68 



STATEMENT AND EXPOSITION OF 



existing angular velocity of M, A, B, (in their revolution of every one of them 

 around its centre of reference) would, in every instance, have the same value. 



(d) But this same angular velocity in the moon's orbit varies inversely as the 

 square of the radius-vector, and the coexisting values of the squares of EA and EB 

 respectively, having (as already shown) constant ratios to that ; their ratios may be 

 substituted for the ratios of the respective coexisting values of the squares of the 

 radii-vectores themselves; and the inversion of the one for the inversion of the other. 



(e.) By substitution, then, the respective squares of EA and EB are inversely 

 as the coexisting angular velocities in the moon's orbit. 



(/.) But the same angular velocity being (as also shown) common to all the three 

 masses in question; every one of those masses will also have its angular velocity 

 inversely as the square of its own radius-vector; and that will imply the principle 

 of the conservation of areas ; and thus maintain not only for the moon, but also for 

 the other masses, in the consentaneous revolution of all, a dynamical equilibrium. 



{g.) Then withal the constancy of the ratios already specified, Avill secure, under 

 the coexisting similar change of angle, the same ratios among the radii-vectores of 

 all the three trajectories here in question; and just all that implies that the same 

 polar equation will apply to all the three. 



(7i.) Hence the trajectories of A and B are both ellipses ; as well as (perturba- 

 tions apart) is the orbit of the moon ; even more than this, under those stringent 

 conditions (common to all); viz. the trajectories are all similar ellipses. 



(82) The positions of the points A and _B, on the supposition that the girdle 

 on the one side, is between the earth and the moon, as in Fig. 13, is exhibited in 

 the following table ; the distances represented being in terms of the earth's equa- 

 torial radius. 



Moon's Distance ' 



{EA) Internal Distance of Girdle 



{EB) Externa! Distance of Girdle 



IN PEKIGEE. 



AT MEAN DISTANCE. 



IX APOGEE. 



5G.964 

 48.30,9 

 56.T90 



G0.273 

 51.11G 

 G0.090 



63.583^ 



53.922i 



63.389 



On the supposition that the girdle encompasses the moon, as in Fig. 14, we 

 have: — 



Moon's Distance 



{EA) External Distance of Girdle 



IN PEEIGEE. AT MEAN DISTANCE. 



IN APOGEE. 



56.964 



66.426 



60.273- 



10,285 



63.583i 

 14.144i 



(83) As A, B, and the moon thus describe similar ellipses with their radii- 

 vectores coincident in the same straight line ; it is manifest that the portions of 

 the girdle in the immediate neighborhood of A and B will expand (the material 



