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In these expressions we must remember that M is a negative quantity, that L+ M and M+ N can neither of them be negative, and that their sum L+ 2M+ N cannot be greater than the largest semidiameter of the condenser. Hence if R is large compared with the dimensions of the condensers, the second term of the values of [AA] and [AB] will be quite insensible, and even if the condensers are placed very near together these terms will be small compared with L, M, or N.

If a, instead of being part of a condenser, is a conductor at a considerable distance from any other conductor, we may put m = n = 0, and if A is also a simple conductor, M=N=0, and we find

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by which the capacities and mutual induction of two simple conductors at a distance R can be calculated when we know their capacities when at a great distance from other conductors. Note 24.

See

NOTE 17, ART. 194.

Theory of the Experiment with the Trial Plate.

Let A and B be the inner, a and b the outer coatings of the Leyden jars.

Let C be the body tried and D the trial plate, M the wire connecting A with C, and N the wire connecting b with D.

Let E be the electrometer with its connecting wires.

Let the coefficients of induction be expressed by pairs of symbols within square brackets, thus, let [(4+ C) (C + D)] denote the sum of the charges of A and C when C and D are both raised to potential 1 and all the other conductors are at potential 0.

First Operation.-The insides of the two jars are charged to potential P, the outsides and all other bodies being at potential 0.

The charge of A is [A (A + B)]P„, and that of b is [b (A + B)] P ̧·

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Second Operation. The outside coating of b is insulated, the charging wire is removed, and the inside of B is connected to earth. The charges of A and of b remain as before.

Third Operation.-A is connected to C by the wire M, and b is connected to D by the wire N.

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 P.

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

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

=

· [A (A + B)] P。,

(1)

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[(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

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Eliminating P, and P, from equations (1), (2) and (3),

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P2 [(C + D + E)2] {[(A + C + M)3] [(b + D + N)3]

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

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

+ [b (A + B)] {[(C + D) (b + D + N ) ][ (4 + C + M)2]

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

3

(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

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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 [b] or [Bb]. We may therefore put [Bb] = - [b], and in equation (4) neglect all terms except those containing the factors [4][b3] or [A2][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 ([C'3] + [D3] + [E2]) = P。 ([C3] − [D3]),

(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

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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] + [E2]) = P. ([C3] −[D′3]).

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

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

(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'2],

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

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