73, 74] NEWTONIAN POTENTIAL FUNCTION. 145 



Let the coordinates of a point in the repelling or attracting* body 

 be a, b, c. 



The potential at any point P, x, y, z, due to the mass dm at Q, 

 a, b, c, is 



FIG. 32. 



where r is the distance of the point x, y, z from the repelling 

 point at a, b, c. The whole potential at x, y, z is the sum of that 

 due to all parts of the attracting body, or the volume integral 



Now we have 



dm = pdr, 



or in rectangular coordinates dr = da db dc, 



dm = p da db dc. 



If the body is not homogeneous, p is different in different parts 

 of the body K, and is a function of a, b, c, continuous or discon- 

 tinuous (e.g. a hole would cause a discontinuity). Since 



(A) V = [([ = [([ pdadbdc 



JjJK r JJ] K J^-a^+ty-by + ^-c)*' 



For every point x, y, z, V has a single, definite value. It is 

 accordingly a uniform function of the point P, x, y, z. 



It may be differentiated in any direction, we may find its 

 level surfaces, its first differential parameter, whose negative is 

 equal to the whole action of K on a point of unit mass, and the 

 lines of force, normal to the level, or equipotential surfaces. 



* In order to save words, and to conform to ordinary usage, we shall say simply 

 attracting, for a negative repulsion is an attraction. 



W. E. 10 



