MR. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
45 
than unity for such values of — 1 as we shall be concerned with), we have 
■i = i -A = 4, 
( 3 .) 
which gives too great a value of e. Now, that we may be able to satisfy equation 
(2.), g must be less than q, i. e. it must diminish as the thickness of the earth’s crust 
increases ; and, therefore, the thickness which corresponds to this approximate value 
of s will be too small ; or the actual thickness of the solid crust of the globe which 
would give the precession P, must necessarily be greater than that for which the 
value of s is -g- q. 
4. We must now proceed to determine the relation between the value of and 
that of — a, the thickness of the solid crust. 
If we assume (as I shall now do) that the fusibility of the matter composing the 
earth is equal at equal depths*, it would seem that the only conceivable causes which 
can affect the degree of solidity or fluidity of the mass, are temperature and pressure. 
It may be doubted by some persons whether solidification be actually promoted by 
the latter cause or not ; but there will be no corresponding uncertainty in our con- 
clusions respecting the minimum thickness of the earth’s crust consistent with the 
observed amount of precession ; because, if they be true, this cause being effective, 
they will easily be seen to be so, a fortiori, if it produce no effect. 
If temperature produced no effect in solidification, the surfaces of equal solidity 
(or fluidity) would be surfaces of equal pressure, and therefore of equal density ; and 
if pressure did not promote solidification, the surfaces of equal solidity would be iso- 
thermal surfaces. Assuming both causes to be effective, conceive two surfaces of 
equal density and temperature respectively, passing through the same point (in the 
axis of the spheroid, for instance) ; then will the surface of equal solidity through the 
same point be intermediate to the two former, the ellipticities of which will therefore 
be limits to that of the surface of equal solidity. It is these limits which we must 
now proceed to determine. 
The greatest difficulty in the determination of the temperature at any point of a 
body cooling by conduction, is that which arises from satisfying the conditions at the 
surface in each particular case. This has been effected only in the sphere, the circu- 
lar cylinder, and a few other simple cases, but not including that of the spheroid, the 
isothermal surfaces of which, consequently, have never been completely determined'!'. 
* This may admit of local exceptions, such as probably exist, without any sensible modifications in our 
general conclusions. 
t An ingenious memoir on this subject by M. Lame is contained in the fifth volume of the ‘ Memoires des 
Savans Etrangers,’ in which he has examined the conditions under which the isothermal surfaces within an 
ellipsoid will also be ellipsoids, when it has arrived at a permanent state of temperature. He has also made the 
general expression for the temperature at any time to depend on the integration of certain differential equa- 
