MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
57 
quadrates ; but in such case we ignore differences, and do not deal with S, but with a 
single heap system of the same number of units (secs. 127, 128). The special form 
of S has its effect as the selector of special quadrates, or collections of quadrates 
(sec. 321). 
325. Thus if S be a group the unified pairs divide up into systems, and we 
naturally select these. The product of two such systems of unified pans is another 
such system. The unified systems (which are represented by the sorts of the pans) 
give us a primitive associative algebra. The equations we obtain are precisely the 
same as those we should get if we considered chains in the manner indicated in 
section 234. In both instances the equations exhibit relations existing between sorts 
of letters, which in both instances represent simple networks. In the one instance we 
arrive at these relations by considering individual component pans of simple networks, 
in the other by considering the whole systems of pans composing those networks. 
326. The unified aspects of any system S are a group. The unified pans of these 
are substitutions by which any aspect of S is substituted for another. The product 
of any two of these is a substitution proper to S (sec. 105), viz., that which results 
from operating with the multiplier and multiplicand in succession. 
327. A substitution proper to S is represented by a two-row constituent of the 
tabular representation of S. Each column, of this constituent may be regarded as a 
quadrate, the whole constituent being thus regarded as a collection of quadrates. 
Thus a substitution may be represented either by a collection of quadrates obtained 
by taking pairs of S, or by a collection of quadrates obtained by taking pairs of the 
group of which the units are unified aspects of S. 
Isolated Collections — Residuals—Satisfied Collections. 
328. A collection of units which is such that each unit of the collection is unique 
with respect to the residue of the collection may be said to be isolated. Each unit of 
the collection may be termed the residual of the rest. 
329. The residual of a pair a, b may be written ( ab) (sec. 139), and similarly in 
the case of triads, &c. 
330. We may graphically represent an isolated triad as in fig. 61, where X is an 
Fig. 61. 
auxiliary graphical unit; a non-isolated triad being represented as in fig. 62, without 
finks or auxiliary units. The same mode of representation. may also be applied in the 
case of isolated n-ads where n =f= 3. 
MDCCCLXXXVI. T 
