MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 493 



of all (^-classes with the class u as a subclass. The symbol cm^'u will denote the 

 common <-subregioii for u. The symbolic definition is 



cm/ it = n'v{(f>\v . uecls'v} Df 



Note. If no class v, with the property <f> and containing u as a subclass, exists, 

 then cm/tt will be the class of all entities. But if a class v exists which has the 

 property <j) and contains u, then cm^'u is a subclass of O^. In the sequel it will be 

 found that this latter is the only relevant case for our purposes. 



Elucidatory Note. -Assuming our ordinary geometrical ideas, let the property of 

 the " flatness" of a class of straight lines be defined thus : A class of straight lines is 

 flat, either, when it is a necessary and sufficient condition for membership that 

 a straight line meets two members of the class, not at their point of meeting, or, 

 when the class is a unit class with one line as its sole member. Thus a plane (as 

 a line-locus) is flat, a three-dimensional space (as a line-locus) is flat, and so on. Now 

 let the property <j) in the above definition be the property of flatness. If then u is a 

 class consisting only of two straight lines, the common (^-subregion for u is either a 

 three-dimensional space or a plane, according as the two lines are not, or are, coplanar. 

 Also in a space of higher dimensions than three, if u be a class consisting of three 

 straight lines, the common <-subregion for u may be either (l) a plane, or (2) a three- 

 dimensional space, or (3) a four-dimensional space, or (4) a five-dimensional space, 

 according to the circumstances of the lines. It will be noticed that, in the application 

 of this theory of the common ^-subregion to the particular case of geometrical flatness, 

 the common <-subregion of any class of lines is itself flat. But this is not, in general, 

 the case when any property not flatness is considered. It is this peculiar property 

 of flatness which has masked the importance in geometry of the theory of common 

 (^-subregions. 



*3'121. Definition. Two classes u and v have ^-equivalence if cm/w = cm/v. 

 The class of those classes (including u itself as a member), which have (^-equivalence 

 with u, is denoted by equiv/w. The symbolic definition is 



equiv/ u = v (cm/ v = cm/ u) Df 



*3'13. Definition. A class u (not the null class) is fy-pvime, when, if v be any 

 proper part (part, not the whole) of u, v is not (^-equivalent to u. The class of those 

 classes which are ^>-prime will be denoted by the symbol prm^,. The symbolic 

 definition is 



prm^ = u{^\u :vcu . ^.(u- v) . r> . cm/v ^ cm/?} Df 



Elucidatory Note. With the assumptions of the elucidatory note on cm/w, it is at 

 once obvious that two straight lines form a ^-prime (where <j> is flatness) class, 

 whether they are or are not coplanar. But if u consist of three straight lines, (1) u 

 is not (-prime if cm/M is a plane, (2) u is not, in general, ^-prime if cm/w is a space 

 of three dimensions, but u is (in this case) ^>-prime if the three lines are concurrent, 



