( -203 ) 



according to their rest figures. It' we use the list of ScHROETER, then 

 the rest figures of the triplets of the system of Netto are resp. : 

 N (01..,/..),,,,, all VII, (0..6.8..\. yc l ) all VIII. 

 Their possible decompositions into principal five-sides are: 



in tin- preceding table 



3 



,, 5, 6, 7 ,, ,, ,, 



o 



>> ° >> J) >> 



Q 



»J ** >> )> )> 



„ 10,11,12,13 



,, 14 and 15 



„ 16,17, 8,19,20 „ 



According to 4 these decompositions now give rise to 20 triple 

 systems, among which occur with certainty all possible system- ; 

 however, they can he identical mutually and to K or to N. We 

 now remark that VI and VII can be completed in hut one 

 way ; so as soon as these rests occur the system is identical to 8, 

 resp. 9; from this follows already that K and 8, and likewise N and 9 

 are of the same type (see table page 292). 



Now rest figure VII appears for 11 according to the triplet 37,/; 

 VI on the contrary for : 



1, 2, 3, 4, 5, 6, 7, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20 

 resp. according to the triplets : 



123, 123, 145, 167, 123, 123, 47,/, 37,/, 35c, 57/>, 5G,/, 79c, 49,/, 

 27//, 18,/, 35c and GOb ; 



so 11 is identical to 9 and K, the remaining are identical to 8 and N 

 and we have proved : 



that except tin* tiro known /onus no triple systems of thirteen 

 elements exist 



At the same time is evident that to recognise a given system it 

 is sufficient to find a few rest figures until one meets cither VII 

 (or VIII in greater number than six) or II, III, V, VI, IX or X. 



The preceding has moreover given a method 3 ) to determine the 



l ) The cyclic order is here: 1 2 15 i ."> D 7 8 '.» :i h c. 



-) Compare de Vries Versl. en Meded. K<>n. Akad. V. Wet. VI, p. 13, where 

 in a more restricted sense the same method is used to trace the (exclusively 

 regular) principal poly-sides of the Cff. -„ . Such ;i principal poly-side however 

 determines a triple system and reversely. Indeed, the cyclic systems of 13 and 1") 

 appear already in the work of in; Vries (I.e. p. 16 ami 17), however without 

 being regarded as such. 



