AND ON THE REMOTE HISTORY OF THE EARTH. 
477 
the tides in the earth, there would be an apparent acceleration of the moon in a 
century of 
1043"-28 E sin 2e-\-232"'50E' sin e'.(48) 
for the moon moves over 0"‘5490 of her orbit in one second of time. 
This apparent acceleration would however be considerably diminished by the effects 
of tidal reaction on the moon, which will now be considered. 
§ 14. Tidal reaction on the moon."' 
The action of the tides on the moon gives rise to a small force tangential to the 
orbit accelerating her linear motion. The spiral described by the moon about the 
earth will differ insensibly from a circle, and therefore we may assume throughout 
that the centrifugal force of the earth’s and moon’s orbital motion round their common 
centre of inertia is equal and opposite to the attraction between them. 
We shall now find the tangential force on the moon in terms of the couples which 
we have already found acting on the earth. Those couples consist of the sum of three 
parts, viz.: that due (i) to the moon alone, (ii) to the sun alone, and (iii) to the action 
of the sun on the lunar tides and of the moon on the solar tides, the latter two being- 
equal inter se. 
Now since action and reaction are equal and opposite, therefore the only parts of 
these couples which correspond with the tangential force on the moon are those which 
arise from (i), and one-half those which arise from (iii). 
We may thus leave the sun out of account if we suppose the earth only to be acted 
on by the couples JL.+iifL*,; these couples will be 
called IL', M, 0, and the part of the change of obliquity which is due to 11', JFV 
will be called yh 
at 
Let r and — fl be the moon’s distance, and angular velocity at any time, and v the 
ratio of the earth’s mass to the moon’s. 
Let T be the force which acts on the moon perpendicular to her radius vector, in 
the direction of her motion. 
From the equality of action and reaction, it follows that Tr must be equal to the 
couple which is produced by the moon’s action on the tides in the earth, acting in the 
direction tending to retard the earth’s diurnal rotation about the normal to the ecliptic. 
Referring to Plate 36, fig. 1, we see that the direction cosines of this normal are 
— sin i cos n, — sin i sin n, cos i ; hence 
Tr= — sin i(l L' COS sin + cos i. 
* This section has been partly rewritten and rearranged since the paper was presented. (Dec. 19,1878.) 
