( 358 ) 



(For » = 3 the system jusl contains the total group and so 

 the theorem of Netto reappears . 



So we have: 



The possibility of the multiplication is fissured for ƒ? = 4 or prime. 



Now by taking ;is factors n l = n i =p, so term //,./• — (>, we 

 obtain : 



l-'nr [> ■==. 4 or prime the existence of S(p,2 , n = />* is assured. 



Such a system, term //, ,v = 1 is, regarded as Cf.-scheme, a 

 ( 'f. j/; 3 /( _|_i, // /?-f-l),.} ; its p-sets can l»c divided into (p 4" 1) principal 

 groups of />, each of which contains all the /> 3 elements. 



We can now give in a more extensive from an other theorem 

 of Netto I.e. § 2), which becomes here: 



Out of <m S(p, 2), // "it ' S '(//, "2), (/> I ) // -f- I can be formed if we 

 have 'if mn' disposal a scheme of (// V f sets <>j p out of />(/> J) 

 elements, having mutually not more t/ian "//<' element m <-t>iinii<>//, 

 which elements must In' able t<> break n/> into p principal groups 



of(p 1). 



For, we can add to the elements a\ t \, a\p . . . . a\ ttl of the given 

 system (p — 2) scries of new ones: 



''2,1 , ''2,2 > • • • ■ "2,-i 



"j.—\,\ . 'i,,—\,i , . . . . 'V-t,» ' 

 mikI moreover a last element a . 



For each set of p of the given system we musl now form outoi 

 the p{p — 1; elements with the same second indices a scheme as 

 the one indicated in the theorem, taking care that always the 

 ( /i 1) elements with equal second index form a principal group. 

 Finally we must add to each principal group the clement a 0} by 

 which also these principal groups are completed to p. It i> clear 

 that the sets of p formed in this way can have two by two at most 

 hut one element in common, while their number amounts to: 



n{n-l) l( p -l) n .\-l\{ip-l)n\ 



(/' — 1 • - + » = > 



U p(p 1)^ P(P-1) 



so that an S(p . '1) . ( p — 1) n -\- 1 i> formed. 



The possibility of the method now depends on the presence of 



a scheme : 



[pip-l^-i , (P— 1)2 j 



or. replacing p by p— 1, of 



\p(p + l)p . P^,}, 

 bul that is just the notation of the above-mentioned S(p , 2), n = p* t 



