I To prove this let us consider the triple integral The method of integration by parts, reduces this to the accents over the quantities indicating, as usual, the values of those quantities at the limits of the integral, which in the present case are on the surface of the body, over whose interior the triple integrals are supposed to extend. f dU" Let us now consider the part \dydzV"~^— due to the greater valnes of x. It is easy to see since dw is every where perpendicular to the surface of the solid, that if d<r" be the element of this surface corresponding to dydz, we shall have and hence by substitution Lyd*V»^--\wpV»TM:. In like manner it is seen, that in the part anl consequently -fdydzV'^^-fda'pV^. Then, since the sum of the elements represented by da-', together with those represented by da", constitute the whole surface of the body, we have by adding these two parts where the integral relative to da is supposed to extend over the whole surface, and dx to be the increment of x corresponding to the increment dw. In precisely the same way we have therefore, the sum of all the double integrals in the expression before given will be obtained by adding together the three parts just found; we shall thus have _ f^y. [dU dx ^dU dy dTT d»\ (foydU. where V and ^ represent the values at the surface of the body. Hence, the integral Idxd dz {dZd-E+^IiE+^dE\ ) [dx dx dy dy dz dz J' by using the characteristic 8 in order to abridge the expression, becomes - j^ — -jdxdydz VBU. Since the value of the integral just given remains unchanged when we substitute V in the place of U and reciprocally, it is clear, that it will also be expressed by x, y, z. Then, if V be the value of this function for any other point p exterior to this surface, we shall have Hence, if we equate these two expressions of the same quantity, after having changed their signs, we shall have Thus the theorem appears to be completely established, whatever may be the form of the functions U and V. In our enunciation of the theorem, we have supposed the differentials of U and V to be finite within the body under consideration, a condition, the necessity of which does not appear explicitly in the demonstration, but, which is understood in the method of integration by parts there employed. In order to show more clearly the necessity of this condition, we will now determine the modification which the formula must undergo, when one of the functions, U for example, becomes infinite within the body; and let us suppose it to do so in one point p' only: moreover, infinitely near this point let U be sensibly equal to -; r being the distance between the point p' T and the element dxdydz. Then if we suppose an infinitely small sphere whose radius is a to be described round p', it is clear that our theorem is applicable to the whole oi the body exterior to this sphere, and since, S U= S ^ = 0 within the sphere, it is evident, the triple integrals may still be supposed to extend over the whole body, as the greatest error that this supposition can induce, is a quantity of the order a*. Moreover, the part of daU-j— , due to the surface of the small sptiere is only an infinitely small quantity of the order a; there only remains f dXJ therefore to consider the part of J daV-^ due to this same surface, which, since we have here when the radius a ia supposed to vanish. Thus, the equation (2) becomes where, as in the former equation, the triple integrals extend over the whole volume of the body, and those relative to da, over its exterior surface: V being the value of V at the point p. In like manner, if the function V be such, that it becomes infinite for any point p" within the body, and is moreover, sensibly equal to -,, infinitely near this point, as U is infinitely near to the pointy, it is evident from what has preceded that we shall have the integrals being taken as before, and U" representing the value of U, at the point p" where V becomes infinite. The same process will evidently apply, however great may be the number of similar points belonging to the functions tfand V. For abridgment, we shall in what follows, call those singular values of a given function, where its differential coefficients become infinite, and the condition originally imposed upon U and V will be expressed by saying, that neither of them has any singular values within the solid body under consideration. (4.) We will now proceed to determine some relations existing between the density of the electric fluid at the surface of a body, and the potential functions thence arising, within and without this surface. For this, let pda be the quantity of electricity on an element da of the surface, and V, the value of the potential function for any point p within it, of which the co-ordinates are ^(f-*)'+(*-*)■+(£-«)■ f, r), £ being the co-ordinates of da, and J v(f - aOa + fa-y)•+(£-*')"' the integrals relative to t2o* extending over the whole surface of the body. It might appear at first view, that to obtain the value of V from that of V, we should merely have to change x, y, z into a', y', z: but, this is by no means the case; for, the form of the potential function changes suddenly, in passing from the space within to that without the surface. Of this, we may give a very simple example, by supposing the surface to be a sphere whose radius is a and centre at the origin of the co-ordinates; then, if the density p be constant, we shall have V=Airpa and V'=- W 1*" + y" +7*' which are essentially distinct functions. With respect to the functions V and V in the general case, it is clear that each of them will satisfy Laplace's equation, and consequently 0 = 8Fand 0 = 8T': moreover, neither of them will have singular values i for any point of the spaces to which they respectively belong^and at the surface itself, we shall have — \ the horizontal lines over the quantities indicating that they belong to the surface. At an infinite distance from this surface, we shall likewise ] F'-O. |