46 Frederick C. Ferry. 



and the linear director. If either point lie on the linear 

 director, the conic is naade up of the linear director and the 

 generator through the other point. If both points lie on 

 the double director in the same or different sheets, the conic 

 consists of the double director. If the two points lie on 

 the linear director, the conic consists of the linear director 



and the generator À = -^ ^ . a. 



The coordinates of any point on the conic can be 

 expressed linearly in terms of the coordinates of any two 

 points on the same conic thus, — 



X = sX^ + tX, 



V = sv^ + tv^ 



Any two conies have a point of intersection; if they 

 are given by 



I a^X -j- b^iJ- -j- V ^^ ^ I 

 I a^X + b„|x -f c.,v = I 



the point of intersection is 



X : u. : V = I b^ c^ I : I c^ a^^ I : I a^ bj^ 



b2 c^ I I c„ a., I I a; b. 



But if a-^-.h^ = a^ : — b^, the two conies have not only 

 the point of intersection given but have also the point 

 X: [i: V = zba^ia^: common; hence all conies having 

 \-\ = S • — ^ h^^^ i^ general two points of intersection, 



X : jx : V = b^c^ — b.c^ : c^a^ — c^a^ : 2a^b2 

 and also = a^ : a^ : 0, 



