Account of pendulum experiments undertaken in the Harton Colliery, for the purpose of determining the mean density of the earth, by G. B. Airy
Divide the whole of the matter into cylindrical rings of which UI is the axis: let the internal and external radii of <s>two</s> one of these rings be p and p + δp. Call the azimuth of any point of the ring θ: the endsurface of the prism included between θ and θ + δθ is p.δp.δθ. Let z be the vertical <s>[?]</s> ordinate measured upwards from the lower plane, the solid contact of <s>[?]</s> the point of the prism included between z and δz is pδp.δθ.δz: <s>and</s> its attraction on the point I, supposing its density to be d, is d.pδp.δθ.δz/[p<sup>2</sup> + z<sup>2</sup>]: and the resolved part of this, in the vertical direction, is d.pδp.δθ.zδz/(p<sup>2</sup> + z<sup>2</sup>)<sup>3/2</sup>. Integrating with respect to z between the limits z = 0 and z = <s>r</s>c = UI, we have d.pδp.δθ.(1/p  1/(p<sup>2</sup> + c<sup>2</sup>)<sup>1/2</sup>) Integrating with respect to θ for the whole circumference we have 2π.d.(δp  pδp/(p<sup>2</sup> + c<sup>2</sup>)<sup>1/2</sup>). Integrating with respect to <s>[?]</s> p, we have 2π.d{p + c  (p<sup>2</sup> + c<sup>2</sup>)<sup>1/2</sup>)}. This is the attraction upwards on the point I. The attraction downwards on the point U will be the same: and thus the difference of attractions on U and I, estimated in the downwards direction, will be 4π.d{p + c  (p<sup>2</sup> + c<sup>2</sup>)<sup>1/2</sup>)}. If the planes be continued without limit, <s>[?]</s> or p
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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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