Equilibrium of Rotating Fluid. 679 



his equations for the bifurcation of the spheroids into sur- 

 faces of revolution may be stated in the following terms : — 

 Let E n be a polynomial in c of degree n, satisfying the 

 equation 



(i+c2) S" +2<;rf £' _n(n+i)R » =o ' 



and such that ~R n (i) = t n . Let S» be a second solution of the 

 equation, connected with E ?l by the relation 



S " =R U R» 2 (iW 



Then the equations of bifurcation * are 



R 1 S 1 = R 2 /iS2n> 



and of these the one with the smallest root is that for which 

 n = 2. ^Yhen written at full length this equation is 



where R 4 = \ [3 + 30 c ! + 35 c 4 ] , 



*'•* «(R/ + c 2 )=^ (119 + 655^ + ^V+1225r«), 



which reduces to equation (10) above. 



4. For the sake of conciseness we shall speak of the scries 

 of figures of equilibrium into which the spheroids pass as 

 " pear-shaped ^ figures,'* or simply ",pears," although the 

 original significance of the name is here entirely lost. 



To determine the form of the pear-shaped figure we 

 re-write equation (2') in the form 



r 2 (l + Rr)=a+ £ q. v a , B r » Pb . . (D) 



*=l 71 = V J 



and we also put 



K = K o +K 1 + K 2 2 + ..., . . . (ll) 

 where is a parameter which vanishes for the particular 

 pear which is also the bifurcating spheroid. Before sub- 

 stituting this value of r in equations (J3) put 

 %=P n /(l + AV)i» 



* Comparison of the two forms of the bifurcation equations shows 

 that rPhndx 



1 ^+^r =R2nS2n/c, 



a result which it does not seem easy to obtain directly, although it is 

 immediately reducible to the simpler form, 



,] V 2 ndx „ 



-rp=(-)SWc. 



o 



