THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
811 
The integrations will be carried out by the method of quadratures, and the process 
will be divided into a series of “periods of integration,” as explained in § 15 and 
§ 17 of the paper on “Precession.” These periods will be the same as those in that 
paper, and the previous numerical work will be used as far as possible. It will be 
found, however, that it is not sufficiently accurate to assume the uniform revolu¬ 
tion of the nodes beyond the first two periods of integration. For these first two 
periods the equations of § 7, Part II., will be used ; but for the further retrospect we 
shall have to make the transition to the methods of Part III. It is important to defer 
the transition as long as possible, because Part III. assumes the smallness of i and j, 
whilst Part II. does not do so. 
By (104) and (86) of Part II . we have, when j' = 0, and fl'/n is neglected, 
d% sin 4 f, . . • f" o / q • .-> • \ i '.1 2 42 0 . . ,, \ 
df =I Pa ^ sm * cos «[nl-fsin“j) + T' —t sec i cos j-TT (1—f sin- j) 
dn sin 4f 
dt no 
(1-1 sin 3 v;)(t 3 +t , 2 )-1(I — f sin 3 7)r 3 sin 3 p 
,42 
— r 3 — cos i cos sin 3 7(1—f sin 3 j’) 
If we put 1— \ sin 3 i— cos i, 1 — § sin 3 j/‘=cos 3 p, and neglect sin 3 i sin 3 j, these may 
be written 
di sin 4f. . . . „ . f n . , 0 « . , 2/2 , . . .1 P 
- =-+ sm i cos ^ cos* r- + t ~ sec- i~tt — —r : - sec i sec -1 ' 
dt na * \ n J 
L 
dn sin 4f 
dt wci 
cos i cos j\ r 3 + t' 2, sec j — t~- +-§tt' sin i tan i cos ~ 2 j 
(256) 
If we treat sec j and cos j as unity in the small terms in r 3 , tt , and fl/n, (256) 
only differ from (83) of “Precession” in that difdt has a factor cos 3 j and dn/dt has 
a fact or cos j. 
Again by (64) and (70) 
1 dj 1 sin 4f , . 
— 7 — d - T ~ i cos 1 sm J 
k dt % cc * J 
1 dt; sin 4f „ i . ., 42 
7 77 =-T- -k COS l COS 7(1- 
k dt g 3 J v n 
If we divide the second of (256) by the second of (257) we get an equation for 
dn/d£, which only differs from (84) of “ Precession” in the presence of sec j in place of 
unity in certain of the small terms. Now j is small for the lunar orbit; hence the 
equation (88) of “Precession” for the conservation of moment of momentum is very 
nearly true. The equation is, with present notation, 
1 
_[•••• ( 257 ) 
i sec j) ! 
