NEWSON: GEOMETRY OF ONE DIMENSION. 109 



JACOBIANS, ETC. 



Let V and W be two quantics, the first of degree d: and the second 

 of degree //. The determinant formed of the first derivatives of V 

 and W with respect to x and with respect to y, viz: 



dV dV 

 dx dy 



dW dAV 



dx dy 



is called the Jacobian of V and W. This expression is often written 

 for brevity J(VW). It is readily seen that the equation J(VW)=o 

 is of the degree (///-j-;;— 2), and therefore the Jacobian of V and W 

 represents a group of {inA^n — 2) points. We wish now to determine 

 the geometric properties of the Jacobian. 



Let us consider the first polars of any point (x^, y^j with respect 

 to V and W. In general tliese first polars of (x,, y^) are distinct 

 groups of points; but for certain positions of (Xj, y^) these two first 

 polars may have a common point. The ecjuation of the locus of the 

 points common to these first polars is readily found. For the first 

 polars of (Xj, y^) with respect to \ and W are given respectively by 



(P„V:3.x,f+y,f=o. 



' M X -^ 'd y 



_ „, dW dW 



VV^e obtain the equation of the common points of these two groups by 

 eliminating x, and y^ from these two equations. But this eliminant 

 is J(V\V); Hence our first geometric property may be stated in the 

 form of 



THEOREM VIII. — The Jacobian of two quanties V and W is 

 the locus of all points cofnmon to tlie first polars of a point with respect 

 to J' ami IV. 



The equation J(VVV)=o may also be obtained from another con- 

 sideration which leads to a second important geometric property of 

 the Jacobian of two quantics. The polar points of (x^, yj) with 

 respect to Y and W respectively are given by 



dxj dy, 



dW, dW, 



