Geometry on the Cubic Scroll of the First Kind. 43 



When (p' , q') and (p" , q") have no multiple points, we 

 have in general that 



1) Any curve, which passes through the intersections of 

 the two curves (p' , q')q'=p- and (p" , q")q,.^p» j whose equations 

 are 'z>^ = and cp^ = , and is itself of the species (p , q) ^ ^ , 

 can have its equation put in the form cp ^ Acp^ -|- Bcp^ = 0, 

 where A == and B = represent curves of the species 

 (p — p' J ^ — ^.' — ^) ^^^ (P — P" > <1 — q" — ''^) respectively. 



It is evident that x > q — q" — (p ^ — p") , else B = 

 would not exist as a curve. 



2) Any curve, which passes through the intersections 

 of the two curves (p' , qOr - <- y and (p" , q")q„ ^ p„ , whose 

 equations are ^^==0 and % = , and is itself of the species 

 (P ) qL ^ f , n . „ .-^ , . , can have its equation put in the 



vr 7 l^q = p— (p'_q'}_(p"_q")+l > ^ f 



form cp ^ Acp^ -f- B<p2 = wliere A = and B = represent 

 curves of the species (p — p' , p — p' — (p" — q") + 1) and 

 (p — p" , p — p" — (p' — q') + 1) respectively. 



When (p' , q') and (p" , q") have multiple points, we have 

 in general that 



3) Any curve, which passes through the intersections 

 of the two curves (p' , q') ,^ , and (p" , q")q„^p-- , whose equa- 

 tions are 0^ = and cp„ = , and where (p' , q') has an r-tuple 

 point at P and (p" , q") an s-tuple point at P , and which is 

 itself of the species (p , q) .^ , will have an (r -[- s — l)-tuple 

 point at P and can have its equation put in the form 

 «p ^ Acp^ -f" Bt2 ""^ ^ where A = represents a curve of the 

 species (p — p' , q — q' — y-) with P an (s — l)-tuple point, and 

 B = represents a curve of the species (p — p", q — q" — x) 

 with an (r — l)-tuple point at P. 



Here evidently x > q — q" — (p — p"). 



