554 
MR, G. H. DARWIN ON PROBLEMS CONNECTED 
and hence the surface normal traction which is competent to produce the tides of the 
first order is — which is equal to ivcd^r sin 2e sin 2 6 sin 2(<£— cot). Hence 
the intensity of this normal traction is estimated by the quantity wc£\t sin 2e, and this 
affords a standard of comparison with the quantity %vdrje sin 2e, which was the 
estimate of the intensity of the secondary tides. The ratio of the two is 2e, and 
since the ellipticity of the mean spheroid is small, the secondary tides must be small 
compared with the primary ones. It cannot be asserted that the ratio of the heights 
of the two tides will be 2e, because the secondary tides are of a higher order of har¬ 
monics than the primary, and because the tangential stresses have not been reduced to 
harmonics and the problem completely worked out, I think it probable that the height 
of the secondary tides would be considerably less than is expressed by the quantity 
2e, but all that we are concerned to know is that they will be negligeable, and this 
is established by the preceding calculations. 
It follows, then, that the precessional and nutational forces will cause no secular 
shifting of the surface with reference to the interior, and therefore cannot cause any 
such geographical deplacement of the poles, as has been sometimes supposed. 
II. The distribution of heat generated by internal friction and secular cooling. 
In the paper on “ Precession” (Section 16) the total amount of heat was found, 
which was generated in the interior of the earth, in the course of its retardation by 
tidal friction. The investigation was founded on the principle that the energy, both 
kinetic and potential, of the moon-earth system, which was lost during any period, 
must reappear as heat in the interior of the earth. This method could of course give 
no indication of the manner and distribution of the generation of heat in the interior. 
Now the distribution of heat must have a very important influence on the way it will 
affect the secular cooling of the earth’s mass, and I therefore now propose to investigate 
the subject from a different point of view. 
It will be sufficient for the present purpose if we suppose the obliquity to the 
ecliptic to be zero, and the earth to be tidally distorted by the moon alone. 
It has already been explained in the first section how we may neglect the mutual 
gravitation of a spheroid tidally distorted by an external disturbing potential wr 1 S, 
if we suppose the disturbing potential to be wr z (& — where r=a-\-cr is the equation 
to the tidal protuberance. 
It is shown in (4) that 
S —sin 2e sin 3 0 sin 2 ((f)— cot). 
If we refer the motion to rectangular axes rotating so that the axis of x is the major 
