247 



quadruplet, is permutable with all its four triads. Vide 

 art. 76 of my memoir 0)i 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=^3 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),=A •517-126=: 

 A-1462537-l243765=zA 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 



