( 114 ) 



suppose moreover that the basepoiiits are situated in sucli a way 

 that not a single curve Cm of the pencil passing through a singular 

 point S of C„ touches one of the branches through S and that not 

 a single Cm has with the fixed curve a contact of higher order then the 

 first ; then only the curves Cm showijig a common contact with Cn {"■ — 'i, 

 t = 1), furnish to ^ [ic-^ — t^) a contribution equal to the number 

 of those touching curves Cm- Hei-e however a restriction ought to 

 be made. It may appear namely that a Cm of the pencil has a double 

 or mullii)le point lying on Cn which then furnishes a contribution 

 to ^ {ii\ — ^i). We can avoid this case by requiring that for an 

 arbitrary situation of the basepoints with respect to 6'„ no singular point 

 of Cm may lie on C'„. This is however no longer possible when by 

 mutual coincidence of basepoints the pencil admits of curves showing 

 an inlinite miniber of doul)le or multiple points, in other words when the 

 pencil contains curves, wiiich consist of or contain two or more coinci- 

 ding parts. Though the ecpiation (3) can still be applied to these cases we 

 shall exclude them for simplicity's sake from our discussion. With these 

 suppositions we tind tiiat ^ {il\ — t^) is equal to the number of 

 curves of the pencil touching Cn- So we find the following theorem : 



Theorem II. For a pencil of curves of order in, none of which 

 contains two or more coincidimi parts, the number of cnrves touchiinj 

 an ahjebraic plane curve C» of class k with respect to which the base- 

 points of the pencil have no particidar situation is represented bi/ 

 ^ + 2 n (m — 1). 



If Cn is a general curve in point-coordinates then k =z 71 (n — 1), 

 and the required number is n (n -j- 2 in —- 'S). If we compare this to 

 the number given in the above theorem we find : 



Theorem III. Erery .singular point S of Cn diininishes the number 

 of curves wJdcJi belong to the pencil mentioned in theorem II and 

 'whicli properly touch C,„ by the .same number as that with v)hich S 

 diminishes the class of C- 



We now imestigate which particularities can present themselves 

 in consequence of a particular situation of the basepoints with respect 

 to Cn- To this end we consider iu the first place an origin .S' of 

 branch T of order t of ('„ supposing ;S' not to be situated in one of the 

 basepoints ; further we suppose that the cui-ve 6',, through S touches the 

 branch T and intersects it in ir = t -\- y coinciding points (so in y 

 points more than when the curve through >S of the pencil were not to 

 touch the branch). Then this point S counts (as far as the branch T 

 is concerned) according to (5) for w- — / = y points of contact proper. 

 If we restore by a small displacement of the basepoints their arbi- 

 trary position with respect to Cr, the curve through <S' of the pencil 



