( 112 ) 



of the pencil), has for erert/ pencil of curves the same value equal 

 to the (jenus of the fixed curri\ 



The remarkable thing iicrc of llie ul)tainc(l expression lor the 

 genus is thai the genus wliieli is invariant wilii respect to rational 

 transformations is i-eally exclusively expressed in (|ua)ilities each 

 invariant in itself over against i-ational transformations. 



1 feel the more Justilied in stating the above theorem having found 

 the theorem conlirmed in \arious simpler cases, e.g. for the case 

 that the given curxe admits of doul)l(' points and cusps onlv whilst 

 the basepoints can assume an\ particular position with respect to the 

 given curve, also for the case that the given curve is provided with 

 higher singularities where however only the simplest |)articidar cases 

 with reference to the ])osition of the i)asepoints have been considered, 

 e.g. the case that one oi' two basepoints fall in a higher singular 

 l)oint. 



But all the same a ri<i(>n>us and sim/ilr jn-oof wliicli renders a 

 subtle distinction of the great uinnber of paiMicular cases which can 

 present themselves superfluous, is very desirable. 



Sneek, July 1904. 



Mathematics. — "On tlw cm-res of a pencil touchiin/ an al</ebraic 

 plane curre with hiqher sin;/ularities" . By Mr. Fkku. Schuh. 

 (Communicated by Prof. Koktkwkg.) 



In the previous paper 1 have stated the following theorem : 

 If an ahjehraic plane curve is cut by a pencil of curves in n 

 movable points (listributiiu/ themselves for the various curves of the 

 pencil over N'^, X.,, . . . brandies of thr fi.et-d eui-ri', then 

 1 _ ,,' _|. 1 V (,,' _ X,) 



(2 {n' — lYj) taken for all curves of the pencil) has for even/ pencil 

 of curves the same value which is ecpial to the (jenus g of the fixed curve. 



Expressed in a formula this runs: 



2(</ + n'- 1) = :£■(;/ -A\) (1) 



With the aid of this theorem the following problem of contact 

 can be solved : 



I'o determine the number of curves (f a pencil touching a plane 

 curve Cn of order n, class k and genus g. 



To this end we tirst substitute in equation (1) for ^ (//' — X^) 

 a summation over the points of Cn or better (as w^e always 

 count a i)oint of r„ through wdiich more branches pass as more 

 than one point) over the origins of branches of C«. Let *S' be 



