104-105] FORMULAE FOR PLANETARY ELLIPSOID. 151 



and q n (?) 



= (-)&quot; [P. (?) cot- r- ^TT^P-. (?) + 3^1)^- (?) - } 



1.3. 5. ,.(2n + l) 1 s 2(2n+3) 



+ &quot; ~~2~.4(2n+3)(2n+5) ^ * ~ &quot;| ^* 



the latter expansion being however convergent only when ?&amp;gt;1. 

 As before, the solution (4) is appropriate to the region included 

 within an ellipsoid of the family f = const., and (5) to the external 

 space. 



We note that 



As particular cases of the formula (7) we have 



?, or) = cot- r, 



The formulae for the stream-function corresponding to (4) and 

 (5) are 



and 



105. The simplest case of Art. 104 (5) is when n = 0, viz. 



(^^cot- 1 ^ ........................... (1), 



where f is supposed to range from oo to + oo . The formula 

 (10) of the last Art. then assumes an indeterminate form, but we 

 find by the method of Art. 101, 



(2). 



* The reader may easily adapt the demonstrations cited in Art. 101 to the 

 present case. 



