38 Ellipsoidal Configurations of Equilibrium [OH. in 



themselves the necessary and sufficient condition that the standard ellipsoid 

 (51) shall be a figure of equilibrium under a rotation &>. 



37. On subtracting corresponding sides of equations (65) and (66) we 

 obtain 



and the elimination of 6 between this and equation (67) leads to 



..................... (68). 



It accordingly appears that equations (65) to (67) can be satisfied in two 

 ways ; first by taking 



a'=6 2 .................................... (69) N 



and second by taking 



= #J c .............. .. ............... (70). C_ 



Maclauriris Spheroids 



38. Let us examine the former alternative first. When a = b, the series, 

 of ellipsoids become a series of spheroids which include the sphere a = b c 

 for which ft) 2 = 0. 



Equation (65) now becomes identical with (66). The elimination of 6 

 between this equation and equation (67) gives 



ft) 2 a 2 - c 2 r \d\ 

 or 



a 2 - c; 2 r 

 a 2 J 



%7rpabc 



Since ft> 2 must be positive, it appears that a 2 must be greater than c 2 ; 

 the spheroids are all oblate. On evaluating the integral in equation (71), 

 the value of ft) 2 is found to be given by 



2-777? &' ^ e * ' 



where e is the eccentricity, defined by e 2 = (a 2 c 2 )/a 2 . 



Thus the eccentricity of the spheroid depends only on the ratio of ft) 2 to p, 

 as it is apparent from a consideration of physical dimensions that it must. 

 The following table of corresponding values of &) 2 /p and e is given by Lamb*, 

 being compiled from values calculated by Thomson and Taitf: 



* Hydrodynamics (4th Ed.), p. 673. I have inserted into this table, Darwin's values for 

 e= -81267, the point of bifurcation, 

 t Nat. Phil. 772. 



