374 Mr. G. H. Bryan. On the Stability of a [Mar. 27, 



( TO _ m ) ( w + m + l)_ > o (1.1.). 



Writing for £" , we see that inequality (11) is a sufficient condition 

 that the expression — 



tm r (0 . Unfit) -*»'(t) • «»'(0 



may he positive in the case of m—r and n—s both even, provided that 

 (11) is satisfied for all values of £ between and oo. 

 11. We have to consider two cases : — 



I. Suppose n = m. Then condition (11) will be satisfied, for all 

 values of f, provided that r > s. Therefore £/(£) • v>n(X) * s always 

 > tn(£) • u n(X) whatever be the value of £, provided that r > .9. In 

 other words, for given values of n, £, the product t n *(& .u n *(g) in- - 

 creases as s increases, and is greatest when s = n (corresponding to 

 the sectorial harmonics). 



II. Suppose n — s = m—r and, therefore, n — m = s—r. Condition 

 (11) may be written — 



(n— m) l^ + w + 1— J > 0. 



The first factor is positive provided that n > m. The second is 

 necessarily positive, for r, s are not greater respectively than m, n ; 

 therefore n-\-m-\-l > s-\-r, and therefore, a fortiori, 



_ s-H* 

 rc + m + l > 



for all values of g\ Hence, putting 5 = n— 2Jc, and therefore r — m— 2&, 

 we have — 



provided that m is < n. Therefore the product t n n ~ u (X) . u n n ~ 2k (£) 

 decreases for any given value whatever of £ as n increases. In 

 particular, t n n (£) . u n n (<£ > ~) decreases as the number n is increased. 



12. Let us now apply these results to the spheroid under con- 

 sideration. From the results of Case II, 



*5 3 (t)-«5 3 (r) < <4 2 (r).% 2 (t), 



and from the Table, £ 4 2 (£) . ^ 4 2 (£) and p 4 (£) . g 4 (t) are each less than 

 pf£) . for this particular value of Therefore, a fortiori, 



and jPiCO-giCO-VCO-wsKr) > o 



where t = '3183. 



