AND ON THE REMOTE HISTORY OF THE EARTH. 
511 
condition. If we neglect the obliquity of the ecliptic, the equation (57) of tidal 
reaction, when adapted to the case of a viscous spheroid, becomes 
^ dt 2r > 
— l 
(fan 
sin 4e 
Now it is clear that the rate of tidal reaction can never be greater than when 
sm 4e 1 =l, when the lunar semi-diurnal tide lags by 22-|°. Then since t=~, we shall 
obtain the minimum time by integrating the equation 
Whence 
£^_ 9 ,.§^ 0£12 
*r w 
~t =\f %i-f is ) 
XO 
Now £= , and we have found by the solution of the biquadratic that the initial con¬ 
dition is given by 21'4320 ; also with the present value of the month /2y=4'38, 
the present year being in both cases the unit of time. Hence it follows that £ is very 
nearly '2, and £ 13 may be neglected compared with unity. Thus— t—jz 
-L O 
y (fao 
2" 
Now/x=4'007 and —^ is 86,844,000 years. 
Hence —t=5S, 540,000 years. 
Thus we see that tidal reaction is competent to reduce the system from the initial 
state to the present state in something over 54 million years. 
The rest of the paper is occupied with the consideration of a number of miscellaneous 
points, which it was not convenient to discuss earlier. 
§ 19. The change in the length of year. 
The effects of tidal reaction on the earth’s orbit round the sun have been neglected ; 
I shall now justify that neglect, and show by how much the length of the year may 
have been altered. 
It is easy to show that the moment of momentum of the orbital motion of the moon 
C 
and earth round their common centre of inertia is —, where C is the earth’s moment of 
sll 6 
v 2 
inertia, and s=f 
av 
l\ff 
(i+y) 
The moment of momentum of the earth’s rotation is obviously C n. The normal 
to the lunar orbit is inclined to the earth’s axis at an angle i. Hence the resultant 
moment of momentum of the moon and earth is 
1 
c V+TTkid 
2 n 
(sJ2if ' sSl 
3 U 
i cos ^ 
MDCCCLXXIX. 
