288 



line p, the remaining null-points xV ' of tlie null-rays n borne by 

 N will describe a curve (A' ')/,. Its order is evidently' equal to the 

 number of rays n, which have a null-point on p and another on q. 



Let us now consider the points that {N')^ has in common with /). 



Each of the ,? null-points of ;ms associated to each of the remaining 

 (/? — 1) null-points, and therefore is a (>' — l)-fold point of the curve 

 (iV '). The remaining points iV^' Iji'ig on p are evidently double 

 null-points on one of the null-rays determined by them. Hence: 



The locus of the double null-points is a carve {N ^) #ƒ order 



a" -|- 2«,i — u -f- ? — 'Ï — <^^' 



The consideration of the curves (/^) and (Q) pi'oduces analogously : 

 The double null-rays envelop a curve in^) of the cluss ^V -\- 2<f/^ -\- 



-j- a—^—a—(7^. 



5. By means of an arhihary conic </" another null-system ma}' 

 be derived from a given null-system. Let .V be one of the null- 

 points of the ray n, A'* the intersection of 7i with the polar line 

 of zV with regard to (f'\ A new null-system arises now if on each 

 straight line 7i the null-points jV are replaced by the coi'responding 

 points A'* '). The number /? remains intact. In order (o find what 

 a passes into, we observe that the null-rays n of the new null- 

 point A^* must have one of their old null-points A'^ on the j)olar 

 line p of A*. The null-rays n of the points of p envelop the curve 

 {p)u-\-3- On each of the {(( -\- i^) tangents which it sends through i\^* 

 is jV* one of the new null-points. 



Bj/ the harmonical iransformation '^1 {<(,^) is there/ore transformed 

 into a ^1* {u + /?, ^). 



If A^ lies on y" while one of its null i-ays touches at y*, A'^* 

 becomes an arbitrary point oï n, and 7i a singular straight line of ?i^*. 



In order to determine the number of these singular rays, we 

 associate to each tangent n of 7' the /? tangents /;, which meet n 

 in its /? null-points N. 



The envelop {p)y.-\-,i determined by ;> has evidently 2 (« -f- /^) 

 tangents in common with 7;'. Besides the straight line /;, which, 

 as /?-fold tangent of the envelope {p), replaces ,? common tangents, 

 (2« -\- /i) rays n are associated to p. The correspondence between 

 p and n has 1{(( + /?) coincidences ; on </" lie therefore 2(« -|- ^) 

 points iV, of which one of- the rays n touches at 7\ In other words 

 ?1R* («-|-/3, /i) has 2(« -j- j5) singular rays more than 5yj(/'<,/S). 



1) The ^'harmonicar' transformation dually corresponding to this I applied 

 formerly to a ?ilO,^3) (vide "Plane Linear Null- Systems'''). 



