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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                                <s>44</s>43. For <s>[?]</s> ascertaining the probable error <u>e</u> I have used
the following process.
     Let E<sub0</sub>, E<sub>1</sub>, E<sub>2</sub>, &c, as before, be the actual
errors of comparison, estimated by the effect which they
produce on a 4-hours' rate, in the 8<sup>th</sup> decimal place
of logarithms: Let R<sub>1</sub>, R<sub>2</sub>, &c, be the successive
individual results for the rate of the lower pendulum
on the upper pendulum: and let  A be the adopted
value of that rate. Now if A <s>[?]</s> were rigorously correct,
we should have the following equations:
E<sub>0</sub> =             E<sub>0</sub>
E<sub>1</sub> =  E<sub>0</sub> + R<sub>1</sub> - A = E<sub>0</sub> + R<sub>1</sub>      - A
E<sub>2</sub> = E<sub>1</sub> + R<sub>2</sub> - A  = E<sub>0</sub> + (R<sub>1</sub> + R<sub>2</sub>) - 2A
E<sub>3</sub> = E<sub>2</sub> + R<sub>3</sub> - A = E<sub>0</sub> + (R<sub>1</sub> + R<sub>2</sub> + R<sub>3</sub>) - 3A
        &c
and, adding all for the First Series, which terminates
with R<sub>26</sub> and E<sub>26</sub>,

E<sub>0</sub> + E<sub>1</sub> + &c + E<sub>26</sub> = 27E<sub>0</sub> +26R<sub>1</sub> + 25R<sub>2</sub> + &c + R<sub>26</sub> - [26.25/2]A
E<sub>0</sub> is yet undetermined. Now the Theory of Probabilities
which we have used requires that the chances of positive
and negative errors be equal, and therefore that
(subject to the irregularities of chance) E<sub>0</sub> + E<sub>1</sub> + &c + E<sub>26</sub> = 0
This gives E<sub>0</sub> = [26.25/2.27]A - 1/27 (26R<sub>1</sub> + 25R<sub>2</sub> + &c + R<sub>26</sub>): 
                            
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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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