30 
MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
172. The replica of a multiple system S may have some single systems in common 
with S ; some of the single systems may be their own replicas. 
173. The letters placed adjacent to graphical units for the purpose of reference 
compose a system which is a replica of the graphical system. 
Independent and Related Systems. 
174. Let S, S' be a system consisting of two detached systems S and S'; let 
a, b, c, d, . . . be the units of S, n the number of those units, and in the number of 
the self-correspondences of S. Let a', b', c , d', . . . be the units of S', n the number of 
those units, and m the number of the self-correspondences of S'. Let y be the num¬ 
ber of self-correspondences of the system S, S': y cannot be > mm', otherwise there 
would be duplicate rows in the tabular representation of S, S'. Consider now a set 
of rows in the tabular representation of S, S' obtained by taking any row It and ail 
others in which the letters a', b', c , d!, . . . remain untransposed from the order they 
have in R. Let the number of rows in the set be k. A transposition indicated by 
two rows of the set if made to operate on any row of the set will clearly give a 
row also of the set. Now take any fresh row not of the set; this must also be 
one of a set of k rows detached from the former set. Thus all the rows divide up 
into m sets of k rows, and we have m’k—y. If k' be the corresponding number in the 
case where a, b, c, d, . . . remain untransposed, we have mk'=y. Thus we have 
m'Jc=mk'=y=mm' or < mm. 
175. Let y — mm', then k=m, k'=m f , and the two systems are independent. 
176. We must have mm'=y unless m and m have a common integral factor; so 
that two systems S and S' must be independent unless the numbers of their respective 
self-correspondences have a common factor. The two systems may of course be 
independent when m and m have a common factor. 
177. If m and m are prime to each other S and S' may be said to be prime to each 
other ; so that systems which are prime to each other are independent. 
178. If mm < P, m and rn must have a common factor, and the two systems S 
and S' will be related. If Vm'n . . ., x'y'z . . . are undistinguished components of 
S', and abed ... be an aspect of the whole system S, then the aspects Vm'n' . . abed . . . 
and x'y'z . . . abed . . . will not in general be undistinguished from each other ; and any 
graphical diagram representing the system S, S' must have lines, or successions of 
lines, connecting units of S to units of S', so that S and S' cannot be drawn on 
separate sheets of paper. 
179. If we have three detached systems, S l5 S 2 , S 3 ; and S : and So are related, and 
also S x and S 3 , it does not follow that So and S 3 are related ; the single consideration 
that m l and m. 2 may have a common factor, and also m 1 and m 3 , without m 2 and to 3 
having one, shows this. 
180. If m — m = y, then S and S' are replicas of each other. 
