874 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
From this it follows, that by supposing the viscosity large enough we may make the 
obliquity and inclination to the invariable plane as small as we please, by the time 
that state is reached in which the month is equal to twice the day. 
Hence, on the present hypothesis, we trace the system back until the lunar orbit is 
sensibly coincident with the equator, and the equator is inclined to the ecliptic at an 
angle of 11° or 12°. 
It is probable that in the still more remote past the plane of the lunar orbit would 
not have a tendency to depart from that of the equator. It is not, however, expedient 
to attempt any detailed analysis of the changes further back, for the following reason. 
Suppose a system to be unstable, and that some infinitesimal disturbance causes the 
equilibrium to break down ; then after some time it is moving in a certain way. Now 
suppose that from a knowledge of the system we endeavour to compute backwards 
from the observed mode of its motion at that time, and so find the condition from 
which the observed state of motion originated. Then our solution will carry us back 
to a state very near to that of instability, from which the system really departed, 
but as the calculation can take no account of the infinitesimal disturbance, which 
caused the equilibrium to break down, it can never bring us back to the state which 
the system really had. And if we go on computing the preceding state of affairs, the 
solution will continue to lead us further and further astray from the truth. Now 
this, I take it, is likely to have been the case with the earth and moon ; at a certain 
period in the evolution (viz.: when the month was twice the day) the system probably 
became dynamically unstable, and the equilibrium broke down. Thus it seems more 
likely that we have got to the truth, if we cease the solution at the point where the 
lunar orbit is nearly coincident with the equator, than by going still further back. 
In § 21, fig. 7, is given a graphical illustration of the distance-rate of change in the 
inclinations of the lunar orbit to its proper plane, and of the earth’s proper plane to the 
ecliptic ; the dotted curves refer to the hypothesis of large viscosity, and the firm- 
curves to that of small viscosity. 
The figure is explained and discussed in that section; I will here only draw 
attention to the wideness apart of the two curves illustrative of the rate of change 
of the inclination of the lunar orbit. This shows how much influence the degree 
of viscosity of the earth must have had on the present inclination of the lunar orbit 
to the ecliptic. 
It is particularly interesting to observe that in the case of small viscosity this curve 
rises above the horizontal axis. If this figure is to be interpreted retrospectively, 
along with our solution, it must be read from left to right, but if we go with the time, 
instead of against it, from right to left. 
Now if the earth had had in its earlier history infinitely small viscosity, and if the 
moon had moved primitively in the equator, then until the evolution had reached the 
point represented by P, the lunar orbit would have always remained sensibly coin¬ 
cident with its proper plane. Then in passing from P to Q the inclination of the 
