28 Mr Basset, On the Stability of [Nov. 28, 



5. A few years ago, I gave an investigation* which purported 

 to show that a Maclaurin's spheroid, which is composed of friction- 

 less liquid, is stable for an ellipsoidal displacement. This result is 

 known to be erroneous, but at the same time the analysis shows 

 that a Jacobi's ellipsoid composed of frictionless liquid, and which 

 differs very slightly from the spheroidal form, is stable for an 

 ellipsoidal displacement. In the light of Poincare's investigations 

 this result is interesting; because it shows that a surfifice of 

 bifurcation does not necessarily involve an exchange of stabilities. 

 On the other hand, we should anticipate that a surface which forms 

 the limit between a stable and an unstable system is a surface of 

 bifurcation, and consequently that the spheroid whose excentricity 

 is "9529 is a surface of this character. 



6. The phrases 07-dinary stability and secular stability have 

 been employed by Poincare in the following sense : let e'^t+>.^t \^q 

 the time factor of a vibration ; then if a is zero, the steady motion 

 (or equilibrium) is ordinarily stable, but if a is negative it is 

 secularly stable. It appears to me that this distinction is un- 

 necessary, and that the employment of these phrases in this sense 

 is an inaccurate use of language ; and that as the physical proper- 

 ties, as well as the conditions of stability, of viscous liquids are 

 different from those of frictionless liquids, the most accurate and 

 intelligible course is to discuss the two species of liquids separately. 

 If a is zero or negative, the system is absolutely stable ; for in both 

 cases it performs small oscillations about its position of steady 

 motion (or equilibrium), but in the one case the oscillations are 

 permanent, whilst in the other they gradually diminish and 

 ultimately die away. It is, however, possible for the steady 

 motion (or equilibrium) to be such that a small displacement 

 causes the system to perform finite oscillations about its undis- 

 turbed configuration. In cases of this kind, the time factor in the 

 beginning of the disturbed motion will involve the exponential 

 term e"*^, where a is positive, and the system will be unstable in the 

 ordinary sense of the word ; but as the complete solution of the 

 equations of disturbed motion must necessarily consist of periodic 

 terms, the system may accurately be described as secularly stable. 



7. Up to the present time we have confined our attention to 

 ellipsoidal displacements ; but the stability of a rotating ellipsoid 

 composed of viscous liquid has been investigated in a very able 

 manner by Poincare -f*, when the disturbance is of the most general 

 character. He has, however, made a slight slip in the work, in 

 consequence of having omitted to recognize the fact that the 

 angular velocity of the disturbed figure is not the same as that of 



* Proc. Lond. Math. Soc, vol. xix. p. 53. 

 t Acta Mathematica, vol. vii. p. 259. 



