194 
ME. A. CAYLEY ON THE DOLELE TAYGENTS OE A PLANE CTETE. 
be expressed in a simplified form, and he thus effected the actual reduction of m _to 
the form where E stiff contained the arbitrary qnantiues 
(a, jS, y) in the degree (m-2)(m-3). In particular for a quaidic curve, the equaUon 
E=0 was shown to be 
3aQ4”Q3=0, 
where the left-hand side is of the degree 2 in (a, (3, y) and the degree 16 m (it, y, z) ; 
and which should therefore by means of the equation U=0 be reducible so as to contam 
the factor (aa;'+/3?/+y2^)^- ^ ^ 
Jacobi’s paper, “ Beweiss des Satzes, dass eine Cmwe w-ten Grades im allpmemen 
in(n-2)(n^-9) Doppeltangenten hat,” Oelle, t. xl. pp. 237-260 (18o0), did not, I 
think, materially advance the solution of the question. In a letter to Jacobi,, dat^ the 
30th December, 1849, published at the conclusion of the last-mentioned paper, KessE 
gave the equation of the curve of the 14th order for the points of contact of the double 
tangents of a quartic, viz. in my notation. 
(a, B, c, jr, e, a,H, 3,Hf-H(a, b, c, jf, <b, iib,, a,, 3 Jh=o. 
and the demonstration is given in Hesse’s paper, “ Ueber die ganzen homogenen 
Functionen von der dritten und vierten Ordnung zwischen diei Variabeln,’ Civl/e, t. - . 
pp. 285-292 (1851), and is reproduced in Mr. Salmon’s Treatise on the Higher Biane 
Curves (1852). Two very interesting memoirs by Hesse and Steinee, Crelle,^ t xhx. 
(1855), relate to the geometrical theory of the double tangents of a quartic, and it is not 
necessary to refer to them more particularly. It is to be observed that the cur^e 
which determines the points of contact of the double tangents is not absolutely deter- 
minate ; for we may, it is clear, in the place of nU=0, write HU-f M.U_0, whereM is 
an arbitrary function of the proper degree; a very elegant transformation in the case o 
the quartic is given in Hesse’s paper, “Transformation der Gleichung der CmTenllteu 
Grades, welche eine gegebene Curve 4ten Grades in den Beruhrmigspmicten ihrei 
Doppeltangenten schneiden,” Crelle, t. Iff. pp. 97-103 (1856). 
Mr. Salmon’s work above referred to, contains the fundamental theorem o t^ e 
qential of a cubic, viz. a tangent to a cubic meets the cubic in a third point which les 
on the second or line polar of the point of contact with respect to the Hessian. In mi 
“ Memoir on Curves of the Third Order I gave an identical equation relatmg to the 
tangential of a cubic, but which is not there exhibited in its proper form ; this 
wards effected by Mr. Salmon, in the paper “ On Curves of the Thu-d Order f- 
equation, as given by Mr. Salmon, is in the notation of the present memoir, 
-?^.U+i».DU-iDH.|3T+H.T=0, 
an equation which in fact puts in evidence the last-mentioned theorem for the tangential 
of a cubic. 
* PliilosopHcal Transactions, vol. cxlvii. (1857), pp. 415-416, art. No. 37. 
t Ibid. vol. cslviii. (1858), pp. 535-541. 
