ON THE DOUBLE TANGENTS OF A PLANE CURVE. 
[From the Philosophical Transactions of the Royal Society of London, vol. cxlix. for the 
year 1859, pp. 193—212. Received March 17,—Read April 14, 1859.] 
It was first shown by Plücker on geometrical principles, that the number of 
the double tangents of a plane curve of the order m was — 2)(m 2 — 9): see the 
note, “Solution d’une question fondamentale concernant la théorie générale des Courbes,” 
Crelle, t. xii. pp. 105—108 (1834), and the “Théorie 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 l’élimination et sur la théorie des Courbes,” Crelle, t. xxxiv. (1847), 
pp. 30—45, [53], I showed how the problem of double tangents admitted of an analytical 
solution, viz. if U= 0 is the equation of the curve, L, M, A the first derived functions 
of U, and 
D = a (Md z - Nd y ) + /3 (Nd v - Ld z ) + 7 (Ld z - Md x ) 
(where a, /3, 7 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 [F] = 0; [F] being a given function of 
D 2 t7, 1?U, ... D m U, 
of the degree m 2 — m — 6 in respect of (a, /3, 7), the degree m? — 2ni 2 — 10/n + 12 in 
respect of (x, y, z), and the degree m 2 + m — 12 in respect of the coefficients of U. 
It was necessary, in order that the points of intersection should be independent of the 
arbitrary quantities (a, /3, 7), that we should have identically