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, 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= in 

 \n{n — 1) points. The system will be (n — l) ply infinite, since 

 there are \(n + 2)(n — 1) parameters subjected to \n{n — \) 

 conditions. Let v u = be one of the curves, and denote its points 

 of contact with Z7= collectively by a v Let the most general 

 (n — l) ic through a x be 



X^u + \ 2 v 12 + X 3 v ]3 + . . . + \ n v m = 0, 



where \ lt \ 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 

 n — 1, n — 2 such that 



v 12 -=v n v, 2 - Uw 1122 . 



In the same way quantics v rs , w nrs must exist such that 



v lr v ls = v u v rs - Uw nrs . 



The (n — 1) 1CS v 22> v 33 , . . ., v nn will each touch Um ^n (n — 1) points and 

 v rs will meet U in the points of contact of v„, v ss . The products 

 v rs Vij, VrfVgj, v. r jV si differ by expressions in which Z7isa factor. The 

 (n - l) ic V= 0, where V = V% + V^ 22 +...'+ 2X 1 \ 2 ^i 2 + 2\ 1 X 3 w I , + . . ., 

 touches U = at \n (n — 1) points and these points lie on each of 



the curves x— = 0, 5— = 0, . . . . 



The determinant of the expressions v, say A, is the discriminant 

 of V as a quadratic in X lt \ 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 U r ~ 1 as a factor. The determinant 

 A itself is then a constant multiple of U 71-1 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 ~- 

 multiplied by a linear expression. Let £/~ n_2 /3 rs be the minor of 

 v rs . The determinant formed by these minors is the (n - l) th 



