10 Frederick C. Ferry. 



The same may be easily shown thus; the linear director has 

 - = -=:oo while -= finite; hence if we substituter-,,— , 



A JJL [X À (X 



and — for X,}x, and v respectively in 



+v^^-\ü a,,+v\U =0 



' p-ci+1 ' p-q 



obtaining 



we have to find the number of intersections of cp = with 

 V=:0; putting v' = in cp = gives U ^ =X'VMJ __ =0 

 where U is a homogeneous polynomial in >/ and jx' 



whence are given p — q values of -^ and so of -, corresponding 



to the p — q points on the linear director. Thus as U =0 

 gives at once the p points in which the curve (p,q) meets 

 the double director, so does U _= give directly the p — q 

 generators passing through the points of the linear director 

 Avhere that line is met by the curve (p , q). 



Any plane through the double director has for its 

 intersection with 2 that double line and a generator. Any 

 such plane meets the curve (p , q) in p points on the double 

 director and in q points on the generator lying in that plane, 

 i. e., in p + ^ points in all. Therefore the order of the 

 curve (p , q) is p -|- q. The same may be shown thus ; any 

 plane ax -(- by -|- cz -f- ds = has for its intersection with 2 

 the curve aXv ~|- bjxv -|- cX' -\- djx"' = and this curve has 

 p -{- q points common to the curve (p , q) whose equation is 

 <p = 0, as their equations show. 



