THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
787 
The equation (187) gives the rate of change of amplitude of oscillation. 
The cases which we have now considered, by the method of variation of parameters, 
are closely analogous to those to be treated below, and have been treated in the same 
way, so that the reader will be able to trace the process. 
They are in fact more than simply analogous, for they are what our equations 
(181) become if the obliquity of the ecliptic be zero and £= 0 , 77 = 0 . In this case 
L=j, and dj/dt = —jy- 
This shows that the secular change of figure of the earth, and the secular changes 
in the rate of revolution of the moon’s nodes do not affect the rate of alteration of the 
inclination of the lunar orbit to the ecliptic, so long as the obliquity is zero. This last 
result contains the implicit assumption that the perturbing influence of the moon on 
the earth is not so large, but that the obliquity of the equator may always remain 
small, however the lunar nodes vary. In an exactly similar manner we may show that, 
if the inclination of the lunar orbit be zero, di/dt—iS. 
This is the result of the previous paper “On the Precession of a Viscous Spheroid,” 
when the obliquity is small. 
According to the method which has been sketched, the equations to be integrated 
are given in (l81), when we write a-\-at for a, afl-aT for a, for /3, b-fb'£ for b, 
and then treat a, a, &c., a , a', &c., y, g, &c., as constants. 
Before proceeding to consider the equations, it will be convenient to find certain 
relations between the quantities a, a, &c., and the two roots /q and k. 2 of the quadratic 
(k+ a) (/c-ff /3) = ab. 
We have supposed the two roots to be such that 
Kl + K 2=— a — P 
*q — k . 2 = — + 4ab 
Then 
K l K 2 =(a l Q —ab) . 
(188) 
(189) 
/q 3 fl- k 2 — a 2 -f /3 2 fl- 2ab 
/q 2 K 3 2 =(a 2 +ab)(/3 3 -f-ab) — ab(a-|-/3) 3 
/p + ab — *q S =0q + K 3 )(#Cj + a) 
/3 3 +ab — k 2 — (/q + k 2 ) (/q fl- a) I 
a 3 -fab — K 1 2 =(K 1 +/q)(/q+/3) | 
a 2 -f ab — k 2 2 =(k 1 -\- K 3 )(#q-f ft) J 
(190) 
( 191 ) 
