# Account of pendulum experiments undertaken in the Harton Colliery, for the purpose of determining the mean density of the earth, by G. B. Airy ```                                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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## Cite as

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