518 
MR. G. H. DARWIN ON THE 
The former of these two suppositions seems more likely to be near the truth than 
the latter. 
Now -|(1 — 677~ 3 j='26138, so that C may lie between ("26138)J/a 2 and ('4 )Ma?. 
The sun’s apparent radius is 961"'82, therefore the unit of distance being the present 
distance of the earth from the sun, a=961'827r/648,000 ; also 714 = 315,511. 
Lastly the sun’s period of rotation is about 25 - 38 m.s. days, so that w=27r/25‘38. 
Combining these numerical values I find that On (the solar rotational momentum) 
may lie between ‘444 and '679. The former of these values seems however likely to be 
far nearer the truth than the latter. 
It follows therefore that the total orbital momentum of the planetary system, found 
above to be 22, is about 50 times that of the solar rotation. 
motion, the rotation of the homogeneous planet had to be made identical with that of the real earth. A 
consequence of this is that the rotational momentum of the earth in my problem bore a larger ratio to the 
orbital momentum of the moon than is the case in reality. Since the consequence of tidal friction is to 
transfer momentum from one part of the system to the other, this treatment somewhat vitiated subsequent 
results, although not to such an extent as could make any important difference in a speculative investiga¬ 
tion of that kind. 
If it had occurred to me, however, it would have been just as easy to have replaced the actual hetero¬ 
geneous earth by a homogeneous planet mechanically equivalent thereto. The mechanical equivalence 
referred to lies in the identity of mass, moment of inertia, and rotation between the homogeneous 
substitute and the real earth. These identities of course involve identity of rotational momentum and of 
rotational energy, and, as will be seen presently, other identities are approximately satisfied at the same 
time. 
Suppose that roman letters apply to the real earth and italic letters to the homogeneons substitute. 
By Laplace’s theory of the earth’s figure, with Thomson and Tait’s notation (‘ Nat. Phil.,’ § 824) 
where / is the ratio of mean to surface density, and 0 is a certain angle. 
Also G=fAfu 2 ( 1 + 
where e is the ellipticity of the homogeneous planet’s figure. 
Then by the above conditions of mechanical identity 
whence 
M= M and C = G 
Now put m=n 2 a/g, m—n^ajg \ where 
condition gives n =n. 
Therefore 
But 
>■, cj are mean pure gravity in the two cases. 
Then the remaining 
Hence 
l-4m<® 
h- Q AL-m 
1 /e* f 
This is an equation which gives the radius of the homogeneous substituted planet in terms of that of 
