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 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 + cx^{2} + dx^{3} + ex^{4}＃ to any n^{o} 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 M^{r} 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

#### Notes:

**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).