358 
ME. A. CAYLEY ON THE DOUBLE TANOENTS 
the deduction of his second form 112=0 from the original form ni=0, which theorem 
is given in his paper “ Transformation der Gleichung der Curven 14ten Grades welche 
eine gegebene Curve 4ten Grades in den Beriihrungspuncten ihrer Doppeltangenten 
schneiden,” Crelle, t. lii. pp. 97-103 (1856), containing the transformation in question; 
I prove this theorem in a different and (as it appears to me) more simple manner; 
2nd, on a theorem relating to a cubic curve proved incidentally in my memoh- “ On the 
Conic of Five-pointic ContacCat any point of a Plane Curve”,* the cubic curve being in 
the present case any first emanant of the given quartic curve : the demonstration occu- 
pies only a single paragraph, and it is here reproduced ; and I reproduce also Hesse’s 
demonstration of the equivalence of the two forms ni = 0 and 112=0. 
Let U = (*X‘L y-> be a quartic function of [x, y, z); (a, h, c,f, g, 1i) its second differen- 
tial coefficients; (a, b, c, f, g, h) the reciprocal system 
cib—h^, — hf-hg^fg-^ch). 
And let II be the Hessian of U, or determinant abc — af^ — hg'^ — cl^-\-1fgli (H is of 
course a sextic function of .r, z) ; (a', b', c',f', g', h') the second differential coefficients 
of H ; (a', b', c', f', g', ii') the reciprocal system 
(b’c'—f''\ c'a'—g'^, gK—af, b!/' —b'g' , f'g' —dh!). 
Then U=0 being the equation of a quartic curve, the equation of the curve of the 
fourteenth order which by its intersections determines the points of contact of the 
double tangents of the quartic curve, may be taken to be (Hesse’s original form) 
n.=(A, B, c, F, G, riX^^H, B^H, B,H)^— 3H(a, b, c, f, g B^)"H=0t. 
Or it may be taken to be (Hesse’s transformed form) 
n2=5(A, B, c, F, G, hXB^H, B,H, BJi)^-3(V, B', o', f', g', h'XB.U, B,U, B,U)^=0. 
And moreover, if Ui=-|(^iB^ + ?/iBy-}-2iBJU, and if Hj be the Hessian of Uj, and 
{a!', V, d',f", g", h") the second differential coefficients of H— 3Hi, where in the diffe- 
rentiations (Xi, y^, Zj) are" treated as constants but after the differentiations are effected 
they are replaced by (x, y, z), and if (a", b'', c’', f'', g", h") be the reciprocal system 
(JV-/^ d'd'-g"\ d'V'-W\ g"W-d'f \ Wf"-V'g\ f’g”-d'd% 
then the equation of the curve of the fourteenth order may be taken to be (Salmon’s 
form) 
^ n3=(A", B", c", F", G", h"xb,u, b^u, b.u) =0. 
I have preferred to wnlte the three equations in the foregoing forms ; but it is clear that 
the terms 
(a, b, c, f, g, nXB„ B^,, BJ^H ; (A, b', c', F, g', h'X^„ B^)^U 
might also have been written 
(a, b, c, f, g, hX«', V, d, 2/', 2g\ W ) ; (A, b', c', f', g', dja, c, 2/, 2g, 2h}. 
* Philosophical Transactions, vol. cxlix. (1859), see p. 335. 
t In quoting this formula in my former memoir, the numerical factor 3 is by mistake omitted. 
