newson: geometry of one dimension. 105 



If this quadratic consists of coincident points, its discriminants 

 must vanish. Therefore 



d^U, d2Uj 

 dx^ dxjdyj 



d-Uj d-Uj 

 dx,dy, dyf 



liut if the point (x^, y^) be taken as a variable point, this last expres- 

 sion (H)Uj is no other than the Hessian of U. We are therefore 

 able to state the following: 



THEOREM II. — The Hessian of a qiiaiitic U is ilie loeiis of aU 

 points 7u /lose polar quadratics with respect to U consist of double points. 



(For the analogous theorem in geometry of two dimensions, see 

 Salmon's Higher Plane Curves, Art. 70). 



It is plain that if U be a quantic of degree ;/, the Hessian of U is 

 of degree 2(// — 2); therefore the Hessian of U represents a group of 

 2(« — 2) points. 



Closely associated with the Hessian of a ([uantic U is another 

 group of points, which from its analogous curve in two dimensional 

 geometry, I have ventured to call the Steinerian of U. 



DEFINITION. — The Steinerian of U is the locus of all points 

 whose first polars with respect to U have a double point : these double 

 points of course constitute the Hessian. 



From this definition of the Steinerian it follows that the Steinerian 

 must be of the same degree as the Hessian, viz; of the degree 2(// — 2). 

 The equation of the Steinerian of U is found by the following pro- 

 cess: The equation of the first polar of (Xj, y^) with respect to U is 

 (P,)U^o. If this first polar have a double point, its discriminant 

 vanishes and (x^, y, ) is a point on the Steinerian. Hence ccjuating 

 to zero the discriminant of (Pj)U, we obtain an equation in x and 

 y of the degree 2(// — 2) which is the equation of the Steinerian. 



It was shown above in Theorem II that the Hessian is the locus of 

 all points whose polar quadratics with respect to U consist of double 

 points. We wish now to find the equation of the locus of these 

 double points. The equation of the polar quadratic of (x,, y,) is 



^ ^ dxf dx^dy^ dyf 



The condition that this should consist of coincident points is the 

 the vanishing of its x and y derivatives, viz: 

 d2U, , d^U, 



r^ + y 



dxf ■' dx^dy^ 

 d^U, d^U, 



dx.dyj dy3 



