( ^16 ) 



cuts those branches successivelj in z\, z\, . . . points S. Tlien we 

 have 



n' z= mn — S z^' . 

 Further we can break up ^ {il\ — t^) into the sliare ^ (?(?/ — t^') 

 of the point S and the siiare of the other points of 6',. The 

 meaning of tu\ is here, tliat the curve of the pencil cnttiii«i; tlie 

 branch T\ in more then z\ points S does this in z\-\-ir\ points. 

 For equation (4) we can then write 



:2{,v,-t,):=:k^2n{m-\)-:z:{w\-\-2z\^t\Y. . (6) 



where -^ ("'i — ^i) must now he taken oiilv for the cur\ei)()ints 

 outside S. If we represent by //', tlie number //^^ -j- c\ of tlie points >S 

 in which tlie brancii T\ is cut by the osculating curve of tiie j'encib 

 equation (6) becomes : 



V (,,^ -t,) = k^ 2n (n - 1) - ^ {»\ 4- z\ - t\). 



It is evident from this equation that the point S, as tar as 

 branch T\ is concerned, absorbs u\ -\- z\ — 1\ points of contact proper. 

 This can be formulated in the following tiieorem: 



Theorem V. If a single or inultiple hasepoinf of a pencil of 

 curves coincides with the origin S of n branch of order t of an 

 alcfebraic curve Cn, whilst that branch cats an arbitrarj/ curve of 

 the pencil in z, the osculatinp curve of the pencil on the contniri/ in 

 u points S, then, the point S absorbs u -\- z — t points of contact of 

 curves of the pencil with C,,, in other words for an arbitrary displace- 

 ment of the basepoints coinciding with S the point S furnishes 

 u -\- z — t points of contact, which are then situated on the considered 

 branch. 



By allowing the basepoints to undergo not an arbitrary displace- 

 ment but a particular one, another theorem in connection with 

 theorem IV can still be deduced from this. We can namely make the 

 basepoints change their places slightly along the osculating curve of the 

 pencil in such a way, that no more basepoints coincide Avith S. In that 

 case the point *S continues to absorb after the displacement of the base- 

 points a certain number of points of contact proper, and according to 

 theorem IV to the amount of u — t\ in fact after that displacement 

 the curve of the pencil through S intersects the branch in ?/ points S, 

 so that for the point .S' now w is equal to u. If we comjiare the 

 number u — t of the absorbed points of contact to the amount given 

 in the theorem V we find: 



Theorem VI. //' the curves of a pencil cut the branch T of an 

 algebraic plane cuive in z fixed points coinciding nnth its origin S, 

 then point S gives to that branch z points of contact ivith curves of the 



