( 359 ) 



if but in the diagram we exchange rows and columns. It satisfies, 

 moreover the further conditions; so we have: 



The deduction of an S(p,2), {p — 1) n -f- 1 out of an S(p,2),nis 

 assured, when p = 5 or a prime number -\-l. 



If we apply this to n = p, we find: 



The existence of S {p, 2), n = p {p — 1) -|- I is assured for p = 5 

 or a prime number -\- 1. 



These systems form the term I, x — I ; as Cf.-schemes they are 

 CfF. \p (p -- 1) + 1,1 



It is clear that their remainders to one element again become 

 S(p — 1,2), n = {p — 1)', that is term II, x = 1 out of the series 



P -i. 



In cyclic form we find : 



ra = 13, p = 4 : (1, 2, ..5, ..7) eye. 



n = 21, p = 5 : (1, 2,. .5,. .15,. .17..) eye. 



n = 31, p = Q : (1, 2, . .5, . .7 . .14, . .22 . .) eye. l ) 



We finally pass on to the generation of an «S'(4, 2,) ?i=25. 

 As Cf. the system is a Cf. (25 8 , 50 4 ), the remainder to each qua- 

 druplet is a Cf. (21 J. The latter must have the property that the 

 non-united elements may be united to triplets, so that out of it arises 

 a Cf. (21 4 , 28,) which breaks up into four principal-7-sides "). By 

 imagining the seven lines of such a 7-side to be every time convergent 

 to one point and the four points of convergence to be collinear, 

 the desired Cf. (25 8 , 50 4 ) is formed. 

 We obtain a solution by submitting 

 1, ..3,.. 9, ..12; 



1, . .8, . .15 , (completed by 22); 



1, . . 2, . . 6 , (in turns completed by 23, 24 and 25); 



to the cycle 



(1, 2, 3, . . . 21) 

 and by finally adding: 

 22, 23," 24, 25. 

 In like manner we shall find that in general an 

 S(p,2), n = 2p(p — l) + l, 

 that is a Cf. \2p (p — 1) -f i 2 , y , 4/> (p — 1) -f 2 f ,\, 



J ) Gomp. E. Malo, (Interm. des Mathém. XVI, p. 63). The 2p cycles given 

 there for each p are mutually identical and so they furnish every time but one 

 type. 



') Gomp. a former paper in these Proceedings (p. 200). In the mean time I have 

 found that the result given there, in as far as it concerns n = 13 is obtained in about 

 the same way by Brunel {Journal de Liouville, 1901), and already m 1899 by 

 de Pasquale, Rendic. R. Inst. Lombardo (2) 32, p 213. 



24 



Proceedings Royal Acad. Amsterdam. Vol. XI. 



