764 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
lunar orbit Iras been found, on the assumption that the nodes of the lunar orbit rotate 
uniformly. 
It is intended to trace the effects of tidal friction on the earth and moon retro¬ 
spectively. In the course of the solution the importance of the solar perturbation of 
the moon, relatively to the influence of the earth’s oblateness, will wane; the nodes 
will cease to revolve uniformly, and the inclination of the lunar orbit and of the equator 
to the ecliptic will be subject to nutation. The differential equations of Part II. will 
then cease to be applicable, and new ones will have to be found. 
The problem is one of such complication, that I have thought it advisable only to 
attempt to obtain a solution on the hypothesis of the smallness both of the obliquity 
and of the inclination of the orbit to the plane of reference or the ecliptic. It seems 
best however to give the preceding investigation, although it is more accurate than 
the solution subsequently used.* 
The first step towards this further consideration is to obtain a clear idea of the 
nature of the motions represented by the analytical solutions (118) or (119) of the 
present problem. 
Assuming then i and j to be small, we have from (112) and (115) 
a 
=oqfi- a. 2 , a= cq, /3=b 1 J r b i , b=b ] . 
( 121 ) 
j cos N=L 1 cos (fcg-bmg+Xo cos (Kg-f-m 2 )~l 
j sin A r =X 1 sin (/cg+nq) +X 3 sin (/cg-j-m 2 ) ^ 
i cos xp — Ly cos (/cg + nq)d-X/ cos (/cg+m 2 ) j 
i sin xp —Ly sin (/cg+nq) +Xg sin (/cg + im) J 
( 122 ) 
Take a set of rectangular axes ; let the axis of x' pass through the fixed point in the 
ecliptic from which longitudes are measured, let the axis of z be drawn perpendicular 
to the ecliptic northwards, and let the rotation from x to y' be positive, and therefore 
consentaneous with the moon’s orbital motion. 
Then N is the longitude of the ascending node of the lunar orbit, and therefore the 
direction cosines of the normal to the lunar orbit drawn northwards are, 
sing cos (N— \tt), sin j sin (N—\tt), cos j ; or since j is small, j sin N, —j cos N, 1. 
And xp is the longitude of the descending node of the equator, and therefore the 
direction cosines of the earth’s axis, drawn northwards are, 
sin i cos (xp-\-^7r), sin i sin (p+thr), cos i ; or since i is small, —i sin ip, i cos xp, 1. 
Now draw a sphere of unit radius, with the origin as centre; draw a tangent plane 
* See the foot-note to § 18 for a comparison of these results with those ordinarily given. 
