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

```
images

## Manuscript details

Author
George Biddell Airy
Reference
PT/54/5
Series
PT
Date
1855
IIIF (What's this?)
This is a link to the IIIF web URL for this item. You can drag and drop the IIIF image link into other compatible viewers

## Cite as

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