KANSAS UNIVERSITY QUARTERLY. 



These points coincide when their eliminant vanishes, i. e., when 

 dV, dVj 

 dx^ dyj 



dW, dWj 



"d^ d^ 



But when (Xj, y^) are taken as the variable point, this is no other 

 than the Jacobian of V and W. Hence our second geometric pro- 

 perty of the Jacobian may be stated as follows: 



THEOREM IX. — The Jacobian of two qualities V a?id JF is also 

 the loeus of all points whose polar points with respeet to V and W 

 coincide. 



In the particular case when \' and W are of the same degree n, 

 their Jacobian has still another important geometric meaning. The 

 equation V-)-/C'W=o, where k is a variable parameter, is the equation 

 of an involution of the //th order. The equation of the double points 

 of this involution may be readily found. For if V-|-/^W=:^o has a 

 double point, its x and y derivaties are simultaneous equations and 

 their resultant vanishes. Thus, 



Eliminating /' we have 



But this is the equation of the Jacobian as otherwise defined; whence 

 we have 



THEOREM X.— lVhen the two qn antics V and W are of the same 

 degree n, they determine an involution of the nth order; the Jacobian 

 of V and W in this case is the equation of the 2{n^i) double points 

 of this involution. 



The Jacobian of any two groups belonging to an involution of the 

 nWi order is the same as that of any other two groups of the same 

 involution. For the Jacobian of the two groups V-L/C'jW=^o and 

 V-L/('2^^'=° is found to differ from the Jacobian of V and W only by 

 a common factor which involves k^ and k^. Thus 



