18 
MR. A. B. KEMPE OH THE THEORY OF MATHEMATICAL FORM. 
collection, m must be a factor of f n, and the aspects break up into '= systems, each 
consisting of m aspects. 
97. It follows immediately from sec. 92 that if we write down the symbols abed . . ., 
&c., representing any aspect and all other aspects which are undistinguished from it, 
and we transpose the letters of abed . . . and make precisely the same transpositions 
in the case of the letters of the symbols representing the other undistinguished 
aspects, we shall get a collection of symbols of undistinguished aspects which may or 
may not be the same as the former. 
98. If abed ... be an aspect of a whole system S of n units, so that all the aspects 
undistinguished from it are aspects of the whole system S, there being in all m un¬ 
distinguished aspects, the different transpositions of sec. 97 will give us all the 
different systems of aspects of S referred to in sec. 96. 
99. If the array of letters abed . . . representing any aspect of a whole system S be 
given, and also those arrays representing all the other aspects of S which are un¬ 
distinguished from abed . . ., then the form of S is given ; i.e., if pqrs . , . be an 
aspect of any component collection of S, we know what other aspects of component 
collections of S are distinguished and what undistinguished from p>qrs . . . For add 
letters to pqrs . . . until we get an array pqrs . . . Iran . . . representing an aspect of 
the whole system S. If this is not the same as one of the given arrays, transpose the 
letters of any one of the latter so that it becomes pqrs . , . hnn . . ., and make pre¬ 
cisely similar transpositions in the case of each of the other given arrays. If now 
wxyz . . . >-< pqrs . . ., it follows from sec. 94 that there must be an aspect 
iux yz . . . ijh . . of the whole system S, such that wxyz . . . ijh . . . >-< pqrs . . . 
Inin . . . ; and if wxyz . . . <-> pqrs ... it follows from sec. 95 that there is no aspect 
■wxyz ... ijh .. . of the whole system S such that wxyz . . . ijh . . . >-< pqrs . . . 
lain . . .; i.e., in the former case there must be a symbol wxyz ... ijh .. . among 
the transposed arrays, in the latter case there can be no such symbol. 
Tabular Representation of Systems. 
100. A convenient mode of arranging the symbols abed . . ., &c., representing a 
system of m undistinguished aspects of a whole system S of n units a, b, c, d, . . . is 
a 
4 
n, 
H'2 
Fis:. 23. 
to place them one above another so that letters occupying “the same position” in the 
different rows may lie in the same column. For example, the system of fig. 23 may 
