1886.] Equilibrium for a Rotating Mass of Fluid. 331 



energy of a sphere of radius a is f m 2 /a ; which is the known result. 

 For an ellipsoid of revolution a=0, and /3=0, and F=7; so that 



1=1— i 7— — 



sin 7 



The function (27) is the quantity tabulated below. It seemed pre- 

 ferable to tabulate a positive quantity, and it is on this account that 

 the intrinsic energy corresponding to the infinitely long ellipsoid is 

 entered as unity. 



Having now obtained all the necessary formulae, we may proceed 

 to consider the solution of the problem. 

 We have to solve 



sec 2 a sec 2 £E-(2F-E)-tan£sec 2 a sec 2 £=0, . . (28) 



where tan g= sin a tan /3 cos 7, tan d = sin /3 = sin <z sin 7, 



P d P 



and F= , E = cosBdy. 



JoCOs/3 Jo 



The axes of the ellipsoid are 



Q> , n xi b a c a „__ x 



-= (sec p sec 7)3, - = -cos/3, - = -0037. . (29) 

 a a a a a 



If e 1? e 2 j e 3 are eccentricities of the sections through ca, cb } ab 

 respectively, we have 



e^sin/S, e 3 =sin7, e 3 = cos a sin 7 sec /?. . (30) 



Having obtained the solution, we have to compute 



w 2 



— -=cot /3 cosec ft cot 7(F— E) + cot 3 /3cos ft secVE — cos 2 /3 cot 2 7 sec 2 a* 



. . . (31) 



Then we next compute ft and e and i from the formulae (25), (26), 

 (27). 



The functions F and E are tabulated in Table IX of the second 

 volume of Legendre's,' Traite des Fonctions Elliptiques,' in a table of 

 double entry for a and 7 for each degree. 



The solution of (28) by trial and error was laborious, as it was 

 necessary to work with all the accuracy attainable with logarithms of 

 seven figures. 



The method adopted was to choose an arbitrary value of 7, and 



* As stated above, some of the computations were actually made from the 

 l'ormula (20). 



