3(U 



Throiigli eacli point S pass (iii'' -{- m -- 6) tangents of the null- 

 curve ö'"+^ of S; as tliey bear a double null-point eacl), S is an 

 (/n* -J- ''^ — 6)-fold point of the com plementary cxirve x. Besides the 

 points S, 7i has moreover the groups of (/?* — 2) null-points in common 

 with (P)"'+^; these points lie on the [m^ -\- m — 2) straight lines t, 

 which meet in P. The two curves have consequently in common 

 (^j)i^ _|_ ni + 1) {ni' + m — 6) + (v/j' + ni.--2) {in.- 2) points. Foi- the 

 order of x we find from this (m* -{- 3 ni^ — 5 7/i* — 9 vi — 2) : (jn -\- 1), 

 i. e. in' + 2 m' — 7m — 2, or {m — 2) (m -\- 4 m -\- i). 



4. The straight lines t envelop a curve t of the class {in -j- 2) 

 (m — 1). 



If a curve c'"-^^ of the net has a node D, DP replaces two of 

 the rays t meeting in P; P is then a point of t and PJ) the 

 tangent in P at that curve. 



If P lies on a binodaJ c"'+\ with nodes I) and D' , PI) and PD' 

 replace each two straight lines t and are tangents in a node of t. 



If a 6""+' has a cusp in A^ PK replaces three straight lines t, 

 and P is a cusp of t. 



Now the net [c""+^] contains according to a well-known proposition 

 I }n{7n—l) (3 ni" -f- 3 )n — 11) binodal and 12 m {in — 1) cuspidal 

 curves. 



If we moreover take into consideration that the base-points >Sare 

 nodes of t, it appears that t possesses h {^ in* — 4()///'-|-35 ni -\- 2) 

 nodes and 12 ni {m — 1) cusps. 



We can now determine the remaining chaiacteristic numbers of t. 



From the formula v = n [n — 1) — ■ 2d — 3r it ensues at once that 

 the order of t is Sm^. 



From 3u — r = 3r — (> we deduce for the number of points oj 

 infiexion '6{ni — 2)(2??i-f-l). 



The gemis of t is equal to that of y^'", \ iz. equal to \m{lni — 11). 



And we now finally arrive from 



.^=:i(r-l)(i;— 2)-(rf+ q) 



at the number ^{m — 2) {m — ^) {nf -\-l ni-\-'^) of hitangents. 



It appears from the results arrived at that 9v(l ;???,) has 3 (??i — 2) 

 (27».+l) rays nntli triple null-point N^^> and | {m — 2) (?/?— 3) 

 {m^ -{-7 ni-\-4:) rays that have turn double null-points each. 



By means of these two numbers it would be possible to determine 

 again the order of the complementary curve. For the curves y and 

 ■a will touch in the triple null-points and must intersect in the 

 coupled double null-points; they have further in each singular point 



