( 357 ) 



Tho term x = has no significance in /; in // it indicates that 

 when writing all // = p letters we have written al the same time 

 all pairs of those letters. This system II, .« = will he of service 

 in what follows. 



We shall now give a more extensive form to the multiplication' 

 theorem of NETTO (I.e. § 3) which heroines : 



Out of >ui S{i>;l)n x and mi S(p,2), n % an. S{p, ,2)»,?», can be 

 deduced, //'one disposes of a si/stem of p(p I) permutations of p 

 elements, m which <i couple of elements never occupies the same place. 



For, if the elements of the two systems are resp. ak{k-= 1,2 . . . .n,) 



and bi(l=l,2 //../) we can then designate n l n i new elements 



by C/-J. < M' these elements we form three kinds of /> sets of />, namely: 



1 st . out of each set of p <t x a a . , . . a» new jy-sets: 



Ci,/, c 2 ,/, . . . , . e„j (/= 1 ,2 //,,) ; 



2" (l . out of each set of /> /^/^ .... A,, new //-sets: 



Cjfc.l, Cic,2, <!,,/, (/' = J ,2 M,)j 



3 rd . out of each set of p a^, . . . . a p , combined with each sel of p 

 bjt^ .... ^ new //-sets : 



Ci,/, , ('2./. <-',,,(/,, where l x /,, are every time the same 



as the indices \ .... p of the set of /> of b, vet dilfer p — 1 times in 

 order of succession according to the p — J permutations of the system 

 of permutations supposed as disposable. 



It is clear that in this way a couple of the new elements can 

 never appear more than once whilst the number of formed sets of 

 p amounts to : 



»iK— 1) . »„(*»»— !) . . n »,K"-l) »>,— 1) »!«,(»,»,— 1) 



».- 4- n. h /' ( P — 1) • = — 



>(P-1) /'(/'-I) /•(/—!) /'(/'-I) p(p-l) ' 



so that really an S(p,2), nji^ is formed. 



Now we dispose of such a permutation-system, when : 



1) p — 4 : the twelve even permutations; 



2) p prime, the system then consists of: 



(1. 2, 3, 4 />),.,,,, 



(1, 3, 5, 7 p—Viajc. 



(1, 4, 7, 10 ),,,. 



(1, p, p— J, p— 2. ..2),,,,, 1 ). 



! ) Comp. e.g. Brunel, Proc. Verb. Soc. de Bordeaux 1894/95, p. 56, or Ahrens. 

 Mathematferhe Unterhaltungen, |>. i'7-J : "Promenaden von u- Personen v.w jen". 

 II is mil decided whether also other values of /> allow a solution. 



