56 Ellipsoidal Configurations of Equilibrium [CH. in 



61. Our first task must be to evaluate the potential from the secondary, 

 now assumed to be an ellipsoid of mass M' and semi-axes a, 6', c ' . 



Let us momentarily take the centre of the secondary for origin, then the 

 potential of the ellipsoid at any external point x, y ', z will be 



where the lower limit X' is the root of 



/2 /i/2 ~'-2 



^x+^x + c^Vx-^ ............ < 109 > 



and A' stands for 



Differentiating, and bearing in mind that the lower limit X' is a function 

 of x, y' and z ', we obtain 



d\ 



L 



y 



,a /2 + X7 

 and similar equations give dV/dy', d 2 V/dy' 2 etc. 



These equations are general. At a point on the axis of #/, we put y' = z' 0, 

 and the value of X' is, from equation (109), X' = x' 2 a'-. The equations 

 become 



.(110). 





To obtain the differential coefficients of the potential V at the centre of 

 the primary, we put x = R, and of course X' = R 2 a' 2 . If F denotes the 

 value of the potential at this point, and dV/dx etc. denote the value of 

 differential coefficients at this point, the general value of the potential of 

 the secondary, referred to the centre of the primary as origin will be 



In this expression the terms in xy, yz, zx have been omitted because 

 they vanish on account of the symmetry of figure of the ellipsoid. Terms of 

 degrees three and higher have also been omitted because they would destroy 

 the ellipsoidal shape of the primary. The approximation to which we are 

 now working is one in which the primary is supposed to remain of ellipsoidal 

 shape, so that all tide-generating terms of degrees three and higher must 



