AND ON THE REMOTE HISTORY OF THE EARTH. 
473 
is one of stability, and when from below to above, of instability. It follows from tins 
that the positions of stability and instability must occur alternately. When e=0 or 
45° (fluidity or rigidity) the curve reduces to the horizontal axis, and every position of 
the earth’s axis is one of neutral equilibrium. 
But in every other case the position of 90° of obliquity is not a position of equi¬ 
librium, but the obliquity tends to diminish. On the other hand, from e = 0° to about 
30° (infinitely small viscosity to tide retardation of two hours), the position of zero 
obliquity is one of dynamical instability, whilst from then onwards to rigidity it 
becomes a position of stability. 
For viscosities ranging from e=0° to about 42^° there is a position of stability which 
lies between about 50° to 87° of obliquity ; and the obliquity of dynamical stability 
diminishes as the viscosity increases. 
For viscosities ranging from e = 30° nearly to about 42^°, there is a second position 
of dynamical equilibrium, at an obliquity which increases from 0° to about 50°, as the 
viscosity increases from its lower to its higher value. But this position is one of 
instability. 
From e= about 42^° there is only one position of equilibrium, and that stable, viz. : 
when the obliquity is zero. 
If the obliquity be supposed to increase past 90°, it is equivalent to supposing the 
earth’s diurnal rotation reversed, whilst the orbital motion of the earth and moon 
remains the same as before; but it did not seem worth while to prolong the figure, as 
it would have no applicability to the planets of the solar system. And, indeed, the 
figure for all the larger obliquities would hardly be applicable, because any planet 
whose obliquity increased very much, must gradually make the plane of the orbit of 
its satellite become inclined to that of its own orbit, and thus the hypothesis that the 
satellite’s orbit remains coincident with the ecliptic would be very inexact. 
It follows from an inspection of the figure that for all obliquities there are two 
degrees of viscosity, one of which will make the rate of change of obliquity a maximum 
and the other minimum. A graphical construction showed that for obliquities of about 
5° to 20°, the degree of viscosity for a maximum corresponds to about e = 17^°*, 
whilst that for a minimum to about e=40°. In order, however, to check this con¬ 
clusion, I determined the values of e analytically when 7=15°, and when the 
fortnightly tide (which has very little effect for small obliquities) is neglected. I 
find that the values are given by the roots of the equation 
£c 3 -4-10x a +13'660a: — 20 , 412 = 0, where x=3 cos 4e. 
This equation has three real roots, of which one gives a hyperbolic cosine, and the 
* I may Fere mention that I found wlien e=l7§-°, that it would take about a thousand million years for 
the obliquity to increase from 5° to 23-|°, if regard was only paid to this equation of change of obliquity. 
The equations of tidal friction and tidal reaction will, however, entirely modify the aspects of the case. 
3 P 2 
