1879.] liquid ellipsoid about its mean axis. - 245 



Therefore, integrating, 



E + i {Ai^ + Bf + Cz') - <» (a. + «■) ^; (a;« - f) 



+ 4'"1?Tp)'-'"1^"' + ^''=''' 



a constant. 



The surfaces of equal pressure can therefore be made similar to 

 the external surface, and a free surface is therefore possible, if 



(If 0) = 0, we have the case considered by Jacobi ; if tu' = 0, we 

 have the case first considered.) 



These conditions may be written 



a'-bY . ,,, a^-b' 



a A — a ft) + ft) -2 — Ta — 4a 6 , „ . ,„,, 



ft) 



ft) — ft) - 



-bY . „, a'^-h 



= b^B-bH(o'-(o -^—r-A +^a?h 



2 7,2 



+ 6V (a^ + 6Y 



or a'A-b'B 



= 2ft) (ft) + ft)') (a' - 6') 



therefore 



2ft)(ft) + ft,)-ft, (^^.-p^.j +ft) = ^2_J. > 



4a''6^ ,, r d\ 



or (a) + ft)) "rj^^-;;-^.^ =j^ 



{a' + by Jo (a' + X)^- (b' + X)^ {c' + \)^' 



Hence a^A> V^B or d^G, and the greatest axis can never be 

 the axis of revolution. 



