MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
Q 
O 
composing them are not. Tims the angular points of a square are undistinguishable 
from each other, and a pair of such points lying at the extremities of a side are undis¬ 
tinguishable from the three other like pairs, but are distinguished from the two pairs 
formed by taking the angular points at the extremities of the diagonals, which pairs 
again are undistinguishable from each other. Further, a pair ab may sometimes be dis¬ 
tinguished from the pair ba though the units a and b are undistinguished. Thus, in 
fig. 1 the three black spots a, b, c, are all undistinguished from each other, each has an 
a 
■arrow proceeding to it and an arrow proceeding from it, but the pair ab is distinguished 
from the pair ba, for an arrow proceeds from a to b, but none from b to a. 
6. It will be convenient to speak of ab and ba as different aspects of the collection 
of two units a, b. Here the terms “ aspect ” and “ collection ” are each to be 
understood as referring to two separate units, and not to those units regarded in the 
aggregate as a single unit. 
7. Again, there are also distinguished and undistinguished triads, tetrads, . . . 
m-ads, . .. n-ads . . .; every m-ad being, of course, distinguished from every n- ad. 
Just as we may have ab distinguished from ba, though a is undistinguished from b, 
so we may have an n -ad pqrst . . . uv distinguished from qusvt . . . rp, though the 
units p, q, r, s, &c., are all undistinguished from each other, and further, though their 
pairs are also undistinguished, as likewise their triads, &c. Here pqrst . . . uv and 
qusvt . . . rp will be termed, as in the case of pairs, different aspects of the collection 
p, q, r, s, t, . . . u, v ; the term “ collection ” being understood to refer to a number of 
separate units without reference to the various “aspects” of the collection. Different 
aspects of the same collection of n units will be regarded as different n-ads. 
8. The terms “pair,” “triad” . . . “ n- ad,” “collection,” “aspect” will always be 
understood to refer to two, three, n, &c., units, and never to aggregations of units 
considered as a single unit. Pairs, triads, w-ads, collections, aspects ma.y, of course, 
be regarded as units, but when they are so regarded the fact will be distinctly 
pointed out. 
9. Every collection of units has a definite form due (1) to the number of its 
Fig. 2. 
component units, and (2) to the way in which the distinguished and undistinguished 
units, pairs, triads, &c., are distributed through the collection. Two collections of 
b 2 
