786 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Next consider the equations 
dz 
it= a y-y z > 
dy 
dt 
az yy 
Where « is constant, but y is a very small quantity compared with a, but which 
may vary slowly. 
Treat y as constant, and differentiate, and we have 
^+<A=_y(| + «y 
dt 
dhj (dy 
|+^=- y (n_« ! 
dt 
Then if we neglect y, we have the first approximation 
z=L cos (at- j-m), y=—L sin (a£-f-m) 
Substituting these values for z, y on the right, we have 
dh 0 
df> +a ‘ 
y'2aL sill (a£-j-m) 
And a similar equation for y. 
The solutions are 
2 = L cos (at-\-m )— yLt cos («£+m) 
y— —L sin (at-\-m)-\-yLt sin (a^fi-m) 
From this we see that, if we desire to retain the first approximation as the solution, 
we must have 
1 dL_ 
L dt~~ y 
This will be true if y varies slowly; hence 
L=L Q e~f ydl 
and the solution is 
2= L 0 e~f ydt cos (a£-}-m) 
y= — L Q e~f ydt sin (ai+m) 
It is easy to verify that these are the rigorous solutions of the equations, when a is 
constant but y varies. 
