Geometry on the Cubic Scroll of the First Kind. 



III. Coordinates on 2. 



Puttinff - = - and - =t- , 2 ffives - =—, whence f or 

 coordinates of a point on ^ we have 



X : y : z : s = Xv : [JLV : X"' : [X . 



In these coordinates the double director is given by v = 0; 

 for each point of this line X : [x has a determinate value, 

 while values of X : [x of the same absolute magnitude but with 

 opposite signs correspond to the same point of the double 

 director but are to be regarded as belonging to different 

 sheets ; i. e., any two points of v = harmonic to the pinch 

 points represent the same point of the double director but 

 belong to different sheets, as do the two generators from 

 that point of the double director. Thus to any value of 

 X ; [X corresponds a generator, and hence any generator Ts 

 given by an equation of the form aX -|- b[x = ; the two 

 generators from the same point of the double director are 

 given by two equations of the form aX -f- bjx = and 

 aX — bij. = 0. 



The linear director is given by -^-=oo; for each 



X [X 



point of this line X and [x are infinitesimal but their ratio is 

 finite and has the value belonging to the generator that 

 passes through the point. The pairs of generators from the 

 different points of v = form an involution of rays which 

 join X = 0, tx = (both infinitesimal) with the point pairs 

 of v = 0. 



