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XI. On the Double Tangents of a Plane Curve. By A. Cayley, Esg., F.B.S. 
Eeceived March 17, — Bead April 14, 1859. 
It was first shown by Pluckee on geometrical principles, that the number of the double 
tangents of a plane curve of the order m was : see the note, “ Solution 
d’une question fondamentale concemant la theorie generale des Courbes,” Crelle., t. xii. 
pp. 105-108 (1834), and the “ Theorie der algebraischen Curven” (1839). The memoir 
by Hesse, “Ueber die Wendepuncte der Curven dritter Ordnung,” Crelle^ t. xxviii. 
pp. 97-107 (1844), contains the analytical solution of the allied easier problem of the 
determination of the points of inflexion of a plane curve. In the memoir, “ Recherches 
sur relimination et sur la theorie des Courbes,” Crelle, t. xxxiv. pp. 30-45 (1847), I 
showed how the problem of double tangents admitted of an analytical solution, viz. if 
U = 0 is the equation of the cuiwe, L, M, N the first derived functions of U, and 
D=«(MB,-NB,)+^(NB,-LB,)+y(LB,-MB.) 
(where a, j3, y are arbitrary), then the points of contact of the double tangents are 
given as the intersections of the curve U = 0, with a curve the equation whereof is in 
the first instance obtained under the form [Y] = 0; [Y] being a given function of 
D*U, D®U, ..D'"U of the degree — m — 6 in respect of («, f3, y), the degree 
TO®— 2m^— lOm+12 
in respect of (^, y, z), and the degree to®+to — 12 in respect of the coefiicients of U. It 
was necessary, in order that the points of intersection should be independent of the 
arbitrary quantities (a, |3, y) that we should have identically 
[Y]=A.U+N.nU, 
N being of the degree to® — to — 6 in (a, j3, y), and consequently HU a function of (s, y, z) 
without (a, (5, y). Guided by Hesse’s investigation for the points of inflexion, I asserted 
that it was probable that N was of the form ; which being so, HU 
would be of the degree (to — 2)(to® — 9) in respect of (^, y, z), and the degree to®4-to — 12 
in respect of the coefiicients, and I was thus led to the theorem, “ On trouve les points 
de contact des tangentes doubles en combinant avec I’equation de la courbe une equation 
nU=0, de I’ordre (to — 2)(to® — 9) par rapport aux variables et de I’ordre to®+to — 12 
par rapport aux coefiicients — c’est a dire, puisqu’il corresponde deux points de contact 
a une tangente double, le nombre de ces tangentes est egal a - 5 to(to — 2)(to® — 9): theo- 
reme demontre indirectement par M. Pluckee.” 
Hesse, in the memoir “ Ueber Curven dritter Ordnung, See.” Crelle, t. xxxvi. pp. 143— 
176 (1848), showed how the components D®U, D’U, .. O’”!! of [Y] could each of them 
MDCCCLIX. 2 D 
