CONSTANCY AND INDEPENDENCE 24/15 



the correspondence continues as before. Thus, A of Figure 



14/10/1 has 



[/] = R R and I**] 



And a? 3 is not independent of x v But if cc 2 becomes inactive, 



[/] = 



[F] = 



R R R 



R 



R R 



R R R_ 



and x 3 is now independent of a^. 



The other diagrams, B, D and E, may be verified similarly. 



24/15. We can now investigate the problem of S. 14/8 : the 

 separation of parts in a dynamic whole. 



Theorem : If the variables of an absolute system are divisible 

 into three sets, A, B, and C such that no f A contains any of the 

 set a&c, and no fc contains any of x A , i.e. so that the diagram of 

 immediate effects is A ^ B ~^_ C, and if variables Xb remain 

 constant, then A is independent of C, and vice versa. 



If all variables of set B are constant, the differential matrix, 

 in partitioned form, will be 



It is idempotent, so this matrix is also the integral matrix. As 

 the elements at the top right and bottom left corners are zero, 

 A and C are independent of each other. 



On the other hand, without further restrictions the constancy 

 is not necessary. Thus, suppose that A and C are independent 

 and that the differential matrix is 



R R (T 

 P R Q 

 .0 R R^ 



where P and Q are to be determined. For the integral matrix 

 to have zeros in the top right and bottom left corners, it is easily 



249 



