140 



MEMOIRS NATIONAL ACADEMY OF SCIENCES. 



[Vol. XIV. 



Logarithmic. 



Table XLIII — Continued. 



V 



Unit= 1" 



T^^Iw^TjPrj'gftCi COS Arg. + (a-6(i)IuSTiPrj'QftC2 sin ATg. + (0-d„Y:^tv^vPv'qf'C3 COS Alg. 

 where C„ Cj, C3 represent the respective coefBcients. 



PERTURBATIONS OF THE THIRD COORDINATE. 



For the third coordinate the developments are limited to perturbations of the first order 

 and of the first degree with the exception of some secular terms of second degree. We can 

 therefore use osculating elements in this section, and use 6 and i? without distinction. 



By Z 8 eq. (39), 41, eq. (83) and 8, eq. (41) the equations Z 115, (192) are given, in which 

 I is defined. 



Since 



dS^SS dS dd^^ 

 de diie de " 



By Z 9, eq. (45) we have, with sufficient accuracy, Z 115, eqs. (193). Within these limits, 



dd w,, , 



_=-(l_,cosO. 



Substituting this relation in the above equation and in eq. (192) in turn, the differential equation 

 to be integrated is (194). 



Since F, G, H are power series in w, it is evident from eqs. (192) that 



g = i-, + 2> + J,w^+ 



where 



-^i ~ ^ i.p.q 'T'U.p.q '' ^^i.p.q 



