( 353 ) 



A geometrical way of putting the question is this: 



which combinatory configurations whose points £ are represented 

 hi/ the combinations </ to </ of u letters, whilst the combinations p 

 to j> represent its Sp^—q . jwssess systems of ' s '/y— ,, containing all 

 points of the Cf each one time and how many types of sack systems 

 appear in each definite case? 



The first question gave rise to investigations for = 3, q = 2, the 

 triple systems (Kirkman, Reiss, Netto, Moore, Heffter, Brunel 1 )); 

 the second question is discussed by J. de Vries 1 ) and Carp*). 



Here some results are communicated for /> ^> 3, q ^> 2. We 

 adhere for this to the first form of the question and we call a 

 system as is demanded there an S(p,q),n. 



If we isolate in an S(p,q),n all sets of p having in common a 

 certain arbitrarily chosen letter, and if we omit that letter from it, 

 an S(p — 1, (j — 1), it — 1 is generated; repetition of (he operation 

 gives rise to an &{p — 2, q — 2), n—2 and so on; the possibility ol 

 an S(p t q) presupposes thus that of a series of systems of lower 

 rank 4 ), which series can be broken oil" at 



S (p - q + 2 , 2), n - q + 2 

 Inversely all imaginable systems are acquired by completion of 

 systems commencing with q = 2. 

 So we can expect : 

 out of £(3,2), w= 7 : £(4, 3), n= 8 .... A 

 out of £ (3,2), n = 9 : £ (4, 3), n = 10 . . . . B 



£(5,4), n — 11 .... C 

 £(6,5), n = V2 . . . . 1) 

 etc. We will show that the four systems mentioned exist each oj 

 them in one type 



b Cambridge and Dublin Moth. Journal II. 1847; Journal f.^J. r. n. a. 

 Mathem. 56, 1859; Mathem. Annalen 42, L893; 43, 1893; 49, L897; 50, L898; 

 Association francaise, Congres 0>' Bordeaux L895; Journal de Lioavdle (5) 

 VII, 1901. 



2) Versl. en Meded. Kon. Akad. v. Wet. 3rd series, VI, p. 13, 1889 ; Mathem. 

 Annalen 34, 35, 1889, '90. 



3 ) Dissertation, Utrecht 1902, p. 38. 



4 ) For each system of thr series the condition must be satisfied : I {divisible 



\<tJ 



/' \ / l : M 



. Thus there will be no S(6,4), n= L5, although I is divisible by 



/UN . 

 , on account of the impossibility of an S(5,3) n 14 : ( J ts not divisible 



