by substituting 200 for 199, which may be done without sensible error. The condensing power would therefore be equal to 50; that is to say, the disc in communication with the source would become, by the mere fact of the induction exercised upon it by the second disc, capable of becoming charged with fifty times more electricity than it could have acquired had it been in communication with the same source, without being under the influence of the second disc.

In order to determine m experimentally with a given condenser, we charge it with electricity; we then separate the two discs, taking care that they do not lose their electric charge. We touch the first with the proof plane of the torsion balance, and on carrying the plane to the balance, we obtain an angle of torsion a, which is proportional, as we know, to the total charge of the disc, namely to A. We have, therefore, a=n A, n being a fraction smaller than unity. We do the same with the second disc, charged with the contrary electricity, B; we have, therefore, an angle b=nв, provided we take the precaution to touch this disc in a point placed in a similar manner to the point that had been touched of the first disc. It is essential to remark that the two discs are in all respects perfectly similar, of the same size, the same form, &c. We have, therefore, a=na and b=nв; therefore,

When we are considering a Leyden jar, whose insulating stratum is very thick, since it is a plate of glass, the quantity m is far removed from unity, and B is much smaller than A. Also, when we unite the two coatings of the jar, in order to produce the discharge, or the neutralisation of the two electricities, there remains, after this discharge has taken place, a considerable quantity of electricity in the coating that had been in communication with the source. We may easily calculate this quantity by

charge, there remains therefore on the coating that had been charged with the quantity A, a quantity A-B, or

NOTE C. (p. 179.)

Note relative to the measure of Magnetic Forces.

We have said that there are two modes of measuring magnetic forces; the first, based upon the employment of the torsion balance; the second, upon the principle of oscillations. In either method we must take into account the action of the terrestrial globe, which contributes, independently of the force whose intensity we are measuring, to the total effect given by direct observation. This action of the terrestrial globe, in fact, is exercised upon the magnetised needle according to laws susceptible of being subjected to rigorous calculation, of which the present note is intended to give a summary description, and the results of which will not only serve for the demonstration of the principles that we have already implicitly admitted, but will be indispensable to us in the sequel in the study that we shall be called upon to make of terrestrial magnetism. We shall divide this note into three parts: 1st. The general expression of the effect of terrestrial magnetism upon a needle in equilibrium; 2d. Law of the movement of the needle when it is deranged from its position of equilibrium, but enabled to turn around its centre of gravity, still remaining in the plane of the magnetic meridian; 3rd. Law of this movement, when the needle goes out of the plane of the magnetic meridian.

§ 1. General expression of the effect of terrestrial magnetism upon the needle in equilibrium in the magnetic meridian.

b

S

M

We will suppose (Fig. 3.) a magnetised needle, ab, of any form, suspended by its centre of gravity. Let μ be the quantity of free magnetism in one of its points, for example in м; this point will be attracted by forces arising from the north pole of the earth, and will be repelled by those arising from the south pole; the former will act in the direction MA, and the latter in the direc

tion MB.

a

B

R

Fig. 3.

These two forces will themselves have a resultant, MR,

situated in the plane formed by their directions; we will call this

resultant g', in order not to confound it with g, which expresses the force of gravity; the total effect on the point м will therefore be expressed by the product g'. Each of the other points of the needle will therefore be solicited by a same force g', since the resultant of the magnetic forces of the globe cannot vary in the small extent that is occupied by the length of the needle; μ only, namely the quantity of free magnetism, varies from one point to the other. The general expression, therefore, of the effect upon each point will be g′μdm, dm representing the element of mass for which μ does not vary.

We may remark that all the resultants of the magnetic force of the globe, represented by g', are parallel; consequently, all the forces gudm are parallel to each other, and their general resultant is equal to their sum. Expressing this sum by s, we shall have the general resultant equal to sg'μdm. But sudm, in this expression, represents the sum of the free magnetisms in each of the points of the needle. Now, as there are as many south magnetisms as there are north magnetisms, and as a certain quantity of south magnetism neutralises an equal quantity of north magnetism, the sum suμdm must be equal to 0; and, consequently, the general resultant g'sudm must be null. This result indicates that, when a needle has taken such a direction as the sum of the forces of terrestrial magnetism tends to impress upon it, these forces cannot impress upon it any motion of translation in space. Experiment confirms this: for a steel needle does not increase in weight by the effect of magnetisation; and, if it is arranged as indicated in Fig. 76. p. 184., it directs itself just as if it had been suspended immediately by its centre, without having been carried either forwards or backwards.

§ 2. Laws of the movement of the needle, when compelled to move around its centre of gravity in the plane of the magnetic meridian.

The needle whose movement will next engage our attention is what has been called the dipping-needle (Fig. 74. p. 161.). We represent it by a simple right line, although in reality it is composed of several parallel right lines; but it is easy to see that the real case may be reduced to the hypothetical elementary case. Let ab (Fig. 4.) be this needle; let MR be the resultant g' of the forces of terrestrial magnetism, necessarily comprised in the plane of the magnetic meridian, and whose direction forms with the

vertical an angle i, constant for the place of observation, and for a given time. Supposing the needle suspended by its centre of

b

G

Fig. 4.

P

a

R

gravity, the resultant g' tends to destroy its horizontality, and to give it an inclination, which, in our hemisphere gives the branch that is di rected towards the north, namely that wherein we suppose the point м to be situated, a tendency to incline. toward the earth. In Fig. 4. the needle ab is no longer represented as horizontal; and, in order to ascertain the direction that it will assume, we decompose the resultant g' or MR into two components; one, MP, drawn perpendicularly to the direction of the needle, the other drawn in the same direction as the length of the needle. The first is evidently the only one that contributes to cause the needle to turn, and it is equal to gcos. PMR; and by letting z represent the variable angle formed by the needle with the vertical, we have PMZ-90°-z; and PMR=90°—z+i; or cos. PMR, or cos. (90°—z+i) = sin. (z−i): so that the perpendicular component will become equal to g' sin. (z-i); and the action exercised upon the point м is μg'sin. (z-i), supposing that at this point there is μ of free magnetism. In order to obtain the real effect of its force, or its movement around the centre of suspension c, we must multiply its expression by the distancer from м to C, which gives urg ́sin. (z−i). Now, z-i is the angle formed by the direction of the resultant with that of the needle, which shows that the force which causes the point м to turn is analogous to that which draws back each of the points of a pendulum to the vertical: which, when this pendulum is drawn out from the vertical to an angle a, is equal to g sin. a, g being the force of gravity; and its momentum is gr sin. a, r being the distance of the extremity of the pendulum from its centre of suspension.

In order to obtain the effect, not only upon the point м, but on the whole extent of the needle, it is necessary to consider each of its halves separately, or each of the two arms of the lever ca and cb. The sum of the moments for each will be surg' sin. (z−i); but g'sin. (z-i) is a quantity that does not change in passing from one point to another, or may therefore be regarded as constant; and the expression become g' sin. (z-i)surdm.

surdm must be obtained separately for each of the halves of the needle for the magnetisms in the two halves are contrary and equal, and if one tends to make the needle descend, the other tends to make it ascend; so that the effects are added together in order to make it turn. Let us call the two sums s' and s"; we obtain, for the total momentum, by which the needle is actuated, g'(s'+s") sin. (z-i); an expression that becomes null when z=i; namely, when the direction of the needle coincides with the resultant of the magnetic forces of the earth. Thus, so long as the needle is out of this position, it is solicited to return to it, and it does return to it by oscillating from one side to the other, about the direction of its fixed inclination, until its momentum is destroyed by the resistance of the air and the inertia of suspension. These oscillations of the needle are perfectly analogous, and are subjected to the same laws as are those executed by a pendulum when drawn from the vertical, in order to return to it by virtue of its gravity. In fact, the terrestrial magnetic forces that solicit the divers points of a needle, have directions parallel to each other like the forces of gravity; and like as those forces have a resultant that is applied to a point which is the centre of gravity of the body, there exists for each part of the needle in which a similar kind of magnetism resides, a magnetic centre, to which we may suppose all the magnetic force of the earth applied. It is true, that here the magnetic forces are not all equal to each other, since, which enters as a factor into the expression μg'dm, varies from one point of the needle to another, but this circumstance makes no change in the analogy that we have established; for the existence of centres of gravity does not depend upon the equality, but upon the parellelism of the forces of gravity. We may therefore calculate the position of the point of application of the resultant of the magnetic forces, as we calculate the place of the centre of gravity in a heavy body. The distance of these points from the centre of suspension of the needle will be, for each of the halves of the needle, equal to the sum of the momentums divided by the sum of the forces; that is to say, by Surdm

sudm

sudm is not null here, since it is relative to only each

of the halves of the needle.

It follows from what has been said that, like as intensities of gravity are measured from the duration of very small oscillations made by a pendulum around the vertical, those of the magnetic forces of the terrestrial globe may be measured by the duration of