AND ON THE REMOTE HISTORY OF THE EARTH. 
507 * 
Throughout all the preceding investigations, the periodic inequalities have been 
neglected. Now a full development of the couples H, JFT, Jl, which are due to the 
tides, shows that there occur terms of speeds n — 2/2, and n— 4/2 in the first two, and 
of speeds 2/2 and 4/2 in the last. The terms in n — 2/2 in It and Jkt will clearly 
give rise to an increasing nutation at the critical point which we are considering, 
but they will be so very much smaller than those arising out of the attraction on 
the permanent equatorial protuberance that they may be neglected. The terms in 
n — 4/2 are multiplied by very small quantities, and I think it may safely be assumed 
that the system would pass through the critical phase where + n = 4/2 with sufficient 
rapidity to prevent the nutation becoming large. 
If we were to go to higher orders of approximation in the disturbing forces, it is 
clear that we should meet with an infinite number of critical phases, but the coefficients 
representing the amplitudes of the resulting nutations would be multiplied by such 
small quantities that they may safely be neglected. 
§ 18. The initial condition of the earth and moon.''' 
It is now supposed that, when the earth’s rotation has been tracked back to where 
it is equal to twice the moon’s orbital motion, the obliquity to the plane of the lunar 
orbit has become zero. Then it is clear that, as long as there is any relative motion 
of the earth and moon, the tidal friction and reaction must continue to exist, and n 
and /2 must tend to an equality. The previous investigation shows also that for small 
viscosity, however nearly n approaches /2, the position of zero obliquity is dynamically 
stable. 
As n is approaching /2, the changes must have taken place more and more slowly in 
time. For if the earth was a cooling spheroid, it is unreasonable to suppose that the 
process of becoming less stiff in consistency (which has hitherto been supposed to be 
taking place, as we go backwards in time) could ever have been reversed ; and if it 
were not reversed, then the lunar tides must have lagged by less and less, as more and 
more time was given by the slow relative motion of the two bodies for the moon’s 
attraction to have its full effect. Hence the effects of the sun’s attraction must again 
become sensible, after passing through a phase of insensibility—a phase perhaps short 
in time, but fertile in changes in the system. I shall not here make the attempt to 
trace the reappearance of these solar terms. 
It is, however, possible to make a rough investigation of what must have been the 
initial state from which the earth and moon started the course of development, which 
has been tracked back thus far. To do this, it is only necessary to consider the equa¬ 
tion of conservation of moment of momentum. 
* For further consideration of this subject, see a paper on the “ Secular Effects of Tidal Friction,” 
‘ Proc. Roy. Soc.,’ No. 197, 1879. The arithmetic of this section has been recomputed since the paper 
was presented. 
