MLATIYES. 337 



If the relative x be divided in this way into X,A",A . eto, so that x * that 

 which is either X or X' orX" or X", etc., then noiwt is thai which is at once nonX 

 and non-X' and non-X", etc.; that is to say, 



non-# = non-X, non-A", non-A '". non-A ', etc.; 



where non-X is such that there is something (Z) such that everything i tiort-JT to 

 Z', and so with non-X', non-X", etc. Now, non-:r may be any relative whatever. 

 Substitute for it, then, y\ and for non-X, non-X', etc., Y, 3 , etc. Then we have 



y=Y, T , F, F", etc. ; 

 and Y'Z' = 1 , Y"Z" = 1 , Y'"Z ' = 1 , etc., 



where Z', Z", Z'" are individual terms which depend for what they denote on Y', Y . Y". 

 Then we have 



1 == Y'Z' = Y' z ' = TV + 0) = Y'z, T ° = Y'Z\ F° , 



• 7/7 



or F* = 1 , r"0 = 1 , 7"'o = l , etc. 



Then y« = ( F, F', F", etc.)° = F<> , F° , F"° , etc = 1 . 



We have by definition, x1 = x . 



Hence, by (6), ax = {a1)x = a(/ar) . 



Now a may express any relation whatever, but things the same way related to 



ything are the same. Hence 



* = / 



We have by definition, 1 = Y . 



Then if X is any individual X,l = X,/ = X,/X 



But 



Hence 



rx = x. 



X,1 = X,X; 



and by (23) 



whence if we take a? = X -f- X' + X" + -* + etc -> 



where X,X' etc. denote individuals (and by the very meaning of a general term this 



can always be done, whatever x may be) 



z,l = (X+X + X'-f etc.),l = X,l + X,l + X,l-fetc. = X+X + X + etc. = x, 



or 



x,l = x . 



or 



We have by (24) x -^ I = x + I — *,1 = * + 1 



. + 1 = 1. 



x 



