To prove this let us consider the triple integral dU dU [ dæ dy dz {(dr) (de) + (dv) (dv) + (1) (1)}. dx dy dy dz 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. Let us now consider the part [dydzV" dU" due to the greater values of x. It is easy to see since du is every where perpendicular to the surface of the solid, that if do" be the element of this surface corresponding to dydz, we shall have due to the smaller values of x, we shall have da Then, since the sum of the elements represented by do', together with those represented by do", constitute the whole surface of the body, we have by adding these two parts where the integral relative to do is supposed to extend over the whole surface, and do to be the increment of a 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 Hence, the integral represent the values at the surface of the body. by using the characteristic & in order to abridge the expression, becomes 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 - [do Ud-fdx dy dz USV. Hence, if we equate these two expressions of the same quantity, after having changed their signs, we shall have dV dU = do dz dz V d+fdx dy de VBU - fde Udv + fdady da U 8 V... (2). 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' and the element dx dy dz. 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 of the body 1 1 exterior to this sphere, and since, SU=8 =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. Mereover, the part of dV doud, due to the surface of the small spnere is only an U dw infinitely small quantity of the therefore to consider the part of face, which, since we have here order a; there only remains dU do V due to this same sur dw when the radius a is supposed to vanish. Thus, the equation (2) becomes dV [dæåydz UdV+fdo Udr = fdædyde VdU+ (do vd - 4π V' ...(8); dr where, as in the former equation, the triple integrals extend over the whole volume of the body, and those relative to do, 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, 1 sensibly equal to, infinitely near this point, as U is infinitely near to the point p', it is evident from what has preceded that we shall have dV dz π [dzdydz U8V+ [do Udv-4#U"=fdædyde VòU+ [do vdu - V"...(S); JdxdydzU8V+fdo dw 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 U and 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 I 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 do of the surface, and V, the value of the potential function for any point p within it, of which the co-ordinates are x, y, z. Then, if V' be the value of this function for any other point p' exterior to this surface, we shall have the integrals relative to do 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 ', 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 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=8V and 0 = 8 V': moreover, neither of them will have singular values 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 have V' = 0. |