836 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
motion gets continually less and less. Meanwhile, the earth’s orbital motion round the 
sun is continually increasing, and the distance from the sun decreasing retrospectively. 
Theoretically this would go on until the sun and moon (treated as particles) revolve 
as though rigidly connected with the earth and with one another. This is the con¬ 
figuration of maximum energy of the system. 
The solution is physically absurd, because the distance of the two bodies from the 
earth would then be very much less than the earth’s radius, and d fortiori than the 
sun’s radius. 
It must be observed, however, that in the retrospect the relative motion of the moon 
and earth would already have become almost insensible, before the earth’s distance 
from the sun could be sensibly affected. 
V. 
SECULAR CHANGES IN THE ECCENTRICITY OF THE ORBIT. 
§ 23. Formation of the disturbing function. 
We will now consider the rate of change in the eccentricity and mean distance of the 
orbit of a satellite, moving in an elliptic orbit, but always remaining in a fixed plane, 
namely, the ecliptic; and the rate of change of the obliquity of the planet’s equator 
when perturbed by such a satellite will also be found. 
Up to the end of Part I. the investigation for the formation of the disturbing 
function was quite general, and we therefore resume the thread at that point. 
In the present problem the inclination of the satellite’s orbit to the ecliptic is zero, 
and we have 
7 z=zr=P= cos \i, k=k— Q= sin \i 
We thus get rid of the zs and k functions, and henceforth zz will indicate the 
longitude of the perigee. 
Then by equations (24-8), 
Mp —M 2 3 =P 4 cos 2 (y— 6)-f2P 2 Q' 2 cos 2 y+(fi ! cos 2 ( x -f -0) 
— 2 M x M 2 =: The same with sines for cosines 
M 2 M 3 = -P*Q cos ( X - 2 0) + PQ{P*-Q*) cos X + PQ* cos ( X + 2 d) 
M 1 M 3 = The same with sines for cosines 
i-M 3 3 =i(P^-4P 3 g 2 +Q 4 ) + 2P 2 $ 3 cos 2(9 
