AND ON THE REMOTE HISTORY OF THE EARTH. 
521 
long time be greater than v, because the moon being a smaller body must have 
stiffened quicker than the earth. Hence unless cj—f2 is very much less than n — fl, 
e'j must be larger than e x . Therefore if in the early stages of development the earth 
had a small viscosity, it is probable that the effects of the moon’s tides on her own 
orbit must have had a much more important influence than had the tides in the earth. 
I shall now show, however, that this state of things must probably have had so short 
a duration as not to seriously affect the investigation of this paper. By (89) and (90) 
we have, as the equation which determines the rate of tidal friction reducing the 
moon’s rotation round her axis, 
da> 
dt 
pVrf\3 
2 g yiv'a'-) 
sin 4e 1 
Now (—rf =12,148 ; and hence, for the same values of e', and e,, the moon’s rotation 
\wa l J 
round her axis is reduced 12,000 times as rapidly as that of the earth round its axis, 
and therefore in a very short period the moon’s rotation round her axis must have 
been reduced to a sensible identity with the orbital motion. As a> becomes very 
nearly equal to ft, sin 4e\ becomes very small. Hence the term in the equation of tidal 
reaction dependent on the moon’s own tides must have become rapidly evanescent. 
Now while this shows that the main body of our investigation is unaffected by the 
lunar tide, there is one slight modification of them to which it leads. 
In Section 18 we traced back the moon to the initial condition, when her centre 
was 10,000 miles from the earth’s centre. If lunar tidal friction had been included, 
this distance would have been increased; for the coefficient of x in the biquadratic 
vj ci! 7 * w a'° 
(viz.: 11,727) would have to be diminished by r(<y — oj 0 ). Now -—r is very nearly 
ToVoth, and the unit of time being the year, it follows that we should have to suppose 
an enormously rapid primitive rotation of the moon round her axis, to make any 
sensible difference in the configuration of the two bodies when her centre of inertia 
moved as though rigidly connected with the earth’s surface. 
The supposition of two viscous globes moving orbitally round their common centre 
of inertia, and one having a congruent and the other an incongruent axial rotation, 
would lead to some very curious results. 
§ 24. Secular contraction of the earth A 
If the earth be contracting as it cools, it follows, from the ■ principle of conservation 
of moment of momentum, that the angular velocity of rotation is being increased. Sir 
William Thomson has, however, shown that the contraction (which probably now 
only takes place in the superficial strata) cannot be sufficiently rapid to perceptibly 
counteract the influence of tidal friction at the present time. 
* Rewritten in July, 1879. 
3x2 
