38 Frederick C. Ferry. 



cp^(x,y,z) = Up+z.Up_, + zlUp_2 + 



cp, (x,y,z)= Up, + z.Up,_,+ zlUp,_2 4- 



P'-q' + l 



I n' — n'-H I Tt' — n' 



then must U _ and U ,_ , have a common factor; and if 

 the contact be of order x, then must U and U , have 

 common factors of degree x. If the corresponding curves 

 on 2 whose equations are 



|cp,(X,}x,v)EEEUp + V.Up_, + vlU^_,+ 



' p'— q'+l ' P'— q' ' 



have a common factor in Ü _^ and U , , then must (p,q) 

 and (p' , qO have a common point on the linear director; and 

 if U and U , , have common factors of decree x, then 



p-q p'— q' - o ' 



must (p , q) and (p' , q') have a total number of x intersections 

 on the linear director. Hence in every case the tivo curves 

 (p,q) and (p', qO have pq' + P'q — <\<^ points of intersection: 

 and of these points x will lie on the linear director when and 

 only when the two corresponding plane curves have contacts 

 of total order x at the multiple point (x,y). 



The maximum number of double points of a plane curve 

 of order p is ^(p — l)(p — 2); and, besides a point of multi- 

 plicity p — q, such a curve can have only 



llp-l)(p-2)-l(^p-q)(p-q-l) = p(q-l)~iq(q-fl) + l 



double points. Hence a cuwe (p , q) can have in general 

 p (q — 1) — ^ q (q -|- 1) + 1 double points : and of this number 

 a point or points of total multiplicity x will lie on the linear 



