496 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
taken, as before, equal to 20 - 9 million feet. Then the last column states that energy 
enough has been turned into heat in the interior of the earth to warm its whole mass 
so many degrees Fahrenheit within the times given in the first column of the same 
table. 
The consideration of the distribution of the generation of heat and the distortion of 
the interior of the earth must be postponed to a future occasion. 
In the succeeding paper I have considered the bearing of these results on the secular 
cooling of the earth, and in a subsequent paper (‘ Proceedings of the Poyal Society,’ 
No. 197, June 19, 1879, p. 168) the general problem of tidal friction is considered by 
the aid of the theory of energy. 
§ 17. Integration in the case of small variable viscosity .* 
In the solution of the problem which has just been given, where the viscosity is 
constant, the obliquity of the ecliptic does not diminish as fast as it might do as we 
look backwards. The reason of this is that the ratio of the negative terms to the 
positive ones in the equation of obliquity is not as small as it might be; that ratio 
Sill 2 6^ 
principally depends on the fraction , which has its smallest value when e is very 
small. 
I shall now, therefore, consider the case where the viscosity is small, and where it 
so varies that e always remains small. 
This kind of change of viscosity is in general accordance with what one may 
suppose to have been the case, if the earth was a cooling body, gradually freezing as 
it cooled. 
The preceding solution is moreover somewhat unsatisfactory, inasmuch as the three 
semi-diurnal tides are throughout supposed to suffer the same retardation, as also are 
the three diurnal tides; and this approximation ceases to be sufficiently accurate 
towards the end of the integration. 
In the present solution the retardations of all the lunar tides will be kept distinct. 
By (40) and (40'), Section 11 , 
. _ 2(n—J2) 2)i 2(n + f2) _ „ 2/2 
tan 2e,=-, tan 2e= —, tan 2e 2 = , tan 2e =—, 
P P P P 
, n — 2fl . n , n + 2/2 
tan e ■, =-, tan e =-, tan e „ =- 
l p P P 
for the lunar tides. 
For the solar tides we may safely neglect fl / compared with n, and we have 
* This section has been partly rewritten and rearranged, and wholly recomputed since the paper was 
presented. The alterations are in the main dated December 19, 1878. 
