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MR. G. H. DARWIN ON THE PRECESSION OP A VISCOUS SPHEROID, 
respect. The two tides which tend to increase the moon’s distance are much larger 
than the others, so that the moon in general tends to recede from the earth. The 
increase of distance is, of course, accompanied by an increase of the moon’s periodic 
time, and hence there is in general a true secular retardation of the moon’s motion. 
But this change is accompanied by a retardation of the earth’s diurnal rotation, and a 
terrestrial observer, taking the earth as his clock, would conceive that the angular 
velocity of an ideal moon, which was undisturbed by tidal reaction, was undergoing a 
secular acceleration. The apparent acceleration of the ideal undisturbed moon must 
considerably exceed the true retardation of the real disturbed moon, and the difference 
between these two will give an apparent acceleration. 
It is thus possible to give an equation connecting the apparent acceleration of the 
moon’s motion and the heights and retardations of the several bodily tides in the earth. 
Now there is at the present time an unexplained secular acceleration of the moon of 
about 4" per century, and therefore if we attribute the whole of this to the action of 
the bodily tides in the earth, instead of to the action of ocean tides, as was done by 
Adams and Delaunay, we get a numerical relation which must govern the actual 
heights and retardations of the bodily tides in the earth at the present time. 
This equation involves the six constants expressive of the heights and retardations of 
the three bodily tides, and which are determined by the physical constitution of the 
earth. No further advance can therefore be made without some theory of the earth’s 
nature. Two theories are considered. 
First, that the earth is purely viscous. The result shows that the earth is either 
nearly fluid—which we know it is not—or exceedingly nearly rigid. The only traces 
which we should ever be likely to find of such a high degree of viscosity would be in 
the fortnightly ocean tide; and even here the influence would be scarcely perceptible, 
for its height would be ‘992 of its theoretical amount according to the equilibrium theory, 
whilst the time of high water would be only accelerated by six hours and a half. 
It is interesting to note that the indications of a fortnightly ocean tide, as deduced 
from tidal observations, are exceedingly uncertain, as is shown in a preceding paper/" 
where I have made a comparison of the heights and phases of such small fortnightly tides 
as have hitherto been observed. And now (July, 1879) Sir William Thomson has 
informed me that he thinks it very possible that the effects of the earth’s rotation may 
be such as to prevent our trusting to the equilibrium theory to give even approximately 
the height of the fortnightly tide. He has recently read a paper on this subject 
before the Boyal Society of Edinburgh. 
With the degree of viscosity of the earth, which gives the observed amount of secular 
acceleration to the moon, it appears that the moon is subject to such a true secular 
retardation that at the end of a century she is 3 "’ 1 behind the place in her orbit which 
she would have occupied if it v T ere not for the tidal reaction, whilst the earth, considered 
as a clock, is losing 13 seconds in the same time. This rate of retardation of the earth 
* See the Appendix to my paper on the “ Bodily Tides,” &c., Phil. Trans., Part I., 1879. 
