MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. .501 
the value of G will be invariable. Hence we may make G'=G, and a,=a', there- 
fore 
y=G'[l+(|m'^-E)(cos’«-|)] ; 
Ii representing the value of 2^4- {o^-\-y^) at any time after the formation of the shell. 
Let 
then neglecting all powers of the ellipticity higher than the first, 
y=r[l-l-(|m'^— E) cos^^] (7.) 
If ^=90°, then y=r, so that F will be the expression for gravity at the equator. 
The expression for Fi, gravity at the poles, will be 
r,=r[i + (|»i:-E);. 
Hence if we designate by S the excess of gravity at the pole above gravity at the 
equator divided by the latter quantity, we shall have the general theorem 
S+E=2^^'j2 (8.) 
If I'=Ii, this expression will be identical with what is generally called Clairaut’s 
theorem. 
If the earth had been originally a homogeneous fluid mass, and if in solidifying it 
5 
continued so, then E=-/w', S=E. 
If the spheroid were heterogeneous, the excess of gravity at the pole above gravity at 
the equator at any period of time divided by the latter quantity, and the excess of the 
equatorial axis above the polar axis divided by the latter axis, would form two fractions, 
the sum of which multiplied by the square of the ratio of the moment of inertia of the 
mass at the same period to its present moment of inertia, would form a constant product 
ivhich would be equal to twice the ellipticity which the spheroid would have in the case 
of homogeneity. 
This and the foregoing article appear to contain all that it may be at present 
possible to deduce from the improved hypothesis of the primitive state of the earth, 
with relation to its figure, and to the variation of gravity at its surface. It may be 
hereafter possible to assign to the quantities entering <p {a) in equation (5.) such values 
deduced from considering the physical properties of the primitive fluid, as would 
enable us to find a numerical value for the ellipticity. 
6. In art. 3, a short general sketch has been given of the consequences which 
may result from the improved hypothesis of the primitive fluidity of the earth to phy- 
sical geology, or to the changes occurring upon the external crust of the earth during 
3 T 
MDCCCLI. 
