AND ON THE REMOTE HISTORY OF THE EARTH. 
519 
effects are evanescent with the obliquity. Since this ring only raises the tides which 
are called sidereal semi-diurnal and sidereal diurnal, and since we see by (57), 
Section 14, that tidal reaction is independent of those tides, it follows that there is 
no tangential force on the ring tending to accelerate its linear motion. If, however, 
the arc of the ring be not of uniform density, there is a slight tendency for the lighter 
parts to gain on the heavier, and the heavier parts become more remote from the 
planet than the lighter. 
§ 23. Double tidal reaction. 
Throughout the whole of this investigation the moon has been supposed to be 
merely an attractive particle, but there can be no doubt but that, if the earth was 
plastic, the moon was so also. To take a simple case, I shall now suppose that both 
the earth and moon are homogeneous viscous spheres revolving round their common 
centre of inertia, and that the moon is rotating on her own axis with an angular 
velocity co, and that their axes are parallel and perpendicular to the plane of their 
orbit. Then the whole of the argument with respect to the earth as disturbed by 
the moon, may be transferred to the case of the moon as disturbed by the earth. 
All symbols which apply to the moon will be distinguished from those which apply 
to the earth by an accent. 
Then from (21) or (43) we have 
7Y=irtSm4ei 
and the equation which gives the lunar tidal friction is 
Now 
and 
So that 
Also 
and therefore 
da) 
dt 
i T • , ' 
4— sm 4e 
~q 
1 
vm* 
w'a,' 3 
T 
2 y v/ 
5 a w 
r ' 2 / wa? \ 3 t 2 
g' \w'a' 2 ) g 
C / _wV 5 
C w « 5 
iu 2 a 
w' 2 a' 
sin 4e\ 
3 x 
(89) 
(90) 
MDCCCLXXIX. 
