CERTAIN HARMONIES OF THE SOLAR SYSTEM. 57 



{t'fJox^^-F i^ itself 



^ = ; whence 



(t)\ for F If 



9 



F 



{t'f^ (0 ; and 



^F 

 9 



( F 1 



If j 



Then if (t'), thus computed, be found to be equal to the moon's own periodic time, 

 the point A will have been accurately ascertained ; the particle, or the insensible 

 mass (in the line EM), completing its revolution at the distance FA, in the same 

 time with the actual revolution of the moon around the common centre of gravity 

 of the moon and the earth. 



But if (f ) differ at all from tJiat, the difference may be exhausted by the con- 

 tinued application of the method of trial and error. 



When A is situated beyond the moon (in accordance with the representation in 

 Fig. 14) the sum of the attractive forces of the two bodies must be made to enter 

 into the equation to determine the value of (f), instead of the difference of those 

 same forces. So also, for the distance from E to B, on the opposite side of the 

 earth. 



(81) Now the division or the extension of EM (rs the case may be) so as to give 

 the distance EA, this depends upon the forces in question, and, ultimately, on the 

 ratio of the masses, and not upon the absolute length of E3£. Hence EA and EB 

 will each have a constant ratio to EM; whether the moon be in apogee, or in 

 perigee, or at the mean or any other distance. The same is true of the distance 

 of the moon from the common centre of gravity of the moon and the earth, i. e. of 

 the radius-vector of the mobn^s orhit ; and for the same reason. 



Now, — (a.) Every other of the quantities in question having, after this manner, 

 a constant ratio to EM ; it will follow that, under all their variations of value, the 

 value of any one of the quantities will preserve a constant ratio to the coexistent 

 value of any other ; and therefore, specifically, to the coexistent value of the moori's 

 radius-vector; or the square of the one, a constant ratio to the square of the other. 



(6.) Next, as M, E, A, and B, under the conditions in question, are preserved in the 

 same straight line; it follows from the doctrine of parallels, that the angular cJiange 

 of direction of M revolving about the common centre of gravity of M and E, or 

 that of A and B revolving about E, will be the same with reference to any fixed 

 direction in space, such as that of EM (at any instant), or with reference to its 

 parallel ; or the same will be true with respect to the first tendency to such change, 

 i. e. its differential. 



(c.) Hence also, especially, the angular chaiige of direction which would take 

 place, were such a tendency preserved during the next xmiit of time, i. e. the co- 



8 January, 1875. 



