My dear Sir,

I hope you got my last. I have lately had more time to consider the application of Abel’s 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 Abel’s.

But I write this to rectify in one particular what I stated before. It would seem that Abel’s theorem applied to integrals of the form

requires some little modification; at all counts, the integral

may be made to depend immediately upon

which is reached at once by Abel’s theorem, & I arrive by this means at your result. So also for example

the equations of condition between, p, q & r bring those which you have employed. å_{1}, å_{2}, å_{3} 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 Dec^{r} 1834

The Hon^{ble} 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.