81 



Irrelevvnt Factors en Bitangentials of Plane Algebraic 



( !l RYES. 



By U. S. Hanna. 



Three years ago I presented a paper to the mathematical section of the 

 Academy dealing with the proof of a formula used by Mr. Heal in an ar- 

 ticle published in the Annals of Mathematics, vol. VI, page 64. This 

 formula was used by Heal in freeing a bitangential of the plane quintic, 

 which he had developed in a previous paper in the Annals, vol. V, page 33, 

 from an irrelevant factor, the square of the hessian of the quintic. Since 

 then I have continued the study of the subject and wish to present an in- 

 teresting result in the light of Heal's work. 



Taking the general equation in the symbolic notation 



(ai xi + a 2 x 2 — - a3 X3) n = a x " = b x " = c x " = ... = o, ( 1 ) 



for the n-ic and deriving the first polar, with respect to the n-ic, of any 

 point y, we have 



(ai xi -f as X2 + a3 X3)"" 1 (ai yi -f a2 Js -f a3 ys) = a x n_1 a y = o, (2) 



Any point on the line through the points x and y may be represented 

 by / x -4- it y, where a and // have a fixed ratio for any particular point. If 

 x be a point on the n-ic and y be a point on the tangent to the n-ic at the 

 point x, then we have equations (1) and (2) satisfied by the points x and 

 y respectively, and equation (2), as an equation in y, represents the tan- 

 gent to the n-ic at x. If, in addition to these conditions, the point 

 a x -\- y j lie on the n-ic, we must have from (1) 



fa "1 " 



! /x+//y j = ( A » x + "ay) 11 = o, 



from which, by virtue of (I) and (2), we get 



n ( n — 1 ) „ „ ., „ , n (n — 1 ) ( n — 2 ) „ . . „ , 



-A— - — - a x "- 2 a y 2 A"-2 -f - ' '- ax"- 3 a y 3 A"- 3 // -f •■• + 



naxay 11 " 1 A//"- 3 -f- a y " /t n - 2 — o. ... (3) 

 Equation (3) is an (n-2)-ic in A and u which gives the positions of the 

 remaining n-2 intersections of the tangent to the n-ic at x with the n-ic 

 itself. In order that this tangent be a bitangent the discriminant of equa- 

 tion (3) must vanish. This discriminant is a function of x and y, and if y 

 6— A. of Science. 



