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. + A B 1 [1 − § (œ° + y°) + ¿ (x* + y1) + { x°y° − } (a° + y®) − x*y° (x° + y°) 1 c 3.5 2 + }(x + y) + 3x2y2 (x* + y) + 'x'y' - &c.] - 4x-3. 4xy3· 3 3.11+ &c.] + AB. € 3.5.7 3.5 11 y' 1 c 2.3.5.7 [1−5_ y2+2.7x+ -x3y2+ -y-&c.] 11.13 4 x 5 x*y° – 4. 3.x°y* + 4y® — &c.] 4.5.7 11 5 7 · x2y2 + 3y* 7 3 In this expression for the energy of the system the coefficients A2, A4, 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, 4, and B. We thus find for the values of these coefficients 22 where A and B are the charges, and P11, P12, and P, are the coefficients of potential, the value of which we now find to be 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 of the cylinder must be greater than that of the ellipsoid, because the electric capacity of any figure is greater than that of any part of that figure. It is easier to state the conditions of the problem than to obtain an exact solution. Let 27 be the length of the cylinder, and let b be its radius. Let the axis of the cylinder be the axis of x, and let the origin be taken at the middle point of the axis. Let y be the distance of any point from the axis. Let Adx be the quantity of electricity on that part of the curved surface of the cylinder for which x is between x and x + dx; we may call λ the linear density of the electricity on the cylinder. Let σ be the surface-density on the flat ends. Let be the potential at a point on the axis for which x = §. the first integral representing the part of the potential due to the curved surface, and the other two the parts due to the positive and the negative flat ends respectively. The condition of equilibrium of the electricity is that must be constant for all points within the cylinder, and therefore for all points on the axis between the two ends. If, by giving proper values to A and σ, we can make the value of constant for any finite length along the axis, then, by Art. 144 of "Electricity and Magnetism," will be constant for all points within the surface of the cylinder. It was shown in Note 2 that the distribution of electricity in equilibrium on a straight line without breadth is a uniform one. We may expect, therefore, that the distribution on a cylinder will approximate to uniformity as the radius of the cylinder diminishes. If we suppose A and σ to be each of them constant, where f and f, are the distances of the point (§) on the axis from the edges of the curved surface at the + and ends of the cylinder respectively. Just within the positive flat end of the cylinder, where έ is just less than 1, CAPACITY OF A CYLINDER. 395 If the electricity were in equilibrium, this would be zero, and if the cylinder is so long that we may neglect the reciprocal of f,, we find or the surface-density on the end must be equal to the surface-density on the curved surface. The whole charge is therefore E =λ (21+b). (5) The greatest value of the potential is at the middle of the axis, where 0. Calling it and putting ƒ=l, (0) This is the smallest value of the potential for any point of the cylinder. The capacity of the cylinder cannot therefore be less than Cavendish does not take into account the flat ends of the cylinder, but in other respects these limits are the same as those between which he shows that the capacity must lie. The approximation, however, is by no means a close one, for when the cylinder is very narrow the upper limit is nearly double the lower. Indeed Cavendish, in Arts. 479, 682, has recourse to experiment to determine the best form of the logarithmic expression. We may obtain a much closer approximation by the following method, which is applicable to many cases in which we cannot obtain a complete solution. Let W be the potential energy of any arbitrary distribution of electricity on a body of any form where e is the charge of any element of the body, and the potential at that element. The charge is E = X(e). (12) Let us now suppose the electricity to become moveable and to distribute itself so as to be in equilibrium. The potential will then be uniform. Let its value bey, and since the charge remains the same the potential energy of the electrification in the state of equilibrium will be Since W, the potential energy due to any arbitrary distribution of the charge, may be greater, but cannot be less than W, the energy of the same charge when in equilibrium, the capacity may be greater, but cannot be less, than This inferior limit of the capacity is greater than that derived from the maximum value of the potential, and, as we shall see, sometimes gives a very close approximation to the true capacity. In the case of the cylinder, if we suppose A to be uniform, and neglect the electrification of the flat ends When the length of the cylinder is more than 100 times the diameter this value of the capacity is sufficiently exact for all practical purposes. The capacity of the inscribed ellipsoid is To obtain a closer approximation let us suppose that the linear is expressed in the form A+ A, + &c. +A, the general term density |