86-87] TESSERAL HARMONICS. 125 



87. When we abandon the restriction as to symmetry about 

 the axis of a?, we may suppose 8 nt if a finite and single- valued 

 function of &>, to be expanded in a series of terms varying as cos so) 

 and sin sw respectively. If this expansion is to apply to the whole 

 sphere (i.e. from G> = to ay = 2?r), we may further (by Fourier's 

 theorem) suppose the values of s to be integral. The differential 

 equation satisfied by any such term is 



If we put 



S n = (l-^)*v, 



this takes the form 



(1-/O J,-2( + l) / . t |; + (n-)( + + l)t; = 0, 



which is suitable for integration by series. We thus obtain 



(n-s-2)(n-s)(n+s+l)(n+s+3) 4 

 1.2.3.4 ** 



5 _ ( . 



'"" 



1.2.3,4.5 



the factor cos sco or sin sco being for the moment omitted. In the 

 hypergeometric notation this may be written 



t*-i,l+J + i. !,/*)} ...... (3). 



These expressions converge when /* 2 < 1, but since in each 

 case we have 



the series become infinite as (1 p?)~ 8 at the limits /*= + !, unless 

 they terminate*. The former series terminates when n s is an 

 even, and the latter when it is an odd integer. By reversing the 



* Lord Rayleigh, Theory of Sound, London, 1877, Art. 338. 



