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 2d 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 4th, 6th, &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 wh prevent my going to work at it.
Believe me dear sir yours ever trly
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, 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 13th 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".