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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                                which seemed to show that the effect of accidental
errors of observation on the ultimate result would
probably be less than in the methods which up to
that time had been used, confirmed me in the
wish to try it. The nature of these preliminary
calculations was nearly the following:
2. Conceive the Earth to be a sphere of
radius R and near density D, surrounded by a
spherical shell of thickness <s>r</s> c and density d,
so that the radius of the external surface is R+c.
As the attraction of the spherical shell upon
a point at or within its inner surface is nothing,
the attraction at the confines of the inner sphere
and shell is represented <s>omitting certain factors</s>
by 4π/3 . R<sup>3</sup>D/R<sup>2</sup> = 4π/3 . RD  : and the attraction at the external
surface of the shell is represented by 4π.R<sup>3</sup>D + ((R+c)<sup>3</sup> - R<sup>3</sup>)d / 3(R+c)<sup>2</sup>,
which to the first power of c/R is 4π.RD/3 .(1 - 2c/R + 3c.d/R.D)
Calling the gravity at those two points respectively G
and g, g/G = 1 - 2c/R + 3c.d/R.D , from which <s>D/d</s> d/D =
R.g/3c.G - (R/3c - 2/3). Considering the fractions d/D and g/G
                            
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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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