288 



entirely free. It only affects the form of the perturbative function 

 S, whici) plays no part in (he definition of the elements. We can 

 eltlier nse ordinary relative co-ordinates (/S being in tiiat case different 

 for each planet), or we can introduce canonical relative co-ordinates, 

 either by the method of Jacobi-Radau ("elimination des noeuds") or 

 by PoiNCüRÉ's "transformation «" (Acta Mathematica, Vol. XXI, 

 page 86). [In these last two cases the body P of course is not the 

 true planet, but a fictitious planet, different according to the choice 

 of co-ordinatesj. Levi-Civita uses Poincake's co-ordinates, but this is 

 not material : the isoenergetic elements may as well be used with 

 any other system of relative co-ordinates. 



Also it is hardly necessary to point out that in all three cases we 

 can introduce new elements by canonical transformations and thus 

 derive from the isoenergetic or the isoprotometric elements the same 

 modifications which have been derived from Delaunay's elements. 

 Thus e.g. we have the three corresponding transformations : 



yi=zL n=L—G W=G — 



I. 



A = Z + f/-fi^ :irz= — g — d- W=z — ^ 



(where we have n = L(l~ \/I^^) , W=2G si?i' ^ i) 



H=U n—U—G W—G-Q 



II 



1] =: u -[- (J -\ d- Jt =z — g — ^ \p =z — {>• 



( 77 = r (1 — \/l~e') , W=2G sin- ^ i) 



W = V n—V - G W= G — 



III. 



(77= F„(l — k^l-e-^) , W—2Gsln'^i), 



from which again we can deri\'e the elements of Poincare-Harzek 

 h = \/2n COS Jt p = \/2W cos \p 



k = V^risiH jr q— \/2Wsin tp. 



If in case III we make the transformation 

 F= V — G Z = G 



we find the elements used by Hill. We have indeed F = m . ^i, 

 Z=m.v, ^ z=^ u (where ?j, u and u are the symbols used by 

 Hill), and the letter / is used by Hili, with the same meaning as 

 in the present paper. 



These elements can also be derived directly from the function 0. 

 The condition (11) must then be omitted : R must be assumed not 

 to contain G. 



