‘246 
We can next form a quadruplet, upon each of the 21 triads 
thus : — 
on (157), 517 on 
(524), 254 
on (563), 653 
126 
517 
517 
(B) 
751 
452 
365 
134; 
563; 
524; 
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 2T6 
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 
164352 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 
1643527 1462537 
6235417 1243765 
2154367 1326547 
1 6 3 4 7 2 5, 
of which the former are the didymous radicals of 0 : ,=6 1 53427 
of the third order, and the latter are those of 04=1362745 of 
the fourth order, where <V=(157), which determines the 
