1887.] On the Theory of Mathematical Form. 



195 



In future when we compare aspects of a collection, or of a number 

 of collections of the same number of units, unless the contrary is 

 manifestly the case, it will be supposed that the same collection of n 

 marks is employed in each case. 



We may choose as our discrete heap of n marks a collection of n 

 " relative positions," or sorts of places, which are distinguished from 

 each other, and from all other relative positions : e.g.,. we may choose 

 the sorts of places — first, second, third, &c, in a row. In such a case 

 the units a, b, c, . . . . may be regarded as occupying the sorts of 

 places which serve as marks. 



74. If two collections of units a, b, c, and p, q, r, . . . . are 



undistinguished from each other, they may be regarded as correspond- 

 ing to each other in one or more ways, in each of which corre- 

 spondences to each unit, pair, triad, &c, of one collection there cor- 

 responds in the other a counterpart unit, pair, triad, &c, undistin- 

 guished from the former in any circumstance. In any one' of those 

 correspondences two corresponding units may be regarded as occupy- 

 ing corresponding places, or, as we may express it, places of the same 

 sort ; and we may if we please regard these sorts of places as dis- 

 tinguished from each other, and from all other sorts of places ; i.e., 

 we may regard the correspondence as giving rise to two aspects, 

 A, B, C, . . . . and P, Q, R, . . . the former an aspect of a, b, c, 

 . . . ., the latter of p, q, r, . . . . The two aspects A, B> C, . . . . 

 and P, Q, R, . . . are clearly undistinguished from each other : and, 

 as the units of each are distinguished from each other, to each unit 

 of one there corresponds one unit, and one only, in the other which 

 is undistinguished from it. 



75. Two aspects will not be undistinguished nnless they can be 

 regarded as derived in the manner indicated in the last section ; and 

 therefore the undistinguishableness of two aspects A, B, 0, . . . . 

 and P, Q, R, . . . . indicates the existence of a definite correspond- 

 ence between the two nndistinguished collections a, b, c, .... and 

 p, q, r, . . . . in which correspondence to each unit, pair, triad, &c, 

 of the one there corresponds in the other a counterpart unit, pair, 

 triad, &c, undistinguished from the former in any circumstance. 



Similarly, the undistinguishableness of two aspects of the same 

 collection a, b, c, . . . . may be said to indicate a self-correspondence of 

 the collection, i.e., a correspondence in which to each unit, pair, triad, 

 &c, of the collection a, b, c, . . . . there corresponds the same, or 

 another unit, pair, triad, &c, of a, b, c, ... . undistinguished from 

 the former in any circumstance. 



76. In future, when a correspondence of two collections, or a self- 

 correspondence of a collection, are spoken of, correspondences suhc as 

 those described in the preceding sections are intended to be referred 

 to, unless the contrary is expressly stated, as in Sections 162 to 169. 



r 2 



