24/11 DESIGN FOR A BRAIN 



element in the k-th row and the j-th column of the integral 

 matrix is or is not zero respectively. 



The advantage of equation (1) is that the differential matrix 

 is often formed with ease (for only zero or non-zero values are 

 required), and often the first multiplication shows that [/] 2 = [/]. 

 When this is so, the integral matrix is at once proved to be equal 

 to [/], and all the independencies are obtained at once. A 

 further advantage is that the theory of partitioned matrices can 

 often be used, with considerable economy of time. The next 

 few sections provide some examples. 



24/11. In an absolute system the independencies cannot be 

 assigned arbitrarily. 



By the theorem of S. 24/8, the integral matrix, being the sum 

 of powers, is idempotent ; and therefore, by S. 24/7, has its zeros 

 patterned. The independencies of an absolute system must 

 always be subject to this restriction. 



What is really the same line of reasoning may be shown in an 

 alternative form. The group property requires (S. 19/10) that 



F k {F x (x« ; /), F 2 (x» ;*),...;«'} = *W4 4 - • - I * + O; 

 so if X/c is independent of Xj then x° will not appear effectively on 

 the right-hand side, and it must therefore not appear effectively 

 on the left. So if, say, F m ( . . . ; t) contains x°p then x ^ must not 

 occur in F k ; so x k must be independent of x m as well. 



24/12. If the variables of an absolute system are divisible into 

 two groups A and B, such that all the variables of A are inde- 

 pendent of B, but not all those of B are independent of 4> then 

 the subsystem A dominates the subsystem B. 



Theorem : The subsystem A is itself absolute. 



Write down the group equations of the A's : 



F A {F^; t), F 2 (x°; /),...; t'} = F A {x° v «& . . .; t + t'} 

 where the subscript a refers to all the members of A in succession. 

 Each F A is independent of a?jj, so, omitting the unnecessary 

 symbols from each side both from the F's and from the x°'s, we get 



F A {F A {xl; t), . . .; t'} = F A {x° A ; t + t'} 

 where the change of subscript means that only the members of 

 A are now included. Inspection shows that these are the equa- 



246 



