Fig. 1.Fig. 1.
Fig. 1.
Now let heat be poured into the medium at constant rate by a single point-sourceP(Fig.1), and drawn off at a smaller rate by a single point-sinkP', 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 sphereSsurrounding 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. Ifabe the radius of the sphere andfthe distance of the source from the centre of the sphere, the heat carried off by the sink is the fractiona⁄fof 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 therearriving 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.43below) 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, orare 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, andan 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. 2.
Fig. 2.
Fig. 3.Fig. 3.
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 sheetsAandB(Fig.2), and regard them for the moment as mirrors. A first image of an objectPbetween the two mirrors is produced directly by each, but the imageI1inAis virtually an object in front ofB, and the imageJ1inBan objectin front ofA, so that a second image more remote from the mirror than the first is produced in each case. These second imagesI2andJ2in 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 imagein 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 fromAand fromB.
Fig. 4.Fig. 4.
Fig. 4.
Now consider the spherical surface referred to at p.37, which is kept at uniform potential by a charge at the external pointP, and a chargeq'at the inverse pointP'within the sphere. IfE(Fig.4) be any point whatever on the surface, andr,r'be its distances fromPandP', it is easy to prove by geometry that the two trianglesCPEandCEP'are similar, and thereforer'=ra⁄f. [Herea⁄fis used to meanadivided byf. 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 chargeqat another point, is equal to the ratio of the chargeqto the distance between the points. Thus the potential atEdue to the chargeqatPisq⁄r, and that atEdue to a chargeq'atP'isq'⁄r'. Thus ifq'= −qa⁄f,q'atP'will produce a potential atE= −qa⁄fr'= −q⁄r, by the value ofr. HenceqatPand −qa⁄fatP'coexisting will give potentialq⁄r+ −q⁄ror zero, atE. Thus the charge −qa⁄f, at the internal pointP'will in presence of +qatPkeep all points of the spherical surface at zero potential. These two charges represent the source and sink in the thermal analogue of p.37above.
Now replaceSby 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 chargeqatPalone, a charge is induced upon it which, in presence ofP, 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 byqatP. 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 allexternalpoints, 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 byN, andsbe the surface-density: then according to the system of units usually adopted 4πs=N.
Let now the rate of diminution of potential per unit ofdistance 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⁄fatP'andqatP, the falling off gradientNof potential outside the spherical surface at any pointE, and to takeN⁄ 4π, to obtainsthe surface-density atE. Calculation of this gradient for the sphere gives 4πs= −q(f2−a2) ⁄ar3. The surface-density is thus inversely as the cube of the distancePE.
If the influencing pointPbe 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 pointP'will now be outside, but given by the same relation. Andawill now be greater thanf, and the density will be given by 4πs= −q(a2−f2) ⁄ar3, where,fandrhave the same meanings with regard toEandPas before.
P'is in each case called the image ofPin the sphereS, and the charge −qa⁄fthere supposed situated is theelectric imageof the chargeqatP. It will be seen that an electric image is a charge, or system ofcharges, 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 distributionDon the other side of the surface, this surface distribution may be equivalent to several different arrangements ofD. Thus the point-charge atP'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 atP'(Fig.4) would do as well, provided this spherical surface were not large enough to extend beyond the surfaceS.
In order to find the potential of the sphere (Fig.4) when insulated with a chargeQupon it, in presence of the influencing chargeqat the external pointP, it is only necessary to imagine uniformly distributed over the sphere, already electrified in the manner just explained, the chargeQ+aq⁄f. Then the whole charge will beQ, and the uniformity of distribution will be disturbed, as required by the action of the influencing point-charge. The potential will beQ⁄a+q⁄f. For a given potentialVof the sphere, the total charge isaV−aq⁄f, that is the charge isaVover and above the induced charge.
If instead of a single influencing point-charge atPthere 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 isthat obtained by superimposing all the surface distributions found for the different influencing points.
The force of repulsion between the point-chargeqand the sphere (with total chargeQ) can be found at once by calculating the sum of the forces betweenqatPand the chargesQ+aq⁄fatCand −aq⁄fatP'.
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 chargeQuninfluenced by an external charge, one representing the energy on a small conductor (not a point) atPexisting alone, and a third representing the mutual energy of the electrification on the sphere and the chargeqatPexisting 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 increasingfby a small amountdf. This will be the productF . dfof the forceFrequired 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 onS2, say, be kept unaltered,while that onS1is allowed to change, the action ofS2onS1is the same as if the chargeq2were at the centreC2ofS2. Thus iffbe the distance between the centresC1,C2, anda1be the radius ofS1, the distribution will be that corresponding toq1+a1q2⁄funiformly distributed onS1together with the induced charge −a1q2⁄f, which corresponds to the image-charge at the pointI1(withinS1), the inverse ofC2with respect toS1. Now let the charge onS1be fixed in the state just supposed while that onS2is freed. The charge onS2will rearrange itself under the influence ofq1+a1q2⁄f( =q') and −a1q2⁄f, considered as atC1andI1respectively. The former of these will give a distribution equivalent toq2+a2q'⁄funiformly distributed overS2, and an induced distribution of amount −a2q'⁄fatJ1, the inverse point ofC1with regard toS2. The image-charge −a1q2⁄fatI1inS1will react onS2and give an induced distribution −a2(−a1q2⁄f)f', (I1C2=f') corresponding to an image-chargea2a1q2⁄ff'at the inverse pointJ2ofP1with respect toC2S2. Thus the distribution onS2is equivalent toq2+a2q'⁄f−a2a1q2⁄ff'at the inverse pointJ2ofP1distributeduniformly over it, together with the two induced distributions just described.
Fig. 5.Fig. 5.
Fig. 5.
In the same way these two induced distributions onS2may now be regarded as reacting on the distribution onS1as would point-charges −a2q1⁄fanda2a1q2⁄ff', situated atJ1andJ2respectively, and would give two induced distributions onS1corresponding to their images inS1.
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 amounta1V1,a2V2in the two cases. The uniform distribution onS1will raise the potential ofS2aboveV2, and to bring the potential down toV2in presence of this distribution we must place an induced distribution overS2, represented as regards the external field by the image-charge −a2a1V1⁄f(at the image ofC1inS2) wherefis the distance between the centres. The chargea2V2onS2will similarly have an action onS1to be compensated in the same way by an image-charge −a1a2V2⁄fat the image ofC2inS1. Now these two image-charges will react on the spheresS1andS2respectively, 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 reacton 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-chargeqatPbetween the inner surface of the outer and the outer surface of the inner, as shown in Fig.7. There the influence ofqatP, 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.6the distancesOP,OP',OQ,OQ'fulfil the relationOP.OP'=OQ.OQ'=a2. ThusP'is (see p.37) the inverse of the pointPwith respect to a sphere of radiusaand centreO(indicated by the dotted line in Fig.6), and similarlyQ'is the inverse ofQwith respect to the same sphere and centre.Ois called the centre of inversion, and the sphere of radiusais called the sphere of inversion. Thus the sphere of Figs.1and4is the sphere of inversion for the pointsPandP', which are inverse points of one another. For any system of pointsP,Q, ..., another systemP',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 ifP',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 ofP, andPis the image ofP', and so on, with regard to the same sphere and centre of inversion.
Fig. 6.Fig. 6.
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 spherepassing through the centre of inversion. Obviously the inverse of a sphere concentric with the sphere of inversion is a concentric sphere.
The lineP'Q'is of course not the inverse of the linePQ, which has for its inverse the circle passing through the three pointsO,P',Q', as indicated in Fig.6.
The following results are easily proved.
A locus and its inverse cut any lineOPat the same angle.
To a system of point-chargesq1,q2, ... at pointsP1,P2, ... on one side of the surface of the sphere of inversion there is a system of chargesaq1⁄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.
IfV,V'be the potentials produced by the whole direct system atQ, and by the whole inverse system atQ',V'⁄V=r⁄a=a⁄r', whereOQ=r,OQ'=r'.
Thus ifVis constant over any surfaceS',V'is not a constant over the inverse surfaceS', unlessris a constant, that is, unless the surfaceS'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, ifqbe distributed over an elementdSof a surface, the inverse chargeaq⁄fwill be distributed over the corresponding elementdS'of the inverse surface. ButdS'⁄dS=a4⁄f4=f'4⁄a4wheref,f'are the distances ofOfromdSanddS'. Thus ifsbe the density ondSands'the inverse density ondS'we haves'⁄s=a3⁄f'3=f3⁄a3.
WhenVis constant over the direct surface, whilerhas different values for different directions ofOQ, the different points of the inverse surface may be brought to zero potential by placing atOa charge −aV. For this will produce atQ'a potential −aV⁄r'which withV'will give atQ'a potential zero. This shows thatV'is the potential of the induced distribution onS'due to a charge −aVatO, or that −V'is the potential due to the induced charge onS'produced by the chargeaVatO.
Fig. 7.Fig. 7.
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-chargeqplaced at a pointPbetween them as shown in Fig.7. We have seen at p.49how the induced distribution, and theamount 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 spheresS1andS2, 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 distanceABis kept finite, the ratiosOA⁄OP,OB⁄OPapproximate to unity, and the charges to −q.PB⁄AB, −q.AP⁄AB, that is, the charges are inversely as the distances ofPfrom 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 fromPas 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 hisElectricity 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.Fig. 8.
Fig. 8.
Again take the case of two parallel infinite planes under the influence of a point-charge between them. This system inverted fromPas 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, wherenis a whole number, and are therefore bounded in one direction by the straight line which is their intersection. The image pointsI1,J1, ..., ofPplaced in the angle between the planes are situated as shown in Fig.3, and aren− 1 in number. This system inverted fromPas 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 twoperpendicularly intersecting planes, for which Fig.9gives a diagram.
Fig. 9.Fig. 9.
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 radiusais required. This can be obtained by considering the diskas a limiting case of an oblate ellipsoid of revolution, charged to potentialV, say. If Fig.10represent the disk andPthe point at which the density is sought, so thatCP=r, andCA=a, the density isV⁄ {2π2√(a2−r2)}.
The ratioq⁄V, of charge to potential, which is called the electrostatic capacity of the conductor, is thus 2a⁄ π, that isa⁄ 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. 10.
Fig. 10.
Fig. 11.Fig. 11.
Fig. 11.
Now invert this disk distribution with any pointQas centre of inversion, and with radius of inversiona. The geometrical inverse is a segment of a spherical surface which passes throughQ. The inverse distribution is the induced distribution on a conducting shell uninsulated and coincident with the segment, and under the influence of a charge −aVsituated atQ(Fig.11). Call this conducting shell the "bowl." If the surface-densities at corresponding points on the disk and on the inverse, say pointsPandP', besands', then, ason page51,s'=sa3⁄QP'3. If we put in the value ofsgiven above, that ofs'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.
Ifv,v'be the potentials at any pointRof space, due to the disk and to its image respectively, −v'=av⁄QR. If thenRbe coincident with a pointP'on the spherical segment we have (since thenv=V)V'=aV⁄QP', which is the potential due to the induced distribution caused by the charge −aVatQas already stated.
The fact that the value ofs'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 pointQin the same plane but outside the bounding circle.
Now consider the two parts of the spherical surface, the bowlB, and the remainderSof the spherical surface.Qwith the charge −aVmay be regarded as situated on the latter part of the surface. Any other influencing charges situated onSwill give distributions on the bowl to be found as described above, and the resulting induced electrification can be found from these by summation. IfSbe uniformly electrified to densitys, and held so electrified, the inducing distribution will be one given byintegrationover the whole ofS, and the bowlBwill be at zero potential under the influence of this electrification ofS, just as ifBwere replaced by a shell of metal connected tothe earth by a long fine wire. The densities are equal at infinitely near points on the two sides ofB.
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 diameterfgreater 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 offbe only infinitesimally greater than the diameter of the spherical surface of whichBis a part; then the bowl is under the influence (1) of a uniform electrification of density −sinfinitely 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 calledSabove. It is obvious that by (1) a densitysis 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 diminishingSmore and more until the shell is a complete sphere with a circular hole in it. Tabulated results fordifferent relative dimensions ofSwill 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'sElectricity and Magnetism, loc. cit.
It is not quite clear whether Thomson discoveredgeometricalinversion 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 desimages." In Thomson and Tail'sNatural 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 hisopinion 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.
Theincumbent of the Chair of Natural Philosophy in the University of Glasgow, Professor Meikleham, had been in failing health for several years, and from 1842 to 1845 his duties had been discharged by another member of the Thomsongens, Mr. David Thomson, B.A., of Trinity College, Cambridge, afterwards Professor of Natural Philosophy at Aberdeen. Dr. Meikleham died in May 1846, and the Faculty thereafter proceeded on the invitation of Dr. J. P. Nichol, the Professor of Astronomy, to consider whether in consequence of the great advances of physical science during the preceding quarter of a century it was not urgently necessary to remodel the arrangements for the teaching of natural philosophy in the University. The advance of science had indeed been very great. Oersted and Ampère, Henry and Faraday and Regnault, Gauss and Weber, had made discoveries and introduced quantitative ideas, which had changed the whole aspect of experimental and mathematical physics. The electrical discoveries of the time reacted on the other branches of natural philosophy, and in no small degree on mathematics itself. As a result the progress of that period has continued and has increased in rapidity,until now the accumulated results, for the most part already united in the grasp of rational theory, have gone far beyond the power of any single man to follow, much less to master.
It is interesting to look into a course of lectures such as were usually delivered in the universities a hundred years ago by the Professor of Natural Philosophy. We find a little discussion of mechanics, hydrostatics and pneumatics, a little heat, and a very little optics. Electricity and magnetism, which in our day have a literature far exceeding that of the whole of physics only sixty years ago, could hardly be said to exist. The professor of the beginning of the nineteenth century, when Lord Kelvin's predecessor was appointed, apparently found himself quite free to devote a considerable part of each lecture to reflections on the beauties of nature, and to rhetorical flights fitter for the pulpit than for the physics lecture-table.
In the intervening time the form and fashion of scientific lectures has entirely changed, and the change is a testimony to the progress of science. It is visible even in the design of the apparatus. Microscopes, for example, have a perfection and a power undreamed of by our great-grandfathers, and they are supported on stands which lack the ornamentation of that bygone time, but possess stability and convenience. Everything and everybody—even the professor, if that be possible—must be business-like; and each moment of time must be utilised in experiments for demonstration, not for applause, and in brief and cogent statements of theory and fact. To waste time in talk that is not to the point is criminal. But withal there is need of grace of expression and vividness of description, ofclearness of exposition, of imagination, even of poetical intuition: but the stern beauty of modern science is only disfigured by the old artificial adornments and irrelevancies.
This is the tone and temper of science at the present day: the task is immense, the time is short. And sixty years since some tinge of the same cast of thought was visible in scientific workers and teachers. The Faculty agreed with Dr. Nichol that there was need to bring physical teaching and equipment into line with the state of science at the time; but they wisely decided to do nothing until they had appointed a Professor of Natural Philosophy who would be able to advise them fully and in detail. They determined, however, to make the appointment subject to such alterations in the arrangements of the department as they might afterwards find desirable.
On September 11, 1846, the Faculty met, and having considered the resolutions which had been proposed by Dr. Nichol, resolved to the effect that the appointment about to be made should not prejudice the right of the Faculty to originate or support, during the incumbency of the new professor, such changes in the arrangements for conducting instruction in physical science as it might be expedient to adopt, and that this resolution should be communicated to the candidate elected. The minute then runs: "The Faculty having deliberated on the respective qualifications of the gentlemen who have announced themselves candidates for this chair, and the vote having been taken, it carried unanimously in favour of Mr. William Thomson, B.A., Fellow of St. Peter's College, Cambridge, and formerly a student of this University, who is accordinglydeclared to be duly elected: and Mr. Thomson being within call appeared in Faculty, and the whole of this minute having been read to him he agreed to the resolution of Faculty above recorded and accepted the office." It was also resolved as follows: "The Faculty hereby prescribe Mr. Thomson an essay on the subject,De caloris distributione per terræ corpus, and resolve that his admission be on Tuesday the 13th October, provided that he shall be found qualified by the Meeting and shall have taken the oath and made the subscriptions which are required by law."
At that time, and down to within the last fifteen years, every professor, before his induction to his chair, had to submit a Latin essay on some prescribed subject. This was almost the last relic of the customs of the days when university lectures were delivered in Latin, a practice which appears to have been first broken through by Adam Smith when Professor of Moral Philosophy. Whatever it may have been in the eighteenth century, the Latin essay at the end of the nineteenth was perhaps hardly an infallible criterion of the professor-elect's Latinity, and it was just as well to discard it. But fifty years before, and for long after, classical languages bulked largely in the curriculum of every student of the Scottish Universities, and it is undoubtedly the case that most of those who afterwards came to eminence in other departments of learning had in their time acquitted themselves well in the oldLitteræ Humaniores. This was true, as we have seen, of Thomson, and it is unlikely that the form of his inaugural dissertation cost him much more effort than its matter.