Lacock Abbey
Feb. 22^d 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 ¨C For instance it does not give, as far as I can discover, any satisfactory solution of the equation the number of terms being two only. Nevertheless such a solution exists & may be readily found by other methods. ¨C 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> ¨C I have found a simple & accurate solution of the integral ¡Ò¦Õ(X)dx + ¡Ò¦Õ(Y)dy + &c where X is any rational polynomial of the form a + bx + cx² + dx³ + ex⁴¡­ to any n^o of terms, but containing at least two distinct powers of x, and where ¦Õ is any function whatever. This I think will be admitted to be an important step gained in Analysis. ¨C 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 M^r Jerrard¡± ^<3> specifying among other things that he has found a finite solution for algebraic equations of the fifth degree ¨C 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. ¨C

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

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Notes:

1. John William Lubbock, ¡®On Some Elementary Applications of Abel¡¯s Theorem¡¯, Philosophical Magazine, v. 7, 1835, pp. 161¨C171.

2. WHFT, ¡®Researches in the Integral Calculus, Part One¡¯ Philosophical Transactions of the Royal Society of London, v.126 part 1, 1836, pp.177¨C215 and WHFT, ¡®Researches in the Integral Calculus, Part Two¡¯ Philosophical Transactions of the Royal Society of London

3. George Birch Jerrard (1804¨C1863), Mathematical Researches (1823¨C1825).