Equilibrium of Vis Viva. Ill 



central motion and the old ^y-plane. We have then 



H = Gsin<9, 



90 — being the angle between the two axes of z ; therefore, 

 since G is constant, 



dK=G cos Odd. 



Finally, let us denote by co the angle between the two 

 t v-axes, seeing that it only differs from the angle previously so 

 designated by a quantity which we are at present regarding 

 as constant. In this case 



z 2 = x l cos 6 sin co + y ± cos 6 cos co + z 1 sin 0, 



w 2 — u x cos 0sin co + v x cos cos co + w x sin 0, 



and these two expressions must both vanish, inasmuch as the 

 xy plane is that of the central motion. By means of these 

 two equations we can introduce 6 and co in place of z x and w ly 

 when x ± y 1 u x and v± are constants, and we thus obtain 



dz x dw ± = (y^-x^) ^0 dOdco - 



Further 



o&2 = Xi cos co —y\ sin co, 



y 2 sin 6 = x 1 sin co + y x cos &>, 



with similar equations for n 2 , v 2 , u, v. From these it follows 

 that 



3/i^i — % u i = sin 6 (y 2 u 2 — x 2 v 2 ) = K sin 0, 



and, if 6 and co are contants, 



dx 2 dy 2 sin 6 = dx 1 dy ± ; du 2 dv 2 sin 6 = du x dv^ 



from which 



dxidy 1 dz 1 du x dv x dw x = K cos 6 dx 2 dy 2 du 2 dv 2 d0 dco> 



Now let us denote, as before, by cr and t the component 

 velocities of the relative motion of shell and nucleus in the 

 direction of p and normal to this direction, respectively ; 

 then for constant values of x 2 and y 2 , 



dcrdT=du 2 dv 2 



dKdlj = r. crp dcr dr. 



m + m" ' 



where L<? is the energy of motion of the centre of gravity, and 



N2 



