( 113 ) 



an onj2;in of ti brimcli of 6„ whilst tlie curve of tlie pencil through 

 S intersects the branch under consideration in in points aS', then 

 ^ {'i' — A^i) = 2l (/rj — 1), so tliat ecpiation (i) becomes 



2(./+ h' - l) = :£{ic^ - 1) (2) 



Here ^ {ii\ — I) represents a summation over all origiiis <S' of 

 brandies of C,„ i. e. only over those origins for which /r^i. 

 If one or more basepoints of the pencil lie on C„ the summation 

 must also be extended to those origins coinciding with a basepoint /?. 

 We must then regard as movable curve thi'ough that origin tiie 

 limiting position of the movable curve through F if we allow P 

 to approach to B along the corresponding branch of C,„ in othei- words 

 that curve of the [)encil intersecting the branch at least in ojie 

 point B more than any curve of the pencil. For such an origin 

 in B the number of movable points of intersection, approaching B 

 along the branch under consideration when P ap[)roaches B along 

 the same branch, is represented by iv. 



The following well known relation 



2 (g -^ n-l) = :^{t^-l)-\.k (3) 



exists between order, class and genus of C,,- 



Here -S" {t^ — 1) is a summation over all the origins of branches 

 of Cn whilst t represents the order of the branch, i. e. the number 

 of points of intersection with an arbitrary straight line through that 

 origin coinciding with that origin. 



From (2) and (3) then follows : 



^ {'^\ - ^) = X: -f 2 in' ~ n) (4) 



Theorem I. // a pencil of curves cuts an aUjebraic plane curve Cn 

 of order n and class k in Ji movable points of ivhich w fall in 

 the oritjin S of a branch of order t of C',, we have the relation 

 -2" [w^ — ti)^ l>' -\- 2 {n —. n), wliere -2" {ii\ — t^ must be taken over 

 all the origins of branches of 6',,, also over those coincidimj with 

 basepoints of the pencil. 



With the equation (4) the given [iroblem of contact for even/ C,, 

 and every particular situation of the basepoints with respect to C^ 

 can, as will be shown, be regarded as solved. 



We have but to discuss the found equation. 



If in, is the order of the curves of the pencil, then //' =: ///// for 

 arbitrary situation of the basepoints with i*espect to the gi\'en curve 



Cn, so : 



:£ {w, _ ij = /• -f 2 M {m — 1) (5) 



Here we understand l)y an arbitrary situation with respect to d 

 in the lirsl j>lace that the basepoint.s do not lie on f],. Let us 



8 

 Proceedings Royal Acad. Amsterdam. Vol. Vll. 



