92 



And on the partition, 



10 = 3 . -f 3' • r + 2 • 2, (M3 = 1 = M/ = M2 = FO 

 we can construct, if we take S3 — 6 8^3= 3 ; 



ttJ^ 



1 • 1 • 2 • 6 • G • 2^ 



woven groups each of G • 3 • 2' • 1 • I • 2 — 144 substitutions, 

 all (iilferent groups from those above enumerated. 



(1 00) If we take for each of the r groups Gn the R cyclical 

 permutations of 123 • • R, and write 



Sji =: R, 4 ={'^r), 

 we obtain the group of 



A" B^' C J' Tra Trb TTC ' ttJ 

 substitutions of which Cauchy has demonstrated the existence 

 in the I Xth section of his "Memoire sur les arrangements," &c., 

 in the Exercises de Malhanalique et de Physique Analytique. 

 It is evident that the largest woven group constructible on 

 our partition of N is of 



(ttA)'* (wBy {vCy ' ' ' Tra-Trh ire • ' ' ' —Q 

 substitutions. This group has no derived derangement, i. e. it 



is a maximum group, and has equivalent groups, itself 



included. 



This theorem M is, as I believe, in that generality, new. 



On the Construction of Functions. 

 (104) Theo. N. Let G be any group (I Ai A2 • * Al-i) of 

 L substitutions, made with N elements (12* • N). 

 Let 



p _- a /3 7 . . . e 



Jl — Xi X.^ X-^ X^ 



be the product of N different powers (7O) of the N vari- 

 ables a?! rcj • • • • ic , 

 Let 



be the sum of the N terms, obtained by performing on the 

 subindices of P the substitutions of G. 



