Professor Baker, On the stability of rotating liquid ellipsoids 193 



§ 3. Consider a series of ellipsoidal shapes satisfying the con- 

 ditions ^1 = a, cl>2 = 0, for all of which abc is the same. By </)2 = 

 we can then regard c as a function of h, subject to 



dc ^ _ </.i2 

 dh (f>22 



For increments SA, Sc, ^; being constant, the increment of K, to the 

 second order, with constant a, is then 



hK = aSh - (<^i8A + cf^^Sc) - 1 (c/.iiSA2 + 2(^i2SA8c + 4>22^c^) 

 while 8(7 = (/)ii8/i + ^la^c = ( </>ii — ^^ - ) S^, 



V 022/ 



and, from co^ = ?>Mah^ = 2>M}i^(f>^, we have 



(<^iiA2 + 2</.i/i) 8A - ^ ^2§^. 



But, from § 1 above, we have 



<^22 < 0, 0ii</>22 - ^li > Oj 



1 rp,, , 3 rPi^ 



*^22 



while 



9 -j/ 



^ -'o y 



i'^^ + z[ 



wherein P, 



so that 



2 '^ + ^^ p ^ _ 



4 Jo ?/^ 

 x^ + a^c 



(^iC 



y 



o Ctt/Cy 



¥ 



cl>,Jl^ + 2c^iA, 



^^2 



= -^fipll, 



jA2 



P 2 



5^ ' 



is necessarily positive. 



Thus we see that, as h diminishes, for a series of shapes of 

 rotational equilibrium, K constantly increases, a also increases, hut co 

 diminishes. 



§ 4. We can however show that as a — 6 increases numerically, 

 h does diminish, that is that a + h increases. [The numerical tables 

 show that along the series of ellipsoids a increases, but h and c 

 both diminish.] 



For 



18 [(a - hf-] = 18 



h^ 



VOL. XX. PART I. 



13 



