354 



VIII. NEWTONIAN POTENTIAL FUNCTION. 



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

 at Q, a, 6, c, is 



dF=^, 



where r is the distance of 

 the point x, y, z from the 

 attracting 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 = ydr, 



or in rectangular coordinates 

 rig. 126. dr^dadbdc, 



dm = gdadbdc. 



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

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

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



53) 



It is 



For every point x, y, 2, V has a single, definite value, 

 accordingly a uniform function of the point P, x,y, 0. 



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

 surfaces, its first differential parameter, whose negative multiplied 

 by y 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. 



If for any point x, y, z outside K, r is the shortest distance to 

 any point of K, and r 2 is the greatest distance, we have for any 

 point in K 





dm dm 



