FOR DETERMINING THE MEAN DENSITY OF THE EARTH. 
353 
Addendum. 
On communicating with Professor Stokes, in reference to the effeet of the Earth’s 
rotation and ellipticity in modifying the numerical results of the Ilarton Experiment, 
I was favoured by that gentleman with an investigation, which, with his permission, 
I subjoin as a valuable addition to my own paper. 
“I shall suppose the surface of the Earth to be an ellipsoid of revolution, and will 
employ the notation made use of in my paper on Clairaut’s Theorem, published in 
the fourth volume of the Cambridge and Dublin Mathematical Journal. In this, 
V is the potential of the Earth’s mass. 
r, 0 are the polar coordinates of any point in or exterior to the Earth’s surface; 
r being measured from the centre, and 0 from the axis of rotation. 
a is the equatorial radius. 
s the elliptieity. 
a the angular velocity. 
m the ratio of the centrifugal force to gravity at the equator. 
E the mass of the Earth. 
V the angle between the normal and radius vector at any point of the surface. 
In the following investigation, small quantities of the second order are neglected, 
£ and m being regarded as small quantities of the first order. 
If U=V+ ^ sin^ 
the differential coefficients 
^ 1.^ 
dr r dd 
will give the components of the force along and perpendicular to the radius vector ; 
and, g being the force of gravity. 
dU 
d\] 
g= — cos V.:-- 4- sin 
° dr r d^ 
d\] 
which becomes, since v and are small quantities of the first order. 
d\] 
S=-1F' 
Let V be measured along the vertical ; then 
dq dq • V dq 
-f= COSt'*-^— SinJ'----^3 
dv dr f afl 
or, to the first order. 
d9_^dg__d'^\] 
dv dr dr^ 
Let c be the depth of the mine ; then if 0^ be neglected, we shall have for the 
MDCCCLVI. 
3 A 
