MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
499 
If m' represent the ratio of centrifugal force f to gravity g' at the equator, then 
If this value of /I be substituted in (4.), and if all the integrations be performed 
according to the given value of the function an equation may be obtained after due 
reduction of the form 
(5.) 
in which equation I is considered as a function of a, and where (p is a functional 
symbol. That I may be considered as a function of a, is evident from the transforma- 
tion of X, y, z into polar coordinates, and from the assumed identity of r and a. The 
above equation appears therefore to be general, or applicable to the consideration of 
the figure of any fluid mass upon the conditions which have formed the groundwork 
for its investigation. 
In applying equation (5.) to the case of the earth’s ellipticity, the values of a' 
and r, known at the present day, may be introduced. If it be supposed that the pre- 
sent ellipticity of the earth is the same as that which it had at the time when the first 
external coat of the shell had solidified from the fluid, I' will be a function of E, which 
can be then found if T should be introduced under such a form by some further trans- 
formations. 
The quantities I and I' are evidently the moments of inertia of the mass with re- 
spect to the axis of ^ during certain states of the arrangement of its molecules about 
the centre of gravity. Hence from the properties of ellipsoids of revolution 
j, 2 M ,«'2 
^ 5a- ’ 
ff being a number depending upon the arrangement of the particles in the interior of 
the mass, Mj representing the earth’s mass, and a' the present polar radius of the 
earth, which, without sensible error, may be used for the equatorial radius. From 
the consideration of the phenomena of precession and nutation, the following expres- 
sion for ff' has been obtained* — 
where P represents a number deduced from the phenomena alluded to. If the above 
values of I' and o- be substituted in (5.), it will be transformed into 
(E-^y 
E=mQ - -P, 
* PoNTEcouLANT, 'Pheorie Analytique du Systeme du Monde, tome ii. p. 475. 
