108 Pear-shaped Configurations of Equilibrium [CH. v 



From this expansion for f, the value of Vi can at once be written down ; 

 it will agree with equation (297) if we take 



The equations of equilibrium (299) which must be satisfied are however 

 of the type 



so that the conditions for equilibrium are that 



for all s's. Thus in order to satisfy the equation of equilibrium it is merely 

 necessary that f s shall be of the form (cf. equation (313)) 



4- ... 4- s-^- 1 + *_2 T 2 + ...... (316). 



To introduce the limitation that the curve shall remain of constant area 

 we must have s^ = 0, as is at once evident on considering the form assumed 

 by FO at infinity. To keep the centre of gravity at the origin we must 

 further have s_ 2 = 0. If we replace s_ 2 , s_ 3 , ... by new symbols s (7_i, S CL 2 , 

 we may write equation (316) in the symmetrical form 



(317), 



in which there is no term in 8 <7 and we know that gCLj must ultimately be 

 zero in order that the centre of gravity may remain on the axis of rotation. 



Thus we have found that the assumed equation (310) will represent a 

 configuration of equilibrium provided the explicit solution for f is of the 

 form (312) in which g g etc. are given by equations of the type of (317). 



110. Let u^ introduce p Q , p lt p 2 , ... defined by 



Jfc -& + &, 



* = So 4 + ^ 2 ?2 + S 4 >. etc. 



Then on substituting for 1 &) 2 /2?r/3 from equation (311), the supposed 

 solution (312) assumes the form 



(318). 



