1886.] Equilibrium for a Rotating Mass of Fluid. 329 



These formulae are, it must be admitted, of but little use, since it 

 would be necessary to take in higher powers of sin 2 ^ to obtain results 

 for a variation of 7 of more than 1°. , ' 



Tf ex. be near 90°, so that cos ex. is small, the approximate equation 

 between ex. and 7 is 



1 + tan 2 7 + cos 2 a( \ 5 + \ 7 tan 2 7 + f tan 4 7 + i tan 6 7 ) 



2_log e cot(i^-i7) . [1 + 1 tan27 + cos 2 a(-L- 5 - + -V-tan27 + tan 4 7)] 



sin 7 



=0. . . . (23) 



And 10 is given by 



+ i cos2a [^ n -^ lo ^ cot (i 7r -h) ■ (| + 2tan 2 7 + tan 4 7)-|-otan 2 7 



-*'tan 4 7 Jj. . (24) 



We shall return later to a modification of (24) which will be 

 applicable to very long ellipsoids of equilibrium. 



Besides the angular velocity and the axes of the ellipsoid, the other 

 important functions are the momeDt of momentum, the kinetic energy 

 of rotation, and the intrinsic energy of the mass. In order to express 

 these numerically we must adopt a unit of length, and it will be con- 

 venient to take a, where 



a 3 = abc — a 3 cos (3 cos 7. 



Thus a =, a(sec /3 sec 7)*. 



Let a be the density of the fluid which has hitherto been treated as 

 unity, and let (iWJia 5 /*, (f7r<r) 2 a 5 e be the moment of momentum and 

 kinetic energy, then 



(4 jra)\^ix=\m{c? + 6 2 ) w= T Vro-a 5 (sec /8 sec 7 )f (1 + cos 2 /3) (4nra)tf — Y 

 Thus fi =|V3(sec/3sec 7 )t(l + cos 2 /3) (—^ • • • • (25) 

 The function (25) is the quantity which will be tabulated. 

 Again (I*™) Ve=JC£ «*)aT| ^ . o,=J A /3(|^)V . ^ / -SiV, 



\h7raj 



(2 \ h 

 • • • • 



z 2 



