WITH THE TIDES OP A VISCOUS SPHEROID. 
593 
temperature gradient at the surface is sensibly the same, at whatever rate the heat is 
generated, provided it is all generated within 1,000 million years ; but the temperature 
gradient can never be quite so steep as if the whole heat were generated instan¬ 
taneously. The gradient, if the changes take place within 1,000 million years, is 
found to be about 1° Fa.hr. in 2,600 feet. Now the actually observed increase of 
underground temperature is something like 1° Fahr. in 50 feet; it therefore appears 
that perhaps one-fiftieth of the present increase of underground temperature may pos¬ 
sibly be referred to the effects of long past internal friction. It follows, therefore, that 
Sir William Thomson’s investigation of the secular cooling of the earth is not 
sensibly affected by these considerations. 
If at any time in the future we should attain to an accurate knowledge of the 
increase of underground temperature, it is just within the bounds of possibility that a 
smaller rate of increase of temperature may be observed in the equatorial regions than 
elsewhere, because the curve of equal heat generation, which at the equator is nearly 
500 miles below the surface, actually reaches the surface at the pole. 
The last problem here treated is concerned with the effects of inertia on the tides of 
a viscous spheroid. As this part will be only valuable to those who are interested in 
the actual theory of tides, it may here be dismissed in a few words. The theory used 
in the two former papers, and in the first two parts of the present one, was founded 
on the neglect of inertia ; and although it was shown in the paper on “Tides” that 
the error in the results could not be important, in the case of a sphere disturbed by 
tides of a frequency equal to the present lunar and solar tides, yet this neglect left a 
defect in the theory which it was desirable to supply. Moreover it was possible that, 
when the frequency of the tides was much more rapid than at present (as was found 
to have been the case in the paper on “ Precession”), the theory used might be 
seriously at fault. 
It is here shown (see (62)) that for a given lag of tide the height of tide is a little 
greater, and that for a given frequency of tide the lag is a little greater than the 
approximate theory supposed. 
A rough correction is then applied to the numerical results given in the paper on 
“ Precession” for the secular changes in the configuration of the system ; it appears 
that the time occupied by the changes in the first solution (Section 15) is overstated 
by about one-fortieth part, but that all the other results, both in this solution and 
the other, are left practically unaffected. To the general reader, therefore, the value 
of this part of the paper simply lies in its confirmation of previous work. 
From a mathematical point of view, a comparison of the methods employed with 
those for finding the forced oscillations of fluid spheres is instructive. 
Lastly, the analytical investigation of the effects of inertia on the forced oscillations 
of a viscous sphere is found to be applicable, almost verbatim, to the same problem 
concerning an elastic sphere. The results are complementary to those of Sir William 
Thomson’s statical theory of the tides of an elastic sphere. 
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