CONSTANCY AND INDEPENDENCE 14/16 



First, what do we mean by ' difficulty of stabilisation ' ? 

 Consider an engineer designing, on the bench, an electronic 

 system. He has before him an apparatus which he wants to 

 be stable at some particular state. The apparatus contains a 

 number of adjustable constants, parameters, and he has to find 

 a combination of values that will give him what he wants. The 

 4 difficulty ' of stabilisation may be defined as, and measured 

 by, the proportion of all possible parameter-values that fail to 

 give the required stability. The definition has the advantage 

 that it is directly applicable to the homeostat and any similar 

 mechanism that has to search through combinations of values. 



With this definition it can be shown that if a system of N 

 part-functions has on the average k of its variables active, then 

 its difficulty of stabilisation is the same, other things being equal, 

 as that of a system of k full-functions. 



The proof is given in S. 24/18, but the theorem is clearly 

 plausible. When a system of part-functions is in a region of 

 the phase-space where k variables are active and where all the 

 other variables are constant, the k variables form a system which 

 is absolute and which is not essentially different from any other 

 absolute system of k variables. The fact that we have been 

 thinking of it differently does not affect the intrinsic nature of 

 the situation. Equally, whenever we have postulated an abso- 

 lute system, we have assumed that its surrounding variables are 

 constant, at least for the duration of the experiment or observa- 

 tion. Yet these surrounding variables are usually not constant 

 for ever. So our ' absolute system ' was quite commonly only 

 a portion of a larger system of part-functions. There is there- 

 fore no intrinsic difference between an absolute system of k 

 full-functions and a subsystem of k active variables within a 

 larger system of part-functions. That being so, there is no reason 

 to expect any difference in their difficulties of stabilisation. 



The theorem is of great importance to us, for it means that 

 the time taken to stabilise a system of N part-functions will, 

 very roughly, be more like T 2 of S. 12/3 than T x ; so the change 

 to part-functions may change the stabilisation from ' impossible ' 

 to ' possible '. The subject will be developed in S. 17/3. 



165 



