112 KANSAS UNIVERSITY QUARTP:RLY. 



DEFINITION.—AIi^VW) is the locus of all points (-v^, _r,) whose 

 first polars 7vith respect to V and IV have a coiiuiioii point; these com- 

 mon points of course constitute the Jacobian. 



The equation of M(VW) is obtained by eliminating x and y from 

 the equations of the first polars of (Xj, y^) with regard to V and W, 

 viz: 



dV , dV 

 ^ dx ^^ ^ dy ' 



dW dW 



"^Tx-+>'^7iy- = °- 

 Since these first polars are respectively of degrees {n — i) and (w — i) 

 in X and y, and since each contains Xj and y^ in the first degree only; 

 it follows that their resultant will contain x^ and y, in the degree 

 (// + ///— 2). Hence M(V\V) represents (//-|-w— 2) points and is 

 always of the same degree as the Jacobian. 



The equation of M(VW) can also be obtained from another elimi- 

 nation, the consideration of which leads us to an important geo- 

 metric property of this group of points. The polar points of (Xj, yj) 

 with respect to V and W are respectively 



dV dVj 



dXj -^ dyj 



dW, dW, 



By eliminating x^ and y, from these two equations, we obtain the 

 locus of all the points at which the polar points of (Xj, y^) with 

 respect to V and W coincide. But the equation thus obtained is no 

 other than M(VW)=o obtained by the former elimination. We are 

 now able to state 



THEOREM XII. — The group of points represented hy JI{J'ir) 

 is the locus of the polar points of ]' which coincide with polar points 

 of W; i. e. , it is the locus of the polar points of the [acohian with 

 respect to cither J^ or IV. 



The whole theory of the Jacobian and the function M(VW) may 

 now be condensed and stated in the following comprehensive theorem; 



THEOREM XIII. — If the first polars of a point A with respect 

 to t7oo groups of points l^ and IV have a common point at B, then the 

 polar points of B with respect to V and W coincide at A; and vice 

 versa. The locus of all points B is the Jacobian of l'^ and JV, and the 

 locus of all points A is the function M(VJV). 



From the relations between J(VVV) and M(VW) pointed out above 

 we see that the equation of M(VW) may be obtained by eliminating 



