A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 9 of 28

89 ej] If a new conductor is brought into the field the coefficient of potential of any one of the others on itself is diminished.

For let the new body, B, be supposed at first to be a non-conductor free from charge in any part, then when one of the conductors, A1 > receives a charge el9 the distribution of the electricity on the con- ductors of the system will not be disturbed by JB, as B is still without charge in any part, and the electric energy of the system will be simply K^i = W^n-

Now let B become a conductor. Electricity will flow from places of higher to places of lower potential, and in so doing will diminish the electric energy of the system, so that the quantity 4 ei2Pn must diminish.

But el remains constant, therefore pli must diminish.

Also if B increases by another body b being placed in contact with it, /?n will be further diminished.

For let us first suppose that there is no electric communication

between B and 6 ; the introduction of the new body b will

,' diminish />n. Now let a communication be opened between B

906.] APPEOXIMATE VALUES OF THE COEFFICIENTS. 107

and b. If any electricity flows through it, it flows from a place of higher to a place of lower potential, and therefore, as we have shown, still further diminishes jOn.

Hence the diminution of jt?n by the body B is greater than that which would be produced by any body the surface of which can be inscribed in B} and less than that produced by any body the surface of which can be described about B.

We shall shew in Chapter XI, that a sphere of diameter 6 at a distance r diminishes the value of ptl by a quantity which is

approximately J -^ •

Hence if the body B is of any other figure, and if b is its greatest diameter, the diminution of the value of jt?11 must be less

than §• — .

Hence if the greatest diameter of B is so small compared with its distance from Al that we may neglect quantities of the order

b5 §• -4- 5 we may consider the reciprocal of the capacity of A± when

alone in the field as a sufficient approximation to pl t .

90 a.~\ Let us therefore suppose that the capacity of A1 when alone in the field is K19 and that of A^ K2, and let the mean distance between Al and A2 be r, where r is very great compared with the dimensions of Al and A2) then we may write

JL - i _!•

F^e.K^ + e.r-i, ?£ t,* firf + fa ^

F2 = e^ + e2K2-\ Hence 9n = K^ (l _J ^^^ ^

i s . *

222 = K2 ( 1 - JTj KI r-2)-1 .

Of these coefficients qn and q22 are the capacities of A± and A2 when, instead of being each alone at an infinite distance from any other body, they are brought so as to be at a distance r from each other.

90 b.~\ When two conductors are placed so near together that their coefficient of mutual induction is large, the combination is called a Condenser.

Let A and B be the two conductors or electrodes of a con- denser.

&f z

• A* r 1

108 SYSTEM OF CONDUCTORS. [906.

Let L be the capacity of A, N that of j5, and M the coefficient of mutual induction. (We must remember that M is essentially negative, so that the numerical value of L + M and M+ N is less than L or N.)

Let us suppose that a and I are the electrodes of another con- denser at a distance R from the first, R being very great com- pared with the dimensions of either condenser, and let the coefficients of capacity and induction of the condenser al when alone be I, m, n. Let us calculate the effect of one of the condensers on the coefficients of the other.

Let D = LN-M2 and d=ln-m2-,

then the coefficients of potential for each condenser by itself are

PAB = — PBB =

The values of these coefficients will not be sensibly altered when the two condensers are at a distance R.

The coefficient of potential of any two conductors at distance R is R~l, so that

The equations of potential are therefore

VA =

V =

Solving these equations for the charges, we find

1 R*-(L+2M+N)(l+2m +

(L qAB =M'=M+ ^-

R(L+M)(l+m)

where If, M\ N' are what L, M, N become when the second con- denser is brought into the field.

91.] APPROXIMATE VALUES OF THE COEFFICIENTS. 109

If only one conductor, a, is brought into the field, m=n=0) and

If T i " - h

Rl(L+M)

If there are only the two simple conductors, A and a, M— N=m = n = 0,

L2l ELI

and qAA = L

R*-Ll'

expressions which are the same as those found in Art. 90 a.

The quantity L + 2 H + N is the total charge of the condenser when its electrodes are at potential 1. It cannot exceed half the greatest diameter of the condenser.

L-t-Mis the charge of the first electrode, and M + N that of the second when both are at potential 1. These quantities must be each of them positive and less than the capacity of the electrode by itself. Hence the corrections to be applied to the coefficients of capacity of a condenser are much smaller than those for a simple conductor of equal capacity.

Approximations of this kind are often useful in estimating the capacities of conductors of irregular form placed at a finite distance from other conductors.

91.] When a round conductor, J.%, of small size compared with the distances between the conductors, is brought into the field, the coefficient of potential of Al on A2 will be increased when A5 is inside and diminished when A3 is outside of a sphere whose diameter is the straight line ALA2.

For if AL receives a unit charge there will be a distribution of electricity on A3, --e being on the side furthest from Alt and — e on the side nearest A±. The potential at A2 due to this distribution A/ on A3 will be positive or negative as +e or —e is nearest to A2, and if the form of A3 is not very elongated this will depend on whether the angle Al A3 A2 is obtuse or acute, and therefore on whether Az is inside or outside the sphere described on Al A2 as ., ^ ^ diameter.

If A3 is of an elongated form it is easy to see that if it is placed with its longest axis in the direction of the tangent to the circle

110 SYSTEM OF CONDUCTORS. [92.

• ^ drawn through the points Alf Az, A2 it may increase the potential of A2, even when it is entirely outside the sphere, and how by placing it with its longest axis in the direction of the radius of ; .the sphere, it may diminish the potential of J2, even when entirely within the sphere. But this proposition is only intended for forming a rough estimate of the phenomena to be expected in a given arrangement of apparatus.

92.] If a new conductor, A3, is introduced into the field, the capacities of all the conductors already there are increased, and the numerical values of the coefficients of induction between every pair of them are diminished.

Let us suppose that A1 is at potential unity and all the rest at potential zero. Since the charge of the new conductor is negative it will induce a positive charge on every other conductor, and will therefore increase the positive charge of Al and diminish the negative charge of each of the other conductors.

93 a.~\ Work done ly the electric forces during the displacement of a system of insulated charged conductors.

Since the conductors are insulated, their charges remain constant during the displacement. Let their potentials be 7^, 7£, . . . ^n before and Tj', 7^', ... V£ after the displacement. The electrical energy is

before the displacement, and

r'-i

after the displacement.

The work done by the electric forces during the displacement is the excess of the initial energy W over the final energy W9 or

This expression gives the work done during any displacement, small or large, of an insulated system.

To find the force tending to produce a particular kind of dis- placement, let $ be the variable whose variation corresponds to the kind of displacement, and let 3> be the corresponding force, reckoned positive when the electric force tends to increase <£, then

where Wt denotes the expression for the electric energy as a quadratic function of the charges.

93C-] MECHANICAL FORCES. Ill

93 a.] To prove that - + = 0.

d(f> d$

We have three different expressions for the energy of the system,

(i) r=i2(«n,

a definite function of the n charges and n potentials

(2) ^=J2S («,«./»„),

where r and s may be the same or different, and both rs and sr are to be included in the summation.

This is a function of the n charges and of the variables which define the configuration. Let (/> be one of these.

(3) rr=iss(^.?r.),

where the summation is to be taken as before. This is a function of the n potentials and of the variables which define the configura- tion of which <f) is one.

Since W=W6=WV,

r-2W= 0.

Now let the n charges, the n potentials, and $ vary in any con- sistent manner, and we must have

Now the n charges, the n potentials, and $ are not all independent of each other, for in fact only n + 1 of them can be independent. But we have already proved that

*--

so that the first sum of terms vanishes identically, and it follows from this, even if we had not already proved it that

dWy _

JTf ~ 89

and that lastly,

4r-e + -771 = 0.

Work done by the electric forces during the displacement of a system whose potentials are maintained constant.

93 c.~\ It follows from the last equation t^at the force 4> = —

112 SYSTEM Or CONDUCTORS. [94.

and if the system is displaced under the condition that all the potentials remain constant, the work done by the electric forces is

or the work done by the electric forces in this case is equal to the increment of the electric energy.

Here,, then, we have an increase of energy together with a quan- tity of work done by the system. The system must therefore be supplied with energy from some external source, such as a voltaic battery, in order to maintain the potentials constant during the displacement.

The work done by the battery is therefore equal to the sum of the work done by the system and the increment of energy, or, since these are equal, the work done by the battery is twice the work done by the system of conductors during the displacement.

On the comparison of similar electrified systems. 94.] If two electrified systems are similar in a geometrical sense, so that the lengths of corresponding lines in the two systems are as L to If, then if the dielectric which separates the conducting bodies is the same in both systems, the coefficients of induction and of capacity will be in the proportion of L to L '. For if we consider corresponding portions, A and A ', of the two systems, and suppose the quantity of electricity on A to be £, and that on A' to be /, then the potentials 7 and 7' at corresponding points B and _Z?', due to this electrification, will be

• /

Y — - and V — - ~ AB ' " A'ff

But AB is to A'B' as L to L' ', so that we must have

e:e'::L7'. L'7'.

But if the inductive capacity of the dielectric is different in the two systems, being K in the first and K' in the second, then if the potential at any point of the first system is to that at the cor- responding point of the second as 7 to 7', and if the quantities of electricity on corresponding parts are as E to E', we shall have

ei<f::L7KiL'7'K'.

By this proportion we may find the relation between the total charges of corresponding parts of two systems, which are in the first place geometrically similar, in the second place com- posed of dielectric media of which the specific inductive capacity

94-] SIMILAR SYSTEMS. 113

at corresponding points is in the proportion of K to K', and in the third place so electrified that the potentials of corresponding points are as V to V .

From this it appears that if q be any coefficient of capacity or induction in the first system, and (£ the corresponding one in the second, q:q':-.LK:L'K'',

and if p and p' denote corresponding coefficients of potential in the two systems, \ \

If one of the bodies be displaced in the first system, and the corresponding body in the second system receive a similar dis- placement, then these displacements are in the proportion of L to Z', and if the forces acting on the two bodies are as F to F', then the work done in the two systems will be as FL to F'L'.

But the total electrical energy is half the sum of the charges of electricity multiplied each by the potential of the charged body, so that in the similar systems, if W and W be the total electrical energy in the two systems respectively,

and the difference of energy after similar displacements in the two systems will be in the same proportion. Hence, since FL is pro- portional to the electrical work done during the displacement,

Combining these proportions, we find that the ratio of the resultant force on any body of the first system to that on the corresponding body of the second system is

F-.F':: 72K: V'2K',

e2 e'2

F: F' ''''T^K'TT*!7'

The first of these proportions shews that in similar systems the force is proportional to the square of the electromotive force and to the inductive capacity of the dielectric, but is independent of the actual dimensions of the system.

Hence two conductors placed in a liquid whose inductive capacity is greater than that of air, and electrified to given potentials, will attract each other more than if they had been electrified to the same potentials in air.

The second proportion shews that if the quantity of electricity on each body is given, the forces are proportional to the squares

VOL. I, I

114 SYSTEM OF CONDUCTORS. [94.

of the charges and inversely to the squares of the distances, and also inversely to the inductive capacities of the media.

Hence, if two conductors with given charges are placed in a liquid whose inductive capacity is greater than that of air, they will attract each other less than if they had been surrounded with air and charged with the same quantities of electricity.

CHAPTEK IV.

GENERAL THEOREMS.

95#.] IN the second chapter we have calculated the potential function and investigated some of its properties on the hypothesis that there is a direct action at a distance between electrified bodies, which is the resultant of the direct actions between the various electrified parts of the bodies.

If we call this the direct method of investigation, the inverse method will consist in assuming that the potential is a function characterised by properties the same as those which we have already established, and investigating the form of the function.

In the direct method the potential is calculated from the dis- tribution of electricity by a process of integration, and is found to satisfy certain partial differential equations. In the inverse method the partial differential equations are supposed given, and we have to find the potential and the distribution of electricity.

It is only in problems in which the distribution of electricity is given that the direct method can be used. When we have to find the distribution on a conductor we must make use of the inverse method.

We have now to shew that the inverse method leads in every case to a determinate result, and to establish certain general theorems deduced from Poisson's partial differential equation

:rr + -j-* + ~T* + 47r/) = °-

dx2 dy* dz2

The mathematical ideas expressed by this equation are of a different kind from those expressed by the definite integral

/ + QO /* + <» /* + « n / / tM •00 J — 00 J — 00 T

In the differential equation we express that the sum of the second derivatives of V in the neighbourhood of any point is related to

I 2

116 GENERAL THEOKEMS. [95 &•

the density at that point in a certain manner, and no relation is expressed between the value of V at that point and the value of p at any point at a finite distance from it.

In the definite integral, on the other hand, the distance of the point (of, y' , 2'), at which p exists, from the point (#, y, z\ at which Y exists, is denoted by r, and is distinctly recognised in the expression to be integrated.

The integral, therefore, is the appropriate mathematical expression for a theory of action between particles at a distance, whereas the differential equation is the appropriate expression for a theory of action exerted between contiguous parts of a medium.

We have seen that the result of the integration satisfies the differential equation. We have now to shew that it is the only solution of that equation satisfying certain conditions.

We shall in this way not only establish the mathematical equi- valence of the two expressions, but prepare our minds to pass from the theory of direct action at a distance to that of action between contiguous parts of a medium.

95$.] The theorems considered in this chapter relate to the properties of certain volume-integrals taken throughout a finite region of space which we may refer to as the electric field.

The element of these integrals, that is to say, the quantity under the integral sign, is either the square of a certain vector quantity whose direction and magnitude varies from point to point in the field, or the product of one vector into the resolved part of another in its own direction.

Of the different modes in which a vector quantity may be dis- tributed in space, two are of special importance.

The first is that in which the vector may be represented as the space- variation [Art. 17] of a scalar function called the Potential.

Such a distribution may be called an Irrotational distribution. The resultant force arising from the attraction or repulsion of any combination of centres of force, the law of each being any given function of the distance, is distributed irrotationally.

The second mode of distribution is that in which the convergence [Art. 25] is zero at every point. Such a distribution may be called a Solenoidal distribution. The velocity of an incompressible fluid is distributed in a solenoidal manner.

^V hen the central forces which, as we have said, give rise to an irrolational distribution of the resultant force, vary according to

95 &•] IRROTATIONAL AND SOLENOIDAL DISTRIBUTIONS. 117

the inverse square of the distance, then, if these centres are outside the field, the distribution within the field will be solenoidal as well as irrotational.

When the motion of an incompressible fluid which, as we have said, is solenoidal, arises from the action of central forces depending on the distance, or of surface pressures, on a frictionless fluid originally at rest, the distribution of velocity is irrotational as well as solenoidal.

When we have to specify a distribution which is at once irrota- tional and solenoidal, we shall call it a Laplacian distribution; Laplace having pointed out some of the most important properties of such a distribution.

The volume integrals discussed in this chapter are, as we shall see, expressions for the energy of the electric field. In the first group of theorems, beginning with Green's Theorem, the energy is expressed in terms of the electromotive intensity, a vector which is distributed irrotationally in all cases of electric equilibrium. It is shewn that if the surface-potential be given, then of all irrotational distributions, that which is also solenoidal has the least energy; whence it also follows that there can be only one Laplacian distri- bution consistent with the surface potentials.

In the second group of theorems, including Thomson's Theorem,' i- the energy is expressed in terms of the electric displacement, a vector of which the distribution is solenoidal. It is shewn that if the surface-charges are given, then of all solenoidal distributions that has least energy which is also irrotational, whence it also follows that there can be only one Laplacian distribution consistent with the given surface-charges.

The demonstration of all these theorems is conducted in the same way. In order to avoid the repetition in every case of the steps of a surface integration conducted with reference to rectangular axes, we make use in each case of the result of Theorem III, Art. 21,* where the relation between a volume-integral and the corre- sponding surface-integral is fully worked out. All that we have to do, therefore, is to substitute for X, Y, and Z in that Theorem the components of the vector on which the particular theorem depends.

In the first edition of this book the statement of each theorem was cumbered with a multitude of alternative conditions which

• This theorem seems to have been first given by Ostrogradsky in a paper read in 1828, but published in 1831 in the Mem. de PAcad. de St. Petersbourg, T. I. p. 39. It may be regarded, however, as a form of the equation of continuity.

118 GENERAL THEOREMS. [96 a.

were intended to shew the generality of the theorem and the variety of cases to which it might be applied, but which tended rather to confuse in the mind of the reader what was assumed with what was to be proved.

In the present edition each theorem is at first stated in a more definite, if more restricted, form, and it is afterwards shewn what further degree of generality the theorem admits of.

We have hitherto used the symbol V for the potential, and we shall continue to do so whenever we are dealing with electrostatics only. In this chapter, however, and in those parts of the second volume in which the electric potential occurs in electro-magnetic investigations, w*e shall use ^ as a special symbol for the electric potential.

Green's Theorem.

96^.] The following important theorem was given by George Green, in his { Essay on the Application of Mathematics to Elec- tricity and Magnetism.5

The theorem relates to the space bounded by the closed surface s. We may refer to this finite space as the Field. Let v be a normal drawn from the surface s into the field, and let I, mt n be the direction cosines of this normal, then

+n — (\

ax dy dz dv

will be the rate of variation of the function y in passing along

dy

the normal v. Let it be understood that the value of -7- is to be

dv

taken at the surface itself, where v = 0. Let us also write, as in Arts. 26 and 77, ' d2y

and when there are two functions, y and <!>, let us write

££**£!;»

dx dx dy dy dz dz

The reader who is not acquainted with the method of Quater- nions may, if it pleases him, regard the expressions V2^ and tf.V^V^ as mere conventional abbreviations for the quantities to which they are equated above, and as in what follows we shall employ ordinary Cartesian methods, it will not be necessary to remember the Quaternion interpretation of these expressions. The

96 a.] GREEN'S THEOREM. 119

reason, however, why we use as our abbreviations these expressions and not single letters arbitrarily chosen, is, that in the language of Quaternions they represent fully the quantities to which they are equated. The operator V applied to the scalar function ^ gives the space-variation of that function, and the expression — xS.V^V^ is the scalar part of the product of two space- variations, or the product of either space-variation into the resolved part of the

dty other in its own direction. The expression -j- is usually written

in Quaternions S. UvW, Uv being a unit- vector in the direction of the normal. There does not seem much advantage in using this notation here, but we shall find the advantage of doing so when we come to deal with anisotropic media.

Statement of Green's Theorem.

Let # and <J> be two functions of x, y, z, which, with their first derivatives, are finite and continuous within the acyclic region s, bounded by the closed surface s, then

= ff ll;-fff

where the double integrals are to be extended over the whole closed surface *, and the triple integrals throughout the field, s, enclosed by that surface.

To prove this, let us write, in Art. 21, Theorem III,

x=, r=*, z=*d, (5)

then

(l); (6)

dX dY dZ

^ ~dx llx dy dy dz dz

= _^v24>— /S'.V^V*, by (2) and (3). (7)

But by Theorem III

dY dZ

120 GENERAL THEOREMS. [966.

or by (6) and (7)

(8)

Since in the second member of this equation ^ and 4> may be interchanged,, we may do so in the first, and we thus obtain the complete statement of Green's Theorem, as given in equation (4).

96£.] We have next to shew that Green's Theorem is true when one of the functions, say ^, is a many-valued one, provided that its first derivatives are single-valued, and do not become infinite within the acyclic region s.

Since V# and V4> are single- valued, the second member of equa- tion (4) is single-valued ; but since ^ is many-valued^ any one element of the first member, as ^ V2 4>, is many-valued. If, however, we select one of the many values of #, as ^0 , at the point A within the region ?, then the value of # at any other point, P, will be definite. For, since the selected value of ^ is continuous within the region, the value of ^ at P must be that which is arrived at by continuous variation along any path from A to P, beginning with the value % at A. If the value at P were different for two paths between A and P, then these two paths must embrace between them a closed curve at which the first derivatives of ^ become infinite. Now this is contrary to the specification, for since the first derivatives do not become infinite within the region s, the closed curve must be entirely without the region ; and since the region is acyclic, two paths within the region cannot embrace anything outside the region.

Hence, if #0 is given as the value of ^ at the point A, the value at P is definite.

If any other value of 3*, say 4^4- »K, had been chosen as the value at A, then the value at P would have been ^ + ^/c. But the value of the first member of equation (4) would be the same as before, for the change amounts to increasing the first member by

n*.

and this, by Theorem III, is zero.

96 c.~\ If the region s is doubly or multiply connected, we may reduce it to an acyclic region by closing each of its circuits with a diaphragm.

Let S-L be one of these diaphragms, and ^ the corresponding cyclic constant, that is to say, the increment of ^ in going once

96 d.] GREEN'S THEOREM. 121

round the circuit in the positive direction. Since the region s lies on both sides of the diaphragm s1} every element of ^ will occur twice in the surface integral.

If we suppose the normal z^ drawn towards the positive side of dslt and r/ drawn towards the negative side,

and ^ = ^ + K,

so that the element of the surface-integral arising from dsl will be

Hence if the region s is multiply connected, the first term of equa- tion (4) must be written

//* £-// ---// *-///«**' w

where the first surface-integral is to be taken over the bounding surface, and the others over the different diaphragms, each element of surface of a diaphragm being taken once only, and the normal being drawn in the positive direction of the circuit.

This modification of the theorem in the case of multiply- connected regions was first shewn to be necessary by Helmholtz *, and was first applied to the theorem by Thomson f.

96 dj\ Let us now suppose, with Green, that one of the functions, say <t>, does not satisfy the condition that it and its first derivatives do not become infinite within the given, region, but that it becomes infinite at the point P, and at that point only, in that region, and that very near to P the value of <I> is 4>0 + e/r%, where 4>0 is a finite and continuous quantity, and r is the distance from P. This will be the case if 4> is the potential of a quantity of electricity e concen- trated at the point P, together with any distribution of electricity the volume density of which is nowhere infinite within the region considered.

Let us now suppose a very small sphere whose radius is a to be described about P as centre ; then since in the region outside this sphere, but within the surface s, 4> presents no singularity, we

• ' Ueber Integrals der hydrodynamischen Gleichungen welche den Wirbelbewe- gungen entsprechen,' Crelle, 1858. Translated by Prof. Tait, Phil. Mag., 1867 (I). t ' On Vortex Motion,' Trans. E. S. Edin. xxv part i. p. 241 (1867). J The mark / separates the numerator from the denominator of a fraction.

122 GENERAL THEOREMS. [96 ^»

may apply Green's Theorem to this region, remembering that the surface of the small sphere is to be taken account of in forming the surface-integral.

In forming the volume-integrals we have to subtract from the volume-integral arising from the whole region that arising from the small sphere.

Now / / / <£V2 ^dxdydz for the sphere cannot be numerically greater than

or ^

where the suffix, ff) attached to any quantity, indicates that the

greatest numerical value of that quantity within the sphere is to be

taken.

This volume-integral, therefore, is of the order #2, and may be neglected when a diminishes and ultimately vanishes.

The other volume-integral

cannot be numerically greater than

and is of the order a3} and may be neglected when a vanishes.

C C dty The surface-integral / / <J> -=- ds cannot be numerically greater

than $>0 II—— ds.

y J J dv

Now by Theorem III

and this cannot be numerically greater than (V2 ^-Jtffl8, and $g

6 C C dty

at the surface is approximately -, so that / / 4> —j- ds cannot be nu- merically greater than

and is therefore of the order a2, and may be neglected when a vanishes.

But the surface-integral on the other side of the equation, namely

-j- ds>

GREEN'S THEOREM. 123

does not vanish, for / / -- — ds = — 4^; J J dv

and if ^0 be the value of # at the point P,

dv Equation (4) therefore becomes in this case

97 «.] We may illustrate this case of Green's Theorem by em- ploying it as Green does to determine the surface-density of a distribution which will produce a potential whose values inside and outside a given closed surface are given. These values must coincide at the surface, also within the surface V2 ¥ = 0, and outside V2*'= 0.

Green begins with the direct process, that is to say, the distribu- tion of the surface density, o-, being given, the potentials at an internal point P and an external point P' are found by integrating the expressions

where r and / are measured from the points P and P' respectively. Now let 4> = 1/r, then applying Green's Theorem to the space within the surface, and remembering that V23> = 0 and V2 * = 0, we find 1

where VP is the value of ty at P.

Again, if we apply the theorem to the space between the surface s and a surface surrounding it at an infinite distance a, the part of the surface-integral belonging to the latter surface will be of the order I/a and may be neglected, and we have

Now at the surface, * = */, and since the normals v and v are drawn in opposite directions,

fy

124 GENERAL THEOREMS. [97 &•

Hence on adding equations (10) and (11), the left-hand members destroy each other, and we have

97 #.] Green also proves that if the value of the potential at every point of a closed surface s be given arbitrarily, the potential at any point inside or outside the surface may be determined.

For this purpose he supposes the function 4> to be such that near the point P its value is sensibly 1/r, while at the surface s its value is zero, and at every point within the surface V2 <J> = 0.

That such a function must exist. Green proves from the physical consideration that if s is a conducting surface connected to the earth, and if a unit of electricity is placed at the point P, the potential within s must satisfy the above conditions. For since s is connected to the earth the potential must be zero at every point of s, and since the potential arises from the electricity at P and the electricity induced on s, V2<J> = 0 at every point within the surface.

Applying Green's Theorem to this case, we find

P =

where, in the surface-integral, # is the given value of the potential at the element of surface ds ; and since, if o> is the density of the electricity induced on s by unit of electricity at P,

47r<rp + ~=0, (14)

we may write equation (13)

(15)

where <r is the surface-density of the electricity induced on ds by a charge equal to unity at the point P.

Hence if the value of a- is known at every point of the surface for a particular position of P, then we can calculate by ordinary integration the potential at the point P, supposing the potential at every point of the surface to be given, and the potential within the surface to be subject to the condition

V2*= 0.

We shall afterwards prove that if we have obtained a value of

• which satisfies these conditions, it is the only value of ^ which satisfies them.

98.] GREEN'S FUNCTION. 125

Green's Function.

98.] Let a closed surface s be maintained at potential zero. Let P and Q be two points on the positive side of the surface s (we may suppose either the inside or the outside positive), and let a small body charged with unit of electricity be placed at P ; the potential at. the point Q will consist of two parts, of which one is due to the direct action of the electricity at P, while the other is due to the action of the electricity induced on s by P. The latter part of the potential is called Green's Function, and is denoted by Gpq.

This quantity is a function of the positions of the two points P and Q, the form of the function depending on the surface s. It has been calculated for the case in which s is a sphere, and for a very few other cases. It denotes the potential at Q due to the electricity induced on s by unit of electricity at P.

The actual potential at any point Q due to the electricity at P and to the electricity induced on s is l/rpq -f Gpq) where rpq denotes the distance between P and Q.

At the surface s, and at all points on the negative side of s, the potential is zero, therefore

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