1890.] Rotating Spheroid of Perfect Liquid. 373 



jpicr).?i(r)-^(r).««*a) 



is essentially positive, follows at once. For, as just proved, this will 

 be the case if tn*(OlPi(£)i tnat is > /£» increases with Now, 



from formula (2) it is evident that, n—s being odd, t n s (£) is divisible 

 by £, and the quotient will be (£f 2 + l) is x a rational algebraic function 

 of £ 2 in which all the terms are positive. This quotient evidently 

 increases with which proves the result. 



10. Let us now revert to the original question, but suppose in 

 addition that both m — r and n — s are even. We have just shown 

 that 



Wo)-%/(r ) > veto). 



provided that t n s (g)/t m r (g) increases with £. 

 This will be the case if 



or multiplying by £ 2 + l. 



Since m — r and n—s are both even, it readily appears that dt n s (£)ldg 

 and dt m r (^)jd^ vanish when £ = 0. 



Hence the left-hand side of the last inequality will be positive when 

 £ > if it increases with £ ; this condition gives on again differen- 

 tiating — 



.,(r)|{(^i)^}- V (r)|{(^i)^}>o. 



Now t m r (£) and satisfy the differential equations 



|{(^ + i)^} = {<» + i)-^},m 



{ s 2__ r 2 T 

 n(n-f-l) — m(m + l) — £~2^7[ J *» r > ^> or > 



since t H '(£) an( i are essentially positive, 



2 e 2 



