64] 



The Double-Star Problem 



61 



CD 

 2 



Fig. 8. 



elongated configurations OP' are unstable. We have seen that at the 

 configuration of closest approach R, increasing elongation goes with increas- 

 ing angular momentum ; it follows 

 that R is on the unstable branch OP' P p 



of the series. 



Thus the configuration of closest 

 approach is always unstable ; it is 

 fairly near to when M and M' are 

 nearly equal, but is far removed from 

 in other cases. Passing to the 

 limit of configurations of greater 

 elongation, it is easily shewn that 

 in the extreme configuration P' in 

 which M = oo , <o = 0, the two bodies 

 must overlap ; thus this configuration, although satisfying the mathematical 

 equations, is physically impossible. At some stage between R and P', there 

 must be a configuration (7, in which the bodies are just in contact, but without 

 overlapping ; this configuration, which we may call the contact configuration, 

 is the last one which is physically possible. It is clear that all contact con- 

 figurations are necessarily unstable. 



Darwin calculates in detail the configurations C, R and in the case of 

 p = 1 or M = M'. In the case of configuration the calculation is not very 

 accurate, for his series do not give good approximations when the masses 

 are in or close to actual contact. 



For the configuration of limiting stability, Darwin finds in this case 



a = a 7 = 0-897, 6 = 6' = 0'771, c = c = 0*723, r = 2'638, 

 the unit being the radius of the sphere formed by rolling the two masses 

 into one, and the cross-section is shewn in fig. 9, which is reproduced from 

 Darwin's Collected Works*. 



c 6=771 



Fig. 9. 



* Vol. in. p. 513. I am indebted to the Syndics of the Cambridge University Press for per- 

 mission to reproduce this figure and also figures 10-14 from the original blocks. 



