808 
MR. G. H. DARWIN - ON THE SECULAR CHANGES IN 
From (244), (237), and (232) we have 
bG - aD =(f i)(fJ(? 
and similarly 
2^1 + ^ sin 2g — 2 sin 2g 1 
sin 4f x 
(246) 
• (247) 
§ 17. Change of independent variable, and formation of equations for integration. 
In the equations (224) the time t is the independent variable, but in order to 
integrate we shall require £ to be the variable. It has been shown above that these 
equations are equivalent to only two of them; henceforth therefore we shall only 
consider the first and last of them. It will also serve to keep before us the physical 
meaning of the L’s, if the notation be changed ; the following notation (which has 
been already used in (127)) will be adopted : — 
J = L y = the inclination of the lunar orbit to the lunar proper plane. 
1 = Ld— the inclination of the earth’s proper plane to the ecliptic. 
I 7 = L\ — the inclination of the equator to the earth’s proper plane. 
d = — L. : = the inclination of the lunar proper plane to the ecliptic. 
Then since J, I, &c., are small, we may write 
—f — d. log tan -gj, — d. log tan ^1 .(248) 
-m As 
This particular transformation is chosen because in Part II., where j and i were not 
small, dj/sin j seemed to arise naturally. 
Also since 
Ly AT-^ -J - OL dj^ Kc, “T CL 
L x a ’ L, a 
we have 
• T _ 4“ CL , y 
sm I = — —— sm J 
' a 
sin J = - sin I 
/Co + CL 
(249) 
These equations will give I / and J), when J and I are found. 
Now suppose we divide the first and last of (224) by df/nkdt, then their left-hand 
sides may be written 
d 
d 
