303 



We find (hen for null-curve of 



(.c, a(2) — x^ a(iJ) ;i;,"'-i + ... — 0, 

 from which it is evident that the null-curve rtA:"'+^ of St has anode 

 in Sh- 



This result was to be expected, but of course holds good onlj in 

 the case of S being single null-point for an arbitrary' ray passing 

 through S. 



2. If a point N describes the straight line p, its nnll-i-aj n 

 envelops a curve of class {m -f 1), which will be indicated by tlie 

 symbol {}S),n-\-\- ^or tiie null-curve of an arbitrary point Q i'dersects 

 p in {m -\- \) points N, of which the null-rays pass through Q. 

 Evidently p is an 7/?-fold tangent of (/>)m-|-i • 



The null-curves (/')/,i+i a'^^ (?)m+i '^^ve a common tangent in the 

 null-ray of the point pq. Each of (he remaining common tangents 

 is a straight line n, of which one of the null-points N' lies on /;, 

 another null-point A^ ' on q. If N describes (he straight line />, the 

 remaining null-points N' describe consequently a curve {N') of 

 order {nf -\- 2 in). 



Each of the null-points of p is to be considered {m — J) times as 

 point iV', so that {N ') in those null-points has m{ni — 1) points in 

 common wi(h p. In each of the remaining Sm intersections of /; 

 with (A^'j a point A'^' coincides with a point A^ into a double 

 null-point A^(2) of \\^q corresponding straight line n. 



In a double null-point the curves (P) of a pencil have a common 

 tangent; one of the pencil-curves has a node (here. The locus of 

 the double null-poin(s [curve of coincidence) coincides with the 

 Jacobiana of the net of the curves {P). As the latter is in general 

 a curve of order 3?/?, the conclusion may be drawn from the above 

 made sta(ement that the null-system possesses in genersil no singular 

 straight lines. For, if a straight line has each of its points as null- 

 point, it is common tangent of nidl-curves (/>),,i-|-i and {q),n^i. 



The curve of coincidence y^'" has, as Jacobiana, {ni^ -\- m -\- 1) 

 nodes Sk- 



This may be confirmed as follows. Through /^ pass {m" -j- m — 2) 

 tangents of (/-*)'"+*: their points of contact are double null-points, 

 consequently points of y^'"- The remaining 3//?v7/? -|- 1) — (?/i' + m — 2) 

 intersections of {!*) with y must lie in the singtdar points, but then 

 y must have a node in each point S. 



3. Let us now consider the locus x of (he groups of (m — 2) 

 null-points, lying on the null-rays ^, which possess a double null-point. 



20* 



