CONSTANCY AND INDEPENDENCE 24/14 



tions of a finite continuous group in the variables A. So the 

 ^4's form an absolute system. 



The fact of dominance may be shown in the integral matrix by 

 finding that the deletion of columns A and rows B leaves only 

 zeros ; but the deletion of columns B and rows A leaves some 

 non-zero elements. (If the second operation also leaves only 

 zeros, then the system really consists of two completely inde- 

 pendent subsystems ; the whole system is 4 reducible '.) 



24/13. If A, B, and C are systems such that they together form 

 one absolute system, and if A dominates B, and B dominates C, 

 then A dominates C. 



On the information given, [F], in partitioned form, can be 

 filled in but for two elements, shown as dots : 



It must be idempotent (S. 24/11). Trying the four possible 

 combinations of R and for the two undefined elements, we find 

 that there must be at the top right corner, and R at the bottom 

 left. A therefore dominates C. 



The theorem illustrates again the importance of the concept of 

 ' absoluteness ' ; for without this assumption the theorem, 

 obvious physically, cannot be proved (for lack of the group 

 property). 



24/14. An account of the primary effects of part-functions on 

 the independencies within an absolute system can now be given. 

 The definition of a part-function x p implies that over finite 

 regions of values of x l9 . . . , x n f p [oc l9 . . . , x n ) becomes zero. 

 Within such a region, i.e. while not activated, the canonical 

 equations include dx p /dt = 0, which can be integrated at once to 

 x p = a£ ; so F p {x° ; t) = x° p ; and x p and F p are both constant. 

 dF p /dxj is therefore zero for all values of j other than p. The 

 effect of a part-function x p being inactive is therefore to make 

 the whole of the p-th row of the differential and integral matrices 

 zero (except for the element in the main diagonal, which remains 

 an R). 



247 



