TALBOT3026

Lacock,
Dec. 20. 1834

Dear Sir,

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. Soc^{y 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
H.F. Talbot

1834 Chippenham December twenty H.F. Talbot
J.W. Lubbock Esq^re
29 Eaton Place
Mildred Court
London