A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 11 of 28

d<&^2 (dy\2 /d$>\2 dz* ^dx' Wy'

__

IV 1

= ^Pi

(14)

r r f /dp~~ dp,.v dp.-

ttenA=JJJ(-Lj? + J^+.J!-

(15) v '

the integration being extended throughout the space within Transforming the volume-integral by Theorem III, Art. 2.1,

A =

(16)

where ds is an element of any closed surface including the whole of El but none of S2, and Imn are the direction cosines of the normal drawn from ds outwards.

For the components of the force on E^ in the directions of y and 2, we obtain in the same way

=JJ

C = (lp

(17) (18)

If the action of the system E2 on El does in reality take place by direct action at a distance, without the intervention of any medium, we must consider the quantities pxx &c. as mere abbreviated forms for certain symbolical expressions, and as having no physical significance.

But if we suppose that the mutual action between E2 and E± is kept up by means of stress in the medium between them, then since the equations (16), (17), (18) give the components of the resultant force arising from the action, on the outside of the surface s, of the stress whose six components are pxx &c., we must consider pxx &c. as the components of a stress actually existing in the medium.

L 2

148 MECHANICAL ACTION. [106.

106.] To obtain a clearer view of the nature of this stress let us alter the form of part of the surface s so that the element ds may become part of an equipotential surface. (This alteration of the surface is legitimate provided we do not thereby exclude any part of E-L or include any part of E2).

Let v be a normal to ds drawn outwards.

city Let E = j— be the intensity of the electromotive force in

the direction of v, then

dty dty dty

-j— = — El, —r- = —Em, -=— = — En. dx dy dz

Hence the six components of stress are

A. = ± ff (I2 -«»-»«), Pt. = J- R*mn,

tn = -^™ K-«"-P), p,x = _L JB»nl,

If a, I, c are the components of the force on ds per unit of area

87T

c = — E*n.

Sir

Hence the force exerted by the part of the medium outside ds on the part of the medium inside ds is normal to the element and directed outwards, that is to say, it is a tension like that of a rope,

and its value per unit of area is E2.

O7T

Let us next suppose that the element ds is at right angles to the equipotential surfaces which cut it, in which case

7 dty dty dty

l-j-^.m~T- + n-7 - = 0. (19)

dx dy dz v '

,r/d*\2 /d*\2 /^\21 Now 8,- = I - - -) - (-g.) J

™ (20)

x dz

Multiplying (19) by 2 -- and subtracting from (20), we. find

I07-] COMPONENTS OF STRESS. 149

*\2 /^fx2

Hence the components of the tension per unit of area of ds are

• = -±m,

Hence if the element ^ is at right angles to an equipotential surface, the force which acts on it is normal to the surface, and its numerical value per unit of area is the same as in the former case, but the direction of the force is different, for it is a pressure instead of a tension.

We have thus completely determined the type of the stress at any given point of the medium.

The direction of the electromotive intensity at the point is a principal axis of stress, and the stress in this direction is a tension whose numerical value is

t =^ ^ (22)

where E is the electromotive intensity.

Any direction at right angles to this is also a principal axis of stress, and the stress along this axis is a pressure whose numerical magnitude is also p.

The stress as thus defined is not of the most general type, for it has two of its principal stresses equal to each other, and the third has the same value with the sign reversed.

These conditions reduce the number of independent variables which determine the stress from six to three, and accordingly it is completely determined by the three components of the electro- motive intensity

~

dx dy dz

The three relations between the six components of stress are

«).)

»l\ (23)

™}' )

107.] Let us now examine whether the results we have obtained

150 MECHANICAL ACTION. [107.

will require modification when a finite quantity of electricity is collected on a finite surface so that the volume-density becomes infinite at the surface.

In this case, as we have shown in Art. 78, the components of the electromotive intensity are discontinuous at the surface. Hence the components of stress will also be discontinuous at the surface.

Let I m n be the direction cosines of the normal to ds. Let P, Q, R be the components of the electromotive intensity on the side on which the normal is drawn, and P'; Q\R' their values on the other side.

Then by Art. (78 a) if <r is the surface-density

: (24)

Let a be the ^-component of the resultant force acting on the surface per unit of area, arising from the stress on both sides, then

O7T

• -L « (PQ-P'Q') + ~ n (PR-PR),

(25)

Hence, assuming that the stress at any point is given by equations (14), we find that the resultant force in the direction of a? on a charged surface per unit of volume is equal to the surface-density multiplied into the arithmetical mean of the x- components of the electromotive intensity on the two sides of the surface.

I08.] FORCE ON A CHARGED SURFACE. 151

This is the same result as we obtained in Art. 79 by a process essentially similar.

Hence the hypothesis of stress in the surrounding- medium is applicable to the case in which a finite quantity of electricity is collected on a finite surface.

The resultant force on an element of surface is usually deduced from the theory of action at a distance by considering- a portion of the surface, the dimensions of which are very small compared with the radii of curvature of the surface."*

On the normal to the middle point of this portion of the surface take a point P whose distance from the surface is very small com- pared with the dimensions of the portion of the surface. The electromotive intensity at this point, due to the small portion of the surface, will be approximately the same as if the surface had been an infinite plane, that is to say 2 IT a- in the direction of the normal drawn from the surface. For a point P7 just on the other side of the surface the intensity will be the same, but in the opposite direction.

Now consider the part of the electromotive intensity arising- from the rest of the surface and from other electrified bodies at a finite distance from the element of surface. Since the points P and y are infinitely near one another, the components of the electromotive intensity arising from electricity at a finite distance will be the same for both points.

Let P0 be the ^-component of the electromotive intensity on A or A arising from electricity at a finite distance, then the total value of the ^-component for A will be

P = P0+27ro-/, and for A' P*= P0-2ir<rl.

Hence PQ=(P+P).

Now the resultant mechanical force on the element of surface must arise entirely from the action of electricity at a finite distance, since the action of the element on itself must have a resultant zero. Hence the ^-component of this force per unit of area must be

a — crP0,

108.] If we define the potential (as in equation (2)) in terms of a distribution of electricity supposed to be given, then it follows

• This method is due to Laplace. See Poisson, • Sur la Distribution de MectriciW &c.' Mem. de I'lmtitut, 1811, p. 30.

152 MECHANICAL ACTION. [108.

from the fact that the action and reaction between any pair of electric particles are equal and opposite, that the ^-component of the force arising from the action of a system on itself must be zero, and we may write this in the form

duo

But if we define ^ as a function of #, y> z which satisfies the equation ^^ — o

at every point outside the closed surface <?, and is zero at an infinite distance, the fact, that the volume-integral extended throughout any space including s is zero, would seem to require proof.

One method of proof is founded on the theorem (Art. 100 a), that if V2v£ is given at every point, and ^ = 0 at an infinite distance, then the value of ^ at every point is determinate and equal to

1 //"/! ' = / / / - V2 dx-dydz. (27)

477 J J J r

where r is the distance between the element dx dy dz at which the concentration of * is given = V2* and the point %' y' z' at which ^' is to be found.

This reduces the theorem to what we deduced from the first definition of #.

But when we consider ^ as the primary function of x, y^ z, from which the others are derived, it is more appropriate to reduce (26) to the form of a surface-integral,

(28)

and if we suppose the surface S to be everywhere at a great distance a from the surface s, which includes every point where V2^ differs from zero, then we know that ^ cannot be numerically greater than e/a, where 4ve is the volume-integral of V2#, and that H cannot be greater than d^f/da or — <?/&2, and that the quantities PxxiPxyyPxz can none °f them be greater than p or R2/%-n or e^/STia*. Hence the surface-integral taken over a sphere whose radius is very great and equal to a cannot exceed £2/2 a2, and when a is increased without limit, the surface-integral must become ultimately zero.

But this surface-integral is equal to the volume-integral (26), and the value of this volume-integral is the same whatever be the size of the space enclosed within S, provided S encloses every point at which V2sP differs from zero. Hence, since the integral is zero

no.] FARADAY'S THEORY. 153

when a is infinite, it must also be zero when the limits of integra- tion are defined by any surface which includes every point at which V2^ differs from zero.

109.] The distribution of stress considered in this chapter is pre- cisely that to which Faraday was led in his investigation of induc- tion through dielectrics. He sums up in the following words : —

'(1297) The direct inductive force, which may be conceived to be exerted in lines between the two limiting and charged con- ducting surfaces, is accompanied by a lateral or transverse force equivalent to a dilatation or repulsion of these representative lines (1224.); or the attracting force which exists amongst the par- ticles of the dielectric in the direction of the induction is ac- companied by a repulsive or a diverging force in the transverse direction.

' (1298) Induction appears to consist in a certain polarized state of the particles, into which they are thrown by the electrified body sustaining the action, the particles assuming positive and negative points or parts, which are symmetrically arranged with respect to each other and the inducting surfaces or particles. The state must be a forced one, for it is originated and sustained only by force, and sinks to the normal or quiescent state when that force is removed. It can be continued only in insulators by the same portion of electricity, because they only can retain this state of the particles.'

This is an exact account of the conclusions to which we have been conducted by our mathematical investigation. At every point of the medium there is a state of stress such that there is tension along the lines of force and pressure in all directions at right angles to these lines, the numerical magnitude of the pressure being equal to that of the tension, and both varying as the square of the resultant force at the point.

The expression ' electric tension ' has been used in various senses by different writers. I shall always use it to denote the tension along the lines of force, which, as we have seen, varies from point to point, and is always proportional to the square of the resultant force at the point.

110.] The hypothesis that a state of stress of this kind exists in a fluid dielectric, such as air or turpentine, may at first sight appear at variance with the established principle that at any point in a fluid the pressures in all directions are equal. But in the deduction of this principle from a consideration of the mobility

154 MECHANICAL ACTION. [ill.

and equilibrium of the parts of the fluid it is taken for granted that no action such as that which we here suppose to take place along the lines of force exists in the fluid. The state of stress which we have been studying is perfectly consistent with the mobility and equilibrium of the fluid, for we have seen that, if any portion of the fluid is devoid of electric charge, it experi- ences no resultant force from the stresses on its surface, however intense these may be. It is only when a portion of the fluid becomes charged that its equilibrium is disturbed by the stresses on its surface, and we know that in this case it actually tends to move. Hence the supposed state of stress is not inconsistent with the equilibrium of a fluid dielectric.

The quantity W, which was investigated in Chapter IV, Art. 99, may be interpreted as the energy in the medium due to the distribution of stress. It appears from the theorems of that chapter that the distribution of stress which satisfies the conditions there given also makes ^an absolute minimum. Now when the energy is a minimum for any configuration, that configuration is one of equilibrium, and the equilibrium is stable. Hence the dielectric, when subjected to the inductive action of electrified bodies, will of itself take up a state of stress distributed in the way we have described.

It must be carefully borne in mind that we have made only one step in the theory of the action of the medium. We have supposed it to be in a state of stress, but we have not in any way accounted for this stress, or explained how it is maintained. This step, however, seems to me to be an important one, as it explains, by the action of the consecutive parts of the medium, phenomena which were formerly supposed to be explicable only by direct action at a distance.

111.] I have not been able to make the next step, namely, to account by mechanical considerations for these stresses in the dielectric. I therefore leave the theory at this point, merely stating what are the other parts of the phenomenon of induction in dielectrics.

I. Electric Displacement. "When induction is transmitted through a dielectric, there is in the first place a displacement of electricity in the direction of the induction. For instance, in a Ley den jar, of which the inner coating is charged positively and the outer coating negatively, the displacement of positive electricity in the substance of the glass is from within outwards.

III.] ELECTRIC POLARIZATION. 155

Any increase of this displacement is equivalent, during the time of increase, to a current of positive electricity from within outwards, and any diminution of the displacement is equivalent to a current in the opposite direction.

The whole quantity of electricity displaced through any area of a surface fixed in the dielectric is measured by the quantity which we have already investigated (Art. 75) as the surface-integral of induction through that area, multiplied by JS/4w, where K is the specific inductive capacity of the dielectric.

II. Surface charge of the particles of the dielectric. Conceive any portion of the dielectric, large or small, to be separated (in imagi- nation) from the rest by a closed surface, then we must suppose that on every elementary portion of this surface there is a charge measured by the total displacement of electricity through that element of surface reckoned inwards.

In the case of the Leyden jar of which the inner coating is charged positively, any portion of the glass will have its inner side charged positively and its outer side negatively. If this portion be entirely in the interior of the glass, its surface charge will be neutralized by the opposite charge of the parts in contact with it, but if it be in contact with a conducting body, which is incapable of maintaining in itself the inductive state, the surface charge will not be neutralized, but will constitute that apparent charge which is commonly called the Charge of the Conductor.

The charge therefore at the bounding surface of a conductor and the surrounding dielectric, which on the old theory was called the charge of the conductor, must be called in the theory of induction the surface charge of the surrounding dielectric.

According to this theory, all charge is the residual effect of the polarization of the dielectric. This polarization exists throughout the interior of the substance, but it is there neutralized by the juxtaposition of oppositely charged parts, so that it is only at the surface of the dielectric that the effects of the charge become apparent.

The theory completely accounts for the theorem of Art. 77, that the total induction through a closed surface is equal to the total quantity of electricity within the surface multiplied by 47r. For what we have called the induction through the surface is simply the electric displacement multiplied by 4ir, and the total displacement outwards is necessarily equal to the total charge within the surface.

156 MECHANICAL ACTION. [ill.

The theory also accounts for the impossibility of communicating an * absolute charge ' to matter. For every particle of the dielectric has equal and opposite charges on its opposite sides, if it would not be more correct to say that these charges are only the manifestations of a single phenomenon, which we may call Electric Polarization.

A dielectric medium, when thus polarized, is the seat of electrical energy, and the energy in unit of volume of the medium is nu- merically equal to the electric tension on unit of area, both quan- tities being equal to half the product of the displacement and the resultant electromotive intensity, or

where p is the electric tension, & the displacement, (£ the electro- motive intensity, and K the specific inductive capacity.

If the medium is not a perfect insulator, the state of constraint, which we call electric polarization, is continually giving way. The medium yields to the electromotive force, the electric stress is relaxed, and the potential energy of the state of constraint is con- verted into heat. The rate at which this decay of the state of polarization takes place depends on the nature of the medium. In some kinds of glass, days or years may elapse before the polar- ization sinks to half its original value. In copper, a similar change is effected in less than the billionth of a second.

We have supposed the medium after being polarized to be simply left to itself. In the phenomenon called the electric current the constant passage of electricity through the medium tends to restore the state of polarization as fast as the conductivity of the medium allows it to decay. Thus the external agency which maintains the current is always doing work in restoring the polarization of the medium, which is continually becoming relaxed, and the potential energy of this polarization is continually becoming transformed into heat, so that the final result of the energy expended in main- taining the current is to gradually raise the temperature of the conductor, till as much heat is lost by conduction and radiation from its surface as is generated in the same time by the electric current.

CHAPTER VI.

ON POINTS AND LINES OF EQUILIBRIUM.

112.] IF at any point of the electric field the resultant force is zero, the point is called a Point of equilibrium.

If every point on a certain line is a point of equilibrium, the line is called a Line of equilibrium.

The conditions that a point shall be a point of equilibrium are that at that point

dV __ dV dV _

dx dy ~~ dz

At such a point, therefore, the value of V is a maximum, or a minimum, or is stationary, with respect to variations of the coordinates. The potential, however, can have a maximum or a minimum value only at a point charged with positive or with negative electricity, or throughout a finite space bounded by a surface charged positively or negatively. If, therefore, a point of equilibrium occurs in an uncharged part of the field it must be a stationary point, and not a maximum or a minimum.

In fact, the first condition of a maximum or minimum is that

d

must be all negative or all positive, if they have finite values.

Now, by Laplace's equation, at a point where there is no charge, the sum of these three quantities is zero, and therefore this condition cannot be satisfied.

Instead of investigating the analytical conditions for the cases in which the components of the force simultaneously vanish, we shall give a general proof by means of the equipotential surfaces.

If at any point, P, there is a true maximum value of F9 then, at all other points in the immediate neighbourhood of P, the value of V is less than at P. Hence P will be surrounded by a series of closed equipotential surfaces, each outside the one before it, and at all points of any one of these surfaces the electrical force will be

158 POINTS AND LINES OF EQUILIBRIUM. [113.

directed outwards. But we have proved, in Art. 76, that the surface- integral of the electromotive intensity taken over any closed surface gives the total charge within that surface multiplied by 4 it. Now, in this case the force is everywhere outwards, so that the surface- integral is necessarily positive, and therefore there is positive charge within the surface, and, since we may take the surface as near to P as we please, there is positive charge at the point P.

In the same way we may prove that if V is a minimum at P, then P is negatively charged.

Next, let P be a point of equilibrium in a region devoid of charge, and let us describe a sphere of very small radius round P, then, as we have seen, the potential at this surface cannot be everywhere greater or everywhere less than at P. It must therefore be greater at some parts of the surface and less at others. These portions of the surface are bounded by lines in which the potential is equal to that at P. Along lines drawn from P to points at which the potential is less than that at P the electrical force is from P, and along lines drawn to points of greater potential the force is towards P. Hence the point P is a point of stable equilibrium for some displacements, and of unstable equilibrium for other displacements.

113.] To determine the number of the points and lines of equi- librium, let us consider the surface or surfaces for which the potential is equal to C, a given quantity. Let us call the regions in which the potential is less than C the negative regions, and those in which it is greater than C the positive regions. Let VQ be the lowest, and V^ the highest potential existing in the electric field. If we make C =^, the negative region will in- clude only the point or conductor of lowest potential, and this is necessarily charged negatively. The positive region consists of the rest of space, and since it surrounds the negative region it is periphractic. See Art. 18.

If we now increase the value of C, the negative region will expand, and new negative regions will be formed round negatively charged bodies. For every negative region thus formed the sur- rounding positive region acquires one degree of periphraxy.

As the different negative regions expand, two or more of them may meet in a point or a line. If n--\ negative regions meet, the positive region loses n degrees of periphraxy, and the point or the line in which they meet is a point or line of equilibrium of the nth degree.

1I4-] THEIR NUMBER, 159

When C becomes equal to J{ the positive region is reduced to the point or the conductor of highest potential, and has therefore lost all its periphraxy. Hence, if each point or line of equilibrium counts for one, two, or n, according to its degree, the number so made up by the points or Knes now considered will be less by one than the number of negatively charged bodies.

There are other points or lines of equilibrium which occur where the positive regions become separated from each other, and the negative region acquires periphraxy. The number of these, reckoned according to their degrees, is less by one than the number of positively charged bodies.

If we call a point or line of equilibrium positive when it is the meeting-place of two or more positive regions, and negative when the regions which unite there are negative, then, if there are p bodies positively and n bodies negatively charged, the sum of the degrees of the positive points and lines of equilibrium will be p — 1 3 and that of the negative ones n — 1. The surface which sur- rounds the electrical system at an infinite distance from it is to be reckoned as a body whose charge is equal and opposite to the sum of the charges of the system.

But, besides this definite number of points and lines of equi- librium arising from the junction of different regions, there may be others, of which we can only affirm that their number must be even. For if, as any one of the negative regions expands, it meets itself, it becomes a cyclic region, and it may acquire, by repeatedly meeting itself, any number of degrees of cyclosis, each of which corresponds to the point or line of equilibrium at which the cyclosis was established. As the negative region continues to expand till it fills all space, it loses every degree of cyclosis it has acquired, and becomes at last acyclic. Hence there is a set of points or lines of equilibrium at which cyclosis is lost, and these are equal in number of degrees to those at which it is acquired.

If the form of the charged bodies or conductors is arbitrary, we can only assert that the number of these additional points or lines is even, but if they are charged points or spherical conductors, the number arising in this way cannot exceed (n — 1) (n — 2), where n is the number of bodies.

114.] The potential close to any point P may be expanded in the series jr= ^ + J2r1 + jEra+&c.;

where H19 H2, &c. are homogeneous functions of #, y, z, whose dimensions are 1, 2, &c. respectively.

160 POINTS AND LINES OF EQUILIBRIUM. [115.

Since the first derivatives of V vanish at a point of equilibrium, H1 = 0, if P be a point of equilibrium.

Let Hn be the first function which does not vanish, then close to the point P we may neglect all functions of higher degrees as compared with Hn.

Now Hn = 0

is the equation of a cone of the degree n, and this cone is the cone of closest contact with the equipotential surface at P.

It appears, therefore, that the equipotential surface passing through P has, at that point, a conical point touched by a cone of the second or of a higher degree. The intersection of this cone with a sphere whose centre is the vertex is called the Nodal line.

If the point P is not on a line of equilibrium the nodal line does not intersect itself, but consists of n or some smaller number of closed curves.

If the nodal line intersects itself, then the point P is on a line of equilibrium, and the equipotential surface through P cuts itself in that line.

If there are intersections of the nodal line not on opposite points of the sphere, then P is at the intersection of three or more lines of equilibrium. For the equipotential surface through P must cut itself in each line of equilibrium.

115.] If two sheets of the same equipotential surface intersect, they must intersect at right angles.

For let the tangent to the line of intersection be taken as the axis of z, then d27/dz2 = 0. Also let the axis of x be a tangent to one of the sheets, then d27/dx2 = 0. It follows from this, by Laplace's equation, that d27/dyz = 0, or the axis of y is a tangent to the other sheet.

This investigation assumes that H2 is finite. If H2 vanishes, let the tangent to the line of intersection be taken as the axis of z, and let x = r cos 0, and y = r sin 0, then, since

d27 d27 _

1.^4-—— — 0

the solution of which equation in ascending powers of r is 7— 70 + Al r cos (0--a) + A2 r2 cos (2 0 -f ct2) + &c. + Anrncos (n6 + an). At a point of equilibrium A1 is zero. If the first term that does not vanish is that in rn, then

V— 70 = An rn cos (n0 + an) + terms in higher powers of r,

1 1 6.] THEIR PROPERTIES. 161

This equation shews that n sheets of the equipotential surface V=. FQ intersect at angles each equal to ir/n. This theorem was given by Rankine*.

It is only under certain conditions that a line of equilibrium can exist in free space, but there must be a line of equilibrium on the surface of a conductor whenever the surface density of the conductor is positive in one portion and negative in another.

In order that a conductor may be charged oppositely on different portions of its surface, there must be in the field some places where the potential is higher than that of the body and others where it is lower.

Let us begin with two conductors electrified positively to the same potential. There will be a point of equilibrium between the two bodies. Let the potential of the first body be gradually diminished. The point of equilibrium will approach it, and, at a certain stage of the process, will coincide with a point on its surface. During the next stage of the process, the equipotential surface round the second body which has the same potential as the first body will cut the surface of the second body at right angles in a closed curve, which is a line of equilibrium. This closed curve, after sweeping over the entire surface of the conductor, will again contract to a point ; and then the point of equilibrium will move off on the other side of the first body, and will be at an infinite distance when the charges of the two bodies are equal and opposite.

Earnshaw's Theorem.

116.] A charged body placed in a field of electric force cannot be in stable equilibrium.

First, let us suppose the electricity of the moveable body (A), and also that of the system of surrounding bodies (B), to be fixed in those bodies.

Let V be the potential at any point of the moveable body due to the action of the surrounding bodies (B), and let e be the electricity on a small portion of the moveable body A surrounding this point. Then the potential energy of A with respect to B will be

M = ^(Ye\ where the summation is to be extended to every charged portion of J.

• 'Summary of the Properties of certain Stream Lines/ Phil. Mag., Oct. 1864. See also, Thomson and Tait's Natural Philosophy, § 780 ; and Rankine and Stokes, in the Proc. R. S., 1867, p. 468 ; also W. R. Smith, Proc. R. S. Edin., 1869-70, p. 79.

VOL. I. M

162 POINTS AND LINES OF EQUILIBRIUM. [ll6.

Let a, 6} c be the coordinates of any charged part of A with respect to axes fixed in A, and parallel to those of x,y, z. Let the absolute coordinates of the origin of these axes be £, rj, £

Let us suppose for the present that the body A is constrained to move parallel to itself, then the absolute coordinates of the point a, 6, e will be

The potential of the body A with respect to B may now be expressed as the sum of a number of terms, in each of which V is expressed in terms of 0, #, c and f, r/, £ and the sum of these terms is a function of the quantities a, b, c, which are constant for each point of the body, and of £, rj, £ which vary when the body is moved.

Since Laplace's equation is satisfied by each of these terms it is satisfied by their sum, or

dM dM d*M " ' ~

drf

Now let a small displacement be given to A, so that d^ = Idr, dri = m dr} d£ = n dr ;

and let dM be the increment of the potential of A with respect to the surrounding system B.

If this be positive, work will have to be done to increase r, and there will be a force It = dM/dr tending to diminish r and to restore A to its former position, and for this displacement therefore the equilibrium will be stable. If, on the other hand, this quantity is negative, the force will tend to increase r, and the equilibrium will be unstable.

Now consider a sphere whose centre is the origin and whose radius is /, and so small that when the point fixed in the body lies within this sphere no part of the moveable body A can coincide with any part of the external system B. Then, since within the sphere V2M = 0, the surface-integral

ffdM g JJ dr ' taken over the surface of the sphere, is zero.

Hence, if at any part of the surface of the sphere dM/dr is positive, there must be some other part of the surface where it is negative, and if the body A be displaced in a direction in which dM/dr is negative, it will tend to move from its original position, and its equilibrium is therefore necessarily unstable.

The body therefore is unstable even when constrained to move

1 1 6.] EQUILIBRIUM ALWAYS UNSTABLE. 163

parallel to itself, and a fortiori it is unstable when altogether free.

Now let us suppose that the body A is a conductor. We might treat this as a case of equilibrium of a system of bodies, the move- able electricity being considered as part of that system, and we might argue that as the system is unstable when deprived of so many degrees of freedom by the fixture of its electricity, it must a fortiori be unstable when this freedom is restored to it.

But we may consider this case in a more particular way, thus —

First, let the electricity be fixed in A, and let A move through the small distance dr. The increment of the potential of A due to this cause has been already considered.

Next, let the electricity be allowed to move within A into its position of equilibrium, which is always stable. During this motion the potential will necessarily be diminished by a quantity which we may call Cdr.

Hence the total increment of the potential when the electricity is free to move will be

and the force tending to bring A back towards its original position will be

dm W"c>

where C is always positive.