I04 



KANSAS UNIVERSITY (JUARTERLV. 



Let U=o be a homogenous eciuation of the //th degree, which 

 represents a group of // points on a line. This group will be called 

 simply an //ic. We know from the theory of poles and polars of 

 binary forms that the polar with respect to U of any point (x^, y^) on 

 the line consists of a group of (;/ - i ) points. Generally these (// — i) 

 points are distinct and separate; but it may happen that for certain 

 positions of (Xj, y^ ) some two of these (// — r) points will coincide. 

 We then say that the polar of (Xj, y^) has a double point. Tt is easy 

 to find the equation of the locus* of these double points on all the 

 first polars of U. For the first polar with respect to U of the point 



d d 



(x,, y^) is given by t\\Q /^olar/zi/ig operator 



(The term polarizing operator is due, 1 

 d 



i/^'dx+>^^7iyj ^^^°- 

 believe, to Klein. The 



operator I x,-^y,^^ 



is represented by P^, so that the successive 



polars of (Xj, y^) with respect to U are written (Pj)U, (Pj)-U, . . . 

 (Pj)^U. In like manner the (/;— i)th, (// — 2)th...etc., polars of 

 (Xj, yj) with respect to U, viz: those of degrees i, 2, 3, etc., are 

 written (P)Uj, (P)-U^,.... (P)aij.) If this polar has a double 

 point D, then the point 1) will belong to each of the groups repre- 

 sented by the x and y derivatives of (Pj)U. Consequently, 



d2U 

 "^dx^-^ 

 d^U 



d^U 



yr 



d^U 



' tixdy dy~ 



are simultaneous expiations and their resultant vanishes. 

 Hence 



d^U d^^U I 

 dx- dxdy I 

 d^U d^U I '""°" 

 dydx dy- 



7'his expression written for brevity ( H)U is called the Hessian of 

 U. If this algebraic expression expression be taken as the definition 

 of the Hessian of a quantic U, the above development enables us to 

 state the following geometric projjerty of the Hessian: 



THEOREM I. — The Hessian of a quantic U is tJie locus of the 

 double points on all first polars of U. 



But the Hessian of U has another important geometric property, 

 which we proceed to develope. The polar quadratic (P)3Ui of the 

 point (Xj. y^) is given by the equation, 

 .0 d"U, 



dx? 



^^^'^ydx^dy,""^' 



d^U. 



>=This u«e of the w(n\l Idl-us 



dy? 



convenient but perhap:s not justiHa 



