485 
AND ON THE REMOTE HISTORY OF THE EARTH. 
When £=1‘01, we find from this that — t = 7 20 million years, and that the length 
of the month is 2 8’15 m. s. days. Hence, if we look back 700 million years or more, 
we might find the obliquity 28° 40', and the month 28 ‘15 m. s. days, whilst the length 
of day might be nearly constant. It must, however, be reiterated, that on account 
of our assumptions the change of obliquity is greater than would be possible, whilst 
the time occupied by the change is too short. In any case, any change in this direction 
approaching this in magnitude seems excessively improbable. 
PART II. 
§15. Integration of the differential equations for secular changes in the variables in 
the case of viscosity* 
It is now supposed that the earth is a purely viscous spheroid, and I shall proceed to 
find the changes which would occur in the obliquity to the ecliptic and the lengths of 
the day and month when very long periods of time are taken into consideration. 
I have been unable to find even an approximate general analytical solution of the 
problem, and have therefore worked the problem by a laborious arithmetical method, 
when the earth is supposed to have a particular degree of viscosity. 
The viscosity chosen is such that, with the present length of day, the semi-diurnal 
tide lags by 17° 30'. It was shown above that this viscosity makes the rate of change 
of obliquity nearly a maximum, t It does not follow that the whole series of changes 
will proceed with maximum velocity, yet this supposition will, I think, give a very 
good idea of the minimum time, and of the nature of the changes which may have 
occurred in the course of the development of the moon-earth system. 
The three semi-diurnal tides will be supposed to lag by the same amount and to be 
reduced in the same proportion; as also will be the three diurnal tides. 
There are three simultaneous differential equations to be treated, viz. : those giving 
(1) the rate of change of the obliquity of the ecliptic, (2) the rate of alteration of the 
earth’s diurnal rotation, (3) the rate of tidal reaction on the moon. They will be 
referred to hereafter as the equations of obliquity, of friction, and reaction respectively. 
To write these equations more conveniently a partly new notation is advantageous, 
as follows :— 
The suffix 0 to any symbol denotes the initial value of the quantity in question. 
2 2 
T T T T 
Let w 3 =—, u ~ = —, uu=—: these three quantities are constant. 
g«o 3 w o S^o ^ 
* This section has been rearranged, partly rewritten, and recomputed since the paper was presented. 
The alterations were made on December 19, 1878. 
t If I had to make the choice over again I should choose a slightly greater viscosity as being more 
interesting. 
