486 WAVES OF EXPANSION. [CHAP. X 



where 



--- 



2(l-2w) 2.4(l-2n)(3-2n) &quot; 

 ......... (10)*. 



The first term of (9) is alone to be retained when the motion 

 is finite at the origin. 



The functions ty n (f), ^ n (f) can also be expressed in finite 

 terms, as follows : 



d sin? ^ 



-. 



(11). 



These are readily identified with (10) by expanding sin f, cos f, and 

 performing the differentiations. As particular cases we have 



sn 



3 IV . 3 cos? 



The formulae (9) and (11) shew that the general solution of the 

 equation 



2 (n + 1) d& + Rn = (12 



which is obtained by writing f for AT in (8), is 



d n Ae*+Ber* 





This is easily verified ; for if ,ft n be any solution of (12), we find that the 

 corresponding equation for R n + 1 is satisfied by 



* There is a slight deviation here from the notation adopted by Heine, Kugel- 

 functionen, p. 82. 



