( 109 ) 



do and into those which do not (l('|)oii(l upon tlie choice of the 

 system of coordinates. 



If t\ ,t\,... are the ordei-s, t'\ , /;', , . . . the chisses of the separate 

 branches having- tlieir origin in P (so tiiat -^ t\ = t') tiien 



2 0,-1) = k—2n + V (^•^_i) j^ n; (,^_i)^ 

 whilst according to tlie quoted tiieoreni 



k = :£{t\-^v\)-]r ^{n-^-t;). (e(|nalion (2) I. c. p. 44) 



In the two hist eiinations the tirst -l""-sign refers to the origins in 

 P, the second -S'-sign to the origins outside F. 



From these equations follows 



2 0/-l) = -2(/.-O + ^(t'\-l) + ^^K-l) (1) 



Here n — t'=::n' represents the numlier of movable points of intersection 

 of \\\e curve with straight lines tlirough 7^ If one draws tln-ough Paw 

 arbitrary straight line / \vhich is not a tangent in P, then the points of 

 intersection of that straight line with the curve furnish to 2£ {w^ — 1) a 

 contribution equal to it! — .V/, where A^- represents the number of origins 

 of branches lying outside 7^ on the straight line // ^), thus the number of 

 branches over wdiich the m' movable points of intersection with the 

 straight line // distribute themselves. If tiien we draw through P a 

 straight line Ij touching .V'; branches tiirougli /■• and if we let Ay 

 be the number of branches over whicii tlie it! movable points of 

 intersection with the straight line 1j distribute tiiemselves, then A'^ — X'j 

 of these Xj branches have their origin outside P. The points of 

 intersection outside P with tliese straight lines give a contribution to 

 -2" {in^ — 1) equal to 



Kn-^ .',) - {Nj-N'j) = (u'-N,) - V (,;_i), 

 in which ^ v'y and .2" (i/^ — 1) are taken oidy over the branches 

 touching the straight line Ij in P. From this ensues: 



2 {w,-l) = 2 (u'-N;) + ^ {u-Nj) - 2 (v'-l), 

 or 



2 (tv-l) = V (,/_A7^) _ :s (,'^_i), 



in which now :£ (v\ — l) denotes a summation over all the branches 

 with P as origin, 2 (n'—X^) a summation over r/// straight lines through 

 /-* for which II is ^ X^ and over as many other straight lines as one likes. 

 If we substitute in equation (1) for JS (n^ — l) the obtained value 

 and replace ;/, — f by ii' we find 



^/=\ - n' + i 2i (n' - X,) (2) 



We can sum up what has l)eeji found in the following theorem: 

 Theork.m I. //' (/It iihii'hrdic jihim; ciiirc is iiitcrsccti'd In/ tlir .straight 



1) Coimtiüg liere also each of llie poiüls ot' iulerseclion as origin of a branch. 



