196 Professor Baher, On the stability of rotatioig liquid ellipsoids 

 - (3 + 14^2 + 3ti) - (3 + 13^2) _ 



(about 54° 22', corresponding to a meridian eccentricity e = -8127, 

 and a value of ^ — equal to -187). For this value ^2 = ^> and, by 



ATTp 



what we have previously shown, the spheroid is a particular one 

 of the ellipsoids of rotational equilibrium. 



Thus Si/» is negative, the proof being vahd for the spheroid for 

 which ^2 == ^' a^^ ^^^ spheroids lying between the sphere and this 

 ellipsoid are therefore stable for ellipsoidal displacements. 



§ 6. The coefficient of {x + y)^ in the above expression for Si/r 

 remains negative however even after a has passed the value (about 

 54° 22') for which ^2 = 0, and (/>2 has become positive, indeed up 



to a = ^ , For from 



c 

 ^=^i + *^2^2' 



using c = iph^, and hence 8c = SphSh, we get 





c c^ c 1 



1^11^* + ^12^^ - + i</'22 ^ + 4^2 </'2j 



leading, for a = &, to 



{x — yY ccf)2 {x + yY da 



Sifj 



on the other hand a = (f)-^ + (j>2 



gives a =2a{a — c) 



Jo 



4 a^ 16a2 (^a ' 



c 



ah^ 



Jb (jLJU 



{x + a)2 (aj + c)^ 

 and hence 

 da r { ,c^ X „ o / , a? + a) 2a;^a; 



]a; (2a + c) + 3ac + 3c (a - c) 



(?a Jo i a; + cj (a; + a)3 (a; + c)^ 



which is evidently positive. 



Thus, putting x = y, we infer that the spheroids are stable for 



spheroidal displacements for all values 0/ « up to ^ . 



We can prove that 



a = 6 Q. (1 + t'^)f y-±^ [^^ (1 + 1^2) _ 1 



say, where (1 + t'^)'^ increases with t, while 



, = %[p{l + t^)f' U, 



