Lacock Abbey

Jan^{ry} 8. 1836

My Dear Sir

I send you a parcel by Lawes’s Bath & Chippenham coach this evening, containing my memoir. ^{<1>}

It ought to reach you tomorrow morning, therefore have the goodness to inform me by return of post if you have received it, as parcels are sometimes mislaid at the coach offices.

My former memoir ^{<2>} on the subject I wish to be considered as withdrawn – Part of it will be incorporated with the present, in its proper place.– I think I have performed rather more than I mentioned, in a former letter to you, which was, to give a method of finding the algebraic sum of a series of integrals, of the form ∫φX.dx X being a polynomial with entire powers of x and φ any function of it. But besides this I have given the solution of when x′ is another polynomial, inferior or superior in degree to X by two unities __at least__, so that they are respectively of the n and n ± 2 degree.

I suppose this last limitation may be got rid of, tho’ I do not at present see how, and in that case we should have the solution of ∫φZ.dx Z being any rational function of x.

I expect to be in Town in the course of a fortnight, & hope I may then have the pleasure of seeing you.

I observe by the Report of the British Association for last year that a sum of money has been appropriated for the purpose of verifying M^{r} Jerrard’s solution ^{<3>} of equations by arithmetical examples. That will be the most satisfactory method of examination. I have not yet seen any __clear__ statement of his method, and until verified in particular examples I shall retain doubts of its accuracy.

In Gergonne’s ^{<4>} Annales des Mathématiques (vol. 11 p. 374) a method is given by M. Berndtson, ^{<5>} a Swede, of solving the equation x^{n} − x = k (n being an odd number).

His solution is erroneous, but it is by no means easy to prove it so, because it is in fact a very close approximation. But I found that by assigning a proper value to k, it might be deduced from his theorem that a and b being whole numbers, which being demonstrably false it follows that his theorem (of which indeed he offers no proof) is erroneous. The Editor (Gergonne) admits that he could neither verify nor disprove the theorem; doubtless because he found on trial that it gave a correct result for the first 3 or 4 decimals in those arithmetical examples which he tried. I hope that M^{r} Jerrard’s formulæ will be soon exhibited in a distinct & intelligible form. It is against him that Ruffini ^{<6>} (and I believe Abel ^{<7>}) have demonstrated that the problem ^{<8>} is impossible.

Believe me to remain Yours very Truly

H. F. Talbot

#### Notes:

**1**. WHFT, ‘Researches in the Integral Calculus, Part One’ *Philosophical Transactions of the Royal Society of London*, v. 126 part 1, 1836, pp. 177–215.

**2**. WHFT’s article entitled ‘On the rectification of Parabolic Curves’, pending at this time before the Royal Society Committee of Papers. Minutes to the meetings of this committee record the paper from 3 July 1834 until 7 January 1836. It was withdrawn at the meeting of 28 January 1836.

**3**. George Jerrard (1804–1863), mathematician, generalized Erland Bring’s 1786 reduction of a quintic equation to show that a transformation could be applied to an equation of a degree of n to remove several of the terms. It is believed that he knew nothing of Bring’s reduction, and the removal of terms was not considered to be an adequate solution of the problem of equations above the quadratic.

**4**. Joseph Diez Gergonne (1771–1859), French mathematician.

**5**. Bernard Berndtson, civil servant, ‘Note sur la Résolution d’une classe particulière d’équations algébriques’, *Annales de Mathématiques pure et appliquées*, v. 11 no. 12, June 1821, pp. 374–375.

**6**. Paolo Ruffini (1765–1822), mathematician.

**7**. Niels Henrik Abel (1802–1829), mathematician.

**8**. The problem of solving equations of the 5^{th} degree (quintic equations) or higher.