104-105] FORMULAE FOR PLANETARY ELLIPSOID. 151 



and g()~jb(?)] r 





_ 

 1.3. 5. ..(2^+1) * 2(2n+3) 



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



"*" 2.4(2w+3)(2+5) b "*j " 



the latter expansion being however convergent only when f>l. 

 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 



.................. (8). 



As particular cases of the formula (7) we have 



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. 



<f> = Acot- 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. 



