1890.1 



Rotating Spheroid of Perfect Liquid. 



371 



sectorial harmonics (n = s) in the Table. Thus in every case in which 

 the exact conditions of stability have not been investigated, the 

 sufficient condition for secular stability given by the inequality (7) is 

 satisfied. It remains to apply the criteria (8) and (9) to the cases 

 where (7) is not satisfied. 



•5. First take the case of n = 2, s = 0. 



The fact that ^(f ) . qi(£) —p%(X) ■ (/2(C) * s negative in the above Table 

 does not indicate that the spheroid in question is secularly unstable 

 for this particular type of displacement. Its meaning is that the 

 spheroid is more oblate than that form for which the angular velocity 

 is a maximum. As pointed out in Poincare's memoir,* the disturbed 

 form is here also a spheroid of revolution, and there is no form of 

 "bifurcation" when pj(^) . q\{'C) — • S^CD changes sign. 



The condition of "ordinary " stability, from inequality (8) is 



^a)-?ia)-^a)-^(r)+f{^(r).va)-Pi(r).2i(r)} > o. 



For the particular value of £ considered, the left-hand member of 

 this inequality is 



= - -0249 4- f ('3456) = - -0249 + '2304 = -2055, 



and is positive ; therefore (8) is satisfied. 



Even in the extreme case when the spheroid becomes flattened out 

 indefinitely, so that £ approaches the limit zero, we find 



pi(0 • ffitt)-A(f) • 2 2 cr)+!{^(p • • &(£)} 



_ 1 7T 2 / 1 7T 1 7T7T 7T 



= -42 + 3l22- } =-8 + 6 = 2i' 



and is positive. This accords with Sir William Thomson's result that 

 Maclaurin's spheroid is essentially stable, however oblate, if it is sup- 

 posed constrained to remain spheroidal. 



6. Next consider the sectorial harmonics. As the displacement 

 corresponding to n s = 1 is a mere shifting of the mass as a whole, 

 and we are dealing with the critical value of £* for displacements 

 determined by the harmonic of degree and rank 2, there are only 

 three cases to consider. Now since 



<i 1 (r)-«i 1 a>- i i'i(r)-si(e) = -3456 > 



we find 



= - -0900-1- -1152 = +-0252; 



* ' Acta Mathemat.,' vol. 7, p. 329. 



VOL. XLVII 



