494 DR - A - N - WHITEHEAD ON 



(3) u is (-prime if cm+'u is of four dimensions, (4) u is <-prime if cm^'u is of five 

 dimensions. 



*3'21. The ^-dimension number (or the <f>-dimension) of a class u is the greatest 

 of the cardinal numbers of ah 1 classes (including possibly u itself) which are both 

 ^-equivalent to u and <-prime. The ^-dimension number of u will be denoted by 

 dimwit. The symbolic definition is 



dim/M . = : (ia) : a eNc"(prm^, n equiv/w) : p e Nc" (prrn^ n equiv/w) . D P . p a Df 



Elucidatory Note. With the assumptions of the previous elucidatory notes (where 

 (f> is flatness), we see that those c^-prime classes, the common ^-subregions for which 

 are spaces of three dimensions (as ordinarily understood), are all pairs of non-inter- 

 secting lines and all trios of concurrent non-coplanar lines ; also no class of four lines 

 in such a space can be prime. Thus three is the greatest cardinal number of any 

 (-prime class of lines for which the common </>-subregion is such a space. Hence, 

 according to the above definition, three is the ^-dimension number of the space. 



*3 - 22. Definition. A class n is <j>-ctxial when (l) it is 0-prime and (2) its cardinal 

 number is equal to its ^-dimensions. The class of ^-axial classes is denoted by the 

 symbol ax^. The symbolic definition is 



ax^, = u{ue prm^, n dim^' u} Df 



Elucidatory Note. With the assumptions of the previous elucidatory notes (where 

 <f> is flatness), we see that two coplanar lines form a ^-axial class, and so also do three 

 concurrent non-coplanar lines. 



*3'23. Definition. A class u is (^-maximal when (l) all those of its subclasses 

 (possibly including u itself), which are both <^>-prime and (^-equivalent to u, are 

 <-axial, and (2) there are such subclasses. The class of ^-maximal classes will be 

 denoted by mx^. The symbolic definition is 



= u { 3 ! (prm^, n equiv^,' u n els ' u) . prm^, n equiv^,' u n els ' u c ax } Df 



Elucidatory Note. Referring to the previous elucidatory notes (where <f> is 

 flatness), we see that any set of coplanar lines form a (^-maximal class ; similarly any 

 set of concurrent lines form a (^-maximal class. 



*3'31. Definition. The ^-concurrence, of u and v, where u and v are classes, is that 

 subclass of u (possibly u itself), such that any couple, formed by any member of it 

 and any member of v, is <-axial. The (^-concurrence of u with v is denoted by the 

 symbol u^v. The definition in symbols is 



uj v = x {x e u : y e v . u y . L'X u I'y e ax,,,} Df 



The (^-concurrence of the (^-region (O^) with any class v will be written O^'v 

 instead of 0$ ' v. 



Elucidatory Note. Referring to the previous elucidatory notes (when (f> is flatness), 



