tricity upon it is known, and by substituting for r and h their values just given, there results Moreover the value of the potential function for the point p' whose polar co-ordinates are r, e, and w, is From which we may immediately deduce the forces acting on any point p' exterior to A. to In tracing the surface A, 0 is supposed to extend from 0=0 27: it is therefore evident, T, and w, from 0 to by constructing the curve whose equation is that the parts about P, where 6 = π, approximate continually in form towards a cone whose apex is P, and as the density of the electricity at P is null, in the example before us, we may make this general inference: when any body whatever has a part of its surface in the form of a cone, directed inwards; the density of the electricity in equilibrium upon it, will be null at its apex, precisely the reverse of what would take place, if it were directed outwards, for then, the density at the apex would become infinite*. • Since this was written, I have obtained formulæ serving to express, generally, the law of the distribution of the electric fluid near the apex of a cone, which forms part of a conducting surface of revolution having the same axis. From these formulæ it results that, when the apex of the cone is directed inwards, the density of the electric fluid at any point p, near to it, is proportional to -1; r being the distance Op, and the exponent n very nearly such as would satisfy the simple equation (4n+2) B=3′′: where 28 is the angle at the summit of the cone. If 28 exceeds π, this summit is directed outwards, and when the excess is not very considerable, n will be given as above: but 28 still increasing, until it be comes 27-27; the angle 27 at the summit of the cone, which is now directed 2 γ outwards, being very small, n will be given by 2n log=1, and in case the conducting body is a sphere whose radius is b, on which P represents the mean density As a second example, we will assume for V', the value of the potential function arising from the action of a line uniformly covered with electricity. Let 2a be the length of the line, y the perpendicular falling from any point p' upon it, a the distance of the foot of this perpendicular from the middle of the line, and a' that of the element de' from the same point: then taking the element da', as the measure of the quantity of electricity it contains, the assumed value of V' will be a−x + √ {y® + (a− x)3} the integral being taken from =—a to x+a. Making this equal to a constant quantity log b, we shall have, for the equation of the surface A, a−x + √ {y® + (a − x)*} which by reduction becomes 0 = y3 (1 − b3)3 + x3. 4b (1 — b)* — a2. 4b (1 + b)'. We thus see that this surface is a spheroid produced by the revolution of an ellipsis about its greatest diameter; the semitransverse axis being a 1+b = By differentiating the general value of V', just given, and substituting for y its value at the surface A, we obtain of the electric fluid, p, the value of the density near the apex 0, will be determined by the formula P= 2 Pbn a being the length of the cone. Now writing for the angle formed by dx and du', we have ds being an element of the generating ellipsis. Hence, as in the preceding example, we shall have On the surface A therefore, in this example, the general value and the potential function for any point p', exterior to ▲, is Making now x and y both infinite, in order that p' may be at an infinite distance, there results and thus the condition determining h, in Q, the quantity of electricity upon the surface, is, since R may be supposed equal to √(x2+y3), These results of our analysis agree with what has been long known concerning the law of the distribution of electric fluid on the surface of a spheroid, when in a state of equilibrium. (13.) In what has preceded, we have confined ourselves to the consideration of perfect conductors. We will now give an example of the application of our general method, to a body that is supposed to conduct electricity imperfectly, and which will, moreover, be interesting, as it serves to illustrate the magnetic phenomena, produced by the rotation of bodies under the influence of the earth's magnetism. If any solid body whatever of revolution, turn about its axis, it is required to determine what will take place, when the matter of this solid is not perfectly conducting, supposing it under the influence of a constant electrical force, acting parallel to any given right line fixed in space, the body being originally in a natural state. Let B designate the coercive force of the body, which we will suppose analogous to friction in its operation, so that as long as the total force acting upon any particle within the body is less than B, its electrical state shall remain unchanged, but when it begins to exceed B, a change shall ensue. In the first place, suppose the constant electrical force, which we will designate by b, to act in a direction parallel to a line passing through the centre of the body, and perpendicular to its axis of revolution; and let us consider this line as the axis of x, that of revolution being the axis of z, and y the other rectangular co-ordinate of a point p, within the body and fixed in space. Thus, if V be the value of the total potential function for the same point p, at any instant of time, arising from the electricity of the body and the exterior force, bx +V will be the part due to the body itself at the same instant: since -be is that due to the constant force b, acting in the direction of x, and tending to increase it. If now we make z=r cos 0, x = r sin 0 cosa, y = r sin sin; the angle being supposed to increase in the direction of the body's revolution, the part due to the body itself becomes br sin cos + V. Were we to suppose the value of the potential function V given at any instant, we might find its value at the next instant, by conceiving, that whilst the body moves forward through the infinitely small angle do, the electricity within it shall remain fixed, and then be permitted to move, until it is in equilibrium with the coercive force. Now the value of the potential function at p, arising from the body itself, after having moved through the angle do (the electricity being fixed), will evidently be obtained by changing into do in the expression just given, and is therefore br sine cos +V+br sin é sin dw dV dw, do adding now the part - bx-br sine cos, due to the exterior bodies, and restoring x, y, &c. we have, since for the value of the total potential function at the end of the next instant, the electricity being still supposed fixed. We have now only to determine what this will become, by allowing the electricity to move forward until the total forces acting on points within the body, which may now exceed the coercive force by an infinitely small quantity, are again reduced to an equilibrium with it. If this were done, we should, when the initial state of the body was given, be able to determine, successively, its state for every one of the following instants. But since it is evident from the nature of the problem, that the body, by revolving, will quickly arrive at a permanent state, in which the value of will afterwards remain unchanged and be independent of its initial value, we will here confine ourselves to the determination of this permanent state. It is easy to see, by considering the forces arising from the new total potential function, whose value has just been given, that in this case the electricity will be in motion over the whole interior of the body, and consequently |