AND ON THE REMOTE HISTORY OF THE EARTH. 
525 
the equator which is always coincident with the nodes : this affects the precession ; 
third, there is a couple about the earth’s axis of rotation, and this affects the length of 
the day (Sections 3, 4, and 5). All these couples vary as the fourth power of the moon’s 
orbital angular velocity, or as the inverse sixth power of her distance. 
These three couples give the alteration in the precession due to the tidal movement, 
the rate of increase of obliquity, and the rate at which the diurnal rotation is being 
diminished, or in other words the tidal friction. The change of obliquity is in reality 
due to tidal friction, but it is convenient to retain the term specially for the change of 
rotation alone. 
It appears that if the bodily tides do not lag, which would be the case if the earth 
were perfectly fluid or perfectly elastic, then there is no alteration in the obliquity, nor 
any tidal friction (Section 7). The alteration in the precession is a very small fraction 
of the precession due to the earth considered as a rigid oblate spheroid. I have some 
doubts as to whether this result is properly applicable to the case of a perfectly fluid 
spheroid. At any rate, Sir William Thomson has stated, in agreement with this 
result, that a perfectly fluid spheroid has a precession scarcely differing from that of a 
perfectly rigid one. Moreover, the criterion which he gives of the negligeability of the 
additional terms in the precession in a closely analogous problem appears to be almost 
identical with that found by me (Section 7). I am not aware that the investigation on 
which his statement is founded has ever been published. The alteration in the pre¬ 
cession being insignificant, no more reference will be made to it. This concludes the 
analytical investigation as far as concerns the effects on the disturbed spheroid, where 
there is only one disturbing body. 
The sun is now (Section 8) introduced as a second disturbing body. Its independent 
effect on the earth may be determined at once by analogy with the effect of the moon. 
But the sun attracts the tides raised by the moon, and vice versd. Now notwith¬ 
standing that the periods of the sun and moon about the earth have no common 
multiple, yet the interaction is such as to produce a secular alteration in the position 
of the earth’s axis and in the angular velocity of its diurnal rotation. A physical 
explanation of this curious result is given in the note to Section 8. I have dis¬ 
tinguished this from the separate effect of each disturbing body, as a combined effect. 
The combined effects are represented by two terms in the tide-generating potential, 
one of which goes through its period in 12 sidereal hours, and the other in a sidereal 
day'"'; the latter being much more important than the former for moderate obliquities 
to the ecliptic. Both these terms vanish when the earth’s axis is perpendicular to the 
plane of the orbit. 
As far as concerns the combined effects, the disturbing bodies may be conceived to be 
* These combined effects depend on the tides which are designated as K 2 and K 3 in the British Asso¬ 
ciation’s Report on Tides for 1872 and 1876, and which I have called the sidereal semi-diurnal and 
diurnal tides. For a general explanation of this result see the abstract of this paper in the ‘ Proceedings 
of the Royal Society,’ No. 191, 1878. 
