Park Square

Friday 16 May

My dear Sir

I have taken it for granted that x is the measure of the abscissa of the spectrum, beginning at the brightest point, and on such a scale that the distance of the remotest corresponding lines shall be a whole circumference or 360º: these lines depending on the effect of any neighbouring pairs of striae. I also suppose the light to be reflected at a moderate obliquity of the general surface (of Mr. Barton’s ^{<1>} coat buttons), so that the fringes shall be nearly equidistant. Then for each value of x I consider that period of time only at which the oscillation of the principal pencil either commences there, or is half performed, the curve representing in the first case the place of the particle, in the second its velocity.

It is then obvious that at the supposed beginning of the oscillation, for the central line, where x = 0, cos 0 will be the value of the velocity for each separate pencil and that at any distance x from this point, the arc x will represent either the state of the first oscillation after a corresponding period or that of the oscillation of the second pencil when the first commences its oscillation at that point and that each successive line on the button will send an oscillation retarded by the arc nx, if n denotes its order reckoned from the first. Hence it follows that the velocities to be combined at that point in the middle of the principal oscillation will always be cos 0, cos x, cos 2x – and so forth.

In the case that you put, of course t is synonymous with x, and the three terms are cos 0, cos t and cos 2t; or 1, cos 120º and cos 240º the sum being 0 – In general the sum of my series is very nearly ^{sin nx} / _{x} .

Your experiment on the yellow light ^{<2>} is very interesting; I think Herschel ^{<3>} mentions it – It is singular how great a difficulty I had to convince so accurate a philosopher as Wollaston ^{<4>} that he was mistaken in asserting that yellow light was __always__ a compound

I am not deeply read in all parts of the Mécanique Céleste: nor do I know who first demonstrated the instability of the ~~Instability~~ equilibrium of the ring of Saturn if at rest but no doubt it was the first person that attempted it; for it scarcely seems to want a demonstration, any more than the Stability of the equilibrium of the sea, which Laplace ^{<5>} thought so clever. It is indeed only to one who has considered that the equilibrium of a spherical __shell__ ~~is~~ would be neither stable nor unstable, but neutral, that it could occur to inquire how far the same neutrality might extend to a ring: but he would immediately decide in the negative

Yours very truly

T. Y.

H. Fox Talbot Esq

31 Sackville Street –

#### Notes:

**1**. Sir John Barton patented a method of making iris metal ornaments, where extremely minute lines refracted the light and created prismatic colours. See P. Grodzinksi, ‘A Ruling Engine Used by Sir John Barton – and Its Products’, *Industrial Diamond Review*, v. 8, February 1948, pp. 37–42.

**2**. WHFT, ‘Some Experiments on Coloured Flames’, *The Edinburgh Journal of Science*, v.5, n.1, July 1826, pp. 77–81.

**3**. Sir John Frederick William Herschel (1792–1871), astronomer & scientist.

**4**. William Hyde Wollaston (1766–1828), physicist.

**5**. Pierre Simon, Marquis de Laplace, *Traité de mécanique céleste* (Paris: Crapelet, 1799–1825).