304 



IV. " On the Double Tangents of a Curve of the Fourth Order." 

 By ARTHUR CAYLEY, Esq., F.R.S. Received May 30, 1861. 



(Abstract.) 



The present memoir is intended to be supplementary to that " On the 

 Double Tangents of a Plane Curve," Phil. Trans, vol.cxlix. (1859) 

 pp. 193-212. I take the opportunity of correcting an error which 

 I have there fallen into, and which is rather a misleading one, viz. 

 the emanants U^ U 2 , . . were numerically determined in such man- 

 ner as to become equal to U on putting (o^, y lt * x ) equal to (z, y, z) ; 

 the numerical determination should have been (and in the latter part 

 of the memoir is assumed to be) such as to render H I} H 2 , &c. equal 

 to H, on making the substitution in question ; that is, in the place 

 of the formulae 



&c ' 



there ought to have been 



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 ; such last-men- 

 tioned curve is not absolutely determinate, since instead of 11 = 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 1^ = 0) of the curve of the fourteenth order obtained by 

 him in 1850, and of his transformed form (say II 2 = 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 

 II 3 = 0. It appears by the paper "On the Determination of the 

 Points of Contact of Double Tangents to an Algebraic Curve," Quart. 

 Math. Journ. vol. iii. 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, 



