86-87] TESSERAL HARMONICS. 125 



87. When we abandon the restriction as to symmetry about 

 the axis of x, we may suppose 8 n , if a finite and single-valued 

 function of &&amp;gt;, 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 o&amp;gt; = to o&amp;gt; = 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 

 this takes the form 



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 M 



1.2.3,4.5 



_ 

 &quot;-&quot; 



the factor cos sa&amp;gt; or sin sea being for the moment omitted. In the 

 hypergeometric notation this may be written 



J5- iw, 1 + J* + iw, f , p?)} ...... (3). 



These expressions converge when p? &amp;lt; 1, but since in each 

 case we have 



the series become infinite as (1 p?}~ 8 at the limits //,= + 1, 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. 



