402 VIII. NEWTONIAN POTENTIAL FUNCTION. 



Accordingly to find the value of any term Y n at any point P, 

 (#', qp') we find the zonal surface harmonic whose axis passes through 

 the point P, multiply its value at every point of the sphere hy the 



value of f for that point, by the element of area, and by n ~ r , and 



integrate the product over the surface. It remains to show that the 

 development is possible, that is that the sum of the series 



n GO sin & d& dg>, 







actually represents the function f(&, qp'). This theorem was demon- 

 strated by Laplace, but without sufficient rigor, afterwards by Poisson, 

 and finally in a rigorous manner by Dirichlet. A proof due to Darboux 

 is given by Jordan, Traite d' Analyse, Tom. II, p. 249 (2 me ed.). 



148. Development of the Potential in Spherical Har- 



monics. In investigating the action of an attracting body at a 

 distant point, and for many other purposes connected with geodesy 

 and astronomy, it is convenient to develop the potential function in 

 a series of spherical harmonics. \$ x,y,z denote the coordinates of 

 the attracted point P, r its distance from the origin, a, b, c the co- 

 ordinates of the attracting point Q, r 1 its distance from the origin, 

 d the distance between them, dr' the element of volume at Q, we have 



and using the value of -=- from 105), when r > r', 



.< + 



which, on removing the powers of r from under the integral signs, 

 is the required development in spherical harmonics, 



126) F= + 5 + 7* + "- 



where the surface harmonics Y n are the volume integrals 



127) T 



taken over the space occupied by the attracting body. Since /i enters 

 into the integrand, and, according to 121), it contains the angular 

 coordinates #, qp of P, the surface harmonics Y n are functions of & 

 and (p. 



