( 555 ) 



case, it is easy (o reason thai PP' coincides with AB, CD or EF 

 and so the locus proper consists of* these three lines and there is 

 no envelope proper. The part improper of the locus however consists 

 of six conies ABODE, ABCDF, ABEFC, ABEFD, CDEFA 

 and CDEFB, the part improper of the envelope of the six points 

 A, B, C, D, E and F. The total locus is thus of order fifteen, the 

 total envelope of class six, so that for arbitrary position of the pencils 

 of conies this same holds for the locus proper and the envelope proper. 



Sneek Nov. 1906. 



Mathematics. - "The locus of the pairs of common points of f our 

 ■pencils of surfaces." By Dr. F. Schuh. (Communicated by 



Prof. P. H. Schoutf,). 



(Communicated in the meeting of December 29, 100G). 



J. Given four pencils of surfaces {Fr), (F s ), {F t ) and (irrespect- 

 ively of order r, s, t and u. The base-curves of those pencils can 

 have common points or they can in part coincide, in consequence 

 of which of three arbitrary surfaces of the pencils {F s ), (F t ) and (F u ) 

 the number of points of intersection differing- from the base-curves 

 can become less than stu; we call this number a, calling it h for 

 the pencils {F r ), {F t ) and (F a ), c for the pencils (f r ) (F s \ and (/<"„) 

 and (/ for the pencils (F,), (F 8 ) and {F,). We now put the question: 



What is the order of the surface formed by the pairs of points 

 P and P' , through which a surface of each of the four pencils is 

 possible v 



If the points P and P' do not lie on the base-enrves we call the 

 locus formed by those points the locus proper L on which of course 

 still curves of points P may lie for which the corresponding point 

 P' lies on one of the base-enrves. If one triplet of pencils furnishes 

 at least several points of intersection which are situated for all sur- 

 faces of those pencils on one of the base-curves, then there is a 

 surface that does satisfy the question but in such a manner that if 

 we assume P arbitrarily on this surface the point P' belonging to 

 it is to be found on one of the base-curves; this surface we call the 

 part improper of the locus, whilst both surfaces together are called 

 the total locus. 



2. To determine the order n of the locus proper h we find the 

 points of intersection with an arbitrary right line /. On / we take 



