:m)7 



to each 6\"'+^ the (in^ -}- ))i — 2 — r' -f- /•)<'m'ves f,"'+', which it iiilersecl.s 

 on 7^'", ontsi(1e the points >S. The tignre produced by the pencils 

 coupled in this way consists of twice the curve y, of (ï/?.' -f- ^'^ — 2 — 

 — y' f /•) times the curve c"'^\ which belongs to both pencils and 

 of the compleinentarj curve x„. We now lind as its order (?//," -f ni 

 — 2 — ;■' -f r) {m + 1) — 6m i. e. {m — 2) (m"-- + 4/// + J j — (/n + l)r 

 (r-1). 



With regard to ^3 we conclude from this that the uidl-ciirve of 

 So is to be considered r {r — 1) tiuies as coni|)onent part of a. 



Applying the method of §6 again, we now find the number of 

 triple null-points from 



3m {m -2) + (m-2) (m* + 4//» + ' ) - ("* f 1 ) >' ('' - 1 )A^n -L') (m" Y m -2 -r' -(- r) 

 i.e. 



(m— 2) (6 m 4- 3) — 3 r (r— 1). 



Aiialogouslj we find foi- the number of null-rays with two double 

 null- points 



i {m- 2) {7n— -0) {m' -{- Im F 4) — i {m - 3) (m + 4) r (r- 1). 



8. A very particular linear null-system is obtained by supposing 

 that the functions A]^^ (^1) only contain j\ and./-,. In that case 



represents an involution of rays of the second rank, of which the 

 00* groups, each of ui rays, correspond projectively to the straight 

 lines of the plane. 



The null-curves have now in S^:=l(>^ an ?/?-fold point, are conse- 

 quently rational; the null-curve of S has degenerated into (???-(- 1) 

 rays, which each contain one of the siuiple singular null-points S. 



If the derivatives of Aj^ with regard to j\ and .*•, are indicated 

 ^y (^ifc)i ^^^^ (-4i)i, vve find for the locus of the double nidl-points 

 the equation 



X, x, .r, I 



(A,), (A,), (A,), \=0 

 UX (^4,), (AX 



This curve of order (2m — 1) has in 6', a i2ni — 2)-fold |)oint. 

 By the [m -\- 1) rays S^Sk it is completed into the Jacobiana of the 

 net of the null-curves. 



The rays i with the double null-points envelop a curve t of class 

 (277?, — 1); for (P)"'+i is now of class (777 + 'l)//7 — tn (w — l) = 2m. 



The triple rays of the above mentioned involution are indicated by 



