JZLECIROSTATICAL THEORY.] ELECTRICITY (for the capacity of a sphere is obviously equal to its radius). Cavendish had arrived by experiment at the value = (see Thomson s Reprint, p. 1 80), a very reraark- 1 57 able result for his time. It is very easy, by taking the limit of the right hand side of (34), to find the expression for the density at a distance r from the centre of the disc ; it is r- 9 J (38). 4T<Va a - r a 2ir ^/a* - r 3 In the case of an ovary ellipsoid, a = b<c; and the loi c + c - from which several limiting cases may be deduced. Formula (34), applied to a very elongated ovary ellip soid, shows us that the density at the pointed ends is very great compared with that at the equator. The ratio of the densities in fact increases indefinitely with the ratio of the longest to the shortest dimension. We have in such an infinitely elongated ellipsoid an excellent type of a pointed conductor. The effect of a point or an edge on a conductor may be very easily shown by drawing a series of level surfaces, the first of which is the surface of the conductor itself, which has, say, an edge on it. The consecutive surfaces have sharpness of curvature corresponding to the edge, which gets less and less as we recede from the conductor. The level surfaces at an infinite distance are spheres. Tracing, then, any tube of force from an infinite distance, where the sections of all are equal, inwards towards the discon tinuity, we see that the section becomes narrower as the curvature of the level surfaces sharpens, and at a mathe matical edge the section is infinitely small, and therefore the force is infinitely great. At a mathematical point this is doubly true. At such places the force tending to drive the electricity into the insulating medium becomes infinite. In practice the medium gives way, and disruptive discharge of some kind occurs. ^Ye can find the distribution on a spherical conductor in fluenced by given forces, such for instance as would arise from rigidly electrified bodies in the neighbourhood. The method of procedure would be as follows : Let U be the potential of the rigidly electrified system alone at an}- point of the sphere. Then the problem is to determine a function V, which shall satisfy the equation V 2 V at every point of space, and have the value C - U at the surface of the sphere, where C is a constant to be determined by the conditions of the problem. Expand C - U in series of surface harmonics, and let the result be Then the value of V is v and &c. ... (a). + . . . inside the sphere . . outside . . (y). For these evidently satisfy Laplace s equation, have the given value (o) at the surface of the sphere, and are finite and continuous everywhere. From (j8) and (7), by means of the surface characteristic equation, we can deduce an expression for the density at any point of the sphere, and for the whole charge. If the latter is given we have a condition to determine C; if, on the other hand, the value of the potential of the sphere were given, then this would be the value of 0. Jaseof The case of two mutually influencing spheres was treated by Poisson in the famous memoir which really began the pheros . oi.sson s mathematical theory of electricity. We regret that we nalysis. cannot afford space for more than a mere sketch of his methods. Consider the potentials due to the distributions on each sphere. Let a and b be the radii of the two spheres, r and r" the distances of any point P from their respective centres, and ft and p. the cosines of the angles r and r 1 make with the line joining the centres of the spheres. Since the distributions are evidently symmetrical about the central line, we can obviously expand the potentials due to each distribution in zonal harmonics relative to the cor responding sphere. Hence, if 4ira< [/*,-) denote potential due to sphere a at any point inside it, we have The potential at any external point is which mav be written 4ir rf> ( u, - r Y rj Similarly we have foi the other sphere ( r r "?! 4irZ*(M. T) -Bo + BiQ/^ +B S Q 2 1-I +. (7) for the potential at any internal, and 4*- *(V ,- for the r rj potential at any external point. The whole potential, then, will be given by at any point external to both spheres. Also V = 4irad>| u, - } + 47T-* ( u, -, } inside a: and
/ / r J
t ,-- ) inside b. b I Now, the conditions of the problem require that the values of V in the two last cases shall be constant. Our functions are, there fore, to be determined by the equations which are to be satisfied with obvious restrictions on r and / in each case. .Reverting, however, to the expressions (a), (ft), (y), &c., we see that we need not solve the problem in the general form thus suggested ; for it will be sufficient if we determine the constants A 0) A 1; &c., B , BL &c. Now, if we make/t = l, //I, that is, consider only points on the central line, then Q J = 1, c a a of > r r &c., and &c. B b BJ, &c., are the coefficients , &c., in the expressions for the potentials inside the spheres a and b. Hence, if /( aj F ( ? } denote the values of <p ( M ,- ) , * (/* ,-} , when u
a / b J
and /x = l, we need only solve the equations b* and where we have replaced r and r by their values c-r and c-r, c being the distance between the centres of a and b. Poisson then eliminates the function F, by choosing a new variable (, such that r" = , and remarks that we may give to { any value between c ~ ? + a and - a, and therefore we may write r for f ; we thus have the same variable in both the equations, and F ( ) which occurs in both may be eliminated. The result is This is the functional equation on which depends the solution i f the problem of two mutually influencing spheres. Poisson treats very fully the case of two spheres in contact ; for which case, taking a = l, the above equation becomes fi-t _ b ,./ I+b- r h - fffr / i. /n i
n/ I ;, , ti i AVI r})~ 1 -(- b r
1 We are, of course, assuming aquaintance with the properties ct
spherical harmonics.