46 
MR. HOPKINS’S RESEARCHES IN PHYSICAL GEOLOGY. 
The approximate solution which I am about to offer is founded on the following 
assumption. Let r' be the distance of any point on the surface of the spheroid from 
the centre, and a 1 the polar or least distance ; it is assumed that the temperature in 
the spheroid at a depth = r' — a x is the same as it would have been at that depth if 
the spheroid had been a sphere with radius = r\ If the ellipticity be small, and 
the process of cooling has been continued a sufficient length of time, this assumption 
will manifestly be almost accurately true. The solution of the problem is thus made 
to depend on formulae which have been given by Poisson in his Thdorie de la Chaleur. 
To this I now proceed. 
Determination of the Forms of Isothermal Surfaces within the Earth. 
5. Adopting Poisson’s notation* *, let u denote, at any time t, the temperature of 
any point of the earth, and let 
u = u x + v ! , 
where u x is such as to satisfy the general differential equation for the propagation of 
heat by conduction, the conditions relative to the original temperature, and that 
which would exist at the surface at any time if the earth were a sphere whose radius 
= a x . Then when t becomes very great, as it is assumed to be in the case of the 
earth, taking the common expression for the temperature in a sphere of large radius, 
as a sufficient approximation to that of the actual case of the earth, we have 
where C is the value of u x at the centre-f-. At the surface the first term vanishes, 
and the value of u x is reduced to the second term, which, however, is so small that it 
may here be altogether neglected. Consequently 
Mi = C — sin — r s 
1 7T V 
Let £ denote the value of u at any point for which r = a x , £ being a function of t 
and of 6, the angle which r makes with the axis of revolution of the spheroid; or 
since ( t being very large) u x = 0, approximately when r — a x , £ may be taken as the 
value of u' for that value of r. It remains to find a value of u ' which shall satisfy 
the general differential equation, and the particular condition u! = £ when r = 
Let 
r u! = U 0 + Uj + + U n + . . . ., 
U 0 XJ n being a series of Laplace’s coefficients, and functions of the polar 
coordinates of the proposed point. Also let 
£=(Z 0 + Z 1 + .... + Z, + ....)«-- i 
tions of one independent variable, and of the second order. The complicated form of these equations, however, 
would seem to render them at present of little avail in the solution of the problem considered in the text. 
* Thdorie de la Chaleur, Art. 173. f Ihid. Art. 171. J Ibid. Art. 173. 
