CONSTANCY AND INDEPENDENCE 24/17 



powerful fluctuating signal from another station. It may effec- 

 tively render the listener independent of the broadcaster. 



If to the original conditions we add the restriction that the 

 system A is to become absolute on being made independent of C, 

 then constancy of the variables Xb becomes necessary. For the 

 possibilities examined in paragraphs (1) and (2) leave system A 

 subject to parameters xr which were assumed to be effective and 

 which are now changing. In such conditions A cannot be 

 absolute (S. 21/1) : constancy of the variables xn is therefore 

 necessary. 



24/16. The statement of S. 14/15, that in an absolute system an 

 inactive variable cannot become active unless some variable 

 directly affecting it is active, will now be proved. 



Theorem : If a variable x a is related to a set xb so that/ (. . .) 

 contains only x a and Xb, and if x a and Xb have all been constant 

 over a finite time, and if x a becomes active while the set xb stays 

 inactive, then the system cannot be absolute. 



We are given that dx a /dt =f a (x a , x B ). As x a remained con- 

 stant (at X a , say) while the set Xb were constant (at Xb), it follows 

 that f a (X a , Xb) = 0. But if x a starts to change value, dx a /dt is 

 no longer zero, nor is/ a ; sof a (X a , Xb) is a double- valued function 

 of its arguments, the system is not state-determined, and it is 

 therefore not absolute. 



24/17. In the ' hour-glass ' system of S. 14/11, every variable 

 may be shown to be dependent on every other variable. As in 

 Figure 14/11/1, let systems A and B each act on, and be acted 

 on, by a variable x. The differential matrix, in partitioned form, 

 is 



Its square contains only R's. So none of the A's are independent 

 of the Z?'s, and conversely. 



The proof is confirmed by the theorem of S. 19/22, which shows 

 that, as far as system B is concerned, the values of the A's can 

 be replaced by the derivatives of x. The behaviours of all A's 



251 



