31 Sackville S^{t} ^{<1>}

Tuesday

Dear Sir

I think Sir W. Hamilton’s statement ^{<2>} is very clear, & he seems decidedly of opinion __now__ that Mr J’s method ^{<3>} is erroneous – But even if the first part of it were correct, I do not see that the Problem would be solved.

Let the given equation be x^{5} + x = a

Let y = P + Qx + Rx^{2} + … + Zx^{9} [1]

Suppose it possible to determine the coëff^{ts} P, Q &c so that the resulting equation in y may be of Demoivre’s ^{<4>} solvible <sic> form, therefore the value of y becomes known.

But even granting this, it is not the thing required, which is to find the value of x.

Now x is only found from y, by solving the eq^{n} of the 9^{th} degree [1]

But I do not at all admit that the coëff^{ts} P, Q &c can be determined in the manner required without the solution of an equation of the same degree with the proposed.

Believe me

Yours very Truly

H. F. Talbot

#### Notes:

**1**. 31 Sackville Street, London residence of the Feildings, often used as a London base by WHFT.

**2**. Sir William Rowan Hamilton (1805–1865), Irish mathematician.

**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**. Abraham de Moivre (1667–1754), mathematician.