1890.] 



Rotating Spheroid of Perfect Liquid. 



375 



and the spheroid is stable for harmonic displacements of the 

 degree 5. 



From the results of Case II we also have, if n be greater than 6, 



*»*(£). < wo ■««*«); 



and from the Table 



* 6 6 (r)-« fi 6 (t) < ft(t).&«3; 



therefore, a fortiori, 



*»*(£) . < Pitt) ■ if » > 6. 



Moreover, by Case I, 

 therefore, a fortiori, 



tn'(£).U n °{£) < Pi(t).?i(r), 



or lh(t) -ft(£)-V(0 > °» 



where = '3183. . . . , and n is equal to or greater than 6. 



Thus the sufficient condition of secular stability is satisfied for all 

 types of displacement, with the exceptions already considered in which 

 the "ordinary "conditions of stability have been proved to hold good. 

 Hence the results of the present paper prove conclusively that 

 Maclaurin's spheroid, if formed of perfectly inviscid liquid, will he abso- 

 lutely stable if its eccentricity be less than 0'9528867. If the eccentricity 

 exceed this limit, the spheroidal form will become unstable, and the liquid 

 will assume the form of an ellipsoid. 



13. The state of steady motion which then ensues is intermediate 

 between the forms known as Jacobi's and Dedekind's ellipsoids. 

 The " spin " of the liquid will be everywhere constant and equal, say, 

 to id, and the form of the liquid free surface will be an ellipsoid, 

 whose principal axes rotate about the least axis with angular velocity 

 That this is initially the case is in accordance with the results 

 of [E, §§ 14, 18], supposing that the roots of the period-equation 

 become complex, for their real part will indicate that the disturb- 

 ance travels round with angular velocity ^w. It is unnecessary to 

 discuss this point at greater length here. 



It is also to be noted that the results of the present paper quite 

 preclude the possibility, under ordinary circumstances, of Maclaurin's 

 spheroid ever passing into the form of one or more rings of rotating 

 liquid. This might probably take place if we imagined the liquid 

 surface constrained to remain a figure of revolution. But such hypo- 

 thetical circumstances are devoid of interest, and, since it appears from 

 the results of the present analysis that, when we consider displace- 



