566 
MR. G. H. DARWIN ON PROBLEMS CONNECTED 
so that the whole series of changes from the day of 5 hours 36 minutes to that of 
24 hours occupied much more than 1,000 million years, then the large amount of heat 
which is generated deep down would have time to leak out, so that finally the 
temperature gradient would be steeper than that just found. But this is not 
the case. 
Consider only the first, and by far the most important, term of the expression for 
the temperature gradient. It has the form I) (1—e~ pT ), when t = T at the end of the 
1_ g-pT 
series of changes. Now 1) varies as T -1 , and —- —— has its maximum value unity when 
T=0. Hence, however slowly the tidal friction operates, the temperature gradient can 
never be greater than if the heat were all generated instantaneously; but the tem¬ 
perature gradient at the end of the changes is not sensibly less than it would be if all 
the heat were generated instantaneously, provided the series of changes do not occupy 
more than 1,000 million years. 
III. The forced oscillations of viscous, fluid, and elastic spheroids. 
In investigating the tides of a viscous spheroid, the effects of inertia were neglected, 
and it was shown that the neglect could not have an important influence on the 
results. 4 ' I shall here obtain an approximate solution of the problem including the 
effects of inertia; that solution will easily lead to a parallel one for the case of an 
elastic sphere, and a comparison with the forced oscillations of a fluid spheroid will 
prove instructive as to the nature of the approximation. 
If W be the potential of the impressed forces, estimated per unit volume of the 
viscous body, then (with the same notation as before) the equations of flow are 
dp , dW (da 
V -a-h—— — w 
dx dx 
da n da da 
— 
\dt dx dy 
d'A 
= 0 
4'+&c. = 0 --+&c. =o 
dy dz 
\ . (30) 
—aJ£a*1—0 
d. j^ dy dz 
The terms — 
clcc \ 
iv[ + &c. ] are those due to inertia, which were neglected in the paper 
on “ Tides.” 
It will be supposed that the tidal motion is steady, and that W consists of a series of 
solid harmonics each multiplied by a simple time harmonic, also that W includes not 
only the potential of the external tide-generating body, but also the effective potential 
due to gravitation, as explained in the first part of this paper. 
# “ Tides,” Section 10, 
