( 115 ) 



intersects the branch T in t points S and in 7 {)()in(s \\'\\\^^ near .S', 

 The preceding holds good invariably when Ihe pencil does contain 

 cnr\'es containing two or more coinciding parts if only those do 

 not pass through the considered point S. 



This gi\es rise to the following theorem : 



Thkorem IV. Let S be tlte orujlti. of a hraiick T of an. alqehmic 

 plane curve Cn- If noin the ba.sepoints of a penc'tl of earves pass 

 from an arbitrary position to a particular one so that no basepoint 

 approaches S ichilst y points of intersection of Cn unth the curve 

 through S of that pencil approach S along the branch T, then as 

 mam/ {so y) points of contact of C,, urith curves of the pencil approach 

 S along the same branch. Here has been supposed th/it if the pencil 

 contains curves containing two or wore coinciding parts these ])arts 

 do not pass through S. 



If the basepoints have the particular position descril)ed in this 

 theorem, then ;S' counts for y points of cojitact proper. So we can 

 formulate the theorem as follows : 



//■ S is a point of Cn not coinciding with one of the basepoints 

 of the pencil, whilst the curre through S of the pencil cuts a branch 

 of order t of Cn ivith S as origin, in t -\- y points S, thejt. S absorbs 

 as far as that branch is concern,ed y points of contact proper. 



Theorem IV is an extension of a theorem of Halphen and Stephen 

 Smith, which I discussed in a paper in a previous number of these 

 Proceedings ^). 



The indicated theorem can be expressed as follows: 



Let S be the origin of a branch T of a curve, I the tangent of 

 that branch in S. If a point P approaches I but not S, then as 

 many points of intersection loith PS as points of contact of tangents 

 through P approach the point S along the branch T. 



This is no other than our theorem IV where the pencil of curves 

 is replaced by a pencil of straight lines. 



Let us further consider the case of a singular point S of Cn, with 

 which coincide one or more of the basepoints. As our only business 

 is to determine the number of points of contact proper coinciding 

 with >S' we can assume for simplicity's sake that no basepoints 

 coincide with other points of Cn- 



Further we exclude the case that the pencil contains curves 

 admitting of coinciding parts. 



Let t\,t\^,... be the orders of the different branches T\/P\,... 

 of Cn having 6' for origin, whilst an arbitrary curve of the pencil 



1) On an expression for the class of an algebraic plane curve with higher 

 singularities. These Proceedings VI, p. 42, 



8* 



