247 
quadruplet, is permutable with all its four triads. Vide 
art. 76 of my memoir On the Theory of Groups and many- 
valued functions, in the volume of the Memoirs of this Society 
for 1861. 
It is thus proved that every pair AB or BA of the 21 
square roots of unity determined by the 21 triads have for 
their product either one of 2‘28 substitutions of the third 
order, or one of 2‘21 substitutions of the fourth order, or one 
of the 21 radicals. 
The values of the triplets possible with the 21 radicals 
have next to be discussed. Some of them will be, and some 
will not be, reducible to a couplet. Let {AB} denote any 
triplet or quadruplet of the 28+21 above formed, in which 
the couplet AB (consecutive or not) appears among the 
didymous radicals. The condition that a triplet ABC should 
be reducible to a couplet is any one of the following : — 
1° That A be permutable either with B or with C, 
2° That {BC} contain A' permutable with A, 
3° That {AB} contain C' permutable with C, 
4° That {AC} contain D' permutable with CBC=D, 
for ABC=A'C , CBC=ACD. D is equidistant with B from C 
in {AC}. Every one A of the 21 radicals has four per- 
mutables, which are those of the quadruplet (A). 
It is easily proved, or can be seen by inspection when the 
quadruplets are all written, that any irredudible triplet PQR 
in which QR=+ of the third order, is identical in value with 
ABC where BC=0i of the fourth order. Hence every irre- 
ducible triplet is of the form of A(BC) i =A • 5 1 7 • 1 2 6= 
A’1462537T243765=:A 1426735. It can be proved or seen 
by inspection of triplets and quadruplets (A), (B), that the 
only values of A which render this ABC irreducible are the 
eight following — 
372 , 273 , 425 , 245 , 356 , 635 , 467 , 647 . 
The circular factors of (BC)i are 2 4 6 3, 7 5, 1. None of 
