Potential Function as Measure of Work. By definition, the work done in moving a unit of mass along a path S against a force F is equal to the product F X S. Let the attraction at any point of the path and opposing the force F be dv , negative since in the opposite direction from the motion, and let AS be an element of the path. Then, if the unit mass moves from Pi to P,, the work done is given by Ps limit nn. pidv GS = Vs Vs, W = AS = 0 P. when V. and V. are the values of the potential function of P, and P, respectively. That is, the difference or loss in potential measures the work done.
Since the difference of potential is the meas ure, we observe that the farther P. is from the attracting mass the less V, becomes. For, in general, V = limit v Am Am= 0 r.si and if r, denotes the distance of the nearest point in the attracting mass from P, then 1 vM , Or V _ M V < < fa ro That is, when P is at infinity r. = m, and the potential is zero.
It follows that when P, is at infinity, W = V1 That is, the potential function or the potential at any point P, due to an attracting system, is equal to the work done in moving a unit mass considered as concentrated at P, from P to infinity along any path. In most modern works on mathematical physics the word ((potential)" does not denote the value of the potential func tion at a point but measures the work done in moving a mass from a given position to infinity in the presence of the system considered. W is used by some authors as potential of the mass M' with reference to the mass M. Others
use the negative of W as the equivalent of ((the mutual potential energy of M and M'.9 Laplace's Equation. It is not difficult to dV show that , are everywhere finitedx' oy ' dz 6V eV and that, as a consequence, the potential tion is always finite and continuous. over, when P is outside * the masses, the dV expression under the radical in dx is every where finite within the limits of integration and it becomes possible to differentiate under the signs of integration. That is, dViff3(x'x)= dx' dy' de, fff3(y' y)1 dx' dy' de, d5s ri f ff3(e z)2 rt dx' dy' dz', ri whence we derive by addition Laplace's tion as one to be satisfied by the potential tion at all points outside the attracting mass 62 viV = dx osv 0.
The generalized form due to Poisson re quires an application of a form of Green's the orem due to Gauss and is applicable to all points. As stated, Poisson's equation is v'V 4irp.
For a clear treatment of the elements of the theory of the potential function, consult 'New tonian Potential Function,' B. 0. Peirce, 1886, from which the present article is in the main derived. Consult also the mathematical papers of George Green, reprinted by Ferrers; Thom son and Tait, 'A Treatise on Natural Philos ophy); Maxwell, Elementary Treatise on Electricity' ; Watson and Burbury, (The Math ematical Theory of Electricity and Magnetism' ; Clausius, (Die Potentialfunktion and das Poten tial.) J. BRACE CRITTENDEN, Pn.D