Here again the motion is described as a Keplerian motion in an 

 ellipse with varying elements. The ellipse has a point of contact with 

 the true orbit, and therefore belongs to the family of ellipses mentioned 

 above. The body P in its orbit, and the tictitious planet in its ellipse, 

 however, have not the same velocity, bnt the same momentum. Since 

 they have different masses, they ha\'e also different velocities, agreeing 

 only in direction. 



The energy is now 



IV. /ƒ=- + k]--S. . . . (34) 



and the living force 



2mT ^ — (---l (35) 



a 



If we put M = ]\[, + Ai¥, 



«0 



then the differential equations become 



dii M f2 \\ /\M{2M^-\-LM) 6r bS 



dt am \r a J amr^ dM dM 



dM_AM{2M, + AM)dr dS 

 dt amr^ of* d(i 



(36) 



In the same way as the systems I, II, and III, we can of course 

 derive other systems of elements. A system in which, as in III, the 

 semi major axis is constant, but with osculation, is obtained as 

 follows. We take the same function 0, given by (5) or (9), but 

 now we put 



72^ = hM — «^ + — — - 



V r r' 



The function R thus now contains four parameters. The elements 

 I, II, III are derived as above by assigning to the fourth parameter 

 a constant value h = 5'^ = \-^m. 



The equation (IJ) now becomes 



