AND ON THE REMOTE HISTORY OE THE EARTH. 
527 
relatively to the planet be slow enough (viz. : the month less than twice the clay), the 
obliquity will diminish. 
This result, taken in conjunction with results given later with regard to the evolu¬ 
tion of satellites, shows that the obliquity of a planet perturbed by a single satellite 
must rise from zero to a maximum and then decrease again to zero. If we regard the 
earth as a satellite of the moon, we see that this must have been the case with the moon. 
Plate 36, fig. 5 (Section 12), contains a similar graphical analysis of the various 
values which may be assumed by the tidal friction. As might be expected, the tidal 
friction always tends to stop the planet’s rotation, unless indeed the satellite's period 
is less than the planet’s day, when the friction is reversed. 
This completes the consideration of the effect on the earth, at any instant, of the 
attraction of the sun and moon on their tides; the next subject is to consider the 
reaction on the disturbing bodies. 
Since the moon is tending to retard the earth’s diurnal rotation, it is obvious that 
the earth must exercise a force on the moon tending to accelerate her linear velocity. 
The effect of this force is to cause her to recede from the earth and to decrease her 
orbital angular velocity. Hence tidal reaction causes a secular retardation of the 
moon’s mean motion. 
The tidal reaction on the sun is shown to have a comparatively small influence on 
the earth’s orbit and is neglected (Sections 14 and 19). 
The influence of tidal reaction on the lunar orbit is determined by finding the dis¬ 
turbing force on the moon tangential to her orbit, in terms of the couples which have 
been already found as perturbing the earth’s rotation ; and hence the tangential force 
is found in terms of the rate of tidal friction and of the rate of change of obliquity. 
It appears that the non-periodic part of the force, on which the secular change in 
the moon’s distance depends, involves the lunar tides alone. 
By the consideration of the effects of the perturbing force on the moon’s motion, an 
equation is found which gives the rate of increase of the square root of the moon’s 
distance, in terms of the heights and retardations of the several lunar tides 
(Section 14). 
Besides the interaction of the two bodies which affects the moon’s mean motion, 
there is another part which affects the plane of the lunar orbit ; but this latter effect 
is less important than the former, and in the present paper is neglected, since the moon 
is throughout supposed to remain in the ecliptic. The investigation of the subject will 
however, lead to interesting results, since a complete solution of the problem of tire 
obliquity of the ecliptic cannot be attained without a simultaneous tracing of the 
secular changes in the plane of the lunar orbit. 
It appears that the influence of the tides, here called slow semi-diurnal and slow 
diurnal, is to increase the moon’s distance from the earth, whilst the influence of the 
fast semi-diurnal, fast diurnal, and fortnightly tide tends to diminish the moon’s dis¬ 
tance ; also the sidereal semi-diurnal and diurnal tides exercise no effects in this 
MDCCCLXXIX. 3 Y 
