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Document number: 3055
Date: 22 Feb 1835
Recipient: LUBBOCK John William
Author: TALBOT William Henry Fox
Collection: Royal Society, London
Collection number: LUB 38 T6
Last updated: 3rd November 2012

Lacock Abbey
Feb. 22d 1835

Dear Sir

I am much obliged to you for sending me a copy of your paper on Abel*s Theorem. <1> This theorem is beautiful from its conciseness and comprehensive nature, but it is not as might be supposed a general solution of the problem. Altho* it seems to contain innumerable solutions of the same problem, yet nevertheless others exist which are not contained in it 每 For instance it does not give, as far as I can discover, any satisfactory solution of the equation mathematical equation the number of terms being two only. Nevertheless such a solution exists & may be readily found by other methods. 每 I find from Abel*s theorem only the 2 following solutions of it, viz. x + y = 0 or xy = 1

I shall endeavour to draw up a paper, or a series of papers, to be submitted to the Royal Society on the subject of the Integral Calculus <2> 每 I have found a simple & accurate solution of the integral ÷耳(X)dx + ÷耳(Y)dy + &c where X is any rational polynomial of the form a + bx + cx2 + dx3 + ex4# to any no of terms, but containing at least two distinct powers of x, and where 耳 is any function whatever. This I think will be admitted to be an important step gained in Analysis. 每 I should be glad to have your advice in drawing up this paper. Two methods present themselves to me, either to present the Theory as I think it ought to be stated in a treatise on the subject, or to show the successive steps by which I arrived at it. The latter would probably be the most satisfactory way, but I should be glad to enquire your opinion before I proceed further in drawing up the paper.

I observe an advertisement announcing ※Mathematical Researches by Mr Jerrard§ <3> specifying among other things that he has found a finite solution for algebraic equations of the fifth degree 每 Do you know anything of the work or of the author? As to the problem in question I am of opinion that its solution is impossible, but I think I shall procure his work and see how he sets about it. 每

Believe me to remain Dear Sir Yours very truly
H. F. Talbot


1. John William Lubbock, &On Some Elementary Applications of Abel*s Theorem*, Philosophical Magazine, v. 7, 1835, pp. 161每171.

2. WHFT, &Researches in the Integral Calculus, Part One* Philosophical Transactions of the Royal Society of London, v.126 part 1, 1836, pp.177每215 and WHFT, &Researches in the Integral Calculus, Part Two* Philosophical Transactions of the Royal Society of London

3. George Birch Jerrard (1804每1863), Mathematical Researches (1823每1825).

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