A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 10 of 28

where the suffix a indicates that a point A on the surface s is taken instead of Q.

Let <Tpef denote the surface-density induced by P at a point JL of the surface s, then, since Gpq is the potential at Q due to the

superficial distribution,

/» f _

^/, (2)

where ds' is an element of the surface s at A', and the integration is to be extended over the whole surface s.

But if unit of electricity had been placed at Q, we should have had by equation (l),

7-— <v w

v

ff ff1a Jo ' (4}

=-JJ^ ' ()

where <rqa is the density at A of the electricity induced by Q, ds is an element of surface, and ra<]i is the distance between A and A'.

126 GENERAL THEOREMS. [99 a.

Substituting this value of l/rqcf in the expression for Gpq, we find

ff"—ffffSrd'' &

Since this expression is not altered by changing p into q and q into p, we find that

Gr,= G,p-' (6)

a result which we have already shewn to be necessary in Art. 87, but which we now see to be deducible from the mathematical process by which Green's function may be calculated.

If we assume any distribution of electricity whatever, and place in the field a point charged with unit of electricity, and if the surface of potential zero completely separates the point from the assumed distribution, then if we take this surface for the surface $, and the point for P, Green's function, for any point on the same side of the surface as P, will be the potential of the assumed dis- tribution on the other side of the surface. In this way we may construct any number of cases in which Green's function can be found for a particular position of P. To find the form of the function when the form of the surface is given and the position of P is arbitrary, is a problem of far greater difficulty, though, as we have proved, it is mathematically possible.

Let us suppose the problem solved, and that the point P is taken within the surface. Then for all external points the potential of the superficial distribution is equal and opposite to that of P. The superficial distribution is therefore centrobaric*, and its action on all external points is the same as that of a unit of negative electricity placed at P.

99«.] If in Green's Theorem we make ^=<l>, we find

If *P is the potential of a distribution of electricity in space with a volume-density p and on conductors whose surfaces are sv *2, &c., and whose potentials are ^u^ &c'> w^n surface-densities cr1} (72, &c., then V2^P =477/3, (17)

where £, is the charge of the surface *r

• Thomson and Tait's Natural Philosophy, § 526.

UNIQUE MINIMUM OF W+. 127

Dividing (16) by —877, we find

• (*x e, + V2e2 + &e.) -f 1 *p dx dydz

The first term is the electric energy of the system arising from the surface-distributions, and the second is that arising from the distri- bution of electricity through the field, if such a distribution exists.

Hence the second member of the equation expresses the whole electric energy of the system, the potential # being a given function of a?, y, z.

As we shall often have occasion to employ this volume-integral, we shall denote it by the abbreviation W^ so that

•+(£ >'*(£)>** (»)

If the only charges are those on the surfaces of the conductors, p=0, and the second term of the first member of equation (20) disappears.

The first term is the expression for the energy of the charged system expressed, as in Art. 84, in terms of the charges and the potentials of the conductors, and this expression for the energy we denote by W.

99 d.~\ Let ^ be a function of a?, y^ z, subject to the condition that its value at the closed surface s is ^f, a known quantity for every point of the surface. The value of * at points not on the surface s is perfectly arbitrary.

Let us also write

the integration being extended throughout the space within the surface ; then we shall prove that if ^ is a particular form of V which satisfies the surface condition and also satisfies Laplace's Equation V2 ^ _ 0 (23)

at every point within the surface, then W[, the value of W corre- sponding to *ls is less than that corresponding to any function which differs from S^ at any point within the surface.

For let ^ be any function coinciding with ^ at the surface but not at every point within it, and let us write

^ = ^ + ^2; (24)

then ^2 is a function which is zero at every point of the surface.

128 GENERAL THEOREMS. [99 b.

The value of W for ^ will be evidently

i By Green's Theorem the last term may be written

The volume-integral vanishes because V2 ^ = 0 within the surface, and the surface-integral vanishes because at the surface ^2=0. Hence equation (25) is reduced to the form

r=r1+r2. (2?)

Now the elements of the integral W^ being sums of three squares, are incapable of negative values, so that the integral itself can only be positive or zero. Hence if W^ is not zero it must be positive, and therefore W greater than Wr But if Wz is zero, every one of its elements must be zero, and therefore

dx dy dz

i7

at every point within the surface, and V2 must be a constant within the surface. But at the surface ^ = 0, therefore 4^ — 0 at every point within the surface, and ^ = Vlt so that if W is not greater than Wv ty must be identical with ^ at every point within the surface.

It follows from this that ^ is the only function of #, y, z which becomes equal to ^ at the surface, and which satisfies Laplace's Equation at every point within the surface.

For if these conditions are satisfied by any other function ^3,

; 4hen W^ must be less than any other value of W. But we have

already proved that W^ is less than any other value, and therefore

V/t ^an Wy Hence no function different from ^ can satisfy the

conditions.

The case which we shall find most useful is that in which the

k field is bounded by one exterior surface, s, and any number of

/interior surfaces, SL, s2) &c., and when the conditions are that the

• value of ^ shall be zero at s} 4^ at *1; ^2 at *2, and so on, where

4r1, ^2, &c. are constant for each surface, as in a system of conductors,

the potentials of which are given.

Of all values of ^ satisfying these conditions, that gives the minimum value of W^ for which V2vf = 0 at every point in the field.

1 00 &.] LEMMA. 129

Thomson's Theorem.

Lemma.

100 #.] Let ^ be any function of #, y, z which is finite and continuous within the closed surface s, and which at certain closed surfaces, *lf sz, sp, &c., has the values V19 ^2) typ, &c. constant for each surface.

Let Uj v, w be functions of #, y, z, which we may consider as the components of a vector (£ subject to the solenoidal condition

r, _n- du dv dw

— 8. V(£ = -j- + -j- + ~r = 0, (28)

dx dy dz

and let us put in Theorem III

X=*«, Y=Vv, Z=3>w, (29)

we find as the result of these substitutions

/du dv dw\ .

• (+ + ) todyd*

the surface-integrals being extended over the different surfaces and the volume-integrals being taken throughout the whole field. Now the first volume-integral vanishes in virtue of the solenoidal condition for u^ v, w, and the surface-integrals vanish in the follow- ing cases : —

(1) When at every point of the surface ^ =.0.

(2) When at every point of the surface lu + mv + nw = 0.

(3) When the surface is entirely made up of parts which satisfy either (l) or (2).

(4) When ^ is constant over the whole closed surface, and

nw)ds = 0. Hence in these four cases the volume-integral

/ /

100 £.] Now consider a field bounded by the external closed surface s, and the internal closed surfaces *1} s2, &c.

Let ^ be a function of #, y, z, which within the field is finite and continuous and satisfies Laplace's Equation

V2*=0, (32)

and has the constant, but not given, values *15 #25 &c. at the surfaces sl3 s2, &c. respectively, and is zero at the external surface s.

VOL. i. K

130 GENERAL THEOREMS. [lOOC.

The charge of any of the conducting surfaces, as slf is given by the surface-integral

the normal v± being drawn from the surface s1 into the electric field.

100 c.] Now let y ff, h be functions of #, y> z, which we may consider as the components of a vector 2), subject only to the conditions that at every point of the field they must satisfy the

solenoidal equation

df da dk . x

— + — + — = 0 (34)

dx dy dz

and that at any one of the internal closed surfaces, as slt the surface- integral

(35)

where I, m, n are the direction cosines of the normal v± drawn outwards from the surface sl into the electric field, and e± is the same quantity as in equation (33), being, in fact, the electric charge of the conductor whose surface is s1 .

We have to consider the value of the volume-integral

(36)

extended throughout the whole of the field within s and without

<?15 s2) &c., and to compare it with the limits of integration being the same. Let us write 1 d<& 1 d3> 1 dy u=f+ -- =- > v = g-\ --- — , w — k-\ ---- =-> (38) ITT dx 4ir dy ITT dz and W% = 2 *(u2 + & + w*)dxdydz-, (39) then since dy TOO c.] THOMSON'S THEOKEM. 131 Now in the first place, n, v, w satisfy the solenoidal condition at every point of the field, for by equations (38) _. da dy dz ~ dx dy dz 4 and by the conditions expressed in equations (34) and (32), both parts of the second member of (41) are zero. In the second place, the surface-integral ^ w) dsl / / but by (35) the first term of the second member is e, and by (33) the second term is — e, so that / / = 0. (43) Hence, since *t is constant, the fourth condition of Art. 1 00 a is satisfied, and the last term of equation (40) is zero, so that the equation is reduced to the form #5,= ^+^. (44) Now since the element of the integral W§ is the sum of three squares, &2+#2+^2, it must be either positive or zero. If at any point within the field %, v, and w are not each of them equal to zero, the integral W® must have a positive value, and W^ must therefore be greater than W*. But the values u = v = w = 0 at every point satisfy the conditions. Hence, if at every point Id* Id* Id* f . -' * =-' ^- then Wv=.W^ (46) and the value of W® corresponding to these values of f, g, ft, is less than the value corresponding to any values of ft g> h, differing from these. Hence the problem of determining the displacement and po- tential, at every point of the field, when the charge on each conductor is given, has one and only one solution. This theorem in one of its more general forms was first stated by Sir W. Thomson*. We shall afterwards show of what gene- ralization it is capable. * Cumbridye and Dublin Mathematical Journal, February, 1848. 132 GENERAL THEOREMS. [lOO d. 100^.] This theorem may be modified by supposing that the vector 5), instead of satisfying the solenoidal condition at every point of the field, satisfies the condition . . where p is a finite quantity, whose value is given at every point in the field, and may be positive or negative, continuous or discon- tinuous, its volume-integral within a finite region being, however, finite. We may also suppose that at certain surfaces in the field lf+ mg + nh + I'f + my + n' h' = <r, (48) when I, m, n and I', m'} n' are the direction cosines of the normals drawn from a point of the surface towards those regions in which the components of the displacement are /*, g^ k and f, g' ', V re- spectively, and or is a quantity given at all points of the surface, the surface-integral of which, over a finite surface, is finite. 100 e.~\ We may also alter the condition at the bounding surfaces by supposing that at every point of these surfaces If+mg+nh = cr, (49) where a- is given for every point. (In the original statement we supposed only the value of the integral of a- over each of the surfaces to be given. Here we suppose its value given for every element of surface, which comes to the same thing as if, in the original statement, we had considered every element as a separate surface.) None of these modifications will affect the truth of the theorem provided we remember that SP must satisfy the corresponding conditions, namely, the general condition, d2* d2* d*y . . TT + T-2- + l~2"+47rP = °> (50) da;* dy* dsP and the surface condition £ + 77 For if, as before, 1 d* I then ^, vt w will satisfy the general solenoidal condition du dv dw _ fa + dj + fa = } and the surface condition lu+mv+nw+l'u'+m'v'+n'w' = 0, IOI &.] INTENSITY AND DISPLACEMENT. 133 and at the bounding surface lu -f mv + nw = 0, whence we find as before that and that #5> Hence as before it is shewn that W^ is a unique minimum when 7/5 = 0, which implies that (£ is everywhere zero, and therefore 1 d* J_^ z_ _!_<** ~' " ~" 101 a.~\ In our statement of these theorems we have hitherto confined ourselves to that theory of electricity which assumes that the properties of an electric system depend on the form and relative position of the conductors, and on their charges, but takes no account of the nature of the dielectric medium between the conductors. According to that theory, for example, there is an invariable relation between the surface density of a conductor and the electro- motive intensity just outside it, as expressed in the law of Coulomb E = 4™. But this is true only in the standard medium, which we may take to be air. In other media the relation is different, as was proved experimentally, though not published, by Cavendish, and afterwards rediscovered independently by Faraday. In order to express the phenomenon completely, we find it necessary to consider two vector quantities, the relation between which is different in different media. One of these is the electro- motive intensity, the other is the electric displacement. The electromotive intensity is connected by equations of invariable form with the potential, and the electric displacement is connected by equations of invariable form with the distribution of electricity, but the relation between the electromotive intensity and the electric displacement depends on the nature of the dielectric medium, and must be expressed by equations, the most general form of which is as yet not fully determined, and can be determined only by ex- periments on dielectrics. 101 b.] The electromotive intensity is a vector defined in Art. 68, as the mechanical force on a small quantity e of electricity divided by e. We shall denote its components by the letters P, Q, E, and the vector itself by ($. In electrostatics, the line integral of @ is always independent 134 GENERAL THEOREMS. [lOI C. of the path of integration, or in other words (£ is the space- variation of a potential. Hence p d* d* d* r = -- 7- > U = -- 7— » A/ = das dy or more briefly, in the language of Quaternions 101 <?.] The electric displacement in any direction is defined in Art. 68, as the quantity of electricity carried through a small area A, the plane of which is normal to that direction, divided by A. We shall denote the rectangular components of the electric displacement by the letters ft g, ft, and the vector itself by 3). The volume-density at any point is determined by the equation _^/, dg dh ~ dx "*" dy "*" dz ' or in the language of Quaternions P= -&V3X The surface-density at any point of a charged surface is deter- mined by the equation o- = lf+ mg + nh + I'f + m'g' + n'h', where f, g, Ji are the components of the displacement on one side of the surface, the direction cosines of the normal drawn from the surface on that side being I, m, n, and f, /, h' and I', m', n' are the components of the displacements, and the direction cosines of the normal on the other side. This is expressed in Quaternions by the equation where Uv, Uv are unit normals on the two sides of the surface, and 8 indicates that the scalar part of the product is to be taken. When the surface is that of a conductor, v being the normal drawn outwards, then since/', /, Ji and £)' are zero, the equation is reduced to the form cr = (If+mg + nh); = -S.Uvto. The whole charge of the conductor is therefore - -//'• 101 d.~\ The electric energy of the system is, as was shown in Art. 84, half the sum of the products of the charges into their respective potentials. Calling this energy V, IOT 6.] PROPERTIES OF A DIELECTRIC. 135 where the volume-integral is to be taken throughout the electric field, and the surface-integral over the surfaces of the conductors. Writing in Theorem III, Art. 21, * we find -///(. Substituting this value for the surface-integral in W we find or = lfff(fP + ffQ + Mt)dx dy dz. 1 01 ^.] We now come to the relation between 2) and @. The unit of electricity is usually defined with reference to experiments conducted in air. We now know from the experiments of Boltzmann that the dielectric constant of air is somewhat greater than that of a vacuum, and that it varies with the density. Hence, strictly speaking, all measurements of electric quantity require to be corrected to reduce them either to air of standard pressure and temperature, or, what would be more scientific, to a vacuum, just as indices of refraction measured in air require a similar correction, the correction in both cases being so small that it is sensible only in measurements of extreme accuracy. In the standard medium 477$ = (g, or 4ir/'= P, 47r^=Q, 47r>^ = -5. In an isotropic medium whose dielectric constant is K There are some media, however, of which glass has been the most carefully investigated, in which the relation between 2) and (£ 136 GENERAL THEOREMS. is more complicated, and involves the time variation of one or both of these quantities, so that the relation must be of the form We shall not attempt to discuss relations of this more general kind at present, but shall confine ourselves to the case in which 2) is a linear and vector function of (£. The most general form of such a relation may be written 477$) =4> ((£), where $ during the present investigation always denotes a linear and vector function. The components of 2) are therefore homo- geneous linear functions of those of (£, and may be written in the form 4 TT/= Kxx P + Kxy Q + KXXE ; where the first suffix of each coefficient K indicates the direction of the displacement, and the second, that of the electromotive intensity. The most general form of a linear and vector function involves nine independent coefficients. When the coefficients which have the same pair, of suffixes are equal, the function is said to be self-conjugate. If we express (£ in terms of 3) we shall have or P = 4 TT (kiexf+ 7cyxg + JcK 101 y.] The work done by the electromotive intensity whose components are Pf Q, R, in producing a displacement whose com- ponents are df, dg> and dk, in unit of volume of the medium, is Since a dielectric under electric displacement is a conservative system, W must be a function of ft g> h, and since f, g, 7i may vary independently, we have aw AW aw = J/' Q='~W' R = ~dh' Hence dP = ®W = ffiW =^ Ag AfAg AgAf Af fl ~P But — = lirfcyx, the coefficient of g in the expression for P, and ~Y— ^v7cxy> the coefficient of f'm the expression for Q, 101 L] EXTENSION OF GREEN'S THEOREM. 137 Hence if a dielectric is a conservative system, (and we know that it is so, because it can retain its energy for an indefinite time), &xv — &Vx, and <j)~l is a self-conjugate function. Hence it follows that $ also is self-conjugate, and 101 gj] The expression for the energy may therefore be written in either of the forms or + 2 Kgx RP+2 KxyPQ] dx dy dz, 0£ = 2 ^///fe/2 + £,„/ + >M2 + 2 krfk + 2£M A/+ ^Jcxyfg\ dxdydz, where the suffix denotes the vector in terms of which TFis to be expressed. When there is no suffix, the energy is understood to be expressed in terms of both vectors. We have thus, in all, six different expressions for the energy of the electric field. Three of these involve the charges and poten- tials of the surfaces of conductors, and are given in Art. 87. The other three are volume-integrals taken throughout the electric field, and involve the components of electromotive intensity or of electric displacement, or of both. The first three therefore belong to the theory of action at a distance, and the last three to the theory of action by means of the intervening medium. These three expressions for W may be written, 101 hJ\ To extend Green's Theorem to the case of a hetero- geneous anisotropic medium, we have only to write in Theorem III, *-.**_£+*•,. - + *". 138 GENERAL THEOREMS. [lO2 a. and we obtain (remembering that the order of the suffixes of the coefficients is indifferent), d ( K d ** dx d%~ KVV dy dy ' M~fa~d^ (d^d$_ ^^\ T7 /_^*^* -+ + A|"" rr r = J J * ^ ^-: -- = 7 v\ doc dy dy -dx v r-j \dxdydz KVX m Using quaternion notation the result may be written more briefly, jfy s. Uv ^ (v<i>) fa- Limits between which the electric capacity of a conductor must lie. 102 #.] The capacity of a conductor or system of conductors has been already defined as the charge of that conductor or system 102 a.] LIMITING VALUES OF CAPACITY. 139 of conductors when raised to potential unity, all the other con- ductors in the field being at potential zero. The following method of determining limiting values between which the capacity must lie, was suggested by a paper ' On the Theory of Resonance/ by the Hon. J. W. Strutt, Phil. Trans. 1871. See Art. 308. Let sl denote the surface of the conductor, or system of con- ductors, whose capacity is to be determined, and s0 the surface of all other conductors. Let the potential of ^ be ¥1} and that of *0, ^0. Let the charge of s1 be ^. That of SQ will be — elt Then if q is the capacity of slt . and if W is the energy of the system with its actual distribution of electricity W=\e^ (^-^0), (2) 2W e^ , . * = (4v=*rp = 2F' To find an upper limit of the value of the capacity. Assume any value of # which is equal to 1 at s1 and equal to zero at s0, and calculate the value of the volume-integral extended over the whole field. Then as we have proved (Art. 99 6) that W cannot be greater than W*, the capacity, q, cannot be greater than 2%. To find a lower limit of the value of the capacity. Assume any system of values of f> g, h, which satisfies the equation ^+^+^-0 (5) dx + dy + dz " ( } and let it make / / (l^f-\- m^g + n-Ji) ds± = elt (6) Calculate the value of the volume-integral = tiff /• extended over the whole field ; then as we have proved (Art. 100 c) that W cannot be greater than #J, the capacity, £, cannot be less than e-f , . 2%' V ; The simplest method of obtaining a system of values of/, g, h, which will satisfy the solenoidal condition, is to assume a distribu- tion of electricity on the surface of slt and another on *0, the sum 140 GENERAL THEOREMS. of the charges being zero, then to calculate the potential, #, due to this distribution, and the electric energy of the system thus arranged, which we may call W^. If we then make 1 d* 1 d* 1 d<V ~~' ~~' ""' these values ofy, g, Ji will satisfy the solenoidal condition. But in this case we can determine W& without going through the process of finding the volume- integral. For since this solution makes V2v£ = 0 at all points in the field, we can obtain W& in the form of the surface-integrals, where the first integral is extended over the surface s± and the second over the surface s0. If the surface s0 is at an infinite distance from s13 the potential at sQ is zero and the second term vanishes. 102 #.] An approximation to the solution of any problem of the distribution of electricity on conductors whose potentials are given may be made in the following manner : — Let Sj. be the surface of a conductor or system of conductors maintained at potential 1, and let SQ be the surface of all the other conductors, including the hollow conductor which surrounds the rest, which last, however, may in certain cases be at an infinite distance from the others. Begin by drawing a set of lines, straight or curved, from Sj tO SQ. Along each of these lines, assume ty so that it is equal to 1 at s1} and equal to 0 at <?0 . Then if P is a point on one of these lines we Ps may take vpj = - — as a first approximation. siso We shall thus obtain a first approximation to ^ which satisfies the condition of being equal to unity at s1 and equal to zero at $0. The value of W* calculated from ^ would be greater than W. Let us next assume as a second approximation to the lines of force The vector whose components are a, b, c is normal to the surfaces for which ^ is constant. Let us determine p so as to make a, 6, c satisfy the solenoidal condition. We thus get IO2&.] CALCULATION OF CAPACITY. 141 ,i ± ~ \dx* ' df " dz* ) dxdx zdz If we draw a line from SL to s0 whose direction is always normal to the surfaces for which *sis constant, and if we denote the length of this line measured from s0 by s, then -ndx _ d^ dy d^ dz d^ K = - E = —' = —> d* where R is the resultant intensity = — -r- , so that , __ p dx dx dy dy dz dz ~ds* and equation (11) becomes f™ = wdJL, /**! ^2^; whence p=Cexp.j^ -^r^15 (15) the integral being a line integral taken along the line «y. Let us next assume that along the line *, d$*<> da! 7 dy dz -- r^ = a-7- +£-f +c~, ds ds ds ds then (17) the integration being always understood to be performed along the line s. The constant C is now to be determined from the condition that ^2 = 1 at sl when also ^ = 1 , so that /*! /** V2^ C exp. -^-d*d*—l. (18) */Q *^ 0 ~^^ This gives a second approximation to *, and the process may be repeated. The results obtained from calculating W*lt W^ #*2, &c., give capacities alternately above and below the true capacity and con- tinually approximating thereto. The process as indicated above involves the calculation of the form of the line s and integration along this line, operations which are in general too difficult for practical purposes. 142 GENERAL THEOREMS. [lO2 C. In certain cases however we may obtain an approximation by a simpler process. 102 c.~\ As an illustration of this method, let us apply it to obtain successive approximations to the equipotential surfaces and lines of induction in the electric field between two surfaces which are nearly but not exactly plane and parallel, one of which is maintained at potential zero, and the other at potential unity. Let the equations of the two surfaces be for the surface whose potential is zero, and *i=/i («*) = * (20) for the surface whose potential is unity, a and 6 being given functions of x and y, of which b is always greater than a. The first derivatives of a and b with respect to x and y are small quan- tities of which we may neglect powers and products of more than two dimensions. We shall begin by supposing that the lines of induction are parallel to the axis of 0, in which case /=0, ,7=0)lg = OJ (21) Hence k is constant along each individual line of induction, and qt = —In Jidz— — ±iih(z—a). (22) J a When z = I, * = 1, hence (23) 4: ir (If — a) and ^' (24) b — a which gives a first approximation to the potential, and indicates a series of equipotential surfaces the intervals between which, measured parallel to z, are equal. To obtain a second approximation to the lines of induction, let us assume that they are everywhere normal to the equipotential surfaces as given by equation (24). This is equivalent to the conditions . . d* d* d* 4 Ttf — A-— j 4 TT q = X. -7— > 4 TT h — A -7— > dx dy dz i/ where A is to be determined so that at every point of the field df dg dli , } + + =°3 102 C.] POTENTIAL BETWEEN TWO NEARLY PLAT SURFACES. 143 and also so that the line-integral dy , dz \ , -+'+** (27) taken along any line of induction from the surface a to the surface #, shall be equal to — 1 . Let us assume A= l+A + £(z-a) + C(z-ay, (28) and let us neglect powers and products of A> B, C, and at this stage of our work powers and products of the first derivatives of a and b. ^ ^ The solenoidal condition then gives/"" "~*£x* **«/*»/. /t 1 £ = _V2a, <?=-4V2^g), (29) where V2 = _ If instead of taking the line-integral along the new line of induction, we take it along the old line of induction, parallel to 2;, the second condition gives Hence A = £(*-«) V2(2« + ^), . (3) and We thus find for the second approximation to the components of displacement, da d(b—a)z—a and for the second approximation to the potential, z a If cra and o-j, are the surface- densities and *a and ^6the potentials of the surfaces a and b respectively, CHAPTEE V. MECHANICAL ACTION BETWEEN TWO ELECTEICAL SYSTEMS. 103.] Let El and E2 be two electrical systems, the mutual action between which we propose to investigate. Let the distribution of electricity in E1 be defined by the volume-density, plt of the element whose coordinates are x^y^z^. Let /o2 be the volume- density of the element of E2 , whose coordinates are #2 , y% , #2. Then the ^-component of the force acting on the element of El on account of the repulsion of the element of E% will be Plp2 r* Xl ^ Zl X* y* ^2' where r2 = fa— ^ and if A denotes the x component of the whole force acting on E1 on account of the presence of E2 A =jJJJJj^^hfl***ifyid*id**fy**'*> (*) where the integration with respect to al9 yl} gt is extended throughout the region occupied by E19 and the integration with respect to x^y^z^ is extended throughout the region occupied by -E2. Since, however, p1 is zero except in the system Elt and p2 is zero except in the system E29 the value of the integral will not be altered by extending the limits of the integrations, so that we may suppose the limits of every integration to be + °°- This expression for the force is a literal translation into mathe- matical symbols of the theory which supposes the electric force to act directly between bodies at a distance, no attention being bestowed on the intervening medium. If we now define V2, the potential at the point alt y^ z-^ arising from the presence of the system E2) by the equation ^2 will vanish at an infinite distance, and will everywhere satisfy the equation V2*2=47rp2. (3) 1 04.] MECHANICAL ACTIO^. 145 We may now express A in the form of a triple integral Here the potential ^2 is supposed to have a definite value at every point of the field, and in terms of this, together with the distribution, p15 of electricity in the first system Elt the force A is expressed, no explicit mention being made of the distribution of electricity in the second system E2. Now let *j be the potential arising from the first system, expressed as a function of #,y, zt and defined by the equation *l*kAk, (5) *! will vanish at an infinite distance, and will everywhere satisfy the equation V2#1 = 47rp1. (6) We may now eliminate ^ from A and obtain in which the force is expressed in terms of the two potentials only. 104.] In all the integrations hitherto considered, it is indifferent what limits are prescribed, provided they include the whole of the system Elf In what follows we shall suppose the systems E± and E2 to be such that a certain closed surface a contains within it the whole of El but no part of E2 . Let us also write P = /VH>2, * = *!+*» (8) then within s, p2 = 0, p =p1, and without s pt = 0, p = /oa. (9) Now A11 = -Plda;i^1^1 (10) represents the resultant force, in the direction #, on the system E^ arising from the electricity in the system itself. But on the theory of direct action this must be zero, for the action of any particle P on another Q is equal and opposite to that of Q on P, and since the components of both actions enter into the integral, they will destroy each other. We may therefore write VOL. I. 146 MECHANICAL ACTION. [105. where ^ is the potential arising from both systems, the integration being now limited to the space within the closed surface s, which includes the whole of the system El but none of E2. 105.] If the action of E2 on El is effected, not by direct action at a distance, but by means of a distribution of stress in a medium extending continuously from E2 to E1, it is manifest that if we know the stress at every point of any closed surface * which completely separates El from E2, we shall be able to determine completely the mechanical action of E2 on E±. For if the force on El is not completely accounted for by the stress through $, there must be direct action between something outside of s and some- thing inside of s. Hence if it is possible to account for the action of E2 on E1 by means of a distribution of stress in the intervening medium, it must be possible to express this action in the form of a surface- integral extended over any surface s which completely separates E2 from Elu Let us therefore endeavour to express d2* A= in the form of a surface integral. By Theorem III we may do so if we can determine X, Y and Z, so that _dX dY dZ " " ~~~~~~~ ~*~ ~ Taking the terms separately, 2 d dx dy2 dy dx dy' dy dxdy d ,d3> d*^ 1 d /^ _. dy ^dx dy ' 2 dx ^dy i/ <L? & „. .. . y f x**\ i ft X*\ Similarly ac-ar - ar («"s) - 1« <ar) If, therefore, we write STKESS IN A MEDIUM. 147