62-64] The Double-Star Problem 59 



so that f also measures the proportional increase in the gravitational attraction 

 between the two bodies produced by their ellipsoidal forms. 



63. The terms in f have now been made to disappear from equation 

 (114) and the condition for equilibrium is seen to be that 



over the boundary. 



Comparing this with equation (90), it appears that the second degree 

 terms which were used in Roche's problem, namely 



^_ (^ _ i f _ 0) or ^ (& _ 1^2 

 must now be replaced by 



in which the lower limit V is put equal to R 2 a' 2 . 



Following the former procedure ( 52) we find that equation (122) may 

 be replaced by the three separate equations 



M' fF d\ 2\ a) 2 _^ 



^ " U Uv K 2 4- X) A 7 ~ AV ~ 27rpa6c ~ a 2 ' 



Jf' f 00 rfX o) 2 _6 



~ M } x > (6 /2 + X)A' Zirpabc ~ 6 2 ' 



e 



* ......... ( * 26) - 



These are the equations of equilibrium for the primary, and there is 

 a similar set for the secondary. On solving the set of six equations we 

 obtain a solution of the problem. There is unfortunately no method of 

 solving these equations exactly except by laborious numerical computations. 

 But they are of the same general form as equations (91) to (93) already 

 discussed in connection with Roche's problem, whence it is readily seen that 

 the general arrangement of figures of equilibrium must be the same as that 

 already found in 53 for Roche's problem. 



Stability 



64. The problem of stability demands a more detailed discussion. 



The angular momentum of the system is still given by equation (100), 

 namely 



(127), 



