305 



2{m'-j^7n — 6) points in common. Taking this into account we tind 

 indeed for the order of x the number arrived at above. 



5. Till now we have supposed that the singular points are ail 

 single and different, but moreover (hat each point S is .s-m///d mill- 

 point on a ray arbitrarii}- drawn through *S. An examj)le of aïï^(l,??i), 

 of which the singular points are \mvt\y double mcll-point'iyis UivmsUed 

 by a pencil of curves C, when each straight line is associated to 

 its points of contact with curves of the pencil. A ray passing through 

 a base-point of (<") is touched outside that point by 2(r — 2) curves, 

 while an arbitrary straight line has 2(r — 1) null-points; so each 

 base-point is to be considered as double null-point. The remaining 

 singular points of this null-system ^1{l,2r — 2) lie in the nodes of 

 the nodal curves c>' ; they are evidently single null-points on the 

 straight lines drawn through (hem. 



We shall now suppose that ^\l,in) has .s-, singular points S^'^\ 

 which are double null-points of their rays. As a ray passing through 

 *S''^' ontside that point bears {m — 2) null-points {\\q null-curve o''^) \n\'!i 

 a triple point in *S('^ . The complementary curve now consists of the 

 s^ ]iull-curves o '') and a curve x* of order {m—2){ni'^-\-4:ni-\-l) — 

 — {ni^\)s^, while the curve r has been rej)laced by a curve t* of 

 class {m-\-2){n( — 1) — .s-, and the s^ class-points S '^\ 



If it is taken into consideration that aji^) contains all singular 

 points Sf-'^^ and /S'^ it is found that v.* passes through each point 

 aS' with {m'^^m — 6 — .s'J branches and with {ni^-^-ni — 8 — s^) branches 

 through each point S^^^\ 



6. In order to arrive at a determination of the number of triple 

 null-points N^^\ we associate to each point jYC^) of a ray t the 

 (?/i— 2) null-points iV' of t, and consider the coriespondence which 

 arises in consequence of this in a plane pencil with centre T. 

 As the points iV(-^) lie on the curve y^'", the points xV' on the 

 curve >c', the characteristic numbers of this correspondence are 

 evidently ^m{rn — 2) and {ni — 2) (;/^^-|-47;?,-|-l) — {in-\-\)s^, while any 

 ray t passing through T prodiu-es an {m — 2)-fold coincidence. The 

 number of the remaining coincidences amounts to 



3m (m— 2j + (?/i— 2) {nf -f 4m + 1) — (m + \)s,—{in -\- ni—l-s,) 

 {m — 2) i.e. {m — 2) (6//i -\- 3) — 3.?,. 



There are consequently 3(m — 2) (2/// -j- 1) — 3,s-, /lud-rdi/s inith a 

 triple null-point. 



In order to find the number of coupled double null-|)oints iV^^' 

 we associate to each point ^V' of a ray t each of the remaining 



