XIV. Account of pendulum experiments undertaken in the Harton Colliery, for the purpose of determining the mean density of the earth. & XV. Supplement to the “Account of pendulum experiments undertaken in the Harton Colliery;" being an account of experiments undertaken to determine the correction for the temperature of the pendulum

Abstract

1. In the spring of the year 1826, the idea occurred to me of attempting a determination of the Mean Density of the Earth by means of pendulum-experiments at the top and the bottom of a deep mine; and rough preliminary calculations, 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 mean density D, surrounded by a spherical shell of thickness 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 by 4π/3, R ³ D/R ² = 4π/3 RD ; and the attraction at the external surface of the shell is represented by 4π/3, R ³ , R ³ D + (R+ c ^{3 ___} - R 3 ) d /(R+ c ) ² , which to the first power of c /R is 4π/3 RD(1 - 2 c /R + 3 c /R. d /D). Calling the gravity at those two points respectively G and g , g /G = 1 - 2 c /R + 3 c /R. d /D; from which d /D = R/3 c . g /G - (R/3 c - 2/3 ). Considering the factions d /D g /G and as the only quantities in this equation liable to sensible numerical error, δ ( d /D) = R/3 c δ ( g /G). As the density of the shell, or d , may be ascertained by actual examination, δ ( d /D) = - d /D. δD/D ; and therefore δD/D = - D/ d . R/3 c δ g /G.