THEORY OF CAVENDISH'S TRIALS.

407

The charge of A is communicated to A, C, and M, and the potential of this system is P1, and the charge of b is communicated to b, D and N, and the potential of this system is P2.

Hence we have the following equations to determine P1 and P2 in terms of P.,

[(A + C + M) (A + C + M)] P2 + [(A + C + M ) (b + D + N)] P2

= [A (A + B)] P。,

(1)

[(A + C + M) (b + D + N )] P1 + [(b + D + N ) (b + D + N)] P2

Fourth Operation. The wires M and N are disconnected from C and D respectively, and the jars A and b are discharged and kept connected to earth.

The charges of C and D remain the same as before.

Fifth Operation.-The bodies C and D are connected with each other and with the electrometer E, and the final potential of the system CDE is observed by the electrometer to be P

Equating the final charge of the system CDE to that of the system CD at the end of the fourth equation,

[(C + D + E) (C + D + E )] P2 = [(C + D) (A + C + M )] P1

3

Eliminating P, and P, from equations (1), (2) and (3), P2 [(C + D + E)2] {[(4 + C + M)2] [(b + D + N)3]

= P。

− [(A + C + M ) (b + D + N)]2}

[A (A + B)] {[(C + D) (A + C' + M )] [(b + D + N)3]

J[4 (4 + ·

− [(C + D) (b + D + N)][(A + C + M) (b + D + N)]}

+ [b(A + B)] {[(C + D) (b + D + N )][(A + C + M)3]

− [(C + D) (A + C + M )] [(A + C + M ) (b + D + N)]}

(4)

By means of his gauge electrometer, Art. 249, Cavendish made the value of P, the same in every trial, and altered the capacity of D, the trial plate, so that P, in one trial had a particular positive value, and in another an equal negative value. He then wrote down the difference of the two values of D as an indication to guide him in the choice of trial plates, and the sum of the two values, by means of which he compared the charges of different bodies.

He then substituted for C a body, C', of nearly equal capacity, and repeated the same operations, and finally deduced the ratio of C to C' from the equation

C: C': D, + D ̧ : D ̧' + D ̧'.

The capacities of the two jars were very much greater than any of the other capacities or coefficients of induction in the experiment, and of these [b (B+b)] was less than half the greatest diameter of the second

jar, and may therefore be neglected in respect of [b2] or [Bb]. We may therefore put [Bb] = − [b3], and in equation (4) neglect all terms except those containing the factors [4] [b2] or [42][Bb].

We thus reduce equation (4) to the form

P ̧[(C + D + E)3] = P。 {[(C + D) (A + C + M )] − [(C + D) (b + D + N)]}

= P。 {[C2] + [C (A + M)] − [C (b + N)]

- [D3] - [D (b + N)] + [D (A + M )]}.

(5)

The bodies to be compared were either simple conductors, such as spheres, disks, squares and cylinders, and those trial plates which consisted of two conducting plates sliding on one another, or else coated plates or condensers.

Now the coefficient of induction between a coated plate and a simple conductor is much less than that between two simple conductors of the same capacity at the same distance, and the coefficient of induction between two coated plates is still smaller. See Note 16.

Hence if both the bodies tried are coated plates, the equation (5) is reduced to the form

P2 ([C3] + [D3] + [E2]) = P。 ([C2] − [D2]),

(6)

so that the experiment is really a comparison of the capacities of the two bodies C and D.

But if either of them is a simple conductor, we must add to its capacity its coefficient of induction on the wire and jar with which it is connected, and subtract from it its coefficient of induction on the other wire and jar. These two coefficients of induction are both negative, but that belonging to its own wire and jar is probably greater than the other, so that the correction on the whole is negative.

Hence in Cavendish's trials the capacity deduced from the experiment will be less for a simple conductor than for a coated plate of equal real capacity.

This appears to be the reason why the capacities of the plates of air when expressed in "globular inches," that is, when compared with the capacity of the globe, are about a tenth part greater than their computed values. See Art. 347.

It would have been an improvement if Cavendish, instead of charging the inside of both jars positively and then discharging the outside of B, had charged the inside of A and the outside of B from the same conductor, and then connected the outside of both to earth, using the inside of B instead of the outside, to charge the trial plate negatively. In this way the excess of the negative electricity over the positive in B would have been much less than when the outside was negative.

With a heterostatic electrometer, such as those of Bohnenberger or Thomson, in which opposite deflections are produced by positive and negative electrification, the determination of the zero electrification may

3

CAVENDISH'S TRIALS OF CAPACITY.

409

be made more accurately than any other, and with such an electrometer P, should be adjusted to zero. But the only electrometer which Cavendish possessed was the pith ball electrometer, in which the repulsion between the balls when at any given distance depends on the square of the electrification, and in which therefore the indications are very feeble for low degrees of electrification. Cavendish therefore first adjusted his trial plate so as to produce a given amount of separation of the balls by positive electrification, and then altered the trial plate so as to produce an equal separation by negative electrification. In each case he has recorded a number expressing the side of a square electrically equivalent to the trial plate, together with the difference and the mean of the two values.

He seems to have adopted the arithmetical mean as a measure of the charge of the body to be tried. It is easy to see, however, that the geometrical mean would be a more accurate value. For, if we denote the values of the final potential of the trial plate by accented letters in the second trial, we have

P ̧' ([C3]+[D'3] + [E3]) = P。 ([C2] − [D ́°3]).

Since P+P=0, we find by (6) and (7)

[C2] ([C2] + [E3]) = [D2] [D ́3] + } [Ea] ([D3] + [D'2]).

(7)

If we neglect the capacity of the pith ball electrometer, which is much less than that of the bodies usually tried, this equation becomes

[C2]2 = [D2] [D'],

or the capacity of the body tried is the geometrical mean of the capacities of the trial-plate in its positive and negative adjustments.

NOTE 18, ART. 216.

On the "Thoughts Concerning Electricity," and on an early draft of the Propositions in Electricity.

The theory of electricity sketched in the "Thoughts" is evidently an earlier form of that developed in the published paper of 1771. We must therefore consider the "Thoughts as the first recorded form of Cavendish's theory, and this for the following reasons.

(1) Nothing is said in the "Thoughts" of the forces exerted by ordinary matter on itself and on the electric fluid. The only agent considered is the electric fluid itself, the particles of which are supposed to repel each other. This fluid is supposed to exist in all bodies whether apparently electrified or not, but when the quantity of the fluid in any body is greater than a certain value, called the natural quantity for the body, the body is said to be overcharged, and when the quantity is less than the natural quantity the body is said to be undercharged.

The forces exerted by undercharged bodies are ascribed, not, as in the later theory, to the redundant matter in the body, but to the repulsion of the fluid in other parts of space.

The theory is therefore simpler than in its final form, but it tacitly assumes that the fluid could exist in stable equilibrium if spread with uniform density over all space, whereas it appears from the investigations of Cavendish himself that a fluid whose particles repel each other with a force inversely as any power of the distance less than the cube would be in unstable equilibrium if its density were uniform.

This objection does not apply to the later form of the theory, for in it the equilibrium of the electric fluid in a saturated body is rendered stable by the attraction exerted by the fixed particles of ordinary matter on those of the electric fluid.

(2) The hypotheses are reduced in the later theory to one, and the third and fourth hypotheses of the "Thoughts" are deduced from this.

(3) In the "Thoughts" Cavendish appears to be acquainted only with those phenomena of electricity which can be observed without quantitative experiments. Some of his remarks, especially those on the spark, he repeats in the paper of 1771, but in that paper (Art. 95) he refers to certain quantitative experiments, the particulars of which are now first published [Art. 265].

The "Thoughts," however, though Cavendish himself would have considered them entirely superseded by the paper of 1771, have a scientific interest of their own, as showing the path by which Cavendish arrived at his final theory.

He begins by getting rid of the electric atmospheres which were still clinging to electrified bodies, and he appears to have done this so completely that he does not think it worth while even to mention them in the paper of 1771.

He then introduces the phrase "degree of electrification" and gives a quantitative definition to it, so that this, the leading idea of his whole research, was fully developed at the early date of the "Thoughts."

Several expressions which Cavendish freely used in his own notes and journals, but which he avoided in his printed papers, occur in the "Thoughts."

Thus he speaks of the "compression" or pressure of the electric fluid.

Besides the "Thoughts," which may be considered as the original form of the introduction to the paper of 1771, there is a mathematical paper corresponding to the Propositions and Lemmata of the published paper, but following the earlier form of the theory, in which the forces exerted by ordinary matter are not considered, and referring directly to the "Hypotheses" of the "Thoughts."

The first part of this paper is carefully written out, but it gradually becomes more and more unfinished, and at last terminates abruptly, though, as this occurs at the end of a page, we may suppose that the end of the paper has been lost. I think it probable, however, that when

CAVENDISH'S FIRST MATHEMATICAL THEORY.

411

Cavendish had advanced so far, he was beginning to see his way to the form of the theory which he finally published, and that he did not care to finish the manuscript of the imperfect theory.

The general theory of fluids repelling according to any inverse power of the distance is given much more fully than in the paper of 1771, and the remarks on the constitution of air are very interesting.

I have therefore printed this paper, but in order to avoid interrupting the reader with a repetition of much of what he has already seen, I have placed it at the end of this Note.

CAVENDISH'S FIRST MATHEMATICAL THEORY *.

α

β

Let a fluid whose particles mutually repel each other be spread uniformly through infinite space. Let a be a particle of that fluid; draw the cone baß continued infinitely, and draw the section bẞ: if the repulsion of the particles is inversely as any higher power of the distance than the cube, the particle a will be repelled with infinitely more force from the particles between a and bẞ than from all those situated beyond it, but if their repulsion is inversely as any less power than the cube, then the repulsion of the particles placed beyond bẞ is infinitely greater than that of those between a and bẞ.

If the repulsion of the particles is inversely as the n power of the distance, n being greater than 3, it would constitute an elastic fluid of the same nature as air, except that its elasticity would be inversely as the n+2 power of the distance of the particles, or directly as the n+2 power of the density of the fluid.

3

But if n is equal to, or less than 3, it will form a fluid of a very different kind from air, as will appear from what follows.

B

COR. 1. Let a fluid of the above-mentioned kind be spread uniformly through infinite space except in the hollow globe BDE, and let the sides of the globe be so thin that the force with which a particle placed contiguous to the sides of the globe would. be repelled by so much of the fluid as might be lodged within the space occupied by the sides of the globe should be trifling in respect of the repulsion of the whole quantity of fluid in the globe.

If the fluid within the globe was of the same density as without, the particles of the fluid adjacent to either the inside or outside

P

surface of the globe would not press against those surfaces with any sensible force, as they would be repelled with the same force by the fluid on each side of them. But if the fluid within the globe is denser

* From MS. bundle 17.