28G 



common point being different from the tangent of the orbit. ^) If a 

 name analogous to those coined by Levi-Civita for the other two 

 systems were required, we might call tiiese elements isoprotometrlc 

 elements, since the quantity a, which here remains constant, is called 

 the protometer by Gylden, who was the first to use a system of 

 elements belonging to this class. 



If at a given point of the true orbit, i.e. for given values of 



dr (h dio , , ■ 1 • 



r, s, ID, — , — , — , Ave wish to determine the instantaneous elements 

 dt dt dt 



in the three cases, the method of procedure is as follows. First we 



determine geometrically the inclination i, and node '"^ of the plane 



containing the origin of coordinates and the velocity of the body P. 



dC 

 With the aid of these we find ?^ and — . Then 



dt 



G = mr^ — O ^= G cos I. 



dt 



For the determination of the third linear element we require the 

 living force, or kinetic energy: 



, dr , 



\dtj 



We have then in the three cases (taking ff^ =^ ^ ^oi' ^'^^ cases 

 I and II): 



/. 2T — 



r . U 



II. 27' ^'^^-V ) . (25) 



'2 n , (F-F„)(2(?+r„-F) 



III. 2T = i?; 



' r a J mr' 



From these formulas we find L,U,V. Next a and ö are determined 

 by (17), (18), (19), (20), (21), (23) and then the ordinary elliptic 

 formulae give /• and the true, excentric or mean anomaly. Finally 

 we have 



.'/ = ? — V- 



The differential equations for the elements are given below for 



G 



the three cases. In the cases I and II 1 take ö^=zO, or y =r -—- 



ym 



and in the cases I and III I put 



1) Hill i.e. p. 176, slates that tlie ellipse has a point of contact with tlie orbit. 

 This, however, is an oversight. 



