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XYII. On fne Double Tangents of a Curve of the Fourth Order. 
By Aethue Cayley, Fsg.., F.R.S. 
Eeceived May 30, — Eead June 20, 1861. 
The present memoir is intended to be supplementary to that “ On the Double Tangents 
of a Plane Curve.”* I take the opportunity of correcting an error which I have there 
fallen into, and which is rather a misleading one, ^dz. the emanants Ui, U 2 , . . were 
numerically determined in such manner as to become equal to U on putting (a^i, ^' 1 , ^i) 
equal to {x, y, z); the numerical determination should have been (and in the latter part 
of the memoir is assumed to be) such as to render Hj, H 2 , &c. equal to H, on making 
the substitution in question ; that is, in the place of the formulae 
there ought to have been 
U2= ^ - 2)(^ - _3) + JU, &C. 
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 11=0; the last-mentioned curve is not absolutely determinate, since instead of 
11=0, we may, it is clear, write n-|-MU = 0, where M is an arbitrary function of the 
tenth order. I have in the memoir spoken of Hesse’s original form (say ni = 0) of the 
curve of the fourteenth order obtained by him in 1850, and of his transformed form 
(say n2=0) obtained in 1856. The method in the memoir itself (Mr. Salmon’s method) 
gives, in the case in question of a c^uartic 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,” f that Mr. Salmon has verified by algebraic transformations the equi- 
valence 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 n 3=0 with the one or other of the equations ni = 0, Ha^O, in virtue of 
the equation U=0. The transformation depends, 1st, on a theorem used by Hesse for 
* PhilosopHcal Transactions, vol. cxlis. (1859) pp. 193-212. 
t Quart. Math. Joum. vol. iii. p. 317 (1859). 
