82 



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

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

 always possible. 



We shall write equation (3) as 



Ao A>i-2 + (n — 2) Ai A»-3 ^ + ^" ~ ^2 !^ ~ ^^ ^' '^""^ ^'^ + " "^ 

 (n — 2) An-3 V"3 + A„-2 u"-2 = o, ... (4) 

 where we have 



_ n(n-l) , 2 „ 2 Ai - "" ^^^:^^ a " •^ a « 



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

 criminant of (4) is 



— ^ (Ao A2 — Ai2) = 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 



- 1^ (G2 + 4 H-') = O, 



where we put H = Ao A2 — Af and G ^ Ag M — 3 Ao Ai A2 + 2 Af , and the 

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

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

 factor. This factor can be discarded without difficulty by putting 



G2 + 4 H- = An ] (Ao All — Ai Ao)^ — 4 (Ao A2 — Af ) (Ai A.3 — A; 



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

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



ff (F - 27J2) = O, 



where I = An A4 — 4 A Ai, A3 -f- 3A2 and A;^ J = Ao H I — G^ — 4H\ 



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

 and therefore, by multiplying and dividing the discriminant by Af,, we 

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

 sults obtained for the quartic and quintic in 



^56 

 A 



5^1 A;i P _ 27 (Ao H I - G2 - 4 H^) I = O. 



