Professor Baker, On the stability of rotating liquid ellipsoids 195 



where 'i'i={cr- </.i) h^ {x + y) - cf>^c (^^ + ^j, 



and 



T, = (- a + <f>,) h^ {X + y)^ + cf>,c (^~; + ^ + fj) . 



+ * [k^' (x + yf + 2cf>,,ch^ {X + y) [I + I) + cf>,,c^ (I + I)' . 



If herein we put a = b we obtain, for an arbitrary ellipsoidal 

 variation from the spheroid, to terms of the second order, of the 

 function ip = — ah + cj), which is a constant negative multiple of 

 the energy H, the following 



Si/f 



(cr - ^i) h^ - </>2 



{x + y) 



c<t>^ 



+ (- o- + c/,i) 7.3 (a; + yf + -£2 ^3 (^ ^ ^)2 + (^ _ y)2] 



4a2 



+ l{x + yf 



2ch^ 



<^ii^* + -^r ^12 + ;^^ 



22 



Thus we obtain, as determining the angular velocity or momentum 

 necessary for the rotational equilibrium of the spheroid, the single 

 equation 



(explaining the reason for the change of variables), and thence 



4a2 



+ 



ch c c I 



4^i# + </'i2 — + *<^22 ^2 + 4^ 'A2J {^ + yf- 



We have however shown that 





2 11 



is negative so long as - > - + y^ , and hence, for a = 6, so long as 



a > c. For the value of (/y^, putting a = b, a == c sec^ a,t= tan a, 

 we easily compute 



^2 = -^ I7 (3 + Ut^ + 3^4) - (3 + m^)\ „ 



which is negative for values of a ranging from « = 0, when the 

 spheroid is a sphere, to the value for which ' 



13—2 



