Hence 12

which is of the order 1020 at least. the repulsion between two grammes of matter entirely deprived of electricity, is of the same order as aa2.

If we consider the attraction of gravitation as something quite independent of the attractions and repulsions observed in electrical phenomena, we may suppose

12

so that two saturated bodies neither attract nor repel each other.

(10)

Now we have adopted as the condition of saturation, that neither body acts on the electric fluid in the other. But since neither body acts on the other as a whole, each has no action on the matter in the other, so that our definition of saturation coincides with that given by Cavendish,

1

2

Lastly, let the two bodies not be saturated with electricity, but contain quantities F1 + E, and F, + E, respectively, where F1 = a,M1, and Fa,M, and E, and E, may be either positive or negative, provided that F+E must in no case be negative.

The repulsion between the bodies is

(F1 + E1) (F2 + E2) − (F1 + E1) M ̧¤ ̧ − (F2 + E2) M ̧α1⁄2 + M1M112 (11) and this by means of equations (3) (4) and (10) is reduced to

In the theory of Two Electric Fluids, let V denote the quantity of the Vitreous fluid and R that of the Resinous.

Let the repulsion between two units of the same fluid be b, and let the attraction between two units of different fluids be c.

Let the attraction between a unit of either fluid and a gramme of matter be a, and let the repulsion between two grammes of matter be r.

If a body contains V, units of vitreous, R, units of resinous electricity, and M, grammes of matter, its repulsion on a unit of vitreous electricity will be

V1b-R ̧c - Ma1,

The definition of saturation is that there shall be no action on either kind of electricity. Hence, equating each of these expressions to zero, we find as the conditions of saturation

The total repulsion between the two bodies is

(V ̧V ̧+R ̧R ̧)b−(V ̧R ̧+V ̧R ̧)c−(V ̧+R ̧)M ̧a ̧−(V ̧+R ̧) M ̧a,+M ̧M ̧r1,

The first term of this expression, with its sign reversed, represents the attraction of gravitation, and the second term represents the observed electric action, but the other terms represent forces of a kind which have not hitherto been observed, and we must modify the theory so as to account for their non-existence.

One way of doing so is to suppose bc and a1 = a, = 0. The result of this hypothesis is to reduce the condition of saturation to that of the equality of the two fluids in the body, leaving the amount of each quite undetermined. It also fails to account for the observed action between the bodies themselves, since there is no action between them and the electric fluids.

The other way is to suppose that S, S, = 0, or that the sum of the quantities of the two fluids in a body always remains the same as when the body is saturated. This hypothesis is suggested by Priestley in his account of the two-fluid theory, but it is not a dynamical hypothesis, because it does not give a physical reason why the sum of these two quantities should be incapable of alteration, however their difference is

varied.

The only dynamical hypothesis which appears to meet the case is to suppose that the vitreous and resinous fluids are both incompressible, and that the whole of space not occupied by matter is occupied by one or other of them. In a state of saturation they are mixed in equal proportions.

The two-fluid theory is thus considerably more difficult to reconcile with the facts than the one-fluid theory.

NOTE 2, ARTS. 27 AND 282.

The problem of the distribution, in a sphere or ellipsoid, of a fluid, the particles of which repel each other with a force varying inversely as the nth power of the distance, has been solved by Green*. Green's method is an extremely powerful one, and allows him to take account of the effect of any given system of external forces in altering the distribution.

If, however, we do not require to consider the effect of external forces, the following method enables us to solve the problem in an elementary manner. It consists in dividing the body into pairs of corresponding elements, and finding the condition that the repulsions of corresponding elements on a given particle shall be equal and opposite.

(1) Specification of Corresponding Points on a line.

Let 4,4, be a finite straight line, let P be a given point in the line, and let Q and Q, be corresponding points in the segments AP and PA, respectively, the condition of correspondence being

It is easy to see that when Q, coincides with A1, Q2 coincides with A2, and that as Q moves from A, to P, Q, moves in the opposite direction from A, to P, so that when Q, coincides with P, Q, also

coincides with P.

1

Let Q,' and Q,' be another pair of corresponding points, then

If the points Q, and Q,' are made to approach each other and ultimately

1

"Mathematical Investigations concerning the laws of the equilibrium of fluids analogous to the electric fluid, with other similar researches," Transactions of the Cambridge Philosophical Society, 1833. Read Nov. 12, 1832. See Mr Ferrers' Edition of Green's Papers, p. 119.

to coincide, QQ,' ultimately becomes the fluxion of Q, which we may write Q, and we have

or corresponding elements of the two segments are in the ratio of the squares of their distances from P.

Let us now suppose that A,PA, is a double cone of an exceedingly small aperture, having its vertex at P; let us also suppose that the density of the redundant fluid at Q, is p,, and at Q, is Pai then since the areas of the sections of the cone at Q, and Q, are as the squares of the distances from P, and since the lengths of corresponding elements are also, by (5), as the squares of their distances from P, the quantities of fluid in the two corresponding elements at Q, and Q, are as p,Q,P to p2PQ,*. If the repulsion is inversely as the nth power of the distance, the condition of equilibrium of a particle of the fluid at P under the action of the fluid in the two corresponding elements at Q, and Q, is

2

1

(6)

We have now to show how this condition may be satisfied by one and the same distribution of the fluid when P is any point within an ellipsoid or a sphere. We must therefore express p so that its value is independent of the position of P.

Multiplying the corresponding members of equations (1) and (7) and omitting the common factor à ̧P. PÅ,

2

Let us now suppose that 4,4, is a chord of the ellipsoid, whose equation is

1

(10)

(11)

then the product of the segments of the chord at Q, is to the product of the segments at Q, as the values of p3 at these points respectively, or

We may therefore write, instead of equation (9),

where C is constant, every particle of the fluid within the ellipsoid will be in equilibrium.

We have in the next place to determine whether a distribution of this kind is physically possible.

Let E be the quantity of redundant fluid in the ellipsoid,

Let p be the density of the redundant fluid if it had been uniformly spread through the volume of the ellipsoid, then

and if p is the actual density of the redundant fluid,

(17)

When n is not less than 2, there is no difficulty about the interpretation of this result.

The density of the redundant fluid is everywhere positive.

When n 4 it is everywhere uniform and equal to P..

=

Po

When n is greater than 4 the density is greatest at the centre and is zero at the surface, that is to say, in the language of Cavendish, the matter at the surface is saturated.

When n is between 2 and 4 the density of the redundant fluid at the centre is positive and it increases towards the surface. At the surface itself the density becomes infinite, but the quantity collected on the surface is insensible compared with the whole redundant fluid.