Equilibrium of Rotating Fluid. 673 



Since z, R is on the surface, 



f=r(.r + icosX, \'l — x 2 ), 



Whence 

 C -f J>?' 2 + (1 — v)r 2 F 2 =\ r-j- (rx)dx — rP : I Xj~ (rx) dx 



+ f r» +1 P„ +1 f -^T (Pn-i-^P„)# (r«) ^ . 



1 J_ 1 nfT ' dx K J r n 



We might, owing to a result of Poincare's*, assume at once 

 that r is an even function of x, since there mu-st be a plane 

 of symmetry perpendicular to the axis of rotation. It will, 

 however, add but little complexity to the work if we do not 

 make this assumption ; we shall thus obtain an independent 

 proof of the theorem in this particular case. 



By integration by parts it is readily seen that 



P -I, (p.-i-«p.) /(«)£.=- P ~p n+ ^;\, 



J_ 1 ?i + l v ' dx r n J_i n ~-1 r 



unless n=l, in which case 



Hence the equation which must be satisfied on the surface 

 may be put into the formf 



G' + vr 2 + Q-v)r 2 F 2 



where the accent attached to the 2 means that the term for 

 which n — 2 is to be omitted. If we suppose that the equa- 

 tion of the surface of equilibrium may be expressed in the 

 form 



r 2 = a + u l rY l + ct 2 7> 2 P 2 + , . . a n r n F n + . . . , . (2) 



* Acta Mathematica, vii. p. 331. 



t The equation in this form may be obtained more directly and more 

 elegantly by forming the potential of a circular ring, and expanding 

 before integration with regard to ax. The only advantage of the pro- 

 cedure adopted above is that it avoids the discussion of a nice, though 

 not at all difficult, question which arises in the course of the direct 

 method- 



