MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
29 
abode 
as 7 7 or 
aeacb 
abode 
acebd’ 
It is, however, to be understood that the term “ self-correspondence ” 
will always be used, as heretofore, with reference to self-correspondences which involve 
the correspondence of undistinguished collections only. 
or. 5 
Fig. 27. 
167. The two systems S and X of sec. 81 are of the same form. In the tabular 
representation of either the unified columns will represent the units of the other. 
Further, S and X are independent systems, no special relation except that of 
similarity of form exists between them. Each aspect of S is arrived at by con¬ 
sidering a correspondence of S and X ; it may in fact be regarded as such a corre¬ 
spondence, so that a unified aspect is a unified correspondence, and the whole system 
of unified aspects of S represent a system of unified correspondences of two systems 
of the same form, in which we are restricted to such correspondences as those referred 
to in sec. 163. 
168. When two systems are regarded as corresponding they may be spoken of as 
being projections of each other in as many ways as there are unified correspondences. 
169. If we represent the units of one of two independent systems of the same form 
by the symbols (\a), (X&), (Xc), &c., we may represent those of the other by the 
symbols (pa), (p&), (pc), &c., where in one correspondence w T e may conveniently sup¬ 
pose that (Xa) corresponds to (pa), (X6) to (p6), and so on ; but it is not to be sup¬ 
posed that this correspondence is distinguished from others. 
Replicas. 
1 70. If a, b, c, d, . . . and a, /5, y, §, . , . be two systems of units such that a and a 
are unique with respect to each other, as also b and /3, c and y, &c., and if, a and b 
being any two units of the first system, when a>—“< b we have also aa < bj3, 
then a, f3, y, 8, . . . may be called a replica of a, b, c, d } . . . 
171. The replica of a system S is of the same form as S, and has the same relations 
to other systems as S ha3. In the tabular representation of S and its replica, what¬ 
ever transpositions of the letters representing an aspect of S takes place in passing 
from one row to another, precisely the same transposition takes place in the case of 
the letters representing an aspect of the replica. 
