Account of pendulum experiments undertaken in the Harton Colliery, for the purpose of determining the mean density of the earth, by G. B. Airy

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                                ε, the ellipticity.
w, the angular velocity.
m, the ratio of the centrifugal force to gravity
at the equator.
Ε, 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, ε and m being regarded as
small quantities of the first order.

If U = V + w<sup>2</sup>/2 r<sup>2</sup> sin<sup>2</sup>θ,
the differential coeficients
     dU/dr, 1/r . dU/dθ,
will give the components of the force along and
perpendicular to the radius vector;
and, g being the force of gravity,

     g = - cos v.dU/dr + sin v. 1/r . δU/δθ:
which becomes, since v and dU/dθ are small quantities
of the first order,

     g = - dU/dr

Let v be measured along the vertical: then

     dg/dv = cos v. dg/dr - sin v. 1/r . dg/dθ
                            
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Manuscript details

Author
George Biddell Airy
Reference
PT/54/5
Series
PT
Date
1855
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Account of pendulum experiments undertaken in the Harton Colliery, for the purpose of determining the mean density of the earth, by G. B. Airy, 1855. From The Royal Society, PT/54/5

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