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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                                Then the errors in the results of the separate Swings,
produced by the errors of comparisons, are
(E<sub>1</sub> - E<sub>0</sub>), (E<sub>2</sub> - E<sub>1</sub>), (E<sub>3</sub> - E<sub>2</sub>),..... (E<sub>n</sub> - E<sub>n-1</sub>):
and, combining these with the weights
w<sub>1</sub> , w<sub>2</sub> , w<sub>3</sub>, ..... w<sub>n</sub>,
the <u>actual</u> error of the final result will be
[w<sub>1</sub> (E<sub>1</sub> - E<sub>0</sub>) + w<sub>2</sub> (E<sub>2</sub> - E<sub>1</sub>) + w<sub>3</sub> (E<sub>3</sub> - E<sub>2</sub>) + ..... w<sub>n</sub> (E<sub>n</sub> - E<sub>n-1</sub>)]/ [w<sub>1</sub> + w<sub>2</sub> + w<sub>3</sub> + ..... w<sub>n</sub>]

= [-w<sub>1</sub>E<sub>0</sub> + (w<sub>1</sub> - w<sub>2</sub>) E<sub>1</sub> + (w<sub>2</sub> - w<sub>3</sub>)E<sub>2</sub> + (w<sub>3</sub> - w<sub>4</sub>) E<sub>3</sub> ..... + w<sub>n</sub>E<sub>n</sub>]/[w<sub>1</sub> + w<sub>2</sub> + w<sub>3</sub> + ..... w<sub>n</sub>]

Hence, by the well known rules of the Calculus of Probabilities,
the square of the <u>probable</u> error of the final result will be
[w<sub>1</sub><sup>2</sup>.e<sub>0</sub><sup>2</sup> + (w<sub>1</sub> - w<sub>2</sub>)<sup>2</sup>.e<sub>1</sub><sup>2</sup> +  (w<sub>2</sub> - w<sub>3</sub>)<sup>2</sup>.e<sub>2</sub><sup>2</sup> + (w<sub>3</sub> - w<sub>4</sub>)<sup>2</sup>.e<sub>3</sub><sup>2</sup> + .....  w<sub>n</sub><sup>2</sup>.e<sub>n</sub><sup>2</sup>]/(w<sub>1</sub> + w<sub>2</sub> + w<sub>3</sub> + .....w<sub>n</sub>)<sup>2</sup>.

And if, for simplicity, we suppose all the comparisons
equally good, so that for e<sub>0</sub>, e<sub>1</sub>, e<sub>3</sub> &c we may put
e, the expression for the square of the <u>probable</u> error <s>[?]</s>  
the final result becomes

e<sup>2</sup> x [w<sub1</sub><sup>2</sup> + (w<sub>1</sub> - w<sub>2</sub>)<sup>2</sup> + (w<sub>2</sub> - w<sub>3</sub>)<sup>2</sup> + ..... w<sub>n</sub><sup>2</sup>]/(w<sub>1</sub> + w<sub>2</sub> + w<sub>3</sub> +..... w<sub>n</sub>)<sup>2</sup> :
and we have now to determine values of w<sub>1</sub>, w<sub>2</sub>, &c, which
will make this square of the final probable error a minimum.  
                            
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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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