48 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND 



FIGURE OF EQUILIBRIUM OF ROTATING MASSES. 



The Jacobian Ellipsoids. 



19. The condition that a single figure shall be a figure of equilibrium for a 

 rotation <a about the axis of z is that the centre of gravity shall lie on the axis 

 of z, and that V(, + i 2 (x 2 + y 2 ) shall have a constant value over the boundary X = 0. 

 In searching for a series of figures of equilibrium we must add a further condition 

 of constancy of mass. 



For the undisturbed ellipsoid, with the notation of 10, V 4 +w 2 (* 2 + ?/ 2 ) 



< 58) 



For this to be constant over the surface, it must be identical with 



where x is a constant. If we put 



-x 



= n, 



the equations obtained by comparing coefficients are 



T d\ e 



-r-r-n = -2> 



Jo AA a 



r 



\ 

 Jo 



d\ 9 



-r-fj ~ n = 7T' 

 AB 6 



AC 



Ellipsoids with Distortions of the First Order. 



20. We proceed to consider which of the distorted ellipsoids can give rise to 

 possible figures of equilibrium. 



The solutions of degrees 0, 1, 2 lead to nothing except new ellipsoids, so that the 

 inclusion of these distortions could only represent a step along the already known 

 series of Jacobian ellipsoids or Maclaurin spheroids. 



Consider next an ellipsoid distorted by the addition of a solution of the type (i) 



of degree 3 ( 18), say 0,, = e-^~- a This distortion, as we have seen ( 18), does 



C(f O C 



not affect the total mass or the position of the centre of gravity. The boundary 

 of the distorted ellipsoid is 



^.i.^-? 2 -! , 



a 2 b 3 c 2 



