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In all these cases the absolute attractive forces are as the 

 masses of the attracting bodies; and if there are a num 

 ber of these, A, B, C, E, &c., of which A attracts all 



the rest with forces as ^ &&amp;gt; &c., (D, d, &c., being the 

 distances from A,) and B also attracts A, C, E, &c., 

 with forces as , 2 , the absolute attraction of A and 



B towards each other are as the masses A and B. Hence 

 in a system, as of a planet and its satellites, if the latter 

 revolve in ellipses, or nearly so, and describe areas pro 

 portional, or nearly so, to the times, the forces are mutually 

 as the masses of the bodies ; and conversely, if the forces 

 are proportional to the masses, and ellipses are described 

 and the areas as the times, the mutual attractions of all 

 are inversely as the squares of the distances. 



It is proved, by reasoning of the same kind, that the 

 disturbing force of S is greatest when M is in the points 

 C and D of the orbit (or the quadratures*), and least when 

 M is in A and B (or the line of conjunction and opposition 

 called the syzygies). When M is moving from C to A and 

 from D to B, the disturbing force accelerates the motion of 

 M, which then moves along with the disturbing force. 

 When M moves from A to D, and from B to C, its mo 

 tion is retarded, because the disturbing force acts against 

 the direction of M s motion. So M moves more swiftly 

 in syzygy than in quadrature, and its orbit is more curved 

 in quadrature than in syzygy. But it will recede further 

 from E in quadrature, unless the eccentricity of the orbit 

 should be such as to counterbalance this recession: for the 

 operation of the combined forces is twofold ; it both makes 

 the line of apsides move forward in one point of the 

 body s revolution and backward in another, but more for 

 ward than backward, and so upon the whole makes it ad- 



K 2 



