44 Ellipsoidal Configurations of Equilibrium [CH. in 



When R is great the ratio of successive terms is of the order of magnitude 

 of r/R, so that when R is very great, the tidal potential reduces to its first 

 term 



or, writing /* for M'/R 3 and transforming to Cartesian coordinates, 



........................... ( 75 >- 



48. When the tide-generating potential reduces to this simple form, it is 

 at once clear that ellipsoidal configurations are possible for the primary. The 

 condition that the standard ellipsoid (51) shall be a figure of equilibrium undei 

 a tide-generating potential (75) is that at every point of the boundary 



F; + /-iOr 2 -4.y 2 -^ 2 ) = cons ...................... (76). 



As in 36, this is equivalent to 



where 6 is a constant. 



Equating coefficients of # 2 , 2/ 2 and z z , we find as the equations to b( 

 satisfied : 



- 



Trpabc a 2 



J A - -~ = - ................... . ....... (78), 



(80) - 



The addition of corresponding members of these equations gives 



while on subtracting corresponding members of equations (79) and (80), w< 

 obtain 



The elimination of between this equation and (81) gives 



It is at once clear that as before ( 37) there are two sets of ellipsoida 

 configurations, and these are obtained by satisfying respectively the tw< 

 equations 



6 2 = c 2 .................................... (83), 



m ......................... (84) ' 



