THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
785 
The terms on the right-hand side of these equations are small, because they involve 
a, and therefore we may substitute in them from the first approximation. 
Hence 
dh 
dt 
—u L sin (a^ + m) — 2 ctcUL cos (ai-f-m) 
and a similar equation for y. 
The solution of this equation is 
2 =Z cos cos cos —~Lt 2 sin (a£-{-m) 
The terms depending on t cut one another out, and 
/ 
z— L cos (a£— (— m) —-Zri sin (a£-f-m) 
Similarly we should find 
/ 
y——L sin (at-\-m) — -Lt 2 cos (a^ + rn) 
The terms in t 2 are obviously equivalent to a change in m, the phase of the oscilla¬ 
tion; but the amplitude L is unaffected. We might have arrived at this conclusion 
about the amplitude if, in solving the differential equations, we had neglected in the 
solutions the terms depending on t 2 , as will be done in considering our equations 
below. In those equations, however, we shall not find that the terms in t annihilate 
one another, and thus there will be a change of amplitude. 
That this conclusion concerning amplitude is correct, may be seen from the fact that 
the rigorous solution of the equations 
dz dy 
is 
z=L cos (Jacfc + m 0 ), y— — L sin (\u.dt- j-m 0 ) 
= L cos (a£-|-m 0 —f a'tdt), = —L sin (a£+m 0 — \atdt) 
Whence L is unaffected, w T hilst 
m=m 0 —| a'tdt 
dm da. 
~dt~ t ~dt 
So that 
