100 Pear-shaped Configurations of Equilibrium [CH. v 



thus completing the figure as far as second order terms. We have now a full 

 knowledge of the second order pear-shaped figure, and so are in a position 

 to determine whether or not it is stable. 



Calculation of the Moment of Inertia 



101. The question of stability of the pear-shaped figures turns, as we 

 have seen, on whether or not the angular momentum M of these figures in- 

 creases or decreases as we pass from the critical Jacobian ellipsoid along the 

 series of pear-shaped figures. 



The moment of inertia Mk* of the pear-shaped figure about its axis of 

 rotation is given by 



Mk 2 = I Ip (x 2 + y' 2 ) dxdydz (275). 



We have determined the coefficient s so that the mass of the pear-shaped 

 figure shall remain always equal to the mass of the original ellipsoid. 



We accordingly have M=%7rpabc, and equation (275) reduces to 



the integral being taken throughout the pear-shaped figure. 

 Transform to new variables x', y', z' given by 



x ax, y = by', z = cz' ..................... (277), 



then equation (276) becomes 



1# = ~(&** + Vy'' 2 ) dx'dy'dz .................. (278), 



and the integral is now to be taken through the volume bounded by the 

 surface 



...... (279), 



which is a distorted sphere of unit radius. 



Let r denote the radius vector to this distorted sphere in any direction, 

 so that r 2 = x'' 2 + y' 2 + z'*. Let us again transform to coordinates x, y, z, 

 given by 



#' = rx, y'-ry, z' = rz, 



so that x, y, z are coordinates of points on a sphere of unit radius. Equation 

 (278) becomes 



S .................. (280), 



where dS is an element of surface on this unit sphere. 



