WITH THE TIDES OE A VISCOUS SPHEROID. 
551 
point distant p-\-Sp shifted through — p~(p-\-Sp)8t ; but this second point does 
shift through ^~(p-\-SpYSt. 
Hence the amount of shear in unit time is 
i w 
¥p X Sv~c 
(p+Sp) : -(p+8p)p- 
w dl 2 
_ 4u C P '' 
Therefore at the equator, at the surface where the shear is greatest, the shear per 
unit time is 
fc o-dt 1 cos 2e -“- 
With the present values of r and co, r|(“) w is a shear of pjy per annum. 
vS/ lu 10 
Hence at the equator a slab one foot thick would have one face displaced with 
reference to the other at the rate of cos' 2 2e of an inch in a million years. 
The bearing of these results on the history of the earth will be considered in 
Part IY. 
The next point which will be considered is certain tides of the second order. 
We have hitherto supposed that the tides are superposed upon a sphere; it is, how¬ 
ever, clear that besides the tidal protuberance there is a permanent equatorial pro¬ 
tuberance. How this permanent protuberance is by hypothesis not rigidly connected 
with the mean sphere; and, as the attraction of the moon on the equatorial regions 
produces the uniform precession and the fortnightly nutation, it might be (and indeed 
has been) supposed that there would arise a shifting of the surface with reference to 
the interior, and that this change in configuration would cause the earth to rotate 
round a new axis, and so there would follow a geographical shifting of the poles. I 
will now show, however, that the only consequence of the non-rigid attachment of the 
equatorial protuberance to the mean sphere is a series of tides of the second order in 
magnitude, and of higher orders of harmonics than the second. 
For a complete solution of the problem the task before us would be to determine 
what are the additional tangential and normal stresses existing between the protu¬ 
berant parts and the mean sphere, and then to find the tides and secular distortion (if 
any) to which they give rise. 
The first part of these operations may be done by the same process which has just 
been carried out with reference to the secular distortion due to tidal friction. 
The additional normal stress (in excess of —giva, the mean weight of an element of 
the protuberance) can have no part in the precessional and nutational couples, and the 
MDCCCLXXIX. 4 B 
