shell to diminish indefinitely while its density increases, we ultimately arrive at the conception of a surface- density varying as the perpendicular from the centre on the tangent plane, and since the resultant attraction of this superficial distribution on any * Thomson and Tait's Natural Philosophy, § 520, or Art. 150 of this book. VOL. I. S 258 ELECTRIC IMAGES. [178. point within the ellipsoid is zero, electricity, if so distributed on the surface, will be in equilibrium. Hence, the surface-density at any point of an ellipsoid undis- turbed by external influence varies as the distance of the tangent plane from the centre. Distribution of Electricity on a Disk. By making two of the axes of the ellipsoid equal, and making the third vanish, we arrive at the case of a circular disk, and at an expression for the surface-density at any point P of such a disk when electrified to the potential V and left undisturbed by external influence. If a be the surface-density on one side of the disk, and if KPL be a chord drawn through the point P, then 7 Application of the Principle of Electric Inversion. 178.] Take any point Q as the centre of inversion, and let R be the radius of the sphere of inversion. Then the plane of the disk becomes a spherical surface passing through Q, and the disk itself becomes a portion of the spherical surface bounded by a circle. We shall call this portion of the surface the bowl. If S' is the disk electrified to potential 7' and free from external influence, then its electrical image S will be a spherical segment at potential zero, and electrified by the influence of a quantity 7'H of electricity placed at Q. We have therefore by the process of inversion obtained the solu- tion of the problem of the distribution of electricity on a bowl or a plane disk when under the influence of an electrified point in the surface of the sphere or plane produced. Influence of an Electrified Point placed on the unoccupied part of the Spherical Surface. The form of the solution, as deduced by the principles already given and by the geometry of inversion, is as follows : If C is the central point or pole of the spherical bowl 89 and if a is the distance from C to any point in the edge of the segment, then, if a quantity q of electricity is placed at a point Q in the surface of the sphere produced, and if the bowl S is maintained at potential zero, the density cr at any point P of the bowl will be l8o.] SPHERICAL BOWL. 259 CQ, CP, and QP being the straight lines joining the points, C> Q, and P. It is remarkable that this expression is independent of the radius of the spherical surface of which the bowl is a part. It is therefore applicable without alteration to the case of a plane disk. Influence of any Number of Electrified Points. Now let us consider the sphere as divided into two parts, one of which, the spherical segment on which we have determined the electric distribution, we shall call the bowl, and the other the remainder, or unoccupied part of the sphere on which the in- fluencing point Q is placed. If any number of influencing points are placed on the remainder of the sphere, the electricity induced by these on any point of the bowl may be obtained by the summation of the densities induced by each separately. 179.] Let the whole of the remaining surface of the sphere be uniformly electrified, the surface-density being p, then the density at any point of the bowl may be obtained by ordinary integration over the surface thus electrified. We shall thus obtain the solution of the case in which the bowl is at potential zero, and electrified by the influence of the remaining portion of the spherical surface rigidly electrified with density p. Now let the whole system be insulated and placed within a sphere of diameter/, and let this sphere be uniformly and rigidly electrified so that its surface-density is pf. There will be no resultant force within this sphere, and therefore the distribution of electricity on the bowl will be unaltered, but the potential of all points within the sphere will be increased by a quantity V where y = 2-npf. Hence the potential at every point of the bowl will now be V. Now let us suppose that this sphere is concentric with the sphere of which the bowl forms a part, and that its radius exceeds that of the latter sphere by an infinitely small quantity. We have now the case of the bowl maintained at potential V and influenced by the remainder of the sphere rigidly electrified with superficial density p + p'. 180.] We have now only to suppose p + />'= 0, and we get the case of the bowl maintained at potential 7 and free from external influence. S 2 260 ELECTKIC IMAGES. [l8l. If cr is the density on either surface of the bowl at a given point when the bowl is at potential zero, and is influenced by the rest of the sphere electrified to density p, then, when the bowl is main- tained at potential V9 we must increase the density on the outside of the bowl by p', the density on the supposed enveloping sphere. The result of this investigation is that if f is the diameter of the sphere, a the chord of the radius of the bowl, and r the chord of the distance of P from the pole of the bowl, then the surface- density or on the inside of the bowl is f2-a* _j /f2-a i^—j—tan /S/ ^— 1 / ./ <•*• J t / -/ ' "• (T = 27T2/ ( and the surface-density on the outside of the bowl at the same point is y In the calculation of this result no operation is employed more abstruse than ordinary integration over part of a spherical surface. To complete the theory of the electrification of a spherical bowl we only require the geometry of the inversion of spherical surfaces. 181.] Let it be required to find the surface-density induced at any point of the bowl by a quantity q of electricity placed at a point Q, not now in the spherical surface produced. Invert the bowl with respect to Q, the radius of the sphere of inversion being R. The bowl 8 will be inverted into its image 8' and the point P will have P' for its image. We have now to determine the density </ at P' when the bowl 8' is maintained at potential V\ such that q = V'Rt and is not influenced by any external force. The density o- at the point P of the original bowl is then ' this bowl being at potential zero, and influenced by a quantity q of electricity placed at Q. The result of this process is as follows : Let the figure represent a section through the centre, 0, of the sphere, the pole, C, of the bowl, and the influencing point Q. D is a point which corresponds in the inverted figure to the unoccupied pole of the rim of the bowl, and may be found by the following construction. Draw through Q the chords EQE' and FQF', then if we sup- SPHERICAL BOWL. 261 pose the radius of the sphere of inversion to be a mean propor- tional between the segments into which a chord is divided at Q, E'F' will be the image of EF. Bisect the arc FCE' in If, so that F'J/= &F, and draw 1/QD to meet the sphere in D. D is the point re- quired. Also through 0, the centre of the sphere, and Q draw HOQH' meeting the sphere in H and H'. Then if P be any point in the bowl, the surface-density at P on the side which is separated from Q by the completed spherical surface, induced by a quantity q of electricity at Q, will be q " Fig. 16. 27T2 QH.QH' {PQ Cl)*-a rPQ where & denotes the chord drawn from C, the pole of the bowl, to the rim of the bowl. On the side next to Q the surface-density is g h 277 HH'.PQ3 CHAPTEE XII. THEOEY OF CONJUGATE FUNCTIONS IN TWO DIMENSIONS. 182.] THE number of independent cases in which the problem of electrical equilibrium has been solved is very small. The method of spherical harmonics has been employed for spherical conductors, and the methods of electrical images and of inversion are still more powerful in the cases to which they can be applied. The case of surfaces of the second degree is the only one, as far as I know, in which both the equipotential surfaces and the lines of force are known when the lines of force are not plane curves. But there is an important class of problems in the theory of electrical equilibrium, and in that of the conduction of currents, in which we have to consider space of two dimensions only. For instance, if throughout the part of the electric field under consideration, and for a considerable distance beyond it, the surfaces of all the conductors are generated by the motion of straight lines parallel to the axis of z, and if the part of the field where thjr ceases to be the case is so far from the part considered that the electrical action of the distant part on the field may be neglected, then the electricity will be uniformly distributed along each gene- rating line, and if we consider a part of the field bounded by two planes perpendicular to the axis of z and at distance unity, the potential and the distributions of electricity will be functions of x and y only. If pdxdy denotes the quantity of electricity in an element whose base is dxdy and height unity, and a-ds the quantity on an element of area whose base is the linear element ds and height unity, then the equation of Poisson may be written = 0. 183.] PROBLEMS IN TWO DIMENSIONS. 263 When there is no free electricity, this is reduced to the equation of Laplace, ^2 y The general problem of electric equilibrium may be stated as follows : — A continuous space of two dimensions, bounded by closed curves Q, <?2, &c being given, to find the form of a function, F, such that at these boundaries its value may be Flt F2, &c. respectively, being constant for each boundary, and that within this space V may be everywhere finite, continuous, and single valued, and may satisfy Laplace's equation. I am not aware that any perfectly general solution of even this question has been given, but the method of transformation given in Art. 190 is applicable to this case, and is much more powerful than any known method applicable to three dimensions. The method depends on the properties of conjugate functions of two variables. Definition of Conjugate Functions. . : ^ - 183.] Two quantities a and p are said to be conjugate functions of x and y, if a 4- \/ — I p is a function of x + v — 1 y. It follows from this definition that da dp da dp -r = -r ' an(* T~ + ~T~ — ° > dx dy dy dx _ A !_ » L_ — o dx* * dy* ~ ' dx* r dy* ' Hence both functions satisfy Laplace's equation. M, £*. (2) da dx da dp dy dx da dx Ta ^ dx dy If x and y are rectangular coordinates, and if ds1 is the intercept of the curve (/3 = constant) between the curves a and a + da, and dsz the intercept of a between the curves p and /3 -f dp, then d*i _ ds* .. J IA\ da~d0--R'