368 



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



On page 210 of my paper, I showed that, if we consider only dis- 

 placements determined by the spheroidal sectorial harmonic of the 

 second degree, the limit of eccentricity consistent with stability 

 as obtained from my period-equations agrees with that obtained by 

 Riemann* and Basset.f This, of course, it should do, for the type 

 of displacement considered in both investigations is the same, viz., 

 one in which the deformed surface becomes an ellipsoid, but does not 

 remain one of revolution. We thus have a necessary condition for 

 stability. But we do not know that it is a sufficient condition. In 

 order that this may be so, it is necessary that the critical form thus 

 obtained shall be stable for all other types of displacement. The 

 object of the present paper is to show that such is, in fact, the case. 

 Were it otherwise, the limit of eccentricity consistent with stability 

 would have to be determined afresh. It is needless to remark that we 

 are here exclusively considering what Poincare calls " ordinary " 

 stability, as distinguished from " secular " stability. 



2. The symbols employed in the present paper are the same as in 

 my former communication, and the results there proved will be here 

 assumed. For the sake of convenience, the notation and results 

 required for the present work are collected below, and references to 

 the paper in question will be denoted by the letter [E]. 



The letters a, are used as defined in [E], § 4, (11), (12), viz., if e 

 be the eccentricity of the spheroid — 



a. = sin "^e, 



£ = cot a = (l- e 2)i/e. 



so that e = (l + £ 3 ) -i and £" is the reciprocal of the quantity denoted 

 by /in Thomson and Tait's ' Natural Philosophy ' (vol. 2, § 771). 



The functions p n (£), g»(£)> t n s (£), %/(£), are defined as in [E, § 5^, 

 equations (24) to (27), viz.: — 



?»(?) = 2^n(|)"(r 3 +i)- = (-i) s »p»(^-i) (i.) 



^ = (^ + i)^|)V(r) = ^(i)"> +1 >" •- ^ 



■ ? - ffl= 4"w (3) * 



(4.)- 



* 'G-ottingen, Abhandlungen,' vol. 9 (1860), Mathemat., § 9. 

 f 'Treatise on Hydrodynamics,' vol. 2, p. 124. 



