MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 485 



the axioms of Concept IV. (cf. *2) and those of Concept V. (cf. *22) are two 

 alternative sets of hypotheses as to the properties of R in connection with which the 

 theory of interpoints, as given in the present *1, assumes importance. Some axioms, 

 involving interpoints in their statements, are identical in Concept IV. and Concept V. 

 These axioms are stated now in *1, and their simple consequences are deduced. The 

 theory of interpoints depends on that of " similarity of position " in a relation. This 

 general idea will only be explained in the special form in which it is here required in 

 respect to the essential relation R. 



*ril. Definition. An entity, y, will be said to have a position in the pentadic 

 relation R, similar to that of the entity x, with a as first term and t as last (fifth) 

 term, if, whenever the relation holds between five terms, a being the first term and t 

 the last term, and either x or y or both occurring among the other terms, the relation 

 also holds when x is substituted for y (whenever y occurs), and also holds when y is 



/a??? A 

 substituted for x (whenever x occurs). The symbol R ; ( " ' j denotes the class of 



entities with positions similar to that of x in the relation R, a being first term and t 

 last term. The definition in symbols is 



. = . R 1 (a&yt)} Df 



/a??? A /a??? A 



*1'12. Proposition. If y is a member of RM ' \' j, then R ; f ' ' ' j is identical 



/a??? A 

 with R ; ( " x I In symbols, 



, 



\ / \ v / 



fa ? ? ? A 

 *1'13. Proposition. x is a member of R'( 



V x 



*T21. Definition. A class P of objective reals is called an intersection-point on a 

 (shortened into interpoint on a), when there exists an objective real x, which is a 

 member of R : (a ;;;t), and P is the class whose members are a together with all the 



members of the class R ; ( ' ' V The symbol R ; (a???) stands for the class of inter - 



\ x j 



points on a at the instant t. The definition in symbols is 



R'a???* = P x . x e R ; a;;;0 . P = i'auB~ Df 



*1'22. Definition. P is called an interpoint of the relation R at the instant t, if 

 there exists an objective real a, such that P is a member of R ; (a ???). The symbol 

 intpnt R( stands for the class of interpoints o/R at the instant t. In symbols, 



intput B( = P{(3) . PcR : (a???OI D f 



