# Account of pendulum experiments undertaken in the Harton Colliery, for the purpose of determining the mean density of the earth, by G. B. Airy ```                                Putting now r = a(1 - εcos<sup>2</sup>θ, w<sup>2</sup> = m E/a<sup>3</sup>, we find
g = E/a<sup>2</sup>(1 + 2εcos<sup>2</sup>θ) - 3E/a<sup>2</sup>(ε - m/2)(cos<sup>2</sup>θ - 1/3) - m E/a<sup>2</sup>(1 - εcos<sup>2</sup>θ)

= E/a<sup>2</sup>{1 + (5m/2 - ε)cos<sup>2</sup>θ + ε - 3m/2}

-dg/dv = 2E/a<sup>3</sup>(1 + 3εcos<sup>2</sup>θ) - 12E/a<sup>3</sup>(ε - m/2)(cos<sup>2</sup>θ - 1/3) + mE/a<sup>3</sup>(1 - cos<sup>2</sup>θ)

= 2E/a<sup>3</sup>{1 + (5m/2 - 3ε)cos<sup>2</sup>θ + 2ε - m/2}

whence
-1/g .dg/dv = 2/a{1 + (5m/2 - 3ε)cos<sup>2</sup>θ + 2ε - m/2
- (5m/2 - ε)cos<sup>2</sup>θ - ε + 3m/2}

= 2/a{1 - 2εcos<sup>2</sup>θ + ε + m}
and therefore
F = 1 + 2c/a{1 - 2εcos<sup>2</sup>θ + ε + m}
Now the method adopted in the "Account of
Experiments In," article 57, gives <s>[?]</s>
-1/g .dg/dr <s>[?]</s> = 2c/r =  2c/a(1 + εcos<sup>2</sup>θ)

whence F = 1 + 2c/a (1 + εcos<sup>2</sup>θ)

Therefore if R be the ratio of the value of F - 1
given above to F - 1 as calculated by the method
of the "Account of Experiments",

R = 1 - [2εcos<sup>2</sup>θ + ε + m]/1 + εcos<sup>2</sup>θ = 1 - 3εcos<sup>2</sup>θ + ε + m.
```
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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