THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
793 
By means of these two rules we see that the solutions of the two alternative 
differential equations 
D 4 z=A x |' //l + ^B 1 |^, 1 + the same with 2 for 1 .(200) 
are, so long as t is very small, 
thy 
1 
K 
(Bij 
Ifl 
Ci 
2*i , 1 
2 *i(*i 3 
~ K d) 
2 
-* 2 2 ) 
O - 0 1 o 
[_ K \ — 
+ the same with 2 and 1 interchanged (201) 
Then putting for z 1} £ x , &c., their values from (182), these solutions may be written, 
1 
2=cos (#c 1 «+mi)'j L x -\ 
L 
, tA \d 
1 2/c 1 (a: 1 2 — k 2 ) 
2/c 1 (/c 1 2 —/c 2 2 ) 
2*i 1 
lAi 2 ' 2 «i_ 
-f the same with 2 for 1 (202) 
Hence we may retain the first approximation 
z—L x cos (/cB+ m i) + Z 3 cos (/c 3 ^-|-m 3 ) 
as the solution, provided that L v and Z 3 are no longer constant, but vary in such a 
way that 
K 
A 
A' 
A 
A 
A 7 
dL x _ 
dt 2/cfi/q 2 —Kn 2 ) 2/c 1 (/<: 1 2 — /c 2 3 ) 
and a similar equation for L 3 
1 ' 
2k 1 
J 
(203) 
It will be found, when we come to apply these results, that the solution of the 
equation for D 4 y will lead to the same equations for the variation of L x and L 2 as are 
derived from the equation for D l z. 
A similar treatment may be applied to the equations for D 4 '£ or D i rj, and we find 
similar differential equations for dLy/dt and dLd/dt. 
These equations will be the differential equations for the secular changes in L 2 and 
L. 2 ', which are the constants of integration in the first approximation. 
We will now apply these theorems to the differential equations (181); but as the 
analysis is rather complex, it will be more convenient to treat the variations of a, a, 
/:!, b and the terms in y, g, 8, cl independently. 
We will indicate by the symbol A the additional terms which arise, and will write 
