100, 101] NEWTONIAN POTENTIAL FUNCTION. 191 



Picking out all the terms for which s -f r = n we get for the 



ir'\ n 

 coefficient of f -J 



n _ *(*-*) 

 nl 1.2(271-1)^ 



n 





The first polynomials have the values 



P.* I, 



P 3 (/.) = ! (5^-3^), 



101. Development in Spherical Harmonics. We may 



use the formula (6) 83 for an internal point, to obtain the 

 development of a function of 0, <f>, on the surface of a sphere in 

 the same manner as in 94 for the case of a circle. Since the 

 polynomials in the development of the reciprocal distance involve 

 only the cosine of the angle between the radii to the fixed and 

 variable points, we have if r' < r, 



(22) i = i 



and differentiating this with respect to r, the internal normal, 



<-> 



Inserting these values in (6), 83, namely 



eft' 



and applying it to the case that F is a spherical harmonic 



V m 

 we obtain, since 



(29) o 



+ mr m ~* T m I ( -Y P s O)j r 2 sin ed0d<f> 

 o \*V 



