363 REV. T. p. KIRKMAN ON THE THEORY OF 



I proceed to give a brief sketch of several investigations, 

 all of interest, and most of them of new interest, in this 

 theory, which I hope soon to discuss completely in a 

 second Memoir. 



76. On the didymous factors of the substitution <^. 



With every substitution ^ made with 

 N=Aa + B5 + Cc+--J/ 

 letters, of the order K, which is the least common multiple 

 of ABC • • J, there is given a certain number of systems of 

 K square roots a^a^a^- -a^, such that the product of any 

 two «TO«^ is a power of (j) ; and every power of (f) can be 

 written as the product of K different pairs of such didy- 

 mous radical factors of the system. 



Thus, with the group of powers 



i, (i+i), {i + 2), {i + 3'-, (e + N-i), =x,<l>,^\f--, 

 there is given the system of N square roots of unity 



{2-i),{l-i)[-i)[-i-l)[-i~2)-. (-z-(N-3)) 



and we have, (putting i-i for Hn a^ for the product a^a^, 

 aia<i{ = )i+i, a-ia^{ = )i + 2, «^fl^^( = )^ + 3, &c. 

 ac^az{ = )i+i, ao,ai{ = )i + 2, «2^5( = )i + 3, &c. 

 asai{ = )i+i, a^a^{ = )i + 2, a^ae,{ = )i + 'J„ &c. 

 It is easy to form the system of such radicals corre- 

 sponding to any substitution when the partition is 



N=N.i=A«, 

 and ^ is of the N"' order. Take, for example, the substi- 

 tution 57683241 of the eighth order. 

 We form the skeletons 



12345678 



12438765 = ^1 



<^=5768324i 



8 



3 



4 



6 



7 



2 



2 



7 , 



6 



4 



3 



8 



5 



