But we have by no means reached the limit of our experimental evidence. For Cavendish shows in Art. 49 that if in any portion of a bent canal the repulsion of overcharged bodies is so great as to drive all the fluid out of that portion, then the canal will no longer allow the fluid to run freely from one end to the other, any more than a siphon will equalize the pressure of water in two vessels, when the water does not rise to the bend of the siphon.

Hence if we could make the canal narrow enough, and the electric repulsion of bodies near the bend of the canal strong enough, we might have two conductors connected by a conducting canal but not reduced to the same potential, and this might be tested by afterwards connecting them by means of a conductor which does not pass close to any overcharged body, for this conductor will immediately reduce the two bodies to the same potential.

Such an experiment, if successful, would determine at once which kind of electricity ought to be reckoned positive, for, as Cavendish remarks in Art. 50, the presence of an undercharged body near the bend of the canal would not prevent the flow of electricity.

But even if the electric fluid were not all driven out of the canal, but only out of a stratum near the surface, the effective conducting channel would thereby be narrowed, and the resistance of the canal to an electric current increased.

Now we may construct the canal of a strip of the thinnest gold leaf, and we may measure its electric resistance to within one part in ten thousand, so that if the presence of an overcharged body near the gold leaf were to drive the electric fluid out of a stratum of it amounting to the ten thousandth part of its thickness, the alteration might be detected. Hence we must admit either that the one-fluid theory is wrong, or that every cubic centimetre of gold contains more than ten thousand million units of electricity.

The statement which Cavendish gives of the action between portions of the electric fluid and between the electric fluid and ordinary matter is nearly, but not quite, as general as it can be made.

Since the mode in which the force varies with the distance is the same in all cases, we may suppose the distance unity. Two equal portions of the electric fluid which at this distance repel each other with a force unity are defined to be each one unit of electricity.

Let the attraction between a unit of the electric fluid and a gramme of matter be a. Since we may suppose this force different for different kinds of matter, we shall distinguish the attraction due to different kinds of matter by different suffixes, as a, and a,. Let the repulsion between two grammes of matter entirely deprived of electricity be r, these two portions of matter being of the kinds corresponding to the suffixes 1 and 2.

Now consider a body containing M grammes of matter and F units of the electric fluid. The repulsion between this body and a unit of the electric fluid at distance unity is

If this expression is zero, the body will neither repel nor attract the electric fluid. In this case the body is said to be saturated with the electric fluid, and the condition of saturation is that every gramme of matter contains a units of the electric fluid. From what we have already said, it is plain that a must be a number reckoned by thousands of millions at least. The definition of saturation as given by Cavendish is somewhat different from this, although on his own hypothesis it leads to identical results. He makes the condition of saturation to be (in Art. 6) "that the attraction of the electric fluid in any small part of the body on a given particle of matter shall be equal to the repulsion of the matter in the same small part on the same particle." Hence this condition is expressed by the equation

But as the essential property of a saturated body is that it does not disturb the distribution of electricity in neighbouring conductors, we must consider the true definition of saturation to be that there is no action on the electric fluid.

Now consider two bodies of different kinds of matter M, and M2, and let each of them be saturated.

The quantity of electric fluid in the first will be

The repulsion between the two bodies will be

FF-FM.a,- F„M ̧α, + M M ̧r1

2

1 2 2

2

(3)

(4)

(5)

or, substituting the values of F, and F, and changing the signs, it will

Now we know that the action between two saturated bodies is an

Hence we must make ага 2 2°

k

12

(7)

(8)

for every two kinds of matter, k being the same for all kinds of

matter.

According to Baily's repetition of Cavendish's experiment for determining the mean density of the earth*,

k = 6·506 × 10-8

(centimetre)3

gramme. second

(9)

This number is exceedingly small compared to the product a12,

Baily's adopted mean for the earth's density is 5.6604, which, with the values of the earth's dimensions and of the intensity of gravity at the earth's surface used by Baily himself, gives the above value of k as the direct result of his experiments.

129

which is of the order 1020 at least. Hence r, the repulsion between two grammes of matter entirely deprived of electricity, is of the same order as a12•

If we consider the attraction of gravitation as something quite independent of the attractions and repulsions observed in electrical phenomena, we may suppose a12-r12 = 0,

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

Lastly, let the two bodies not be saturated with electricity, but contain quantities F1 + E, and F2+ E, respectively, where F1 = a,M1, and F,a,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

2

2

(F1+ E1) (F2+ E2) - (F1 + E2) M2α, − (F2+ E2) M2α, + M1M112 (11) and this by means of equations (3) (4) and (10) is reduced to

E1 E,

Theory of Two Fluids.

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

1

V1b-R1c-Ma1,

and the repulsion on a unit of resinous electricity

-V1c+ Rb-Ma ̧.

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 ̧R1) c−(V ̧+R ̧) Ma−(V2+R ̧) M ̧a,+M ̧M ̧","

If we now put

1

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 a, 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 S1 = S2 = 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 A, Q2 coincides with A, 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.

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

* "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.