It may, however, be expressed in the finite form* that is to say P'(v) is a zonal harmonic of the first kind with all its terms positive, and Z2 (v) is a rational and integral function of v of 2n − 1 degrees, which is such as to cancel all the terms of P'2, (v) tan1 which do not vanish when v becomes infinite. 2n The expression (11) is applicable to small as well as great values of v. Thus we find when v is 0, as it is at the surface of the disk, The potential at any point of the disk is therefore the sum of a series of terms, the general form of which is π (2n !)9 Van = 42 2α 21" (re!)* A 2a P(). (14) On the axis, = 1 and av=z, and the potential is the sum of a series of terms, the general form of which is Since we have to determine the value of the potential arising from the first disk at a point in the second disk for which z=c at a distance r from the axis, and if we write where b is the radius of the second disk, and p is a quantity corresponding to u in the first disk, then the most convenient expression for the potential due to the first disk at a point (p) in the second, is where U denotes the value of the potential at the axis, and where, after the differentiations, va is to be made equal to c. To investigate the mutual action of the two disks, let us assume that the surface-density on the second disk is the sum of a number of terms of which the general form is The potential at the surface of the second disk arising from this distribution will be the sum of a series of terms of the form π 1 (2n!)2 Ban Pan (P). 2 b 2 (n!) (19) The potential arising from the presence of the first disk is given in equation (17). Having thus expressed the most general symmetrical distribution of electricity on the two disks and the potentials thence arising, we are able to calculate the potential energy of the system in terms of the squares and products of the two sets of coefficients A and B. If W denotes the potential energy, when the integration is to be extended over every element of surface ds. Confining our attention to the second disk, the part of W thence arising is and the part arising from the term in the density whose coefficient is B is The part of the value of V which arises from the electricity on the second disk itself is the sum of a series of terms of the form (19). The surface-integral of the product of any two of these of different orders is zero, so that in finding the potential energy of the disk on itself we have to deal only with terms of the form The energy arising from the mutual action of the disks consists of terms whose coefficients are products of A's and B's, and in calculating these we meet with the integral* We have, therefore, for the harmonic of order zero. Surface-density on the first disk, σ = A 1 where A is the charge of the first disk. I am indebted for the general value of this integral to Mr W. D. Niven, of Trinity College. Potential at the surface of the second disk, arising from this distribution of electricity on the first, We have next to calculate the energy arising from this distribution on the first disk, together with a corresponding distribution on the second disk, the coefficients of the harmonics for the second disk being B, B2, B1, &c. It will consist of three parts, the potential energy of the first disk on itself, of the first and second on each other, and of the second on itself. The first part will involve only terms having for coefficients the squares of the coefficients A, for those involving products of harmonics of different orders will vanish on integration. The third part will, for the same reason, involve only squares of the coefficients B. The second part will involve all products of the form AB. + AB — [1 − § (x2 + y2) + } (x* + y *) + } x2y2 − ‡ (x® + y°) − x2y2 (x2 + y3) In this expression for the energy of the system the coefficients A, A,, B2, B are treated as independent of A and B. To determine the nearest approach to equilibrium which can be obtained from a distribution defined by this limited number of harmonics, we must make W a minimum with respect to A2, B2, A ̧ and B.. We thus find for the values of these coefficients We are now able to express the energy in the form W = P1,4 + P12AB+ P„B", where A and B are the charges, and Pu P12? 23 and P2 are the coefficients of potential, the value of which we now find to be 22 = 2a π 32.5 c [1 - 4 a3 12 62 2 a2 + b2 a1 + b1 +ᄒ = = 1 с π 26 1 23 a3 π 32.5 c 4 NOTE 12, ART. 151. On the electrical capacity of a long narrow cylinder. The problem of the distribution of electricity on a finite cylinder is still, so far as I know, in the state in which it was left by Cavendish. It is sometimes assumed that the electric properties of a long narrow cylinder may be represented, to a sufficient degree of approximation, by those of the ellipsoid inscribed in the cylinder. The electrical capacity |