242 Mr Greenhill, On the rotation of a [Mar. 24, 



If we put c® = a^ 



I^=-a^a'- h') (a' + Sh') {a" + \) \; 

 and if we put c^ = If, 



N= h'(a'-b') {Sa' + h') {b' + \)\', 

 and therefore for some value of c between a and b, the integral 



If the ellipsoid had been rotating as if rigid with angular 

 velocity co', we should have the equation 



or ^-V-\(o'\x''-\-f) = H; 



and the equation of the surfaces of equal pressure would bq 

 {A - 0)'^) x^+{B- 0)'^) 2/' + C^ = constant ; 



and therefore if a free surface can exist, these ellipsoids must be 

 similar to the external surface, and 



OJ" {A - oi") = ¥(B- <o") = c'G, 



„ a'A-c'C ¥B-c'C 



or (o' = 2 = ^2 > 



a^ b^ ' 



and therefore c must be the least axis ; this is the case considered 

 by Jacobi. 



In Jacobi's case 



a'b' {A-B) + {a' - b') c'C=0, 



'''■ "" Jo K + X) (6'VX)P"'Jo(c^ + X)p-^' 



where 



P'=(a' + X){b' + \){c'+X), 



r{a'b'-a'c'-b-'G')\-c'X' ^ 



or I i^ — ■ — p3 — dX = 0. 



If c = 0, the integral is positive, and if c^ = —^ — rz , the integral 



is negative ; consequently c must have some value between and 

 ab 



