48 
MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
Substitutions. 
275. Each row of the tabular representation of a system is derived from each of the 
others by definite substitutions. Instead of writing down the various rows, we may 
give one only, and state the law or laws according to which the other rows are derived. 
Thus the form of a system a, b, c, cl, . . . is given by merely writing down the letters 
representing the units of the system (since each letter appears only once, we may 
represent the units by the letters in place of their sorts), and stating the laws of 
substitution, in other words, the substitutions proper to the form (sec. 105). 
276. Conversely, in considering a system of n letters, or other things, admitting of 
certain substitutions, we are considering a system of n units of a definite form. 
277. The various arrays of letters considered when dealing with substitutions are 
thus aspects of a system. 
278. When the arrays are regarded as units the substitutions will be represented 
by pairs ; substitutions of the same sort by undistinguished pairs ; similar substitutions 
by similar pairs (sec. 230). 
279. The substitution of one aspect of a system for another, substitutes for a 
component collection of any form another component collection of the same form. 
Algebras. 
280. Consider the system Y of 3 n units arrived at by regarding as units the pairs 
connecting the units a, b, c, cl, .. . of a system S of n units with the three dis¬ 
tinguished units \, g, v. If we adopt the method of representation of secs. 139 and 
152 the units of Y will be represented by— 
(ci\) 
(6X) 
(c\) . . 
(ag) 
(bg) 
(eg) . . &C. 
(av) 
(M 
(cv) 
It will, however, conduce to clearness in 
the present instance to write them thus- 
a K 
h 
C 
% 
K 
c M . . &c. 
C 
The symbols of the three rows represent the units of three different systems, each of 
which systems is a replica of the others and of S. We may denote the three systems 
when regarded as units by S A , S M , S„, respectively. 
281. Now consider the system E arrived at by regarding the triads connecting the 
three systems as units. This will contain ri d units, which may be written thus— 
(«AV-0, («Ac), &c. 
