PART 5. 



SEQUENCES AND INDICES FOR ALL GROUPLESS TRIAD SYSTEMS ON 



15 ELEMENTS. 



By L. D. CuMMiNGS. 



Tlio 80 systems described in this paper separate into the three distinct types: 



(a) Twenty-three systems with a group and with a head. 



(b) Twenty-one systems with a gi'oup and with no head. 



(c) Thirty-six systems with no group and with no head. 



The indices for type (a) and the Heffter system of type (b) have been fully discussed in an 

 earlier paper ' and therefore are omitted here. The indices for types (b) and (c) are given in 



' Cummings, L. D. Transactions of the American Mathematical Society, vol. XV, No. 3, pp. 311-327. 



the following tables 1 (b) and 1 (c), respectively. Now in a given system the index of a triad 

 is invariant under all the substitutions of the symmetric group ; two systems, therefore, which 

 differ either in their indices or in the niunber of triads eniunerated imder each index are cer- 

 tainly incongruent. Hence tables 1 (b) and 1 (c) show conclusively the noncongruency of these 



56 systems. 



Table 1 (b). 



54061°— 19- 



81 



