*i46 



We can next form a quadruplet upon each of the 2\ triads 

 thus : — 



on (157), 517 on (524), 254 on (563), 653 

 126 517 517 



751 452 365 



134; 563; 524; 



(B) 



on (715), 732 on (134), 126 on (261), 126 on (612), 126 

 517 341 237; 635 



746 157 621- 261 



157; 431; 245; 647 



In the first of these, 157 126 134 are three triads which 

 have the same capital, and 157 517 751 are the three triads 

 which have the same figures ; &c. 



The twenty-one quadruplets thus formed exhaust the 21 '6 

 duads not formed in the 28 triplets, so that we have once 

 and once only employed every duad possible with the 21 

 triads 157 571, &c., in these 21 quadruplets and 28 triplets. 



We may interpret the three triads 157 126 134 as the 

 three substitutions 



16435 2 7, 124376 5, 163472 5, 

 which obviously determine each other, having all three 

 elements undisturbed, and all being of the second order. 

 We have thus twenty-one similar substitutions, each defined 

 by a distinct triad. 



The first of the above written triplets and quadruplets are 

 found to be 



1643 5 27 1462537 



6235417 1243765 



2154 3 67 13 26547 



16 3 4 7 2 5, 



of which the former are the didymous radicals of ^,=6153427 



of the third order, and the latter are those of ^4=1362745 of 



the fourth order, where ^/=(157), which determines the 



