MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
27 
157. In the same way it may be shown that the pairs which ( abc . ... ) makes 
with a, b, c, . . . respectively are all distinguished from the pairs which (abc . . . ) 
makes with any other units x, y, z, . . . not components of the collection a, b, c, . . . 
158. A unified aspect ( abc . . . ) is accordingly a discrete pair associate of the 
collection a, b, c, . . . (sec. 14G). 
159. It should be observed that in the case of a number of undistinguished aspects 
no unit can be said to be the unit arrived at by regarding any particular one of 
those aspects as unified. All we can say is that any one of a number of units may 
be so regarded. If, however, one of that number is regarded as representing a 
particular one of the unified aspects, some other definite unit of the number must 
be regarded as the unit which represents another given unified aspect, i.e., the two 
unified aspects will be such that they can only be represented by certain pairs of the 
units and not by any pair. 
160. When we represent aspects by single letters, those letters really represent 
the unified aspects. Whatever relations as to distinguishableness or undistinguish- 
ableness exist between the aspects, there will be precisely the same relations between 
the unified aspects ; so that vve may deal with either the unified or non-unified 
aspects. Thus if A, B, C, D, be the aspects abed, efgli, ijhl, uvwx, when regarded 
as units, then if abcdefgh > < ijkluvwx, we have AB > < CD. 
161. The conception of a unified aspect is an “accidental” one; for the units 
representing unified aspects of components of a system S may be regarded as 
representing any other things holding similar relations to the units of S. The 
method of defining systems by regarding their units as unified aspects of components 
of other systems is, however, so convenient and simple that it will be frequently 
employed, and the accidental part of the definition being borne in mind, no danger 
can arise from the employment of this method of arriving at systems. 
Correspondences of Systems of like Forms. 
162. We may consider correspondences of any two independent systems Sj and S 3 
of the same form, in which every component of S : corresponds to a component of 
S 2 of the same form, and we may regard these correspondences as units. The 
number of such correspondences is in general greater than the number of self¬ 
correspondences of each system, as in the case of the latter we only suppose 
undistinguished components to correspond, and do not admit correspondences of 
distinguished components of like form. 
163. If, however, we start with a correspondence of S : and S 3 and then restrict 
ourselves as regards other correspondences to those in which each component of S : 
corresponds only with such components of S 3 as are undistinguished from that 
with which it corresponded in the first instance, we shall get the same number of 
correspondences as there are self-correspondences of each collection S x , S 3 . 
