30 



Frederick 0. Ferry. 



VI. Singularities of the Curves (p , q) in terms of 



p and q. 



-Using m ,D , r,a , ß,x , y , g, h and H in this connection 

 to denote those characteristics of the section of the develop- 

 able and the cone which Cayley and Salmon [Geometry of 

 Three Dimensions, pp. 291^ — 3] have represented by those 

 letters, we will, from the point of view of (p,q) as the 

 partial or total intersection of S and 2, proceed to find 

 values of three of the above quantities, which will suffice 

 to determine them all. 



We have at once m = p -|- q ; aud, since in general the 

 curve of intersection will have no cusps arising from 

 stationary contact of S and 2, we may put ß = 0. 



For the third singularity we will find r^the ranJc 

 thus; — Given two surfaces S and 2 of orders m' and 3 

 respectively, the condition in general that a tangent to the 

 curve of intersection meet an arbitrary line given by 



( «1^ + Piy + TiZ + B^s = I 



I «2^ + ?2y + To^ + s^ = 1 



is O 



Ï1 



=.0. 



s P2 Ï2 °2 



SS oS oS ôS 

 ux By Sz 5s 

 §2 Ô2 o2 o2 

 ÔX oy ÔZ OS 



(p = is of degree m' -[- 1 and denotes a surface which 

 is the locus of points the intersections of whose polar planes 

 qua S and 2 meet the arbitrary line; and hence the points 

 common to S , 2 , and O are the points in general for which 

 the tangents to the curve of intersection of S and 2 meet 

 the arbitrary line. [cf. Salmon: Geometry of Three Dimen- 

 sions, § 342], 



