THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
875 
orbit to its proper plane would have increased, but the whole increase could not have 
amounted to more than a few minutes of arc. At the point P the day is 7 hrs. 47 m. 
in length, and the month 3‘25 m. s. days in length ; at the point Q the day is 
8 hrs. 36 m., and the month 5'20 m. s. days. From Q down to the present state this 
small inclination would have always decreased. 
If then the earth had had small viscosity throughout its evolution, the lunar orbit 
would- at present be only inclined at a very small angle to the ecliptic. But it is 
actually inclined at about 5° 9', hence it follows that while the hypothesis of small 
viscosity is competent to explain some inclination, it cannot explain the actually 
existing inclination. 
It was shown in the papers on “ Tides ” and “ Precession ” that, if the earth be not 
at present perfectly rigid or perfectly elastic, its viscosity must be very large. And it 
was shown in “ Precession ” that if the viscosity be large, the obliquity of the ecliptic 
must at present be decreasing. Now it will be observed that in resuming the inte¬ 
gration with the hypothesis of large viscosity, the solution of the first method with 
the hypothesis of small viscosity was accepted as the basis for continuing the inte¬ 
gration with large viscosity. This appears at first sight somewhat illogical, and to 
be strictly correct, we ought to have taken as the initial inclination of the earth’s 
proper plane to the ecliptic, at the beginning of the application of the methods of 
Part III. to the hypothesis of large viscosity, some angle probably a little less than 
23^°" instead of 17°. This would certainly disturb the results, but I have not thought 
it advisable to take this course for the following reasons. 
It is probable that at the present time the greater part, if not the whole of the 
tidal friction is due to oceanic tides, and not to bodily tides. If the ocean were friction¬ 
less, it would be low tide under the moon ; consequently the effects of fluid friction 
must be to accelerate, not retard, the ocean tides.! Then in order to apply our present 
analysis to the case of oceanic tidal friction, that angle which lias been called the lag 
of the tide must be interpreted as the acceleration of the tide. 
We know that the actual friction in water is small, and hence the tides of long 
period will be less affected by friction than those of short period ; thus the effects of 
fluid tidal friction will probably be closely analogous to those resulting from the 
hypothesis of small viscosity of the whole earth and bodily tides. On the other hand, 
it is probable that the earth was once more plastic than at present, either superficially 
or throughout its mass, and therefore it seems probable that the bodily tides, even if 
small at present, were once more considerable. I think therefore that on the whole 
* In the present configuration of the earth, moon, and sun, the obliquity will deci’ease, if the viscosity 
be very large. But if we integrate backwards this retrospective increase of obliquity would soon be con¬ 
verted into a decrease. Thus at the end of “the first period of integration,” the obliquity would be a 
little greater than 2.3-p, but by the end of the “ second period ” it would probably be a little less than 
23|°. It is at the end of the “ second period ” that the method of Part III. is first applied. 
t Otherwise the lunar attraction on the tides would accelerate the earth’s rotation—a clear violation of 
the principles of energy. 
