44 Frederick C. Ferry. 



4) Any curve, which passes through the intersections of 

 the two curves (p' , q')f . < y and (p" , q"),v- <p- > whose equations 

 are 9-, = and (d^ ---= 0, and where (p' , q') has an r-tuple point 

 at P, and (p" , q") has 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 cp ^ Acp^ -j- Bcp^ = where A = represents 

 a curve of the species (p ■ — p' , p — ^ p' — (p" — q") -|- 1) with 

 an (s — l)-tuple point at P, and B = represents a curve of 

 the species (p — p" , p — p" — (p' — q') + 1) with an (r — 1) - 

 tuple point at P. 



Thus from 1), if of the pq' -[- p'q — qq' intersections of 



a (P;q)q=p_x ^^^ ^ (P''q\=p" pq" + p"q — qq" He on a 

 (p" , q") „^ „ , the remaining points of intersection lie on a 

 (p — p" , q — q" — ■/-)• Or if of the 2pq — q- intersections of 

 two (p , q\^ /s pp" lie on a (p" , q")q»=p'- > the remaining points 

 of intersection lie on a (p — p",q — q"); thus, if of the 9 

 points of intersection of 2(3,3ys 6 are situated on a (2,2), the 

 remaining 3 lie on a (1,1); and again, if of the 30 intersec- 

 tions of a (6,5) with a (5,5) 17 lie on a (3,2), the remaining 

 13 lie on a (3,2). 



And from 2), if of the pq' + p'q — qq' points of intersec- 

 tion of a (p,q)q<p and a (p',q')q-<p- pq" + P"q — qq" He 

 on a (p" , q")q.^p._(p_,^)+fp-_q,)_i,the remaining points of inter- 

 section He on a (p — p",p — p" — (p' — qO + 1); evidently 

 we must have (p' — q') < (p — q) + 1 ; thus if of the 2pq — q' 

 intersections of two (p,q)q<p's pq" 4" p"q — qq" He on a 

 (p",q") „^ „_^ the remaining points of intersection lie on a 

 (p — p",q — p"-[- 1); thus if of the 8 points of intersection 

 of two (3,2)'s 5 lie on a (2,1), the remaining 3 lie on a (1,1); 

 and if of the 30 points of intersection of a (6,3) with a (6,4) 

 21 He on a (4,3), the remaining 9 He on a (2,1). 



