82 



can be expressed in terms of x, then the discriminant becomes a bitan- 

 gential of the n-ic. It has been shown by Jacobi and Clebsch that this is 

 always possible. 



We shall write equation (3) as 



Ao ?."-* + (n — 2) Ai lr* /t + (n ~ 2 g \ U ~ S) A 2 ^ \fl + • • + 

 (n — 2) A„-3 fy n ~ 3 + An-2 m"- 2 = o, ... (4) 

 where we have 



. n(n — 1) oo» n (n — 1) „ „ 



Ao = \ 2 a.x" 2 a y 2 , Ai = ^ g ' a x " « a/, 



Ar -rr + D(r + 2) ax % + - 



If equation (4) is a quadratic, that is, if the n-ic is a quartic, the dis- 

 criminant of (4) is 



— -^ (An A 2 — Ai 2 ) = O, 



and after y is expressed in terms of x there is no irrelevant factor. 

 If the n-ic be the quintic, the discriminant of (4) is 



_ |I (G 2 + 4 H») = O, 



where we put H = Ao A 2 — A? and G = A 2 , A3 — 3 Ao Ai A 2 + 2 Af , and the 

 y can easily be expressed in terms of x for the functions G and H, but the 

 result contains the square of the hessian of the quintic as an irrelevant 

 factor. This factor can be discarded without difficulty by putting 



G 2 + 4 H 3 = A 2 j (Ao A3 — Ai A2) 2 — 4 (Ao A 2 — A 2 ) (Ai A3 — A 2 ; 



and then expressing y in terms of x for each parenthesis separately. 

 If the n-ic be the sextic, the discriminant of (4) is 



7° (P — 27J 2 ) = O, 



■^■0 



where I = Ao A 4 — 4 A Ai, A3 + 3A 2 and A* J = Ao H I — G 2 — 4H 3 . 



There is no difficulty in expressing y in terms of x for the function I, 

 and therefore, by multiplying and dividing the discriminant by A n , we 

 can immediately write a bitangential of the sextic by substituting the re- 

 sults obtained for the quartic and quintic in 



256 



41 



~ { A„ F — 27 (Ao H I — G 2 — 4 H») 1=0. 



