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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                                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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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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