AND ON THE REMOTE HISTORY OF THE EARTH. 
475 
The computation of the ordinates was done by Crelle’s three-figure multiplication 
table, and thus the figure does not profess to be very rigorously exact. 
This family of curves differs much from the preceding one. For moderate 
obliquities there is no degree of viscosity which tends to make the obliquity diminish, 
and thus there is no position of dynamically unstable equilibrium of the system 
except that of zero obliquity. Thus we see that the decrease of obliquity for small 
obliquities and large viscosities in the previous case was due to the attraction of the 
sun on the lunar tides and the moon on the solar tides. 
In the present case the position of zero obliquity is never stable, as it was before. 
The dynamically stable position at a large obliquity still remains as before, but in 
consequence of the largeness of the ratio fl -An (yth instead of yytli), this obliquity of 
dynamical stability is not nearly so great as in the previous case. As the ratio fl~n 
increases, the position of dynamical stability is one of smaller and smaller obliquity, 
until when fl-~n is equal to a half, zero obliquity becomes stable,—as we shall see 
later on. 
§ 12. Rate of tidal friction when the earth is viscous. 
If in the same way the equations (37) and (38) be applied to the case where the 
earth is purely viscous, when the semi-diurnal and diurnal tides are grouped together, 
we have 
jl£) = ( u ~+ u ,~)lU C0S ~ i+t'sm' 1 ' i) sin 4e+lshr i( 1-f 
-\-uu[^ sin 4 ' i sin 4e + ^ sin 3 i cos 3 i sin 2e'] 
sin 3 i) sin 2e'] 
(43) 
Plate 36, fig. 5, exhibits the various values of — (—'j for the various obliquities and 
cU\n 0 J 
degrees of viscosity, just as the previous figures exhibited —. The calculations were 
(XL 
done in the same way as before, after the various functions of the obliquity were 
expressed in terms of cos 2 i and cos 4 i. 
The only remarkable point in these curves is that, for the higher degrees of 
viscosity, the tidal friction rises to a maximum for about 45° of obliquity. The tidal 
friction rises to its greatest value when e=22-|° nearly; this is explained by the fact 
that by far the largest part of the friction arises from the semi-diurnal tide, which has 
its greatest effect when sin 4e is unity. 
§ 13. Tidal friction and apparent secular acceleration of the moon. 
I now set aside again the hypothesis that the earth is purely viscous, and return to 
that of there being any kind of lagging tides. 
I shall first find at what rate the earth is being retarded when it is moving with its 
© O 
