Although in what precedes we hare spoken of one body only, the reasoning there employed is general, and will apply equally to a system of any number of bodies Whatever, in those cases even, where there is a finite quantity of electricity spread over their surfaces, and it is evident that we shall have for a point p in the interior of any one of these bodies 0 = BV+i-rrp (1). Moreover, the force tending to increase a line q ending in any point p within or without the bodies, will be likewise given by - (-j- J; the function V representing the sum of all the electric particles in the system divided by their respective distances from p. As this function, which gives in so simple a form the values of the forces by which a particle p of electricity, any how situated, is impelled, will recur very frequently in what follows, we have ventured to call it the potential function belonging to the Bystem, and it will evidently be a function of the co-ordinates of the particle p under consideration. (2.) It has been long known from experience, that whenever the electric fluid is in a state of equilibrium in any system whatever of perfectly conducting bodies, the whole of the electric fluid will be carried to the surface of those bodies, without the smallest portion of electricity remaining in their interior: but I do not know that this has ever been shown to be a necessary consequence of the law of electric repulsion, which is found to take place in nature. This however may be shown to be the case for every imaginable system of conducting bodies, and is an immediate consequence of what has preceded. For let x, y, a, be the rectangular co-ordinates of any particle p in the interior -I-J be the force with which p is impelled in the direction of the co-ordinate x, and lending to increase it. In the same way — ^— and — -j- will be the forces in y and z, and since the fluid is in equilibrium all these forces are equal to zero: hence n dV , dV, dV, ,„ which equation being integrated gives Fa* const. This value of V being substituted in the equation (1) of the preceding number gives and consequently shows, that the density of the electricity at any point in the interior of any body in the system is equal to zero. The same equation (1) will give the value of p the density of the electricity in the interior of any of the/bodies, when there 'th**t[ are not perfect conductors, provided we can ascertain the value of the potential function V in their interior. (3.) Before proceeding to make known some relations which exist between the density of the electric fluid at the surfaces of bodies, and the corresponding values of the potential functions within and without those surfaces, the electric fluid being confined to them alone, we shall in the first place, lay down a general theorem which will afterwards be very useful to us. This theorem may be thus enunciated: Let U and V be two continuous functions of the rectangular co-ordinates x, y, z, whose differential co-efficients do not become infinite at any point within a solid body of any form whatever; then will /", \ -|'.*,,- i»\ Jdxdzdt/mV+jdaUf^\=JdxdydzVBU+jdaV(^); cf/J-f/ the triple integrals extending over the whole interior of the body, and those relative to da, over its surface, of which da represents an element: dio being an infinitely small line perpendicular to the surface, and measured from this surface towards the interior of the body. 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. Let ns now consider the part IdydzV" -=— due to the greater values of x. It is easy to see since dto 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 **—as*** and hence by substitution J J ax J dto dx In like manner it is seen, that in the part due to the smaller values of x, we shall have d*dz=+dl»d<r', and consequently Then, since the sura 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 J \ dx dx J J dw dx 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 _ \dcV{^^+*—-&+~—\ = -[d*V—. where V and -*— represent the values at the surface of the body. Hence, the integral [dxd dz{—d— ^iE+iliEX J [dx dx dy dy dz dz)' by using the characteristic 8 in order to abridge the expression, becomes -jdffV~-\dxdydz VhU. 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 Hence, if we equate these two expressions of the same quantity, after having changed their signs, we shall have jd*V— + fdzdydzVSU=(d*U^+jdxdi,dzU&V... (2). Thus the theorem appears to be completely established, whatever may be the form of the functions 17 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, V 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 dxdydz. Then if we suppose an infinitely small sphere whose radius is o to be described round p', it is clear that our theorem is applicable to the whole of the body exterior to this sphere, and since, 8 U— <S - = 0 within the Bphere, 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 a1. Mereover, the part of dV daU-j- , due to the surface of the small spnere is only an infinitely small quantity of the order c; there only remains therefore to consider the part of \d<rV-^— due to this same surface, which, bince we have here f |