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 = px^{2} + qy^{2} + rz^{2} + 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

^{<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,

__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

#### Notes:

**1**. Hawkhurst, Kent.

**2**. Niels Henrik Abel (1802-1829) and Carl Gustav Jacobi (1804-1851), both mathematicians who worked on the solution of equations of higher functions by the introduction of integrals.

**3**. John Frederick William Herschel, *Note on an application of the inverse theory of functions to the integral calculus. (From the works of the late W. Spence)* (London: 1819). A commentary on William Spence (1778-1815), *An essay on the theory of the various orders of logarithmic transcendents; with an inquiry into their application to the integral calculus and the summation of series* (London: John Murray, 1809).

**4**. The alchemical symbol for silver.

**5**. Herschel's Breath Printing was described at the 13^{th} meeting of the British Association for the Advancement of Science in Cork in 1843, in an article entitled "Notice of a remarkable Photographic Process by which dormant Pictures are produced capable of development by the Breath or by keeping in a Moist Atmosphere".