292 KEV. T. p. KIKKMAN ON THE THEORY OF 



12345 25314 41352 54321 = G\ 



24153 



51243 



15423 



42513 



45231 



14532 



52134 



21435 



53412 



43125 



23541 



13254 



31524 



32451 



34215 



35142 



12345 



23541 



51243 



35 142 = 0'.^, 



25413 



31425 



13452 



52431 



53124 



15234 



32514 



21354 



34251 



54312 



24135 



14523 



41532 



42153 



45321 



43215 



12345 



24351 



51324 



45312 = G', 



25134 



41235 



14532 



52431 



54213 



15423 



42153 



21543 



43521 



53142 



23415 



13254 



31452 



32514 



35241 



34125 



12345 



25143 



31542 



53241 =G-'s 



23514 



51324 



15234 



32154 



35421 



13452 



52413 



21435 



54132 



34215 



24351 



14523 



41253 



42531 



43125 



45312. 



Each of these consists of a group of powers of a substi- 

 tution of tlie fifth order and of its derived derangements, 



19. Whatever N may be we have always 

 (N- 1)^-1=0 (mod. N). 

 If then we add to G = S (i + c) the derived 



Q'G = (N-i)i-G=-iG (mod. N), 

 we have (theorem E) a group of 2N substitutions. 



The first and last of the four groups of five in G'(i8) 

 make such a group of ten. 



All the substitutions of the derived Q'G have the form 



q=- {i-^c) = -i-c (mod. N), 

 and 



92=- (^4.^). _.(i + c)=i + c-c = i=(i) ; 



