Io6 KANSAS UNIVERSri'Y (QUARTERLY. 



Eliminating x, and yj from these two equations, we have the e([ua- 

 tion of the desired locus. But this result is exactly the same as that 

 obtained by equating to zero the discriminant of (Pj)U, except that 

 X and y are replaced by x^ and y^. Hence the eliminant of the pair 

 (2) gives us the Steinerian of U. This result may be formulated in 

 the following: 



THEOREM III.— The Steinerian of U is also t)ie locus of all 

 points tiihicli are double points on polar quadratics of U, i. e. it is the 

 locus of the polar quadratics of the Hessian. 



All the above results may be combined and the whole theory of 

 the Hessian and Steinerian stated in the following general theorem. 



THEOREM IV.— If the first polar of a point A has a double 

 point at B, then the polar quadratic of B has a double point at A and 

 vice versa. The locus of all points B is the Hessian of U, and t lie locus 

 of all points A is the Steinerian of L\ 



The Steinerian may be defined in yet another way; since the polar 

 of any point with respect to a double point coincides with that 

 double point, and since the polar quadratics of the ])oints of the 

 Hessian are double points on the Steinerian, it follows that the polar 

 points with respect to U of all ])oints on the Hessian constitute the 

 Steinerian. From this we see that we can obtain the equation of the 

 Steinerian by eliminating x, and y^ from (H)Uj and (P)U,, i. e. 

 from the Hessian and the polar point of (x^, y,). Or we may obtain 

 the same result by eliminating x and y from the Hessian (HW and 

 the first polar of (x^, y^ ), (P)U. This result is equivalent to the 

 statement that the Steinerian is the locus of all points whose first 

 polars have a point common with the Hessian. 



Another group of points closely associated with the Hessian and 

 the Steinerian, designated by (1 for want of a suitable name, is de- 

 fined as follows: 



DEEIA'ITION. -The oronp G is the locus of all first polar points 

 of the Steinerian which do not belong to the Hessian. 



The equation of G is found by eliminating x^ and y ^ between the 

 Steinerian, (S)U— o, and (PjH'^^-o. Since (S)U is of the degree 

 2(// — 2) in Xj and y,, and since ( P,)U is of the first degree in Xj and 

 yj and of the [n — i) degree in x and y, it follows that this eliminant 

 is of degree 2(;/ — 2)(// — 1); but since it contains each point of the' 

 Hessian as a double point, it will contain the square of the Hessian as 

 a factor. Dividing the eliminant by the square of the Hessian, we 

 obtain the equation of (i. The tlegree of (r is therefore 2{n — 2) 

 (« — i) — 4(// — 2)=:=2(« — 2)(;/ — 3). Hence the degree of G is always 

 .(^«— 3) times the degree of the Hessian. 



