ME. A. B. KEMPE ON" THE THEORY OP MATHEMATICAL FORM. 19 
be represented thus : — 
abode 
b c d a e 
c d a b e 
d a b c e 
adobe 
d c b a e 
c b a d e 
b a d c e 
101. The symbol thus arrived at, consisting of nm letters arranged in mrows and 
n columns, will be termed the tabular representation of S. Each of the nm letters 
will be termed an element. 
102. The order of the rows is clearly immaterial. Any alteration in the order of 
the columns merely substitutes for one system of undistinguished aspects of S another 
system of undistinguished aspects of S, and any one of these systems of undistin¬ 
guished aspects defines the form of S, thus the order of the columns is also imma¬ 
terial. The material point is that certain elements are all in the same row, and 
certain elements all in the same column. 
103. Each sort of letter represents a unit of S ; each column regarded as a unit 
represents a sort of place; each row represents an aspect of S, or, regarded as a unit, 
a unified aspect of S ; and each letter, or element, an aspect of a unit of S. 
104. Any row R. 2 may be regarded as derived from another row K 1 by a substitu¬ 
tion of the letters of the row R 1 . If the same substitution be effected upon any 
row E 3 , we get a row E t which is also a row of the table. 
105. The substitutions by which any row of the tabular representation of S is 
derived from another may be said to be substitutions proper to S. 
106. If we confine our attention to certain rows and columns only of the tabular 
representation of S, we get a table which will be termed a constituent of the whole 
table. The order of the rows and columns will be disregarded in a constituent as 
in a complete table. 
107. If a <-> b, a and b can never appear in the same column. Thus the 
columns break up into lots, each lot belonging to a single system. 
108. If a >-< b } then a appears in every column that b does, and b in every 
column that a does. 
109. Generally if abed . . . >- <C.pqrs . . . we have in the table a constituent 
C p\°r s' ' an( ^ ^ a ^ c d • • • <-^ pqrs ... we have no such constituent. Thus 
the forms of the various components can readily be ascertained from the table. 
110. If we desire to consider the forms of portions only of a system, we may 
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