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Document number: 3021
Date: 18 Dec 1834
Recipient: TALBOT William Henry Fox
Author: LUBBOCK John William
Collection: British Library, London, Manuscripts - Fox Talbot Collection
Collection number historic: LA34-48
Last updated: 29th April 2012

My dear Sir,

I hope you got my last. I have lately had more time to consider the application of Abelís theorem <1> to your question & some others, having been before very much taken up with other subjects.

Your have no doubt ere this seen that your theorem about the arms of a hyperbola is one of the simplest cases of Abelís.

But I write this to rectify in one particular what I stated before. It would seem that Abelís theorem applied to integrals of the formmathematical equation
requires some little modification; at all counts, the integral
mathematical equation
may be made to depend immediately upon
mathematical equation
which is reached at once by Abelís theorem, & I arrive by this means at your result. So also for example
mathematical equation
the equations of condition between, p, q & r bring those which you have employed. Ś1, Ś2, Ś3 being either +1 or −1. These integrals however of course in which the radical does not exceed four dimensions may be transformed into elliptic integrals by known methods.

I am, my dear Sir, Yours faithfully
J W Lubbock

18 Decr 1834

The Honble H F Talbot
Lacock Abbey
Chippenham


Notes:

1. Written in 1824, lost in Paris and re-written in 1828, this theorem proved the impossibility of solving equations of the fifth degree (quintic equations) through the use of radicals.

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