(44) 



tliiit llieii //' — t ('oiitiinios fo roprcsont the Jiiinihci' (luiinely zero) of' the 

 points of' ('Oiilacl coiiicidiiiii willi S. 



If' iS' is a point of coiilnct willi a laii.iiciil out of /'* wiiere nothing 

 remarkable lakes place, tlieii for I hat / = J ajid ir = 2, so that 

 ui — ^=1, whilst N now also (•()nnts foi' one point of eonhiet. 



By suniinin^ii- up the xalues in — / for all the |)oints >S' of theeurve 

 for whieh ■w'^t we keep getting for sum / Avith respect to everv 

 ])oint P not lying on the cnrve, thus 



/•= >■ (/r,-^/,). (1) 



We formulate this in Ihe following way : 



Theorem I. Tjct I* he a jtoint not sitindi'd on an alijehra'n- 

 curve and S an arhitrai';/ point of that curve. Let us suppose that 

 the curve cuts the stra/</ht live PS in ?/.', an arhitrary straight line throuc/h 

 iS however in t points coineidimi initJt S ; then the chiss of the rurrc 

 is equal to w — t summcd up for all the points S of tJie curve for irhieh 

 //' ^ / and for <(s manij other curvepoints as onr Hl'es. 



To continue we suppose that P lies on the curve namely in a 

 point of the order t\ \. e. /' is the smallest iinmltei' of coinciding 

 points of intersection of the curve with a straight line through P. 

 For a point S of the cui've not coinciding with /^ the nundier of 

 points of contact coinciding with S is still indicated by //' — t. 

 Moreover a certain nundier of points of contact coincides with P, 

 namely accoi'ding to the HAi.PHEN-theorem to the number of /' -j- 21' r'j, 

 where 2^ v\ rei)resenls a sunnnation with resj>ect to the different 



curvelangents intersecting the cnrve in /' -f- /•',, f -\- r'.^ coinciding 



points /-•. So we get 



/^• = /' + ^^.V + ^^K^/0> (2) 



where 2£ (?r, — t^) re[)resents a summation with respect to all the 

 l)oints N of the cur\e outside P. 



However we can also include the point 7^ among the points >S'. The 

 line connecting P and S becomes in that case indefinite. If we take 

 for PS a line which is not a tangent in P, then we get n' = t'. 

 If however we take for PS a tangent intersecting in t' -\- v^' points 

 coinciding with J^, then we get ^/; = /' -j- r,', so ir — t' = v^'. So for 

 (2) we can write 



k = t'Jt^\'^\-a (3) 



if only we extend the snmmation also to the point >S' lying in i^ itself, 

 in which case we have but to take for 7''>S' those straight lines through 

 P contributing to 2^ {u\ — t^) (thus the taligenls in P) and as many 

 other straight lines through P as one likes. 



The equation (1) is a special case of (3). If namely P is not on 



