1886.] Equilibrium for a Rotating Mass of Fluid. 335 



equal to 7, and that a becomes very nearly equal to 90°. Hence we 

 may put a. = 90°, and /3=7. 

 Thus, approximately, 



— = (sec {3 sec 7)?== (sec 7)'; 



and cos 7==^-^, 7=1^—^- 



The axes of the ellipsoid are 



- & (*)'• 



Now if in formula (24) we only retain the higher powers of tan 7, 

 we have 



u z sm 7 cos 



4<va tan 



^r_ ; L.iog,cofc(i.-i 7 )-fl 



7 |_ sin 7 



=f im^y [ f logeCot ^"-iri ~ sin ^] 



But 



log e cot (Jr-fr) =log e 1 "|" Bm7 =log e 2 +f log, 

 cos 7 a 



Therefore writing 1— §log e 2=C, so that C= 0*3573, we have 



If we put a/a=5*042, this formula gives <i; 2 /47r<7= 0*01264. The full 

 value in the preceding tables was 0'0131 ; thus even with so short an 

 ellipsoid as this, the results agree within 4 per cent. With rougher 

 approximation we have 



'" 2 s /ay, a 

 i^ =4 W l0g «a' 



of which the limit, when a is large, is zero. 

 For the moment of momentum we have 



fi = y (sec p sec 7 )f (1 + cos 2 /3) 



-iM3'K)K-°)' 



