486 WAVES OF EXPANSION. [CHAP. X 



where 



2(2n+3) ' 2.4(2?i+3)(2rc+5) 

 1.3...(2w-l 



2(1.-*!) 2.4(l-2n)(3-2n) '") 



(10)*. 



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

 is finite at the origin. 



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

 terms, as follows : 



Y^ ' 



(11). 



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

 performing the differentiations. As particular cases we have 



3 1\ . 3 cos? 

 ^-^ sm ^--^~' 



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

 equation 



-0 



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



/ d \ 





This is easily verified ; for if R 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. 



