AND ON THE REMOTE HISTORY OF THE EARTH. 
523 
of fl as given in that section must be multiplied by ; and the periodic time must be 
C 
multiplied by —. But in this initial state C is greater than O 0 ; hence the periodic 
'-'o 
time when the two bodies move round as a rigid body is longer, and the moon is 
more distant from the earth, if the earth has sensibly contracted since this initial 
configuration. 
If, then, the theory here developed of the history of the moon is the true one, as I 
believe it is, it follows that the earth cannot have contracted since tins initial state by 
so much as to considerably diminish the effects of tidal friction, and it follows that 
Sir William Thomson’s result as to the present unimportance of the contraction must 
have always been true. 
If the moon once formed a part of the earth w r e should expect to trace the changes 
back until the two bodies were in actual contact. But it is obvious that the data at 
our disposal are not of sufficient accuracy, and the equations to be solved are so com- 
plicated, that it is not to be expected that we should find a closer accordance, than has 
been found, between the results of computation and the result to be expected, if the 
moon was really once a part of the earth. 
It appears to me, therefore, that the present considerations only negative the 
hypothesis of any large contraction of the earth since the moon has existed. 
PART III. 
Summary and discussion of results .* 
The general object of the earlier or preparatory part of the paper is sufficiently 
explained in the introductory remarks. 
The earth is treated as a homogeneous spheroid, and in what follows, except where 
otherwise expressly stated, the matter of which it is formed is supposed to be purely 
viscous. The word “ earth ” is thus an abbreviation of the expression “ a homogeneous 
rotating viscous spheroid also wherever numerical values are given they are taken 
from the radius, mean density, mass, &c., of the earth. 
The case is considered first of the action of one tide-raising body, namely, the moon. 
To simplify the problem the moon is supposed to move in a circular orbit in the 
ecliptict—that plane being the average position of the lunar orbit with respect to the 
* This part has been altered in accordance with the several additions and alterations occurring above. 
The results of subsequent investigations have modified the interpretation to be put on several of the results 
here obtained. I have, moreover, had the advantage of discussing several points with Sir William 
Thomson. —July 9, 1879. 
j The effect of neglecting the eccentricity of the moon’s orbit is, that we underestimate the efficiency 
of the tidal effects. Those effects vary as the inverse sixth power of r the radius vector, and if T be the 
