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Fig. 1.

Now let heat be poured into the medium at constant rate by a single point-source P (Fig. 1), and drawn off at a smaller rate by a single point-sink P', while the remainder flows to more and more remote parts of the medium, supposed infinite in extent in every direction. After a sufficient time from the beginning of the flow a definite system of lines of flow and isothermal surfaces can be traced for this case in the manner described above. One of the isothermal surfaces will be a sphere S surrounding the sink, which, however, will not be at the centre of the sphere, but so situated that the source, sink, and centre are in line, and that the radius of the sphere is a mean proportional between the distances of the source and sink from the centre. If a be the radius of the sphere and f the distance of the source from the centre of the sphere, the heat carried off by the sink is the fraction a ⁄ f of that given out by the source.

In the electrical analogue, the source and sink are respectively a point-charge and what is called the "electric image" of that charge with respect to the sphere, which is in this case an equipotential surface. And just as the lines of flow of heat meet the spherical isothermal surface at right angles, so the lines of force in the electrical case meet the equipotential surface also at right angles. Now obviously in the thermal case a spherical sink could be arranged coinciding with the spherical surface so as to receive the flow there arriving and carry off the heat from the medium, without in the least disturbing the flow outside the sphere. The whole amount of heat arriving would be the same: the amount received per unit area at any point on the sphere would evidently be proportional to the gradient of temperature there towards the surface. Of course the same thing could be done at any isothermal surface, and the same proportionality would hold in that case.

Similarly the source could be replaced by a surface-distribution of sources over any surrounding isothermal surface; and the condition to be fulfilled in that case would be that the amount of heat given out per unit area anywhere should be exactly that which flows out along the lines of flow there in the actual case. Outside the surface the field of flow would not be affected by this replacement. It is obvious that in this case the outflow per unit area must be proportional to the temperature slope outward from the surface.

The same statements hold for any complex system of sources and sinks. There must be the same outflow from the isothermal surface or inflow towards it, as there is in the actual case, and the proportionality to temperature slope must hold.

This is exactly analogous to the replacement by a distribution on an equipotential surface of the electrical charge or charges within the surface, by a distribution over the surface, with fulfilment of Coulomb's theorem (p. 43 below) at the surface. Thomson's paper on the "Uniform Motion of Heat" gave an intuitive proof of this great theorem of electrostatics, which the statements above may help to make clear to those who have, or are willing to acquire, some elementary knowledge of electricity.

Returning to the distribution on any isothermal surface surrounding the sink (or sinks) we see that it represents a surface-sink in equilibrium with the flow in the field. The distribution on a metal shell, coinciding with the surface, which keeps the surface at a potential which is the analogue of the temperature at the isothermal surface, while the shell is under the influence of a point-charge of electricity—the analogue of the thermal source—is the distribution as affected by the induction of the point-charge. If the shell coincide with the spherical equipotential surface referred to above, and the distribution given by the theorem of replacement be made upon it, the shell will be at zero potential, and the charge will be that which would exist if the shell were uninsulated, that is, the "induced charge."

The consideration of the following simple problem will serve to make clear the meaning of an electric image, and form a suitable introduction to a description of the application of the method to the electrification of spherical surfaces. Imagine a very large plane sheet of tinfoil connected by a conducting wire with the earth. If there are no electrified bodies near, the sheet will be unelectrified. But let a very small metallic ball with a charge of positive electricity upon it be brought moderately close to one face of the tinfoil. The tinfoil will be electrified negatively by induction, and the distribution of the negative charge will depend on the position of the ball. Now, it can be shown that the field of electric force, on the same side of the tinfoil as the ball, is precisely the same as would be produced if the foil (and everything behind it) were removed, and an equal negative charge of electricity placed behind the tinfoil on the prolonged perpendicular from the ball to the foil, and as far from the foil behind as the ball is from it in front. Such a negative charge behind the tinfoil sheet is called an electric image of the positive charge in front. It is situated, as will be seen at what would be, if the tinfoil were a mirror, the optical image of the ball in the mirror.

Fig. 2.

Fig. 3.

Now, suppose a second very large sheet of tinfoil to be placed parallel to the first sheet, so that the small electrified sphere is between the two sheets, and that this second sheet is also connected to the earth. The charge on the ball induces negative electricity on both sheets, but besides this each sheet by its charge influences the other. The problem of distribution is much more complicated than in the case of a single sheet, but its solution is capable of very simple statement. Let us call the two sheets A and B (Fig. 2), and regard them for the moment as mirrors. A first image of an object P between the two mirrors is produced directly by each, but the image I₁ in A is virtually an object in front of B, and the image J₁ in B an object in front of A, so that a second image more remote from the mirror than the first is produced in each case. These second images I₂ and J₂ in the same way produce third images still more remote, and so on. The positions are determined just as for an object and a single mirror. There is thus an infinite trail of images behind each mirror, the places of which any one can assign.

Every one may see the realisation of this arrangement in a shop window, the two sides of which are covered by parallel sheets of mirror-glass. An infinite succession of the objects in the window is apparently seen on both sides. When the objects displayed are glittering new bicycles in a row the effect is very striking; but what we are concerned with here is a single small object like the little ball, and its two trails of images. The electric force at any point between the two sheets of tinfoil is exactly the same as if the sheets were removed and charges alternately negative and positive were placed at the image-points, negative at the first images, positive at the second images, and so on, each charge being the same in amount as that on the ball. We have an "electric kaleidoscope" with parallel mirrors. When the angle between the conducting planes is an aliquot part of 360°, let us say 60°, the electrified point and the images are situated, just as are the object and its image in Brewster's kaleidoscope, namely at the angular points of a hexagon, the sides of which are alternately (as shown in Fig. 3) of lengths twice the distance of the electrified point from A and from B.

Fig. 4.

Now consider the spherical surface referred to at p. 37, which is kept at uniform potential by a charge at the external point P, and a charge q' at the inverse point P' within the sphere. If E (Fig. 4) be any point whatever on the surface, and r, r' be its distances from P and P', it is easy to prove by geometry that the two triangles CPE and CEP' are similar, and therefore r' = ra ⁄ f. [Here a ⁄ f is used to mean a divided by f. The mark ⁄ is adopted instead of the usual bar of the fraction, for convenience of printing.] Now, by the explanation given above, the potential produced at any point by a charge q at another point, is equal to the ratio of the charge q to the distance between the points. Thus the potential at E due to the charge q at P is q ⁄ r, and that at E due to a charge q' at P' is q' ⁄ r'. Thus if q' = − qa ⁄ f, q' at P' will produce a potential at E = − qa ⁄ fr' = − q ⁄ r, by the value of r. Hence q at P and − qa ⁄ f at P' coexisting will give potential q ⁄ r + − q ⁄ r or zero, at E. Thus the charge − qa ⁄ f, at the internal point P' will in presence of + q at P keep all points of the spherical surface at zero potential. These two charges represent the source and sink in the thermal analogue of p. 37 above.

Now replace S by a spherical shell of metal connected to the earth by a long fine wire, and imagine all other conductors to be at a great distance from it. If this be under the influence of the charge q at P alone, a charge is induced upon it which, in presence of P, maintains it at zero potential. The internal charge − qa ⁄ f, and the induced distribution on the shell are thus equivalent as regards the potential produced by either at the spherical surface; for each counteracts then the potential produced by q at P. But it can be proved that if a distribution over an equipotential surface can be made to produce the same potential over that surface as a given internal distribution does, they produce the same potentials at all external points, or, as it is usually put, the external fields are the same. This is part of the statement of what has been called the "theorem of replacement" discovered by Green, Gauss, Thomson, and Chasles as described above.

Another part of the statement of the theorem may now be formulated. Coulomb showed long ago that the surface-density of electricity at any point on a conductor is proportional to the resultant field-intensity just outside the surface at that point. Since the surface is throughout at one potential this intensity is normal to the surface. Let it be denoted by N, and s be the surface-density: then according to the system of units usually adopted 4πs = N.

Let now the rate of diminution of potential per unit of distance outwards (or downward gradient of potential) from the equipotential surface be determined for every point of the surface, and let electricity be distributed over the surface, so that the amount per unit area at each point (the surface-density) is made numerically equal to the gradient there divided by 4π. This, by Coulomb's law, stated above, gives that field-intensity just outside the surface which exists for the actual distribution, and therefore, as can be proved, gives the same field everywhere else outside the surface. The external fields will therefore be equivalent, and further, the amount of electricity on the surface will be the same as that situated within it in the actual distribution.

Thus it is only necessary to find for − qa ⁄ f at P' and q at P, the falling off gradient N of potential outside the spherical surface at any point E, and to take N ⁄ 4π, to obtain s the surface-density at E. Calculation of this gradient for the sphere gives 4πs = − q (f² − a²) ⁄ ar³. The surface-density is thus inversely as the cube of the distance PE.

If the influencing point P be situated within the spherical shell, and the shell be connected to earth as before, the induced distribution will be on its interior surface. The corresponding point P' will now be outside, but given by the same relation. And a will now be greater than f, and the density will be given by 4πs = − q (a² − f²) ⁄ ar³, where, f and r have the same meanings with regard to E and P as before.

P' is in each case called the image of P in the sphere S, and the charge − qa ⁄ f there supposed situated is the electric image of the charge q at P. It will be seen that an electric image is a charge, or system of charges, on one side of an electrified surface which produces on the other side of that surface the same electrical field as is produced by the actual electrification of the surface.

While by the theorem of replacement there is only one distribution over a surface which produces at all points on one side of a surface the same field as does a distribution D on the other side of the surface, this surface distribution may be equivalent to several different arrangements of D. Thus the point-charge at P' is only one of various image-distributions equivalent to the surface-distribution in the sense explained. For example, a uniform distribution over any spherical surface with centre at P' (Fig. 4) would do as well, provided this spherical surface were not large enough to extend beyond the surface S.

In order to find the potential of the sphere (Fig. 4) when insulated with a charge Q upon it, in presence of the influencing charge q at the external point P, it is only necessary to imagine uniformly distributed over the sphere, already electrified in the manner just explained, the charge Q + aq ⁄ f. Then the whole charge will be Q, and the uniformity of distribution will be disturbed, as required by the action of the influencing point-charge. The potential will be Q ⁄ a + q ⁄ f. For a given potential V of the sphere, the total charge is aV − aq ⁄ f, that is the charge is aV over and above the induced charge.

If instead of a single influencing point-charge at P there be a system of influencing point-charges at different external points, each of these has an image-charge to be found in amount and situation by the method just described, and the induced distribution is that obtained by superimposing all the surface distributions found for the different influencing points.

The force of repulsion between the point-charge q and the sphere (with total charge Q) can be found at once by calculating the sum of the forces between q at P and the charges Q + aq ⁄ f at C and − aq ⁄ f at P'.

This can be found also by calculating the energy of the system, which will be found to consist of three terms, one representing the energy of the sphere with charge Q uninfluenced by an external charge, one representing the energy on a small conductor (not a point) at P existing alone, and a third representing the mutual energy of the electrification on the sphere and the charge q at P existing in presence of one another. By a known theorem the energy of a system of conductors is one half of the sum obtained by multiplying the potential of each conductor by its charge and adding the products together. It is only necessary then to find the variation of the last term caused by increasing f by a small amount df. This will be the product F . df of the force F required and the displacement.

Either method may be applied to find the forces of attraction and repulsion for the systems of electrified spheres described below.

The problem of two mutually influencing non-intersecting spheres, S₁, S₂ (Fig. 5), insulated with given charges, q₁, q₂, may now be dealt with in the following manner. Let each be supposed at first charged uniformly. By the known theorem referred to above, the external field of each is the same as if its whole charge were situated at the centre. Now if the distribution on S₂, say, be kept unaltered, while that on S₁ is allowed to change, the action of S₂ on S₁ is the same as if the charge q₂ were at the centre C₂ of S₂. Thus if f be the distance between the centres C₁, C₂, and a₁ be the radius of S₁, the distribution will be that corresponding to q₁ + a₁q₂ ⁄ f uniformly distributed on S₁ together with the induced charge − a₁q₂ ⁄ f, which corresponds to the image-charge at the point I₁ (within S₁), the inverse of C₂ with respect to S₁. Now let the charge on S₁ be fixed in the state just supposed while that on S₂ is freed. The charge on S₂ will rearrange itself under the influence of q₁ + a₁q₂ ⁄ f ( = q') and − a₁q₂ ⁄ f, considered as at C₁ and I₁ respectively. The former of these will give a distribution equivalent to q₂ + a₂q' ⁄ f uniformly distributed over S₂, and an induced distribution of amount − a₂q' ⁄ f at J₁, the inverse point of C₁ with regard to S₂. The image-charge − a₁q₂ ⁄ f at I₁ in S₁ will react on S₂ and give an induced distribution − a₂ (− a₁q₂ ⁄ f ) f', (I₁C₂ = f' ) corresponding to an image-charge a₂a₁q₂ ⁄ ff' at the inverse point J₂ of P₁ with respect to C₂S₂. Thus the distribution on S₂ is equivalent to q₂ + a₂q' ⁄ f − a₂a₁q₂ ⁄ ff' at the inverse point J₂ of P₁ distributed uniformly over it, together with the two induced distributions just described.

Fig. 5.

In the same way these two induced distributions on S₂ may now be regarded as reacting on the distribution on S₁ as would point-charges − a₂q₁ ⁄ f and a₂a₁q₂ ⁄ ff', situated at J₁ and J₂ respectively, and would give two induced distributions on S₁ corresponding to their images in S₁.

Thus by partial influences in unending succession the equilibrium state of the two spheres could be approximated to as nearly as may be desired. An infinite trail of electric images within each of the two spheres is thus obtained, and the final state of each conductor can be calculated by summation of the effects of each set of images.

If the final potentials, V₁, V₂, say, of the spheres are given the process is somewhat simpler. Let first the charges be supposed to exist uniformly distributed over each sphere, and to be of amount a₁V₁, a₂V₂ in the two cases. The uniform distribution on S₁ will raise the potential of S₂ above V₂, and to bring the potential down to V₂ in presence of this distribution we must place an induced distribution over S₂, represented as regards the external field by the image-charge − a₂a₁V₁ ⁄ f (at the image of C₁ in S₂) where f is the distance between the centres. The charge a₂V₂ on S₂ will similarly have an action on S₁ to be compensated in the same way by an image-charge − a₁a₂V₂ ⁄ f at the image of C₂ in S₁. Now these two image-charges will react on the spheres S₁ and S₂ respectively, and will have to be balanced by induced distributions represented by second image-charges, to be found in the manner just exemplified. These will again react on the spheres and will have to be compensated as before, and so on indefinitely. The charges diminish in amount, and their positions approximate more and more, according to definite laws, and the final state is to be found by summation as before.

The force of repulsion is to be found by summing the forces between all the different pairs of charges which can be formed by taking one charge of each system at its proper point: or it can be obtained by calculating the energy of the system.

The method of successive influences was given originally by Murphy, but the mode of representing the effects of the successive induced charges by image-charges is due to Thomson. Quite another solution of this problem is, however, possible by Thomson's method of electrical inversion.

A similar process to that just explained for two charged and mutually influencing spheres will give the distribution on two concentric conducting spheres, under the influence of a point-charge q at P between the inner surface of the outer and the outer surface of the inner, as shown in Fig. 7. There the influence of q at P, and of the induced distributions on one another, is represented by two series of images, one within the inner sphere and one outside the outer. These charges and positions can be calculated from the result for a single sphere and point-charge.

Thomson's method of electrical inversion, referred to above, enabled the solutions of unsolved problems to be inferred from known solutions of simpler cases of distribution. We give here a brief account of the method, and some of its results. First we have to recall the meaning of geometrical inversion. In Fig. 6 the distances OP, OP', OQ, OQ' fulfil the relation OP.OP' = OQ.OQ' = a². Thus P' is (see p. 37) the inverse of the point P with respect to a sphere of radius a and centre O (indicated by the dotted line in Fig. 6), and similarly Q' is the inverse of Q with respect to the same sphere and centre. O is called the centre of inversion, and the sphere of radius a is called the sphere of inversion. Thus the sphere of Figs. 1 and 4 is the sphere of inversion for the points P and P', which are inverse points of one another. For any system of points P, Q, ..., another system P', Q', ... of inverse points can be found, and if the first system form a definite locus, the second will form a derived locus, which is called the inverse of the former. Also if P', Q', ... be regarded as the direct system, P, Q, ... will be the corresponding inverse system with regard to the same sphere and centre. P' is the image of P, and P is the image of P', and so on, with regard to the same sphere and centre of inversion.

Fig. 6.

The inverse of a circle is another circle, and therefore the inverse of a sphere is another sphere, and the inverse of a straight line is a circle passing through the centre of inversion, and of an infinite plane a sphere passing through the centre of inversion. Obviously the inverse of a sphere concentric with the sphere of inversion is a concentric sphere.

The line P'Q' is of course not the inverse of the line PQ, which has for its inverse the circle passing through the three points O, P', Q', as indicated in Fig. 6.

The following results are easily proved.

A locus and its inverse cut any line OP at the same angle.

To a system of point-charges q₁, q₂, ... at points P₁, P₂, ... on one side of the surface of the sphere of inversion there is a system of charges aq₁ ⁄ f₁, aq₂ ⁄ f₂, ... on the other side of the spherical surface [OP₁ = f₁, OP₂ = f₂]. This inverse system, as we shall call it, produces the same potential at any point of the sphere of inversion, as does the direct system from which it is derived.

If V, V' be the potentials produced by the whole direct system at Q, and by the whole inverse system at Q', V' ⁄ V = r ⁄ a = a ⁄ r', where OQ = r, OQ' = r'.

Thus if V is constant over any surface S', V' is not a constant over the inverse surface S', unless r is a constant, that is, unless the surface S' is a sphere concentric with the sphere of inversion, in which case the inverse surface is concentric with it and is an equipotential surface of the inverse distribution.

Further, if q be distributed over an element dS of a surface, the inverse charge aq ⁄ f will be distributed over the corresponding element dS' of the inverse surface. But dS' ⁄ dS = a⁴ ⁄ f⁴ = f'⁴ ⁄ a⁴ where f, f' are the distances of O from dS and dS'. Thus if s be the density on dS and s' the inverse density on dS' we have s' ⁄ s = a³ ⁄ f'³ = f³ ⁄ a³.

When V is constant over the direct surface, while r has different values for different directions of OQ, the different points of the inverse surface may be brought to zero potential by placing at O a charge − aV. For this will produce at Q' a potential − aV ⁄ r' which with V' will give at Q' a potential zero. This shows that V' is the potential of the induced distribution on S' due to a charge − aV at O, or that − V' is the potential due to the induced charge on S' produced by the charge aV at O.

Fig. 7.

Thus we have the conclusion that by the process of inversion we get from a distribution in equilibrium, on a conductor of any form, an induced distribution on the inverse surface supposed insulated and conducting; and conversely we obtain from a given induced distribution on an insulated conducting surface, a natural equilibrium distribution on the inverse surface. In each case the inducing charge is situated at the centre of inversion. The charges on the conductor (or conductors) after inversion are always obtainable at once from the fact that they are the inverses of the charges on the conductor (or conductors) in the direct case, and the surface-densities or volume-densities can be found from the relations stated above.

Now take the case of two concentric spheres insulated and influenced by a point-charge q placed at a point P between them as shown in Fig. 7. We have seen at p. 49 how the induced distribution, and the amount of the charge, on each sphere is obtained from the two convergent series of images, one outside the outer sphere, the other inside the inner sphere. We do not here calculate the density of distribution at any point, as our object is only to explain the method; but the quantities on the spheres S₁ and S₂, are respectively − q.OA.PB ⁄ (OP.AB), − q.OB.AP ⁄ (OP.AB).

It may be noticed that the sum of the induced charges is − q, and that as the radii of the spheres are both made indefinitely great, while the distance AB is kept finite, the ratios OA ⁄ OP, OB ⁄ OP approximate to unity, and the charges to − q.PB ⁄ AB, − q.AP ⁄ AB, that is, the charges are inversely as the distances of P from the nearest points of the two surfaces. But when the radii are made indefinitely great we have the case of two infinite plane conducting surfaces with a point-charge between them, which we have described above.

Now let this induced distribution, on the two concentric spheres, be inverted from P as centre of inversion. We obtain two non-intersecting spheres, as in Fig. 5, for the inverse geometrical system, and for the inverse electrical system an equilibrium distribution on these two spheres in presence of one another, and charged with the charges which are the inverses of the induced charges. These maintain the system of two spheres at one potential. From this inversion it is possible to proceed as shown by Maxwell in his Electricity and Magnetism, vol. i, § 173, to the distribution on two spheres at two different potentials; but we have shown above how the problem may be dealt with directly by the method of images.

Fig. 8.

Again take the case of two parallel infinite planes under the influence of a point-charge between them. This system inverted from P as centre gives the equilibrium distribution on two charged insulated spheres in contact (Fig. 8); for this system is the inverse of the planes and the charges upon them. Another interesting case is that of the "electric kaleidoscope" referred to above. Here the two infinite conducting planes are inclined at an angle 360° ⁄ n, where n is a whole number, and are therefore bounded in one direction by the straight line which is their intersection. The image points I₁, J₁, ..., of P placed in the angle between the planes are situated as shown in Fig. 3, and are n − 1 in number. This system inverted from P as centre gives two spherical surfaces which cut one another at the same angle as do the planes. This system is one of electrical equilibrium in free space, and therefore the problem of the distribution on two intersecting spheres is solved, for the case at least in which the angle of intersection is an aliquot part of 360°. When the planes are at right angles the result is that for two perpendicularly intersecting planes, for which Fig. 9 gives a diagram.

Fig. 9.

But the greatest achievement of the method was the determination of the distribution on a segment of a thin spherical shell with edge in one plane. The solution of this problem was communicated to M. Liouville in the letter of date September 16, 1846, referred to above, but without proof, which Thomson stated he had not time to write out owing to preparation for the commencement of his duties as Professor of Natural Philosophy at Glasgow on November 1, 1846. It was not supplied until December 1868 and January 1869; and in the meantime the problem had not been solved by any other mathematician.

As a starting point for this investigation the distribution on a thin plane circular disk of radius a is required. This can be obtained by considering the disk as a limiting case of an oblate ellipsoid of revolution, charged to potential V, say. If Fig. 10 represent the disk and P the point at which the density is sought, so that CP = r, and CA = a, the density is V ⁄ {2π²√(a² − r²)}.

The ratio q ⁄ V, of charge to potential, which is called the electrostatic capacity of the conductor, is thus 2a ⁄ π, that is a ⁄ 1.571. It is, as Thomson notes in his paper, very remarkable that the Hon. Henry Cavendish should have found long ago by experiment with the rudest apparatus the electrostatic capacity of a disk to be 1 ⁄ 1.57 of that of a sphere of the same radius.

Fig. 10.

Fig. 11.

Now invert this disk distribution with any point Q as centre of inversion, and with radius of inversion a. The geometrical inverse is a segment of a spherical surface which passes through Q. The inverse distribution is the induced distribution on a conducting shell uninsulated and coincident with the segment, and under the influence of a charge − aV situated at Q (Fig. 11). Call this conducting shell the "bowl." If the surface-densities at corresponding points on the disk and on the inverse, say points P and P', be s and s', then, as on page 51, s' = sa³ ⁄ QP'³. If we put in the value of s given above, that of s' can be put in a form given by Thomson, which it is important to remark is independent of the radius of the spherical surface. This expression is applicable to the other side of the bowl, inasmuch as the densities at near points on opposite sides of the plane disk are equal.

If v, v' be the potentials at any point R of space, due to the disk and to its image respectively, − v' = av ⁄ QR. If then R be coincident with a point P' on the spherical segment we have (since then v = V) V' = aV ⁄ QP', which is the potential due to the induced distribution caused by the charge − aV at Q as already stated.

The fact that the value of s' does not involve the radius makes it possible to suppose the radius infinite, in which case we have the solution for a circular disk uninsulated and under the influence of a charge of electricity at a point Q in the same plane but outside the bounding circle.

Now consider the two parts of the spherical surface, the bowl B, and the remainder S of the spherical surface. Q with the charge − aV may be regarded as situated on the latter part of the surface. Any other influencing charges situated on S will give distributions on the bowl to be found as described above, and the resulting induced electrification can be found from these by summation. If S be uniformly electrified to density s, and held so electrified, the inducing distribution will be one given by integration over the whole of S, and the bowl B will be at zero potential under the influence of this electrification of S, just as if B were replaced by a shell of metal connected to the earth by a long fine wire. The densities are equal at infinitely near points on the two sides of B.

Let the bowl be a thin metal shell connected with the earth by a long thin wire and be surrounded by a concentric and complete shell of diameter f greater than that of the spherical surface, and let this shell be rigidly electrified with surface density − s. There will be no force within this shell due to its own electrification, and hence it will produce no change of the distribution in the interior. But the potential within will be − 2πfs, for the charge is − πf²s, and the capacity of the shell is ½f. The potential of the bowl will now be zero, and its electrification will just neutralise the potential − 2πfs, that is, will be exactly the free electrification required to produce potential 2πfs.

To find this electrification let the value of f be only infinitesimally greater than the diameter of the spherical surface of which B is a part; then the bowl is under the influence (1) of a uniform electrification of density − s infinitely close to its outer surface, and (2) of a uniform electrification of the same density, which may be regarded as upon the surface which has been called S above. It is obvious that by (1) a density s is produced on the outer surface of the bowl, and no other effect; by (2) an equal density at infinitely near points on the opposite sides of the bowl is produced which we have seen how to calculate. Thus the distribution on the bowl freely electrified is completely determined and the density can easily be calculated. The value will be found in Thomson's paper.

Interesting results are obtained by diminishing S more and more until the shell is a complete sphere with a circular hole in it. Tabulated results for different relative dimensions of S will be found in Thomson's paper, "Reprint of Papers," Articles V, XIV, XV. Also the reader will there find full particulars of the mathematical calculations indicated in this chapter, and an extension of the method to the case of an influencing point not on the spherical surface of which the shell forms part. Further developments of the problem have been worked out by other writers, and further information with references will be found in Maxwell's Electricity and Magnetism, loc. cit.

It is not quite clear whether Thomson discovered geometrical inversion independently or not: very likely he did. His letter to Liouville of date October 8, 1845, certainly reads as if he claimed the geometrical transformation as well as the application to electricity. Liouville, however, in his Note in which he dwells on the analytical theory of the transformation says, "La transformation dont il s'agit est bien connue, du reste, et des plus simples; c'est celle que M. Thomson lui-même a jadis employée sous le nom de principe des images." In Thomson and Tail's Natural Philosophy, § 513, the reference to the method is as follows: "Irrespectively of the special electric application, the method of images gives a remarkable kind of transformation which is often useful. It suggests for mere geometry what has been called the transformation by reciprocal radius-rectors, that is to say...." Then Maxwell, in his review of the "Reprint of Papers" (Nature, vol. vii), after referring to the fact that the solution of the problem of the spherical bowl remained undemonstrated from 1846 to 1869, says that the geometrical idea of inversion had probably been discovered and rediscovered repeatedly, but that in his opinion most of these discoveries were later than 1845, the date of Thomson's first paper.10

A very general method of finding the potential at any point of a region of space enclosed by a given boundary was stated by Green in his 'Essay' for the case in which the potential is known for every point of the boundary. The success of the method depends on finding a certain function, now called Green's function. When this is known the potential at any point is at once obtained by an integration over the surface. Thomson's method of images amounts to finding for the case of a region bounded by one spherical surface or more the proper value of Green's function. Green's method has been successfully employed in more complicated cases, and is now a powerful method of attack for a large range of problems in other departments of physical mathematics. Thomson only obtained a copy of Green's paper in January 1845, and probably worked out his solutions quite independently of any ideas derived from Green's general theory.