1890.] Rotating Spheroid of Perfect Liquid, 369 



The quantities q n (0 and u n s (£) are expressible in a finite form in 

 exactly the same way as the ordinary spherical harmonics of the 

 second kind.* We have, in fact, 



*.(?) = (-l)»{f»(t) eoi-if-B} (5.), 



«.-(r) = (-D-'^j { «tp f-^js } • • ( 6 -)> 



where R, R' are known rational algebraic functions of degree « — 1 

 and m+s— 1 respectively in which all the coefficients are positive. 

 For example : — 



i>i(?) = r. 2i(?) 

 ft(n = i(3r 2 +i), ft (o 



«f) = 3^+1)', V(?) 



tfCf) = 3(r 3 + l), « 2 2 (f) 



and the corresponding functions of the third, fourth, and fifth degrees 

 can be readily written down from my table in the ' Cambridge Philo- 

 sophical Proceedings' (loc. cit.), by introducing the necessary changes 

 in the signs, and putting "cot _1 " in place of " coth 



3. In [E, § 20] I showed that if we consider only displacements of 

 the surface determined by a spheroidal harmonic of degree n and 

 rank s, the condition of secular stability, which, in the present nota- 

 tion, is 



jpiOt) .2i(r)-*»'(r) >o (7.), 



is a sufficient, albeit not a necessary, condition for stability when the 

 liquid forming the spheroid is perfect. That the left-hand member of 

 this inequality is essentially positive when n—s is odd has been proved 

 by Poincare,f and another proof is given below (§ 9). 



In [E, § 16] I showed that, in the case of the zonal harmonic dis- 

 placements of even degree n, the necessary and sufficient condition for 

 ordinary stability is 



n(shkao-Mo • 2»(r)+^i) Wio ■ <h\o-m) ■ ffi<m 



>0 (8.), 



» ' Cambridge Philosophical Society Proceedings,' 1888, p. 292. 



t 'Acta Mathemat.,'Tol. 7, p. 326. Write R; for- «„'(£) and Sj for (2* + !)«„»(?). 



= -{fcotTlf-l}, 



= i(3r 2 +i)cot-ir-ir, 



