Dec. 20. 1834
After finishing my letter to you yesterday, I examined the results you had obtained for the equation
My method is so simple and easy of comprehension that I resolved to try to find whether I could find the solution of this problem without the help of pen ink and paper, and I found no difficulty in doing so, and in ascertaining that the equation was satisfied by the following equations of condition viz.
This solution is however imaginary.
Your solution is that the problem is satisfied by the equations of condition which I found for the Hyperbola, which were
This solution is therefore totally different from the other, which is an interesting fact, if correct, which I see no reason to doubt.
Will you have the goodness to examine what solution is deducible from Abel’s theorem, <1> for the Hyperbola, in the case when only two integrals are combined. viz. or if you please
The solution which I found, was contained in my last paper presented to the R. Socy <2> – It was rather complicated
You seem to think that my theorem for the Hyperbola might have been deduced from the ordinary theory of Elliptic Integrals – If so, there must be some analogous property, belonging to Ellipses and hyperbolas in general, of which this is a particular case (where the hyperbola is equilateral). Should [illegible deleletion] be such a property, it would be very important to have it stated in precise terms. But I am inclined to doubt it. At any rate I should be extremely obliged to you if you will show me how the theorem in question can be demonstrated by the properties of Elliptic Integrals contained in the first work of Legendre <3> on that subject, which is what I suppose is meant by the “ordinary theory” or “known methods”.
I have tried my method on integrals with cubic radials, and have found a very simple equation which satisfies the following –
I don’t think Abel’s theorem will help us here.
Believe me to be
Yours very truly
1834 Chippenham December twenty H.F. Talbot
J.W. Lubbock Esqre
29 Eaton Place
1. Niels Henrik Abel (1802–1829), mathematician. His theorem, written in 1824, lost in Paris and then re-written in 1828, proved the impossibility of solving equations of the fifth degree (quintic equations) through the use of radicals.
2. WHFT, ‘On the Arcs of Certain Parabolic Curves’ was read before the Royal Society 12 June 1834, and published, Proceedings of the Royal Society of London, v. 3 no. 16, 1833–1834, pp. 287–288.
3. Adrien Marie Legendre (1752–1833), Traité des fonctions elliptiques et des intégrals eulériennes avec des tables pour en faciliter le calcul numérique (Paris: Huzard Courcier, 1825–1828).