POTENTIAL. In the third volume of the Celeste,) Laplace, in 1784, deter mined the attraction exerted by a spheroid on a particle outside the mass, in the course of which he discovered the so-called potential function as the Emit of the sum obtained by dividing every element of the mass of the attracting body by its distance from the point upon which the force is exerted. That is, in the case of gravitation, if AM denotes the element of mass, r the distance from the particle at P and V the potential func tion, then limit Dm (1) V = Ans = 0 r where the numeration includes all the elements that compose the mass M. Laplace then pro ceeded to show that if the potential function of a mass was known at any external point, the attraction exerted in any direction by the mass upon that point could be found at once by per forming a differentiation of the function in the required direction and, finally, that such a poten tial function would always be a solution of the differential equation d'v etv r axe dy2 when v' is a symbol generally employed to denote the operator +7- x2 r This is called Laplace's equation and V is a potential function.
Poisson (1813), a student of Laplace, gave the form of the equation which V must satisfy for all points, situated either within or without the attracting mass, as V2 V - 4 rp.
This equation must be satisfied by the potential function for every conceivable distribution of attracting matter at any point P, where pis the density of the attracting matter at P. Since p =0 for points outside, Poisson's equation reduces to that of Laplace in such a case.
In 1827, George Green, a self-educated mathe matician, before he took his degree at Cam bridge in 1837, noted the peculiar property of V; namely, that it is a function of the initial and the final position only, thereby recognizing its universal application in dynamics in the treatment of a conservative system of forces, that is, a system that is independent of all inter mediate conditions. For example, the work done in moving a mass against the force of gravity i from P to Q is the same whether it is lifted directly up from P to Q or moved along any other path, as an inclint d plane, provided al ways there is no friction. That is the earth, and every mass connected therewith constitutes a conservative system with respect to gravity. The work done is independent of the path. But if, as in practice, friction is involved, some energy is dissipated in overcoming this force and the system is no longer conservative. Green
recognized, therefore, that, so far as natural phenomena are concerned, forces that are func tions of distances only constitute a conserva tive system to which the theory of the potential is applicable and, in particular, that in addition to gravitation, forces exerted by electrified and magnetized bodies upon each other are of this nature. Green was the first to call V the poten tial function. Its peculiar property may be said to be this: that it measures the work done in moving a unit of mass, electricity or magnet ism from one position to another by virtue of the forces in action.
Attraction. Newton's law states that every particle of a body attracts every particle of an other body with a force that varies directly as the product of the masses of the attracting particles and inversely as the square of the dis tance between them. Let P be any particle, or point, M any mass, p the average density of an element of mass, Cm, Q a point of the element whose co-ordinates are x, y, z' and take x, y, as the coordinates of P. By definition the at pLiv traction at P in the direction PQ will be PQ' and its components in the direction of the axes pat, become s a, cos P, PQ' ccis V, where a, Q, y are the direction cosines of PQ. But =-- (x' (y' y) (s' whence cos a , oos p , COS Denoting the components of the total attraction by X, Y, Z, we have, by Newton's law: x x)dx' dy' de ri y ff p( y y)dx'dy'ds' re z f ffp(s' s)de dy' when the integration is taken over the mass M. It follows that the resultant attraction at P due to M will be R + V. But by definition, the potential function of P due to M, assuming as the element of mass dm =pdx'dy'dz' will be v = f f fodx'dy' de r f Y' and fix 1 fd pdx' , soali f f dz' differentiation under the sign of integration being possible when P is outside the mass M, since r, the radical in the denominator, can in that case never become infinite. By comparison observe that the last expression is the attraction X. That is d u di) dv dx At = Z; whence R =- x2+y2+z,._,__ CI) I ply) ph\ ctx koy ktfzi That is, to find the component of the attraction in any direction for a point P, find the potential function at P and differentiate this function partially in the required direction.