THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
783 
dz 
dt 
ay+^ — iyZ+gQ 
%= - (« 2 +aC) - (yy+gy) 
j= fiy+by+SC+dz 
y 
C ^=-{PC+bz)+Sy+dy 
(181) 
If the viscosity be not small we have r, G, A, D in place of y, g, 8, d. As it is 
more convenient to write small letters than capitals, in the whole of the next section 
the small letters will be employed, although the same investigation would be equally 
applicable with T, G, &c., in place of y, g, &c. 
The terms in y, g, 8, d are small compared with those in a, a, /3, b, and may be 
neglected as a first approximation. Also a, a, /3, b vary slowly in consequence of tidal 
reaction, tidal friction, and the consequent change of ellipticity of the earth, but as a 
first approximation they may be treated as constant. 
Then if we put 
z l —L l cos (/qi + nq), 
yi = A sin (/qi-j-nq), 
C\ = L\ cos (qG m i), 
Vi=Li sin (fqf + nq), 
z 2 =L. 2 cos (/c^+mn) 
2/a =L Z sin (^ + m 2 ) 
£,=A/cos (/<V+m 2 ) 
V 2= L 2 sin ( K y+ m 2 ) 
By (122) or (118) the first approximation is 
(182) 
z=z 1 +z 2 , y= Vl +y 2 , C=Ci-b t 2 , V=Vi+V 2 
where 
L x fCy + cl Id JO/ ~j~ cl b 
L y a /c 1 + /3 5 L 2 a + 
Before considering the secular changes in the constants L of integration, it will be 
convenient to take one other step. 
The equation of tidal friction (173) may be written approximately 
dn 
dt 
t 2 + t ' 2 
* S 
sin 4f x 
(181) 
because sin 4f will be nearly equal to sin 4^ as long as t ' 2 is not small compared 
with t 2 . (See however § 22, Part IV.) 
5 h 2 
