58 Ellipsoidal Configurations of Equilibrium [OH. in 



then equation (117) gives %/j;' = M'/M, whereas equations (115) and (116) 



give i; ; "ff L ' 



f'~ M X /' 



The difference between the two values of f/f ' consists of a multiplying 

 factor I'/ J. 



Now / can be put in the form 



/=f -, (119) 



JB u? [u - (a 2 - b 2 )Y [u - (a 2 - .c 2 )]* 



while /' is obtained on replacing a, b, c by a, b', c'. Since R 2 is large in 

 comparison with a 2 b 2 and a 2 c 2 , it is clear that the two integrals do not 

 differ by much from one another, both approximating to 



* du 2 



The integrals agree more closely with one another than with this limiting 

 integral, and when the ellipsoids are nearly equal, / and /' become very 

 nearly equal to each other while differing considerably from 2/'3R 3 . Let us 

 suppose that 



+ ?) ................... (120) 



then it is clear that / and /' may, without serious error, be taken separately 

 equal to the quantity on the right*. 



Using this approximation for /, equation (115) becomes 



*f-3P<*0 



while equation (116) assumes the similar form 



These equations are now consistent with equation (117) and by addition 

 we readily find 



'l + t) ........................ (121). 



This determines a>, and is now seen to measure the proportional increase 

 in o> 2 produced by the ellipsoidal shape of the bodies. 



To balance centrifugal force', the gravitational attraction between the two 

 bodies must be a> 2 RMM'/(M + M') or, by equation (121) 



* It will be readily verified that our is identical with the f used by Darwin. 



