676 Dr. J. R. Wilton on Figures of 



by the single letter R w , equations (B) become 



Ka!=| R 1 [l + i(a 1 Ri + a 2 R 2 + ...a2R» + ...)]dx, 



Ka 3 + (1 + & 2 )(K -K)=| P P 2 (a 1 R 1 + a 2 E 2 + ...a*R u + ...)<J 

 and, n > 2, 



K«„= | f 1 P,(l + fo ! )i"- 1 (-^+(i 1 R 1 + ...a„R» + ...)^- 



In these equations every integral containing an odd power 

 of (l + k 2 x 2 )* vanishes, and they plainly reduce to 



Kax = I Ri^Rx + a 3 R 3 + a 5 R 5 + . . . )dx, 



Jo 



Ka 2 + (.1+ F) (K - K) = f P 2 (a 2 R 2 + * 4 R, + a 6 R 6 + . . . ) ^, 



Jo 

 and, n >■ 2, 



Ka n =( P M (H-^ 2 )* ?l - 1 KRn + a n+2 R ?l+2 + ...)<fe. 

 Jo 



It follows that the Hessian (equation 5) reduces to the 

 product of the terms of the leading diagonal, and the points 

 of bifurcation are given by the system of equations 



(«*2) K^fggi, (6) J 



in which K has been written for K, because the points of 

 bifurcation belong to the series of spheroids. 



We proceed to show that equation (6), considered as an 

 equation to determine k, has one and only one solution if n 

 is even, and no solution if n is odd, unless n = l, in which 

 case it is an identity. 



Putting £ = 1/P, and remembering equation (4), we see 

 that equation (6) may be put into the form 



1-yicot-W^fgJ. ... (7) 



To solve this equation in z we consider the two curves 



y^l-^/'zcot-Wz, (8) 



in which y is the ordinate, s the abscissa. 



