834 
ME. G. H. DARWIN ON THE SECULAR CHANGES IN 
§ 22. The effects of solar tidal friction on the 'primitive condition of the earth 
and moon. 
In the paper on “Precession,” §16, I found, by the solution of a biquadratic 
equation, the primitive condition in which the earth and moon moved round together 
as a rigid body. 
Since writing that paper certain additional considerations have occurred to me, 
which seem to be important in regard to the origin of the moon. 
It was there remarked that, as we approach that critical condition of dynamical 
instability, the effects of solar tidal friction must have become sensible, because of the 
slow relative motion of the moon and earth. I did not at that tune perceive the full 
significance of this, and I will now consider it further. 
Suppose the moon to be moving orbitaliy nearly as fast as the earth rotates. Then 
the tidal reaction, which depends on the lunar tides alone, must be very small, and 
therefore the moon’s orbital motion increases retrospectively very slowly. On the 
other hand, the relative motion of the earth and sun is great, and therefore if we 
approach the critical condition close enough, the solar tidal friction must have been 
greater than the lunar, however great the viscosity of the planet. The manner in 
which this will affect the solution of the previous paper may be shown analytically as 
follows. 
If we neglect the obliquity, and divide the equation of tidal friction by that of tidal 
leaction, and suppose the viscosity small, we have from (176) 
= 1 + 
/ r '\ 2 n 
\ t ) n — fl 
t '\ 2 n 
t / n — fl 
Then integrating we have 
n=n Q +- 
(W)+tMu ) (i-P) 
If we do not carry the integration to near the critical phase, where n is equal to fl, 
the last integral is small, but it tends to become large as n becomes nearly equal to fl; 
it has always been neglected in our integration. When however we wish to apply 
this equation to find the values for which n is equal to fl, it cannot be neglected. 
Suppose the integral to be equal to Iv. Then in the first part of the above expres¬ 
sion we may put n=fl=x z and we may neglect fzf'/rfflf — £ 13 ). Hence the equation 
for finding the angular velocity of the two bodies at the critical phase, when n = fl, is 
