( 356 ) 



The further constitution is now determined and must run as follows : 



J 2 6 ( .» 1 4 (I 9 a 2 3 6 a 2 5 6 7 9 3 5 6 7 



1 2 7 8 a 14 7 8 2 3 7 8 9 2 5 8 a 3 5 8 i» a 



1 3 6 8 rt J 5 6 8 2 4 6 7 rz 3 4 6 7 8 4 5 6 8 % 



I 3 7 9 1 5 7 9 a 2 4 <s 9 o 3 4 <> a 4 5 7 a 



1 2 3 (5 7 I 2 7) 6 a J 3 5 6 9 2 3 4 6 9 2 4 5 6 



L 2 3 8 12 5 7 i 3 5 7s 2347024 5 7 8 



I 2 3 9 a 1 2 5 89 1 3 5 a \ 2 348a 2 4 5 i» a 



I 2 4 6 8 I 3 4 (5 ! l 4 5 6 7 2 I! 5 6 8 .'5 4 5 (5 „ 



l 2 4 7 ( .l I :! 4 7 <i I 4 5 s u 2 3 5 7 r, :; 4 5 7 i> 



I 240a l :5 4 s i» 14 5 9 o 2 3 5 9 o 3 4 5 8 o 



A simple form of the system 1 ) is obtained by submitting the 

 quintuplets 



12 3 4 0; 12 'A 5 8; l ■) :\ 6 7: 

 1 2 4 5 9; 1 2 4 6 8; 1 2 5 7 

 to the cycle : 



(1 -J :; 4 5 (; 7 8 i> a). 

 />. By a reasoning analogous i<» thai in section .1 is evident thai 

 .5(6,5 , // = 12. that isaCf. 12,, ; . I32J must appear by complementary 

 completion of S 5.4». // = 1 1 . so ii appears only in one type. 

 In general oul of each 



s /' — 7 + '• 7 ■'. w ~ -7 + •> 

 is formed by complementary completion an 



S(l> = q + 2,ffH- I . /< = 2 7 + 4. 

 Passing i<» systems »S 0,^ = 2), 72 for arbitrary /< J . we observe 

 that for their existence is necessary ai least : 



n(n-l) /'(/'-I) 



divisible by - -, 



i ._ i .— 



and 



// — 1 divisible by p—^-y 

 which conditions arc fulfilled only by the two series of numbers 



/ n = P(P-V- x + 1 \a, = 0,l,2, 



11 it =p(p— l).#-fpj 



1 ) A 6(5,4), » = 11 was given by Lea {Educ. Times IXi. the method in which 

 this was generated is unknown to me, as this publication was not attainable 

 for me; however, the system must be of the same type. We remark moreover, 

 that to prove this, it is not sufficient to assert that all remainders have a fixed 

 type, comp. Martinetti (Annali di Matem. (2) XV). The same holds for A, B 

 and D. 



2 ) These S(p,*2) are to be distinguished from the systems of Bru.nel, in which 

 each set of p is regarded as containing only the p pairs of elements succeeding 

 «ach other [Proc. Verb. Soc. de Bordeaux, 1895/96, p. 58 and 1898/99, p. 59, 71). 



