A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 15 of 28

154.] If in the general equations we transfer the origin of co- ordinates to a point on the axis of x distant t from the centre of the system, and if for #, A, b, and c we substitute t + x, t + A, t + b, and t+c respectively, and then make t increase indefinitely, we obtain, in the limit, the equation of a system of paraboloids whose foci are at the points x = b and x = c, viz. the equation is

^_ + f! = 0. (49)

224 CONFOCAL QUADRIC SURFACES. [J54-

If the variable parameter is A for the first system of elliptic paraboloids, //, for the hyperbolic paraboloids, and v for the second system of elliptic paraboloids, we have A, 3, /u, c, v in ascending order of magnitude, and

os = A + jui-fy— c— b,

,

e-6 (50)

_

e_« J

In order to avoid infinite values in the integrals (7) the cor- responding integrals in the paraboloidal system are taken between different limits.

We write in this case

~~

["• dy.

h --

From these we find

~ ** \ • / * \ j - - - --j

(51)

1 / T\ 1 / 1\ ^

V =

a? = i (<? + £) + 4 (c— 3)(coshy — cos/3 — cosha),

= 2 (c— J) sinh - sin - cosh - > 22 2

= 2 (c— 3) cosh - cos - sinh - •

a 2 * u

(52)

When ^ = c we have the case of paraboloids of revolution about the axis of xy and x = a fe2a—e2^),

y — 20 d^+vcosft (53)

z—

The surfaces for which ft is constant are planes through the axis, ft being the angle which such a plane makes with a fixed plane through the axis.

The surfaces for which a is constant are confocal paraboloids. When a= -— oo the paraboloid is reduced to a straight line terminat- ing at the origin.

1 54-] CYLINDERS AND PARABOLOIDS. 225

We may also find the values of a, /3, y in terms of r, 0, and $, the spherical polar coordinates referred to the focus as orgin, and the axis of the parabolas as axis of the sphere,

a = log (r* cos J 6),

£ = <*>, (54)

y = log (r^ sin J0).

We may compare the case in which the potential is equal to a, with the zonal solid harmonic rt Q{. Both satisfy Laplace's equa- tion, and are homogeneous functions of a?, y, z, but in the case derived from the paraboloid there is a discontinuity at the axis, and i has a value not differing by any finite quantity from zero.

The surface-density on an electrified paraboloid in an infinite field (including the case of a straight line infinite in one direction) is inversely as the square root of the distance from the focus, or, in the case of the line, from the extremity of the line.

VOL. I.

CHAPTER XI.

THEORY OF ELECTRIC IMAGES AND ELECTRIC INVERSION.

155.] WE have already shewn that when a conducting sphere is under the influence of a known distribution of electricity, the distribution of electricity on the surface of the sphere can be determined by the method of spherical harmonics.

For this purpose we require to expand the potential of the in- fluencing system in a series of solid harmonics of positive degree, having the centre of the sphere as origin, and we then find a corresponding series of solid harmonics of negative degree, which express the potential due to the electrification of the sphere.

By the use of this very powerful method of analysis, Poisson determined the electrification of a sphere under the influence of a given electrical system, and he also solved the more difficult problem to determine the distribution of electricity on two con- ducting spheres in presence of each other. These investigations have been pursued at great length by Plana and others, who have confirmed the accuracy of Poisson.

In applying this method to the most elementary case of a sphere under the influence of a single electrified point, we require to expand the potential due to the electrified point in a series of solid har- monics, and to determine a second series of solid harmonics which express the potential, due to the electrification of the sphere, in the space outside.

It does not appear that any of these mathematicians observed that this second series expresses the potential due to an imaginary electrified point, which has no physical existence as an electrified point, but which may be called an electrical image, because the action of the surface on external points is the same as that which would be produced by the imaginary electrified point if the spherical surface were removed.

156.] ELECTRIC IMAGES. 227

This discovery seems to have been reserved for Sir W. Thomson, who has developed it into a method of great power for the solution of electrical problems, and at the same time capable of being pre- sented in an elementary geometrical form.

His original investigations, which are contained in the Cambridge and Dublin Mathematical Journal, 1848, are expressed in terms of the ordinary theory of attraction at a distance, and make no use of the method of potentials and of the general theorems of Chapter IV; though they were probably discovered by these methods. Instead, however, of following the method of the author, I shall make free use of the idea of the potential and of equipotential surfaces, when- ever the investigation can be rendered more intelligible by such means.

Theory of Electric Images.

156.] Let A and B, Figure 7, represent two points in a uniform dielectric medium of infinite extent. Let the charges of A and B be el and e% respectively. Let P be any point in space whose distances from A and B are r± and r2 respectively. Then the value of the potential at P will be TT ei ez

The equipotential surfaces due to

this distribution of electricity are represented in Fig. I (at the end of this volume) when e1 and e2 are of the same sign, and in Fig. II when they are of opposite signs. We have now to consider that surface for which V '= 0, which is the only spherical surface in the system. When eI and e2 are of the same sign, this surface is entirely at an infinite distance, but when they are of opposite signs there is a plane or spherical surface at a finite distance for which the potential is zero.

The equation of this surface is

Its centre is at a point C in AB produced, such that

AC:£ and the radius of the sphere is

AB

The two points A and B are inverse points with respect to this

228 ELECTRIC IMAGES. [157'.

sphere, that is to say, they lie in the same radius, and the radius is a mean proportional between their distances from the centre.

Since this spherical surface is at potential zero, if we suppose it constructed of thin metal and connected with the earth, there will be no alteration of the potential at any point either outside or inside, but the electrical action everywhere will remain that due to the two electrified points A and B.

If we now keep the metallic shell in connection with the earth and remove the point B, the ^otentiaLwithin the .sphere .jrilLbecome_ everywhere zero^ but outside it will remain the same as before. For the surface of the sphere still remains at the same potential, and no change has been made in the exterior electrification.

Hence, if an electrified point A be placed outside a spherical conductor which is at potential zero, the electrical action at all points outside the sphere will be that due to the point A together with another point £ within the sphere, which we may call the . electrical image of A.

In the same way we may shew that if IB is a point placed inside the spherical shell, the electrical action within the sphere is that due to B, together with its image A.

157.] Definition of an Electrical Image. An electrical image is an electrified point or system of points on one side of a surface which would produce on the other side of that surface the same electrical action which the actual electrification of that surface really does produce.

In Optics a point or system of points on one side of a mirror or lens which if it existed would emit the system of rays which actually exists on the other side of the mirror or lens, is called a virtual image.

Electrical images correspond to virtual images in Optics in being related to the space on the other side of the surface. They do not correspond to them in actual position, or in the merely approximate character of optical foci.

There are no real electrical images, that is, imaginary electrified points which would produce, in the region on the same side of the electrified surface, an effect equivalent to that of the electrified surface.

For if the potential in any region of space is equal to that due to a certain electrification in the same region it must be actually produced by that electrification. In fact, the electrification at any point may be found from the potential near that point by the application of Poisson's equation.

1 5 7.] INVERSE POINTS. 229

Let a be the radius of the sphere.

Lety be the distance of the electrified point A from the centre C.

Let e be the charge of this point.

Then the image of the point is at B, on the same radius of the

sphere at a distance — - , and the charge of the image is — e — • '. ' •—

We have shewn that this image will produce the same effect on the opposite side of the surface as the actual electrification^ of the surface does. We shall next determine the surface-density of this electrification at any point P of the spherical sur- face, and for this purpose we shall make use of the theorem of Coulomb,

Art. 80, that if R is the resultant force at the surface of a con- ductor, and o- the superficial density,

R = 47TO-,

R being measured away from the surface. .

We may consider R as the resultant of two forces, a repulsion

s> (1. \

—^ acting along AP, and an attraction e -^ -^— acting along PB.

Ar j -L.D

Resolving these forces in the directions of AC and CP, we find that the components of the repulsion are

along AC, and - along CP.

Those of the attraction are

BC along AG' and ~~e * along CP'

Now BP = j AP, and BC = y , so that the components of the attraction may be written

along AC, and ~* along CP.

The components of the attraction and the repulsion in the direction of AC are equal and opposite, and therefore the resultant force is entirely in the direction of the radius CP. This only confirms what we have already proved, that the sphere is an equi- potential surface, and therefore a surface to which the resultant force is everywhere perpendicular.

230 ELECTRIC IMAGES. [158.

The resultant force measured along CP, the normal to the surface in the direction towards the side on which A is placed, is

E- cf2-"* * /3)

a AP*

If A is taken inside the sphere f is less than at and we must measure R inwards. For this case therefore

^=_e^-/^_l

a APZ In all cases we may write

AD. Ad 1

where AD, Ad are the segments of any line through A cutting the sphere, and their product is to be taken positive in all cases.

158.] From this it follows, by Coulomb's theorem, Art. 80, that the surface-density at P is

AD. Ad 1

The density of the electricity at any point of the sphere varies inversely as the cube of its distance from the point A.

The effect of this superficial distribution, together with that of the point A, is to produce on the same side of the surface as the point A a potential equivalent to that due to e at A, and its image

— e -j at B, and on the other side of the surface the potential is

tj

everywhere zero. Hence the effect of the superficial distribution by itself is to produce a potential on the side of A equivalent to

that due to the image — e -^ at B, and on the opposite side a potential equal and opposite to that of e at A.

The whole charge on the surface of the sphere is evidently — e — since it is equivalent to the image at B.

We have therefore arrived at the following theorems on the action of a distribution of electricity on a spherical surface, the surface-density being inversely as the cube of the distance from a point A either without or within the sphere.

Let the density be given by the equation

where C is some constant quantity, then by equation (6)

1 59.] DISTRIBUTION OF ELECTRICITY. 231

The action of this superficial distribution on any point separated from A by the surface is equal to that of a quantity of electricity — '» or 1-naC

AD. Ad concentrated at A.

Its action on any point on the same side of the surface with A is equal to that of a quantity of electricity

fAD.Ad concentrated at B the image of A.

The whole quantity of electricity on the sphere is equal to the first of these quantities if A is within the sphere, and to the second if A is without the sphere.

These propositions were established by Sir W. Thomson in his original geometrical investigations with reference to the distribution of electricity on spherical conductors, to which the student ought to refer.

159.] If a system in which the distribution of electricity is known is placed in the neighbourhood of a conducting sphere of radius a, which is maintained at potential zero Jby connection with^ thejearth^then the electrifications due to the several parts of the system will be superposed.

Let A13 AI, &c. be the electrified points of the system, f-^f^ &c. their distances from the centre of the sphere, elt e2, &c. their charges, then the images J313 B^ &c. of these points will be in the

O Q

same radii as the points themselves, and at distances — > -^ , &c.

/I /2

from the centre of the sphere, and their charges will be

/I /2

The potential on the outside of the sphere due to the superficial electrification will be the same as that which would be produced by the system of images J?15.Z?2, &c. This system is therefore called the electrical image of the system A± , Az, &c.

If the sphere instead of being at potential zero is at potential F, we must superpose a distribution of electricity on its outer surface having the uniform surface-density

7

4-Tra The effect of this at all points outside the sphere will be equal to

232 ELECTRIC IMAGES. [l6o.

that of a quantity 7a of electricity placed at its centre, and at all points inside the sphere the potential will be simply increased by 7.

The whole charge on the sphere due to an external system of influencing points, A19 A.2, &c. is

d a

E= ra-elT -e,T -&*., (9)

Jl J-2

from which either the charge E or the potential V may be cal- culated when the other is given.

When the electrified system is within the spherical surface the induced charge on the surface is equal and of opposite sign to the inducing charge, as we have before proved it to be for every closed surface, with respect to points within it.

*160.] The energy due to the mutual action between an elec- trified point e, at a distance /from the centre of the sphere greater than a the radius, and the electrification of the spherical surface due to the influence of the electrified point and the charge of the sphere, is

1/= f^ '" 5 /« (/»-')' (10)

where 7 is the potential, and E the charge of the sphere.

The repulsion between the electrified point and the sphere is therefore, by Art. 92^

• ["The discussion in the text will perhaps be more easily understood if the problem

be regarded as an example of Art.^t Let us then suppose that what is described as an electrified point is really a small spherical conductor, the radius of which is & and the potential v. We have thus a particular case of the problem of two spheres of whiclroiie solution has already been given in Art. 146, and another will be given in Art. 173. In the case before us however the radius 6 is so small that we may consider the electricity of the small conductor to be uniformly distributed over its surface and all the electric images except the first image of the small conductor to be disregarded.

We thus have F = - + •£,

// f

The energy of the system is therefore, Art. 85,

2a

By means of the above equations we may also express the energy in terms of the potentials : to the same order of approximation it is

a72 a&T_ 1 ,, a&2 \ o-i _F,+ _(»+)„.]

l6o.] IMAGE OF AN ELECTRIFIED SYSTEM. 233

• e (E ^-' , }

~* ( 2-«22 >

Hence the force between the point and the sphere is always an attraction in the following cases —

(1) -When the sphere is uninsulated.

(2) When the sphere has no charge.

(3) When the electrified point is very near the surface.

In order that the force may be repulsive, the potential of the

/3 sphere must be positive and greater than e -r—^ — 2x2 > an<^ ^e

charge of the sphere must be of the same sign as e and greater

.

At the point of equilibrium the equilibrium is unstable, the force being an attraction when the bodies are nearer and a repulsion when they are farther off.

When the electrified point is within the spherical surface the force on the electrified point is always away from the centre of the sphere, and is equal to

The surface-density at the point of the sphere nearest to the electrified point where it lies outside the sphere is

The surface-density at the point of the sphere farthest from the

electrified point is

a(f—a}}

• —      r 
  

)

When E, the charge of the sphere, lies between

a*(3f-a) a*

and -e

the electrification will be negative next the electrified point and

234

ELECTRIC IMAGES.

[161.

positive on the opposite side. There will be a circular line of division between the positively and the negatively electrified parts of the surface, and this line will be a line of equilibrium.

the equipotential surface which cuts the sphere in the line of equi- librium is a sphere whose centre is the electrified point and whose radius is A//2 — #2.

The lines of force and equipotential surfaces belonging to a case of this kind are given in Figure IV at the end of this volume.

Images in an Infinite Plane Conducting Surface.

161.] If the two electrified points A and B in Art. 156 are electrified with equal charges of electricity of opposite signs, the surfaces of zero potential will be the plane, every point of which is equidistant from A and B.

Hence, if A be an electrified point whose charge is <?, and AD a perpendicular on the plane, produce AD to B so that DB = AB, and place at B a charge equal to —e, then this charge at B will be the image of A, and will produce at all points on the same side of the plane as A, an effect equal to that of the actual electrification of the plane. For the potential on the side of A due to A and B fulfils the conditions that y277"= 0 everywhere except at A, and that V = 0 at the plane, and there is only one form of V which can fulfil these conditions.

Fig.

To determine the resultant force at the point P of the plane, we observe that it is compounded of two forces each equal to ,

one acting along AP and the other along PB. Hence the resultant of these forces is in a direction parallel to AB and equal to

e AB_ AP2 ' IP'

Hence R, the resultant force measured from the surface towards the space in which A lies, is

2eAD

162.] IMAGES IN AN INFINITE PLANE,

and the density at the point P is

235

On Electrical Inversion.

162.] The method of electrical images leads directly to a method of transformation by which we may derive from any electrical problem of which we know the solution any number of other problems with their solutions.

We have seen that the image of a point at a distance r from the centre of a sphere of radius R is in the same radius and at a distance / such that rr — JR2. Hence the image of a system of points, lines, or surfaces is obtained from the original system by the method known in pure geometry as the method of inversion, and described by Chasles, Salmon, and other mathematicians.

If A and £ are two points, A' and B' their images, 0 being the centre of inversion, and R the radius of the sphere of inversion,

OA.OA'=R2 = OB. OB'. Hence the triangles OAJB, OB' A' are similar, and AB : A'B' : : 0 A : OB' ::OA.OB: R2.'

If a quantity of electricity e be placed at A,

a its potential at B will be 7 = —r=^ •

If e' be placed at A' its potential at B' will be

In the theory of electrical images

e:e'::OA:R::R'. OA'.

Hence 7: V : : R : OB, (17)

or the potential at B due to the electricity at A is to the potential at the image of B due to the electrical image of A as R is to OB.

Since this ratio depends only on OB and not on OA, the potential at B due to any system of electrified bodies is to that at B' due to the image of the system as R is to OB.

If r be the distance of any point A from the centre, and / that of its image A', and if e be the electrification of A, and e' that of A', also if L, S, K be linear, superficial, and solid elements at A, and L', £', K ' their images at A', and X, o-, />, A', o-', p the corresponding line surface and volume densities of electricity at the two points,

236 ELECTRIC IMAGES. [163.

V the potential at A due to the original system, and V the potential at A' due to the inverse system, then

v / If R2 r'2 S' R* /4 K' R& r'6 -\ ~r = ' L ' '~~ ' ~^E^ ~S^~~W ~K"~r*~~R^

e- = ?L- !L - = -L ?L

/ l= & ~~~ & ^*(18)

7' r ° R

7~!$"V

If in the original system a certain surface is that of a conductor, ' and has therefore a constant potential P, then in the transformed

system the image of the surface will have , a potential P — . But

by placing at 0, the centre of inversion, a quantity of electricity equal to —PR, the potential of the transformed surface is reduced to zero.

Hence, if we know the distribution of electricity on a conductor when insulated in open space and charged to the potential P, we can find by inversion the distribution on a conductor whose form is the image of the first under the influence of an electrified point with a charge — PR placed at the centre of inversion, the conductor being in connexion with the earth.

163.] The following geometrical theorems are useful in studying cases of inversion.

Every sphere becomes, when inverted, another sphere, unless it passes through the centre of inversion, in which case it becomes a plane.

If the distances of the centres of the spheres from the centre of inversion are a and of, and if their radii are a and of, and if we define the power of a sphere with respect to the centre of in- version to be the product of the segments cut off by the sphere from a line through the centre of inversion, then the power of the first sphere is az — a2, and that of the second is a2— of2. We have in this case

/ / "7~»O /•> /*>

• = - = ^ = T> (19)

a a a2 — a2 R2

or the ratio of the distances of the centres of the first and second spheres is equal to the ratio of their radii, and to the ratio of the

• See Thomson and Tait's Natural Philosophy, § 515,.

GEOMETRICAL THEOREMS. 237

power of the sphere of inversion to the power of the first sphere, or of the power of the second sphere to the power of the sphere of inversion.

The image of the centre of inversion with regard to one sphere is the inverse point of the centre of the other sphere.

In the case in which the inverse surfaces are a plane and a sphere, the perpendicular from the centre of inversion on the plane is to the radius of inversion as this radius is to the diameter of the sphere, and the sphere has its centre on this perpendicular and passes through the centre of inversion.

Every circle is inverted into another circle unless it passes

•7 A

through the centre of inversion, in which case it becomes a straight line.

The angle between two surfaces, or two lines at their intersec- tion, is not changed by inversion.

Every circle which passes through a point, and the image of that point with respect to a sphere, cuts the sphere at right angles.

Hence, any circle which passes through a point and cuts the sphere at right angles passes through the image of the point.

164.] We may apply the method of inversion to deduce the distribution of electricity on an uninsulated sphere under the in- fluence of an electrified point from the uniform distribution on an insulated sphere not influenced by any other body.

If the electrified point be at A, take it for the centre of inversion, and if A is at a distance f from the centre of the sphere whose radius is a, the inverted figure will be a sphere whose radius is a and whose centre is distant/', where

^/'^t_ (20)

" ~"

The centre of either of these spheres corresponds to the inverse point of the other with respect to A, or if C is the centre and B the inverse point of the first sphere, C' will be the inverse point, and B' the centre of the second.

Now let a quantity / of electricity be communicated to the second sphere, and let it be uninfluenced by external forces. It will become uniformly distributed over the sphere with a surface-

Its action at any point outside the sphere will be the same as that of a charge / placed at I? the centre of the sphere.

238 ELECTRIC IMAGES. [165.

At the spherical surface and within it the potential is

r=?> - (22)

a constant quantity.

Now let us invert this system. The centre Bf becomes in the inverted system the inverse point B, and the charge / at If

-n

becomes e' -^ at B, and at any point separated from B by the

J surface the potential is that due to this charge at B.

The potential at any point P on the spherical surface, or on the same side as B, is in the inverted system

e' R

a' AP' If we now superpose on this system a charge e at A, where

e = -g7R, (23)

the potential on the spherical surface, and at all points on the same side as B, will be reduced to zero. At all points on the same side as A the potential will be that due to a charge e at A, and a charge

(24)

as we found before for the charge of the image at B.

To find the density at any point of the first sphere we have

7?3

Substituting for the value of or' in terms of the quantities be- longing to the first sphere, we find the same value as in Art. 158,

<7 = —

(26)

On Finite Systems of Successive Images.

165.] If two conducting planes intersect at an angle which is a submultiple of two right angles, there will be a finite system of images which will completely determine the electrification.

For let AOB be a section of the two conducting planes per- pendicular to their line of intersection, and let the angle of inter- section AOB = -, let P be an electrified point, and let PO = r, ti

and POB = 6. Then, if we draw a circle with centre 0 and radius

165.] SYSTEMS OF IMAGES. 239

OP, and find points which are the successive images of P in the two planes beginning1 with OS, we shall find Q1 for the image of P in OB, P2 for the image of Ql in OA, Q3 for that of P2 in OB, P3 for that of Q3 in OA, and Q2 for that of P3 in OB.

If we had begun with the image of P in AO we should have found the same points in the reverse order Q2, P3, Q3, P2, Qlt provided AOB is a submultiple of two right angles.

For the alternate images P1,P2> ^s are range^ round the circle at angular intervals equal to 2 AOB, and the intermediate images Qlt Q2, Q3 are at inter- vals of the same magnitude. Hence, if 2 AOB is a submultiple of 2 IT, there will be a finite number of images, and none of these will fall within the angle AOB. If, however, AOB is not a submultiple of TT, it will be impossible to represent the actual electrification as the re- sult of a finite series of electrified points.

If AOB = -, there will be n negative images Qlt (J)2, &c., each n

equal and of opposite sign to P, and n— 1 positive images P2, P35 &c., each equal to P, and of the same sign.

The angle between successive images of the same sign is — •

If we consider either of the conducting planes as a plane of sym- metry, we shall find the positive and negative images placed symmetrically with regard to that plane, so that for every positive image there is a negative image in the same normal, and at an equal distance on the opposite side of the plane.

If we now invert this system with respect to any point, the two planes become two spheres, or a sphere and a plane intersecting

at an angle - , the influencing point P being within this angle.

The successive images lie on the circle which passes through P and intersects both spheres at right angles.

To find the position of the images we may make use of the principle that a point and its image are in the same radius of the sphere, and draw successive chords of the circle beginning at P and passing through the centres of the two spheres alternately.

240 ELECTRIC IMAGES. [166.

To find the charge which must be attributed to each image, take any point in the circle of intersection, then the charge of each image is proportional to its distance from this point, and its sign is positive or negative according as it belongs to the first or the second system.

166.] We have thus found the distribution of the images when any space bounded by a conductor consisting of two spherical surfaces

meeting at an angle - , and kept at potential zero, is influenced by

VL

an electrified point.

We may by inversion deduce the case of a conductor consisting

of two spherical segments meeting at a re-entering angle - , charged

to potential unity and placed in free space.

For this purpose we invert the system with respect to P. The

circle on which the images formerly lay now becomes a straight

line through the centres of the spheres.

If the figure (ll) represents a section through the line of centres AB, and if D, D' are the points where the circle of in- tersection cuts the plane of the paper, then, to find the suc- cessive images, draw DA a radius of the first circle, and draw DC, DB, &c., making

angles -, — , &c. with DA. fe n n '

The points C, B, &c. at which they cut the line of centres will be the positions of the positive images, and the charge of each will be represented by its distances from D. The last of these images will be at the centre of the second circle.

To find the negative images draw DP, DQ, &c., making angles

1L , — &c. with the line of centres. The intersections of these n n

lines with the line of centres will give the positions of the negative images, and the charge of each will be represented by its distance from D.

The surface-density at any point of either sphere is the sum of the surface-densities due to the system of images. For instance, the surface-density at any point S of the sphere whose centre is A, is

1 67.] TWO INTERSECTING SPHERES. 241

where A, B, C, &c. are the positive series of images.

When 8 is on the circle of intersection the density is zero.

To find the total charge on each of the spherical segments, we may find the surface-integral of the induction through that segment due to each of the images.

The total charge on the segment whose centre is A due to the image at A whose charge is DA is

where 0 is the centre of the circle of intersection.

In the same way the charge on the same segment due to the image at B is \ (DB + OB), and so on, lines such as OB measured from 0 to the left being reckoned negative.

Hence the total charge on the segment whose centre is A is

0^4- &c.),

167.] The method of electrical images may be applied to any space bounded by plane or spherical surfaces all of which cut one another in angles which are submultiples of two right angles.

In order that such a system of spherical surfaces may exist, every solid angle of the figure must be trihedral, and two of its angles must be right angles, and the third either a right angle or a submultiple of two right angles.

Hence the cases in which the number of images is finite are —

(1) A single spherical surface or a plane.

(2) Two planes, a sphere and a plane, or two spheres intersecting

at an angle - •

99

(3) These two surfaces with a third, which may be either plane or spherical, cutting both orthogonally.

(4) These three surfaces with a fourth cutting the first two

orthogonally and the third at an angle -—, . Of these four surfaces

one at least must be spherical.

We have already examined the first and second cases. In the first case we have a single image. In the second case we have 2n—l images arranged in two series in a circle which passes through the influencing point and is orthogonal to both surfaces.

VOL. I. R

242

ELECTRIC IMAGES.

[168.

In the third case we have, besides these images, their images with respect to the third surface, that is, 4#— 1 images in all besides the influencing point.

In the fourth case we first draw through the influencing point a circle orthogonal to the first two surfaces, and determine on it the positions and magnitudes of the n negative images and the n— 1 positive images. Then through each of these 2n points, including the influencing point, we draw a circle orthogonal to the third and fourth surfaces, and determine on it two series of images, n' in each series. We shall obtain in this way, besides the influencing point, 2nn'— 1 positive and 2nn' negative images. These 4nn' points are the intersections of n circles with Mother circles, and these circles belong to the two systems of lines of curvature of a cyclide.

If each of these points is charged with the proper quantity of electricity, the surface whose potential is zero will consist of n -f n' spheres, forming two series of which the successive spheres of the

7T

first set intersect at angles - , and those of the second set at angles

— , while every sphere of the first set is orthogonal to every sphere of the second set.

Case of Two Spheres cutting Orthogonally. See Fig. IV at the end of this volume.

168.] Let A and B, Fig. 12, be the centres of two spheres cutting