238-240] Spherical Harmonics 207 



(?i + 1) are accordingly 2?i + l in number. Thus the only harmonic of 

 this kind and of degree 1 is - . 



Consider now the various expressions of the type 



where s + 1 + u n. 



These, as we know, are harmonics of degree (n + 1), and from 235 



y 

 it is obvious that they must be of the form -~^, where V n is a rational 



integral harmonic of degree n. Since - is harmonic, V 2 ( ) = 0, so that 



The most general harmonic obtained by combining the harmonics of 

 type (137) is 



but by equation (138) this can be reduced at once to the form 



fa* dy* \r ** dx* 8^' r 



where p + q = n 1 and p' + q' = n. This again may be replaced by 



m *5 R , d n (i\ 



\r) 7 , p dxP dy n ~*> \r) ' 



so that there are 2n + 1 arbitrary constants in all, and it is obvious 

 on examination that the harmonics, multiplied by all the coefficients 



B p , ... Bp,... are independent. Thus, by differentiating - n times, we have 



arrived at 2^ + 1 independent rational integral harmonics, and it is known 

 that this is as many as there are. 



Expansion in Rational Integral Harmonics. 



240. THEOREM*. The value of any finite single-valued function of 

 position on a spherical surface can be expressed, at every point of the 

 surface at which the function is continuous, as a series of rational integral 

 harmonics, provided the function has only a finite number of lines and points 

 of discontinuity and of maxima and minima on the surface. 



Let F be the arbitrary function of position on the sphere, and let the 

 sphere be supposed of radius a. Let P be any point outside the sphere at a 



* The proof of this theorem makes no pretence at absolute mathematical rigour : see 303 

 below. 



