AND ON THE REMOTE HISTORY OF THE EARTH. 
515 
Now according to the theory of viscous tides/'' 
tan of— 
2-43 + 1 (3 m ) _ 2 
gwa 
v= 
“19 
where v is the coefficient of viscosity. 
But throughout the previous work we have written p 
2 gwa 
19v‘ 
Hence tan Bf= yf—, and similarly tan f =ff-. 
AT „• 2 CO 
Also tan 2e,=—. 
P 
I will now consider two cases :— 
1st. Suppose the viscosity to be small, then ff', e x are all small, and 
sin 6/ _ tan 3/ _ 2 2 3 sin 2/ _ tan f 2 2 , 
sinTej tan 2e x * 9 2, sin4e 1 131126! 19 2 
Therefore 
i®i 19 W 
2nd. Suppose the viscosity very great, then 3 ff', 2e 1 are very nearly equal to 
y, and tan ( y — 3/) = y§-~-, tan ( ~—f i = Mn fan (~ — 2e ) = ./“, so that we have approxi¬ 
mately 
and similarly 
So that 
sin 6/ sin (tt-6/) __ t 9 2 
sin 4e x sin (7r—4e x ) 22 3 
sin 2f' — i 9 y o 
sin46 1 _22XZ 
^=(f)*xH(¥+2)=« 
Hence it follows that the terms of the second order may bear a ratio to those of the 
first order lying between 
2 2 
1 9 
-) , or 1T6 
, and 
1 9 
33 
or -576 (- 
Now at the end of the fourth period of integration in the solution of Section 15, - or 
the moon’s distance in earth’s mean radii was 9 ; hence the terms of the second 
order in the equation of tidal friction must at that epoch lie in magnitude between 
yoth and -f 4 i s f °f those of the first order. It follows, therefore, that even at that 
stage, when the moon is comparatively near the earth, the effect of the tides of the 
second order (i.e., of the third degree of harmonics) is insignificant, and the neglect of 
them is justified. 
In the case of those terms of this order, which affect the obliquity, a very similar 
relationship to the terms of the lower order would be found to hold good. 
* “ Bodily Tides,” &c., Phil. Trans., 1879, Part I., Section 5. 
