Feb. 22d 1835
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 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).