( 290 ) 



This case is represented in tig. 10 for the edge (21, 24), whore 

 3,49,50,57,58 arc the five adjacent points, whilst 50,3,49,57,58 

 appears as (he pentagon P 1 P 1 P 1 P 4 / > „ 50, 49, 58, 3, 57 as the 

 starpentagon 1\1\P.J\1\. Really P t P t projects itself into a point; 

 moreover l\l\ and I\I\ on one hand and l\l\ and P 2 P t on the 

 other coincide in projection, which is closely connected with this 

 that X is the same for the two constituents of each pair. 



Mathematics. - - a 0n triple systems, particularly those of thirteen 

 elements." By Dr. J. A. Barrau. (Communicated by Prof. 

 D. J. Kortkweg). 



(Communicated in the meeting of September 26, 1908). 



In a paper to this Academy 1 ) Prof. J. DE Vkies gave a triple 

 system of '13 elements of a different type than the cyclic system of 

 Prof. Netto 5 ); he added however the observation, thai no proofhas 

 been furnished of these types being the only ones. 



Mr. K. ZuLAtJF shows in his dissertation') that the systems given 

 formerly by Kikk.man (1853) and Reisz (1859) are identical to that 

 of de Vries, so that the number of known systems is two-, neither 

 is anything- here decided about die number of "possible systems. 



It seemed desirable to decide upon this point by means of a special 

 investigation A ). To this end some facility is offered by using those 

 expressions which are used in the theory of the configurations, by 

 regarding the 13 elements as points, the 26 triplets as lines which 

 bear three of the points; the whole of the triple system then becomes 

 (he scheme of a diagonalless Cf. (13„, 26 8 ) where it is irrelevant 

 whether this Cf. can be geometrically realized or not. A classification 

 of these Cff. is now our aim in view. 



The rest figure of the second order of a line of such a Vt, i.e. 

 what remains if we leave out that line with its three points and 

 the 3 X 5 lines passing through these points, is of necessity a 

 Cf. (10.,), the JO points of which are in three ways perspective and 

 that according to the three points left out. 



But then reversely each imaginable Cf. (13,., 26,) of the desired 

 type is obtained by : 



1 st . starting from all possible Cff. (10,), 



2 nd . by constructing for each Cf. (10,) the Cf. (10„15 8 ) of its diagonals, 



i) Versl. Kon. Akad. v. Wet. Ill, p. 64, 1894. 

 -i Substitutionentheorie, p. 220; Math. Annalen, Vol. 42. 

 '■'■) "Ueber Trlpeisysteme rot/ 13 Elementen", Giessen, 1897. 

 l ) I subsequently find this question treated also by de Pasquale (Rendic. R. 1st. 

 Lombardo, 2nd Ser., 82, 1899). 



