81 



Irrelevvnt Factors in Bitangentials of Plane Algebraic 



Curves. 



By U. S. Hanna. 



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

 Academy dealing with tlie 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 lie 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 + aa X2 -^ a.s xs)" = a.x" = bx° = Cx" = • • = o, ( 1 ) 



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

 point y, we have 



(ai xi + as X2 + as xs)" ■ (ai yi -j- as ys + a,3 ys) £Z ax" ^ ay = o, (2) 



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

 by ^ X -)- // y, where /I 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 tlie n-ic at the 

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

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

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

 ?!. X -f /' y lie on the n-ic, we must have from (1) 



pAx-(-//yj = ('^ax -|- /'ay)" = o, 



from which, by virtue of (1) and (2), we get 



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



ax"-2 ay2 A»-2 -I ^ L ax""'^ ay'' A"-3 // -f ••• -f- 



2 ! "'^ "^ ' 3 ! 



naxay"-! ^"-3 _|_ ayU ^n-2 — q _ _ (3) 



Equation (3) is an (n-2)-ic in / 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. 



