EQUILIBRIUM FOR CONSTANT MOMENTUM. 



149 



where K is a constant depending upon m and M. From these equations we 

 have 



dd 



dX 



dl 



>'=» 



Then we find 





dX = 



1 K dd 



3J o^ dX' 



da 



^r=+.^4^^. 



3J a^ dX 



(12) 



where 



J = 



dX 



dX' 



d<p 



w 



Since X and X' enter d and (p symmetrically, it follows that when the 

 ellipsoid branches from the Maclaurin spheroid, i.e., when X = X\ we have 

 dd dd , 



JX^W' "^^^^^^ ^^ *^^ ellipsoid branches from the spheroid because of 



increasing density, the eccentricity of one principal axial section increases 

 while that of the other decreases. This continues indefinitelv unless either 



~dX ^^ 'dX' ^^^^'^^^' which IS extremely unlikely. This means the figure 



tends to become cigar-shaped. At a certain elongation (see figure 19) the 

 so-called pear-shaped figure branches. Certainly in homogeneous masses 

 there can be no fission before this elongation, with its corresponding den- 

 sity, is attained. 



