Mathematics. — ''Linear Nidl-Sysieius in the Plane ' . Bj Professor 

 Jan dk Vries. 



(Communicated in the meeting of April 26, 1918). 



1. A liiieai' iiull-syslein ^)^ (1,///) may be determined by two 

 equations of the form 



^1 '^i ^ ^j .^, + 'êt^t = ^ 



where Ak indicates a function of order m, in .r/^. 



When the straight line n revolves round the point P{yk), its m 

 null-points N, viz. the intersections of ^r = with ' the curve 

 ^hA/c = 0, describe a curve of order {)it -f 1). As ^.y = 0, this imll- 

 curve (/-*)'" + ^ has as eqnation, 



.Vi .Vf .Vi 



A, A, A, 



Tfie cnrves (/■')"' + ^ form a net that is represented on the point- 

 lield by the points P; for each netcurve belongs (o a definite point P. 



The net has (w^ -}- in + 1) base-points. For, if for the sake of 

 brevity its eqnation is written in the form 



yi ^. + .V, B, 4- t/, 5, = 0, 

 it appears that the curves B, := and />, = have in the first 

 place the points indicated by .<,•, = 0, yl,-=0 in common, which, 

 however, do not lie on the curve /^, = 0. For the {m* -j- ni -j- J) 

 points Sji, which they have moreover in common, we have the relation 



A^ :.?;, =:A, : .r, =A, : a-,. 



These points lie consequently at the same time on i?, = 0. 



Each of the base-points Sk bears ooi null-i-ays ?i, is therefore a 

 singular point of the null-system. 



Two null-curves (P)'» + ' and (<?)'" + ' have in the first place the 

 ??? null-points of the straight line PQ in common; the remaining 

 intersections must be singular as they bear each two null-rays; they 

 are therefore identical with the {ni'' -{- m -{- i) singular points S. 



If the point 0^ is laid in one of the singular points we have to 



— Mk) 



write Ajc 

 of .r, and .r,. 



.f,'"- 1 -j~- • • » where a'*' indicates a linear function 



