110 Pear-shaped Configurations of Equilibrium [CH. v 



112. Equation (324) is a quadratic equation giving f in terms of 77. 

 Writing a = 1 for convenience, the solution is 



(329) 



Inserting this value for f into equation (325), this equation becomes an 

 equation in 77' having c 3) c lt c_ 1} ... as coefficients. Since this equation must 

 be an identity we may equate the coefficients' of different powers of 77 and 

 obtain 



- - - fCi - f <?_! = 



, etc. 



The first equation is satisfied automatically, as it ought to be. We may 

 assign any value we please to c 3 , this merely determining the scale on which 

 the parameter 6 is measured. Taking c 3 = 1 we find in succession 



c 3 =l, c^-y, 0^ = 0, c_ 3 = 



The vanishing of c_i shews that the centre of gravity of the curve is, as it 

 ought to be, at the origin. . 



The value of is now given in the form 



Inserting this value for f x in equation (326) and equating coefficients, we 

 obtain the equations 



Solving, we obtain in succession, 



< = -*+* ( . 



This completes the solution as far as second order terms, and we find, 

 precisely as in the three-dimensional problem, that there is an ambiguity in 

 the solution, in that S 2 nas n t been determined and cannot be until we pro- 

 ceed to terms of higher order. 



