Items related to MATHMATICAL PAPERS
Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1891 Excerpt: ...determination should have been (and in the latter part of the memoir is assumed to be) such as to render H1 H2, &c. equal to H, on making the substitution in question; that is, in the place of the formulae Ui =-(«$ + y$y + zfiz) U, U2 =, _ j-v (akdx + y$y + z,dz)2 U, &c, there ought to have been = JnJ) (®ldx + Vldy + Z U' U = (n-2)(n-S) + yidy + Z U' &c' this error is corrected ante p. 189. The points of contact of the double tangents of the curve of the fourth order or quartic U = 0, are given as the intersections of the curve with a curve of the fourteenth order II = 0; the last-mentioned curve is not absolutely determinate, since instead of II = 0, we may, it is clear, write II + MU = 0, where M is an arbitrary function of the tenth order. I have in the memoir spoken of Hesse's original form (say Ili = 0) of the curve of the fourteenth order obtained by him in 1850, and of his transformed form (say II2 = 0) obtained in 1856. The method in the memoir itself (Mr Salmon's method) gives, in the case in question of a quartic curve, a third form, say n3 = 0. It appears by his paper " On the Determination of the Points of Contact of Double Tangents to an Algebraic Curve (Quart Math. Journ. vol. in. p. 317 (1859))," that Mr Salmon has verified by algebraic transformations the equivalence of the last-mentioned form with those of Hesse; but the process is not given. The object of the present memoir is to demonstrate the equivalence in question, viz. that of the equation n3 = 0 with the one or other of the equations nx = 0, n2 = 0, in virtue of the equation U=Q. The transformation depends, 1st, on a theorem used by Hesse for the deduction of his second form n2 = 0 from the original form nx = 0, which theorem is given in his paper &qu...
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