1886.] Equilibrium for a Rotating Mass of Fluid. 



321 



In the present case &=sina, &'=cos «, A = cos B. Thus (8) and (9) 

 enable ns to complete the required transformation to elliptic in. 

 tegrals of (6). 



Substituting then from (6) (8) (9) in the expression 



y_ 3 f^. x^d* y*d* ^_^\ 

 4 \ a da b db c dc J ' 



where m = \irabc =-f 7ra 3 cos 3 cos 7, we have 



* asm 7 a^isin ^ Lsm * \cos~acos/3 sm-« 



-o— + -tan 7 cos^l . (10) 



sir«cosV cos-jt J 



Now suppose the ellipsoid to be rotating about the axis of z with 

 an angular velocity and let us choose the axes a, a cos j3, a cos 7, 

 and the angular velocity w, so that the surface may be a surface of 

 equilibrium. 



For this purpose V + |w 2 (a? 2 + 2/ 2 )= constant, must be identical with 



£8 



-5 + 



a? ' a 2 cos 2 /3 1 a a cos a 7 



Now in (10) we have Y in the form 



V=Lx* + Mif + Nz* + P, . . . 

 whence a\L + \ w 2 ) = a 2 (Jf + J w 2 ) cos 2 /3=cvWcos L V 

 Hence Jv-3f +^ C os 2 7 tan 2 /3=0, ") 



Jw 2 = N cos 2 7 — i , 

 or Jw 2 sin 2 /3= If cos 2 £— i. 



There are two kinds of solutions of these equations (12). 

 First, since 



(11) 



(12) 



T h d * 3 a 2 f sin2r O 



2jz=7TbC—r— = — TTQ^COS B COS 7 . 5 . .. - #7 



ofa a, 3 sm^<v in A ' 



cos 8 cos 7 f 7 sin 2 7 



— — Z7T — — d< 



sm°7 J A 



d^f 



M— irac—rr — — 7ra 3 cos B cos 

 db 



2 TsiuV 



> ■ (13) 



= -2: 



cos /3 



cos 7 f Y sin 2 7 , 



it is obvious that L—M vanishes when x—Q. 



