THE REV. T. P. KIRKMAN ON NON-MODULAR GROUPS. 215 



ohgYcHU 

 aKi?d\]jM. 



/E;T/I/3B 



tYkCnRwY. 



tJ)p¥yJdS 



ivA^VqSnT 

 a^MkJsGbR 



vDfCc/GuV 



vA/SVpQmY 



vYlIzKbJj 



rBsKeYiJ) 



vTlCmNyM 



r^qGwJjdOi 



uAyGiUhl 



uPpJjk^sB 



aEcHwKmS 



yBuRUqV 



yYgAjKeN 



yDclsMnQ 



aytxvra ydzhkmf xwcfpli rhujnpk 

 au^ywzs thsimqg vcyeqkj u^ledgn. 



Every duad of the forty-nine elements is once and once 

 only employed. If these be read as systems of didymons 

 factors_, it is easy to prove by inspection that every triplet 

 made with the forty-nine elements is reducible ; for there is 

 no mnltiplet which does not contain a letter permutable 

 with a letter of every other mnltiplet, and every octuplet 

 has a letter 6 permutable with every letter 6' not in the 

 octuplet, such that &6=.&' = 66' , The key to the above 

 multiplets is the following system of 14-1-21 triplets, each 

 being a set of three mutually permutables, in which system 

 every capital is found four times, and every small letter 

 thrice. 



AQR 



SFH 



NTP 



ILE 



GDM 



VBJ 



UCK 





ASN 



QIG 



RUV 



FEK 



HDJ 



TBL 



PCM, 





ahA 



acB 



adC 



ylY 



7iS 



75G 



vgM 





efA 



ghB 



ijC 



uqY 



ffwS 



cnGc 



fuM 





vpK 





vz^ 



tfh 



tnV 



ty^ 



xhK 





^mU 





IbB 



jl3h 



kwV 



pdH 



se^ 





xq'N 



^/3d; 



aiQ 



akT 



auF 



reJ 



rmV 



rwl 



zriN 



kbJ) 



yhQ 



psT 



crriF 



ziJ 



ly? 



qdl. 



We have now to construct this sr 



oup of 8 . 7 . 6. 



We 



observe that the first octuplet has its elements all permu- 



