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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                                or, to the first order,
dg/dv = dg/dr = -d<sup>2</sup>U/dr<sup>2</sup>

Let c be the depth of the mine; then if (c/d)<sup>2</sup> be
neglected, we shall have for the value of the fraction
gravity below/gravity above (which I will call F), calculated on
the supposition that all the attracting mass is internal
to both stations,
     F = 1 - c/g . dg/dv,
where, after differentiation, r is to be put equal to the
radius vector of the surface, namely a(1 - ε cos<sup>2</sup>θ).
Now the value of V (Article 5 of the paper
referred to) is

     V = E/r - (Eε/a - 1/2w<sup>2</sup>a<sup>2</sup>) a<sup>3</sup>/r<sup>3</sup> (cos<sup>2</sup>θ - 1/3)
which is true independently of any particular hypothesis
respecting the distribution of matter in the interior of
the Earth; so that

U = E/r - (Eε/a - 1/2w<sup>2</sup>a<sup>2</sup>) a<sup>3</sup>/r<sup>3</sup> (cos<sup>2</sup>θ - 1/3) + w<sup>2</sup>/2 r<sup>2</sup> sin<sup>2</sup>θ
and g = -dU/dr

   = E/r<sup>2</sup>  - 3(Eε/a - 1/2w<sup>2</sup>a<sup>2</sup>) a<sup>3</sup>/r<sup>4</sup> (cos<sup>2</sup>θ - 1/3) - w<sup>2</sup> r sin<sup>2</sup>θ

whence - dg/dv = -dg/dr

    = 2E/r<sup>3</sup>  - 12(Eε/a - 1/2w<sup>2</sup>a<sup>2</sup>) a<sup>3</sup>/r<sup>5</sup> (cos<sup>2</sup>θ - 1/3) + w<sup>2</sup> sin<sup>2</sup>θ

                            
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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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