370 



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



and in [E, § 18] that, for a sectorial harmonic displacement, the neces- 

 sary and sufficient condition is that 



Pitt) • ffiG:) - *»"(£) .un*tt)+l • V(r)-Pitt) • 2i(0} >o • (9.) ; 



while [E, § 20] if w = 2, the last condition leads to exactly the same 

 results as Riemann's and Basset's investigations (as already men- 

 tioned), and gives for the critical form 



l/£ = 3-1414567, 



whence £ == '3183236, approximately (10.), 



and the eccentricity = sin 72° 20' 33" = -9528867. 



4. To prove that the spheroid is " ordinarily " stable until this 

 critical form is reached, we only have to show that conditions (7), 

 (8), or (9) (as the case may be) are satisfied by this value of £ for 

 every value of n and s. For this purpose I have calculated the 

 numerical values of the products p»(£) . q n (g), and t n *(£) .u n 8 (%) for 

 values of n up to 4, and, in the case of the sectorial harmonics (s = n), 

 up to n = 6 inclusive, taking £ = '3183236. The results calculated 

 to four places of decimals are as follows, the last figure being 

 only approximate : — 



n. 



s. 





«*(£)• 



Pl {l).q x {K)-t n s{V)-^)- 



1 







i?i(£) •<?>(?) 



= -1904 





1 



1 (sectorial) 



•5360 



- '3456 



2 









•2153 



- -0249 



2 



2 (sectorial) 





•3632 



- -1728 



3 



1 





•1566 



+ -0338 



3 



3 (sectorial) 





•2803 



- -0900 



4 









•1116 



+ -0788 



4 



2 





•1266 



+ -0638 



4 



4 (sectorial) 





•2303 



- -0400 



5 



5 (sectorial) 





•1967 



- -0063 



6 



6 ..„ 





•1743 



+ -0161 



From this Table it appears that the expression 

 is positive, except when n — 2, s = 0, and in the case of the first five 



