288 



a dR 

 ,/,,; — — dr; (15) 



a' dR 



M = E — e süi E , du --= — d7' . . . (16) 



^' [/m da ^ ' 



In all these tbrmulcis a and d are written as abbreviations for 

 certain fnnctions of a, /?, y defined by (10). 



All this is independent of the choice of the third pair of canonical 

 elements. We must now specialize the values of the parameters 

 ct, /?, ff, which were so far left entirely indeterminate. Now we can 

 distinguish three cases. In each case two of these parameters are 

 constant, while the third is variable, and a function of it is taken 

 as the element q^. 



Case I. ^ -■=/?„ = const. , (f=:dg = const. 



The third linear element is a function of a and will be called L. 

 Therefore the conjugated variable / is given by 



_d<P d(P da __ da rdR _ ,'?„' [/m da r 

 dL da dL dL J da a^ dL J 



Thus, if we wish to get 



/ ^ M := mean anomaly, 



we must take 



dL ^^'^ \/m 



da a'' 



from which 



Z/— =/J„|/m. \/a ...... (17) 



Since /i^ and m are constants, the semi major axis « is variable. 

 We find at (jnce from (10) 



du. 



L[^l-e' = G + (fym (18) 



Case IT. a = a^ = const., d" ^ d^ =; const. 



The third linear variable U is a function of /i. Therefore the con- 

 jugated variable is 



d0 d^ rdR 2^[/m d^ r 



u ^= —— = — I -— dr ^z I dE. 



dU dUJ d/i «„ dUJ 



Thus in order to get 



u = V. = e.vcentric anomaly, 

 we must take 



dU _2^\/in 

 d^i «0 



