240-242] Spherical Harmonics 211 



where S n is a surface harmonic of order n. Comparing with equation (144), 

 and remembering that a in this equation is the same as the r of equation 

 (147), we see that P n , regarded as a function of the position of Q, is a surface 

 harmonic of order n, and we have already seen that it is a series of powers of 



cos#, or of -, the highest power being the nth, so that r n P n is a rational 

 integral harmonic of order n. It follows that 



Fr n P n dS, 



being the sum of a number of terms each of the form r n l%, is also a rational 

 integral harmonic of order n, say V n . On the surface of the sphere 



so that equation (146) becomes 



which establishes the result in question. 



241. THEOREM. The expansion of an arbitrary function of position on the 

 surface of a sphere as a series of rational integral harmonics is unique. 



For if possible let the same function F be expanded in two ways, say 



(149), 

 (150), 

 where W n , W n ' are rational integral harmonics of order n. Then the function 



is a spherical harmonic, which vanishes at every point of the sphere. Since 

 V 2 w = at every point inside the sphere it is impossible for u to have either 

 a maximum or a minimum value inside the sphere (cf. 52), so that u = 

 at every point inside the sphere. Since W n Wn is a harmonic of order n, 

 it must be of the form r n S n where S n is a surface harmonic, so that 



Thus u is a power series in r which vanishes for all values of r from r = to 

 r a. Thus S n = for all values of n. Hence T^ = T^', and the two expan- 

 sions (149) and (150) are seen to be identical. 



242. It is clear that in electrostatics we shall in general only be con- 

 cerned with functions which are finite and single-valued at every point, and 

 of which the discontinuities are finite in number. Thus the only classes of 

 harmonics which are of importance are rational integral harmonics, and in 

 future we confine our attention to these. We have found that 



142 



