MR. A. B. KBMPB ON THE THEORY OF MATHEMATICAL FORM. 
17 
represented by dbcclef. . . . might equally well be represented by bdacfe . , yet, if 
this be done, the aspect originally represented by bdacfe . . . must be represented by 
dcbaef. . . , and that originally represented by pqrstu ... by qsprut. ... , and similarly 
in the case of other aspects and other transpositions of the sorts of the positions 
occupied by the letters in the arrays. Whatever change be effected in the order of 
the letters in one array, a precisely similar change must be made in the order of the 
letters in each of the other arrays of the same number of letters. The relative order 
in arrays of different numbers of letters is immaterial, as such arrays represent 
collections of different numbers of units which are therefore distinguished, and are not 
regarded as corresponding. 
Elementary Properties of Aspects. 
89. The symbols >-< and <-> will be used to represent “is undistinguished 
from ” and “ is distinguished from ” respectively; and such expressions as “let a 
> -< b ” must be read, “let a be undistinguished from b”; and similarly in other 
cases. 
90. Symbols, such as a, b, c, d, in which commas separate the letters, will always 
be supposed, as heretofore, to represent- a collection of units without reference to 
particular aspects, the order of the letters being accordingly supposed to be im¬ 
material. 
91. A statement such as abed .... >——< pqrs .... implies that the components 
represented by taking corresponding letters on each side of the >-< are undis¬ 
tinguished ; e.y., be >-< qr. A statement such as abed .... <- pqrs . . . . 
does not imply that a, b, c, d, . . . . < - > p, q, r, s, ... . for it is consistent with 
the statement that abed . . . >-< s>'qp . . . which implies that a, b, c, d, . . . >-< 
p, q, r, s, . . . 
92. If abed . . . >-< pqrs . . ., then bdea . . . >-< qsrp . . . ; and similarly 
in the case of any other transposition of letters. 
93. If pqrs ... be merely abed ... in a different order, so that we are considering 
two aspects of the same collection, and if we repeat on pqrs . . . the transposition by 
which it is derived from abed . . ., we get another aspect which by the preceding 
section is undistinguished from pqrs . . ., and therefore from abed . . . 
94. If abed . . >-< pqrs . ., and if l, m, n, o, ... be units other than a, b, c,d,. . ., 
there must be units iv, x, y, z, . . other than p , q, r, s, such that abed . . . Imno . . . 
> -< pqrs . . . wxyz . . . 
95. If abed . . . <;-> pqrs . . . each collection having the same number of units, 
and if l, m, n, o, ... he, units other than a, b , c, d, . . . there cannot be any units 
iv, x, y, z, . . . such that abed . . . Imno . . >-< pqrs . . . ivxyz .... 
96. Every collection of n units has | n aspects. If m aspects of the collection are 
undistinguished from each other, but are distinguished from all other aspects of the 
MDCCCLXXXYI. D 
