# Account of pendulum experiments undertaken in the Harton Colliery, for the purpose of determining the mean density of the earth, by G. B. Airy ```                                For convenience, call this fraction e<sup>n</sup> x N/S<sup>2</sup>.

Differentiate with respect to w<sub>1</sub>, and make the differential coefficient = 0;
S x (2w<sub>1</sub> - w<sub>2</sub>) - N = 0, or 2w<sub>1</sub> - w<sub>2</sub> = N/S.

Differentiate with respect to w<sub>2</sub>, and make the differential coefficient = 0;
S x (2w<sub>2</sub> - w<sub>1</sub> - w<sub>3</sub>) - N = 0, or 2w<sub>2</sub> - w<sub>1</sub> - w<sub>3</sub> = N/S

Similarly 2w<sub>3</sub> - w<sub>2</sub> - w<sub>4</sub> = N/S

...................
2w<sub>n-1</sub> - w<sub>n-2</sub> -w<sub>n</sub> = N/S
2w<sub>n</sub> - w<sub>n-1</sub> = N/S

Let N/S = 2b, the value of b being at present unknown
or perhaps arbitrary. Then

w<sub>1</sub> = w<sub>1</sub>   = w<sub>1</sub>
w<sub>2</sub> = 2w<sub>1</sub> - 2b     = 2w<sub>1</sub> -2b
w<sub>3</sub> = 2w<sub>2</sub> - w<sub>1</sub> - 2b = 3w<sub>1</sub> - 6b
w<sub>4</sub> = 2w<sub>3</sub> - w<sub>2</sub> - 2b     = 4w<sub>1</sub> - 12b
...............
w<sub>n-1</sub> =    (n-1)w<sub>1</sub> - (n-1)(n-2)b
w<sub>n</sub> =    nw<sub>1</sub> - n(n-1)b

Substituting the two last in the equation 2w<sub>n</sub> - w<sub>n-1</sub> = 2b,
we obtain  w<sub>1</sub> = nb;
and, substituting this in the other expression,
w<sub>2</sub> = (2n-2)b
w<sub>3</sub> = (3n-6)b
w<sub>4</sub> = (4n-12)b
w<sub>5</sub> = (5n-20)b
&c
```
images

## 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