52 Mr Bryan, Note on the stability [Feb. 13, 



Therefore w^ is a maximum when 



which proves the correctness of my statement. 



The same thing may also be shown by examining Poincare's 

 alternative method of finding the criteria of stability in § .9 of his 

 paper*. He there considers that the ellipsoid will be in critical 

 equilibrium if, when it has been slightly displaced, the displaced 

 form is also in equilibrium. But he takes the displaced form 

 to have the same angular velocity as the original form wherea,s 

 it really ought to have the same angular momentum. If the 

 displacement of a spheroid is determined by a spheroidal har- 

 monic other than the zonal harmonic of the second degree, the 

 original and the displaced spheroid have the same moment of 

 inertia (to the first order of small quantities) and therefore their 

 angular momenta are equal if their angular velocities are equal 

 and Poincare's method gives correct results for the stability. But 

 in the case of a zonal harmonic displacement of the second degree, 

 the moments of inertia in the two forms are no longer equal. 

 And since the displaced form is also spheroidal, the condition 

 obtained by Poincard is that which must hold when two spheroids 

 with the same angular velocity coalesce, i.e. when the angular 

 velocity is a maximum or minimum. And we know from other 

 considerations that it is a maximum, not a minimum. 



Mr Basset's paper suggests another point which it would be 

 well to inquire into more fully. He says that the spheroid is 

 secularly stable if the motion when slightly displaced is deter- 

 mined by terms of the form e^""*"^'' * where a is negative. But 

 is it quite certain that the small motions of viscous rotating mass 

 of liquid can always be expressed by means of terms of this 

 form ? In electrodynamics we have instances of dissipative 

 systems in which the general equations and boundary conditions, 

 although linear, are not satisfied by a single solution of this form, 

 and those systems in which the solution can be so expressed 

 have been called "self inductive "-|-. It would be interesting to 

 inquire whether any such conditions have to be satisfied in the 

 case of a mass of viscous rotating liquid, or whether such a 

 system is always, so to speak, "self inductive" in this sense. It is 

 certain that the equations of the small motion about equilibrium 

 of a Maclaurin's spheroid of viscous liquid do not admit of any 

 such simple solution as those which I have found for the spheroid 

 of perfect liquid, and the cylinder of viscous liquid. 



It is, moreover, highly desirable that the general conditions of 

 stability of rotating liquids should be proved by an alternative 



* Acta Mathematica, Vol. vii. pp. 319 — 321. 



t Watson and Burbury, Electricity and Magnetism, Vol. ii. 



