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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                                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 end-surface 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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