678 Dr. J. K. Wilton on Figures of 



equations (6) which has the smallest root* is found by taking 

 n = 4, and therefore that the first point of bifurcation on the 

 series of spheroids is given by the positive root of the 

 equation 



^o-J o l + AV' 

 or, putting k= tan a, c= cot a, 



which on simplification becomes 



5075 

 c(119 + 655c 2 + ^—^-f 1225 c 6 ) 



-a(l + c 2 )(9 + 235 c 2 + 875 ,; 4 + 1225 c 8 ) = 0. (10) 

 I find that the solution of this equation is 



a=80°8' 19";7 

 = 1*3986858 . . in circular measure. 



Whence we obtain for the bifurcating spheroid, 

 — = tan :«= 5-7528, 



-=sec« = 5-8391, 



c ' 



- = cos a= *17126, 

 a 



e= sina= '985226, 



9 — = '17452, 

 2irp 



where c is the length of the semi-axis of rotation, and a is 

 the radius of the circular section through the centre of the 

 spheroid. 



3a. It is of interest to compare the bifurcation equations 

 with those given by Poincare (Acta Mathematical vii. pp. 319 

 and 329, 1885). Poincare gives no numerical results, but 



* It is, at first sight, not quite evident why we should not take n=2. 

 But if, with this value of K , we assume an expression of the form (D) 

 for r, we shall find that all the coefficients vanish, except that of r 2 P a ; 

 ». e. the surface is still a spheroid. It is in fact the spheroid for which 

 w is a maximum. 



