NEWSOX: GEOMETRY OF ONE DIMENSION. 



If one of the quantics, W for example, be of the first degree, it 

 represents a single point; the Jacobian of V and ^V must now have 

 some new meaning, for we can no longer properly speak of the polar 

 of a point with respect to W. 



The equation of the first polar of (x^, y^) with respect to V is 



;iven by the operation 



tYi — V=o, or — --, — ^--- V 



dx^^^dy J I yj dx ' dy J 



:^o. If the equation of W be given in the form aXj-pby,=-o, the 



equation of the first polar of W with respect to V may be obtained 



by substituting for -^i in the above operation for the polar. 



3- V , 



Making this substitution we have 



dW , , dW 

 -- and b=-— — 

 dx dy 



dW 

 dx 



Whence the last equation 



dV dW dV 



dy dy dx 



V=o; but 



which is the Jacobian of V and W. Whence we have 



THEOREM XL— When one of the quantics V and W, If /or 



example, is of tJie first degree, /(. T/r) becomes the first polar of the 



point W with respect to /'. 



When the two quantics V and W are so related that one of them 



is the Hessian of the other, e. g., W=H(V), then the Jacobian of 



the two groups of points V and H is called for brevity the Jacobian 



of ^^ Thus the Jacobian of any quantic U whose Hessian is H, is 



given by 



(See Harkness and Morley's Theory of Functions, Chap. I. ) 



If V be of degree n, its Hessian as we know is of degree 2(// — 2); 

 therefore the Jacobian of V is of degree [(« — 1)-|-2(« — 2) — i] or 

 Z{n-2). 



THE FUNCTION M(VW . 

 Closely associated with the Jacobian of two groups of points V and 

 W is another group which I shall designate by M(VW). The func- 

 tion M bears much the same relation to the Jacobian that the Stein- 

 erian bears to the Hessian. This new group of points may be 

 defined as follows: 



