350 Dr Dixon , Note on the Reduction of a Ternary Quantic 
Note on the Reduction of a Ternary Quantic to a Symmetrical 
Determinant. By Dr A. C. Dixon. 
[ Received 25 January 1902.] 
The reduction of a ternary quartic to the form of a symmetri- 
cal determinant of the fourth order with linear constituents is 
important in the theory of the bitangents. The object of this note 
is to discuss the corresponding problem for a plane curve of any 
degree. 
Let U be an n-ic in x, y y z. The problem is to express U as a 
symmetrical determinant of order n whose constituents shall be 
linear in x, y, z. 
Consider the system of curves of degree n — 1 touching U=0 in 
\n (n — 1) points. The system will be (n — l) ply infinite, since 
there are \ (n 4 2) (n — 1) parameters subjected to ^ n (n — 1) 
conditions. Let v n = 0 be one of the curves, and denote its points 
of contact with U= 0 collectively by ct x . Let the most general 
(n — l) ic through oti be 
^1^11 + ^2^12 + ^-3^13 4 • • • 4 h. n V m = 0, 
where \ u \ 2 , ..., \ n are the arbitrary parameters. 
Then the curve v 12 2 =0 passes through all the intersections of 
v n = 0, U=0. Hence quantics v 22 , w n22 must exist of degrees 
w — 1 , n — 2 such that 
Vyi — ^ 11^22 Dw n22 . 
In the same way quantics v rs , iu urs must exist such that 
V lr V ls = v u v rs - Uw nrs . 
The (n — 1) 1CS v 22 , v^, . . ., v nn will each touch U in \n (n — 1) points and 
v rs will meet U in the points of contact of v rr , v ss . The products 
v rs Vij, v r iV S j, v r jVsi differ by expressions in which U i sa factor. The 
(n — l) ic V— 0 , where V = XfVu + \ 2 v & 4 . . .+ 2\ik 2 v 12 4 2\ik 3 v 13 4 . . 
touches U= 0 at \n {n — 1) points and these points lie on each of 
dV n dv _ 
the curves = 0, r— = 0, . . . . 
U A/j o\ 2 
The determinant of the expressions v, say A, is the discriminant 
of V as a quadratic in X lt X 2 , ... , and we have seen that each of its 
minors of the second order contains U as a factor. Hence each 
minor of order r contains Z7 r_1 as a factor. The determinant 
A itself is then a constant multiple of U n ~ l and its first minors 
contain the factor U n ~ 2 , whose degree is n (n — 2). But each first 
minor is of the degree (n — l) 2 and is therefore equal to U n ~ 2 
multiplied by a linear expression. Let U n ~ 2 /3 rs be the minor of 
v rs . The determinant formed by these minors is the (n — l) th 
