528 
MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
achieved, but an approximate solution, founded on an hypothesis to which no weighty 
objections can be urged, has been given by Mr. Hopkins*. This solution will suf- 
fice for my present purpose, particularly when I only refer to the general result of 
his analysis. The result in question is, that the oblateness of the isothermal surfaces 
increases with their distance from the earth’s surface. This result, combined with 
the first of the three conclusions in the foregoing paragraph, seems to show that if 
the earth solidified from its surface to its centre without changing its law of density, 
the ellipticity of the shell’s inner surface would be greater than that of its outer 
surface. The truth of the second conclusion would be considerably weakened, while 
that of the third would be strengthened to the same amount. 
13. It will immediately appear that this last is the conclusion which must be de- 
finitely adopted, if we admit that the matter composing the nucleus becomes denser 
in assuming the solid state. 
Using the notation of art. 6, and neglecting the effect of isothermal surfaces, the 
ellipticity of the surface of the perfect fluid may be generally expressed thus, 
o!), 
the double sign being placed before rrv^ia^a!) to show that, according as the ellipticity 
of the shell’s inner surface is greater or less than that of its outer surface, this term 
should be added or subtracted, as must appear from the theory of the attraction of 
spheroids. If, when e, is supposed to increase, the small term alluded to be neglected, 
the truth of any conclusion as to the rapid increase of deduced from an examina- 
tion of the remaining term will be only rendered still more manifest. 
Let at any period of the shell’s existence the surface of the nucleus be supposed to 
coincide with the shell’s inner surface, then mi=m, and consequently, from the 
expressions given in art. 6, we may deduce 
ma\fQq^a^da mcLi 
ffa being a number depending on the law of density of the nucleus, and analogous to 
a of article 5. But as the centrifugal force at the surface of the spheroid is propor- 
tional to the square of its angular velocity, and as the angular velocity is inversely 
proportional to the moment of inertia of the mass, we shall have 
I' representing the moment of inertia of the mass in its state of entire fluidity, I the 
moment of inertia corresponding to m, and m' the value of m corresponding to I'. 
From the general expression for the moment of inertia of a spheroid, we can write 
(o -2 being the value of when a^=-d). 
* Philosophical Transactions, 1842, p. 45. 
