285 



In the cases I and TI il is advantageous to take (f„z=zO. 



In the cases II and III the value of ((„ is of course immaterial, 

 tlie first term of H may as well be omilted. As to the value of (?o 

 in the cases I and III, it is customary in classical celestial mechanics 

 (case I) to take ^gZ=]/k. This however is not at all necessary, and 



the term can be taken advantage of by an appropriate choice 



of i^o ^0 cancel a term in *S'. This is also advocated by Hilt, in the 

 paper already quoted. Though Hill does not say so (and doubtlessly 

 does not intend to say), a casual reader may easily be led to assume 

 that the possibility of this device is one of the advantages of the 

 system of elements of case III. It is therefore well to point out that 

 it does not depend on the choice of elements, and can as well be 

 applied in case 1. 



By each of the three sets of elements 



the motion of the body P is described as a Keplerian motion 

 in an ellipse with varying parameters. In the cases I and II the 

 variable instantaneous ellipse has a point of contact with the true 

 orbit, and can therefore be called an osculating ellipse. But the 

 definition of this osculating ellipse is different in each case. In fact at 

 every point of the orbit there is an infinity of ellipses having that point 

 and the tangent at that point in common with the orbit and all having 

 one and the same given point as a focus. In case I we choose from this 

 family of ellipses that ellipse that would be described by a body 

 of mass m starting from the given point with the given velocity 



under the action of a central force -^ emanating from the common 



focus. The constant ^g" here has a prescribed value, the same for 

 all points of the orbit. The elements thus derived are those of 

 Delaunay. They are called by Levi-Civita isodynamic elements. 



In the second case we choose that ellipse in which the energy of 

 a Keplerian motion of a body of mass m starting with the given 

 velocity from the given point has a prescribed fixed value A^ = — 2«o*« 

 The elements which we then get are those of Levi-Ch-ita, and are 

 by ijiim called isoenergetic elements. 



In the third case the ellipse has a prescribed semi major axis 



a = — . There is no osculation, the tangent of the ellipse in the 



