MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. :495 



we see that, when u and v are classes of straight lines, u+v (i.e., the ^-concurrence of 

 u with v) is that complete set of lines of u which is such that any member of it is 

 coplanar with every member of v. 



*3'32. Definition. A class u is a self-<j>-concurrence if the ^-concurrence of u with 

 itself is the whole class u. The class of those classes which are self-^>-concurrences 

 will be denoted by conc^,. The symbolic definition is 



conc^, = u {u = iifi'u} Df 



*3'33. Definition. x will be said to be ^-concurrent with y, if the class composed 

 of a; and y only (i.e., the class I'x u <,'?/) is <-axial. 



*3'41. Definition. A <f>-plane is a class u such that there exists a class v, which 

 (1) is <-axial, (2) is composed of two members only, and (3) is such that u is the class 

 cm^v. The class of those classes which are <-planes is denoted by pie,,,. The 

 symbolic definition is 



le, = u V .ve2na,x.u = cm'v Df 



Note. It requires an axiom to establish that a ^-plane is a self-^-concurrence 

 (cf. *16'11). 



*3'42. Definition. A class u is a <j>-point, if there exists a class v, which (1) is 

 <-axial, (2) is composed of three members only, and (3) is such that u is the (^-con- 

 currence of the (^-region with v. The class of those classes which are ^-points is 

 denoted by the symbol pnt^. The symbolic definition is 



. v e 3 n ax, . w = Qv Df 



Note. It requires axioms to establish that a <-point is (^-maximal and is a self-<- 

 concurrence (cf. *14'11'12). Also note that this definition does not apply unless the 

 number of dimensions of O^ is at least three, but then applies unchanged however 

 great this number may be. 



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

 flatness), we see that a ^-point now becomes simply that class of straight lines 

 concurrent at a point. The analogy with KLEIN'S " ideal," or " protective," points is 

 obvious. Only when the present theory is applied, it will be found that the original 

 " descriptive " point has entirely vanished. 



*3'43. Definition. A class is <j>-coplanar if there exists a <-plane of which it is a 

 subclass. The symbol cople^.u denotes that the class u is ^-coplanar. The definition 



in symbols is 



cople^,! u . = . (ftp) . p e ple^ . u e cls'j? Df 



*3'44. Definition. A class is <$>-copunctual if there exists a (-point of which it is a 

 subclass. The symbol copnt^,!w denotes that the class u is <-copunctual. The 

 definition in symbols is 



copnt ! u . = . (3?) . P e pnt^, . u e cls'P Df 



