372 Mr. G. H. Bryan. On the Stability of a [Mar. 27 ; 



= - -0400 + -0864 = + -0464 ; 



= - -0063 + -0691 = +-0628. 



The values of these expressions are all positive ; therefore condition 

 (9) is satisfied in each case, and the spheroid is " ordinarily " stable 

 for the corresponding types of displacement. 



It is therefore stable for all types of displacement considered in 

 the foregoing table, except that for which it is, by hypothesis, 

 ' : critical." 



7. On examining the valnes of t n s (£) . given in the Table, it 

 appears probable that as we proceed to harmonics' of higher degrees 

 this product diminishes in value, and that condition (7) is satisfied 

 universally in all the cases not considered above. That such is 

 actually the case we now proceed to demonstrate. The results are a 

 slight extension of those obtained in § 10 of Poincare's paper, the 

 method here employed being very similar. 



8. Consider the expression — 



tr/^o) • ^ r (£o)-V(£o) • «.*(ta). 



and let us examine under what circumstances it is essentially positive. 

 Prom formula (4) we have 



• Mm*(&>) ■ «»'(&)) 



bO bo 



bo 



This will be essentially positive if the quantity to be integrated is 

 always positive, that is, if for all values of £ lying between £" and cc, 



War)/ Wcr)/ 



> o, 



*m r (0 Wo)' 



where £ > £ , 



which will be the case if t H s (£)lt m r (£) increases with £*. 



9. The result proved by Poincare, and assumed in the preceding 

 investigations, namely, that if n— s be odd — 



