289 

 If (hen we write Z for G, we find ju once 



The element F now is a function of y, and consequently 

 Ö0 f/y /'oil' ^f^y 



c)0 dy rdli -dy r 



Therefore 



from which 



dF 



-— =: - ]/m, 

 dy 



F z= const. — yVm =^ V^ — y \/ni. 



1*0 



Now, bj (10) we have y = ^^ l/l— «% therefore, with the value 



Of. 



(21) of F„, we find 



1%'^'^ 



To the elements I corresponds the classical development of the 

 perturbative function according to the sines and cosines of multiples 

 of the mean anomalies. The development of 6\according to excentric - 

 anomalies, which is required for the elements II, has been given 

 by Nkwcomb in Vol III of the Astron. Papers of the Am. Eph. For 

 the development in function of true anomalies, which is needed 

 when using the elements III, the foundations have been laid down 

 by Hill in the paper already quoted. 



Ca.^e IV. a = «J, = const., ]3 = /?„ =: const., ff = ö^ = Q. 



The third linear element is a function of x. It will be called M. 

 "We have 



Ö0 d-a 



\ rdR /?/ dx r 



- I ^^ — dr = I du. 



tj dK «„ dMj ^ 



Consequently we must take 



^ dM dM. 



dM _ /?/ 



dx «„ 

 from which 



3 *•>€ 

 M = ^-^— = /?„« l/a =: a,xa (32) 



«n 



The semi major axis a is constant, as it was in case III, and a 

 is variable. The meaning of x is however different from that of v 

 in formula (22\ From (10) we find 



M \/' r-"7 —G (33) 



19 

 Proceedings Royal Acad. Amsterdam. Vol. XVI. 



