AND ON THE REMOTE HISTORY OE THE EARTH. 
483 
If these values be substituted in the equation giving the rate of variation of the 
obliquity, it will be found that the obliquity must be decreasing at the rate of ‘00197° 
per million years, or 1° in 500 million years. Thus in 100 million years it would only 
decrease by 12'. So, also, it may be shown that the moon’s sidereal period is being 
increased by 2 hours 20 minutes in 100 million years. 
Lastly, the earth considered as a clock is losing 13 seconds in a century. 
There is another supposition as to the physical constitution of the earth, which will 
lead to interesting results. 
If the earth be elastico-viscous, then for the semi-diurnal and diurnal tides it might 
behave nearly as though it were perfectly elastic, whilst for the fortnightly tide it 
might behave nearly as though it were perfectly viscous. With the law of elastico- 
viscosity used in my previous paper,' 5 ' it is not possible to satisfy these conditions very 
exactly. But there is no reason to suppose that that law represents anything but an 
ideal sort of matter ; it is as likely that the degradation of elasticity immediately after 
straining is not so rapid as that Jaw supposes. I shall therefore take a limiting case, 
and suppose that, for the semi-diurnal and diurnal tides, the earth is perfectly elastic, 
whilst for the fortnightly one it is perfectly viscous. This hypothesis, of course, will 
give results in excess of what is rigorously possible, at least without a discontinuity in 
the law of degradation of elasticity. 
It is accordingly assumed that the semi-diurnal and diurnal bodily tides do not lag, 
and therefore e=e'= 0 ; whilst the fortnightly tide does lag, and E"=cos 2e". 
Thus by (38) there is no tidal friction, and by (60) there is a true acceleration of the 
moon’s motion of \ of 7‘042 sin 4e" seconds of arc in a century. Then if we take the 
most favourable case, namely, when £" = 22° 30', there is a true secular acceleration of 
3"‘521 per century. 
It follows, therefore, that the whole of the observed secular acceleration of the moon 
might be explained by this hypothesis as to the physical constitution of the earth. 
On this hypothesis the fortnightly ocean tides should amount to sin 22° 30', or ‘38 
of its theoretical height on a rigid nucleus, and the time of high water should be 
accelerated by 1 day 17 hours. Again, by (35) ~ = —py- ir sin H, from whence it may be 
shown that the obliquity of the ecliptic would be decreasing at the rate of 1° in 
128 million years. 
The conclusion to be drawn from all these calculations is that, at the present time, 
the bodily tides in the earth, except perhaps the fortnightly tide, must be exceedingly 
small in amount; that it is utterly uncertain how much of the observed 4" of acce¬ 
leration of the moon’s motion must be referred to the moon itself, and how much to 
* Namely, that if the solid be strained, the stress required to maintain it in the strained configuration 
diminishes in geometrical progression as the time, measured from the epoch of straining, increases in 
arithmetical progression. See Section 8 of the paper on “ llodiiy Tides,” &c., Phil. Trans., Part I., 1879. 
