398 



Mr. A. B. Kempe. 



[June 18, 



systems of a double system are independent, is considered; and the 

 relation of the systems, which are such that one is derived from the 

 other by ignoring differences, is discussed. 



§§ 130—136. Sets. 



Sets are collections of units akin to systems (which are always sets), 

 but may differ from them in each being one only of a number of 

 undistinguished sets. 



§§ 137 — 142. Aspects Unique with respect to Collections. 



If the aspects xyz .... abc .... and uvw .... abc .... are undis- 

 tinguished from each other, the aspects xyz .... and uvw .... may 

 be said to be duplicates with respect to the collection a, b, c . . . . If 

 there is no duplicate of xyz .... with respect to a, 6, c .... , then 

 xyz .... may be said to be unique with respect to a, b, c, . . . . 



§§ 143 — 151. Associates. 



(16.) If a, 6, c .... be any collection of units, and if X be another 

 unit, such that the pairs \a, Xb, \c, . . . . are distinguished from the 

 pairs which A. makes with units which are not components of the 

 collection, then X may be said to be an associate of the collection 

 a, b, c . . . . 



Many of our conceptions and definitions of systems of units involve 

 the idea of associates. 



§§152—161. Unified Aspects. 



These are the units arrived at by regarding the aspects of collec- 

 tions as single units. They are associates of the collections of which 

 they are unified aspects, and are unique with respect to them. 



§§ 162 — 169. Correspondences of Collections which are distinguished 

 but of like Forms. 



The nature of these correspondences, and the relations which they 

 bear, when regarded as units, to the units of the collections are con- 

 sidered. 



§§ 170—173. Beplicas. 



Two systems are replicas of each other when they are of the same 

 form and bear precisely similar relations to all other collections of 

 units. 



§§ 174 — 188. Independent and Related Systems. 



The conditions which must be satisfied in order that systems may 

 not be independent are here investigated, and systems which are 

 factors of others are defined and discussed. 



