TALBOT5065

Collingwood ¹
Sep. 13/44.

My dear Sir,

I suppose the most general definition of an Abelian integral might be taken to be this that between ⌠(x) and ⌠(_(x)) there shall subsist an algebraical relation or between several such functions. However in Abels & Jacobis ² discoveries I am not learned. As to above of the second order I suppose the most general definition of a Cone is a Surface the locus of all Straight lines passing thro' a fixed point according to a given law. Now if this law be that some other point in the line shall lie in a fixed curve of the second order (which may be a curve of double curvature by the bye) then will the cone be a cone of the second order. Now it does not of necessity follow that any section of such a cone must of necessity be an ellipse or circle - for instance if the directing curve be a common hyperbola or parabola the surface will have infinite sheets & can no way be cut into a reentering curve.-

The condition of the Cone being one of an the second order gives an equation
ʃ(x,y,z) = 0
when ʃ(xyx) is the general one of the 2^d order.- The other condition gives that when x is charged in the same ratio as y and z this equation shall still hold so that
ʃ(ax, ay, az) = 0
whatever a may be. This of course excludes all powers and products of x y z of dimension 1 or 0 so that
0 = px² + qy² + rz² + sxy + txz + uyz
will be the most general equations of such a cone. Is this

As to the integrals, I quite agree with you that the subject is not exhausted & is very tempting Your integrals if I remember right are a particular (but very extensive and interesting - in fact perhaps the most interesting) case of a general theorem of mine in the notes to Spence's Logarithmic Transcendents ³ - Perhaps you have not seen those notes. I will look out if I have a copy of them & send you. - In my specific applications of the principle only explicit functions satisfying a symmetrical equation Ẋ {x,y} = 0 when Ẋ is the sign of a symmetrical function, are used. - You, by extending your views most happily to implicit functions such as satisfy for example algebra in symmetrical equations of the 4^th, 6^th, &c, degree gave tapped a fresh spring out of which have welled forth some very sparkling & delicious theorems There yet remains another rock to be struck to which a slight knack has been given in the "notes" above mentioned (of which I have found a copy - my only remaining one) and from which I have no doubt a very abundant supply of elegant and valuable formula would burst out and I regret my disuse of mathematics & fifty other reasons w^h prevent my going to work at it.

Believe me dear sir yours ever trl^y
JFW Herschel

PS. - a very curious photographic novelty occurred to me a day or 2 ago (I have however long given up any but occasional and accidental experiments of this kind) - When a paper is washed with a mixture of nitrate of ⁴ and Ferrotartaric acid it becomes very sensitive & gives negative dormant pictures brought out by breathing on ⁵ - Now when such a negative picture is impressed on this paper, turn the back of it for 20 or 30 seconds to the sun after withdrawing it, by slow degrees in a few minutes, a positive impression arises at the back, which before shewed no inclination of photographic action. -

P.S. To prepare Ferro tartaric acid precipitate Tartrate of Ammonia & Iron by lead and decompose the lead salt by weak sulphuric acid. -

H. F. Talbot Esq