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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                                Of which the law is evident. The second difference is constant,
and = -2b.

Substituting these in the expressions for N and S, we find
(after all reductions)
     N = b<sup>2</sup>/3 . n . (n+1).(n+2)
     S = b/6 . n. (n+1).(n+2)
and the equation N/S = 2b becomes identical. Therefore
b is arbitrary. For convenience, make b = 1. Then the
weights for the results of the successive Swings are,
n, 2n-2, 3n-6, 4n-12, &c.

     The square of the probable error of the final result was
found = e<sup>2</sup> x N/S<sup>2</sup>. Substituting, this becomes

e<sup>2</sup> x 12/n.(n+1).(n+2): or the probable error = e x √ 12/n.(n+1).(n+2).

41. It will be instructive to contrast this result with
the result obtained on two other suppositions.
     First, suppose that the Swings had been continuous,
but that there had been no intermediate comparisons of
clocks. The probable error of the first comparison being e,
and that of the last comparison being also e, the probable
error in their combination by substitution will be e√2:
and as this applies to n Swings, the probable error
on the mean = 1/n e √2 = e√2/n<sup>2</sup>.  Comparing this
with the probable error found above, it appears that the 
                            
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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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