22/5 THE EFFECTS OF CONSTANCY 



two, three or four units are in use, the critical surfaces will form 

 (to a first approximation) a square, cube, or tesseract respectively 

 in the phase-space around the origin. The critical states will fill 

 the space outside this surface. As there is some backlash in the 

 relays, the critical surfaces for opening are not identical with 

 those for closing. 



Systems with multiple fields 



22/4. If, in the previous example, someone unknown to us were 

 occasionally to break and sometimes to replace the elastic, and if 

 we were to test the behaviour of the system cc lt x 2 over a prolonged 

 time including many such actions, we would find that the system 

 was often state-determined with a field like A of Figure 22/2/1, 

 and often state-determined with a field like B; and that from time 

 to time the field changed suddenly from the one form to the other. 

 Such a system could be said without ambiguity to have two 

 fields. Similarly, if parameters capable of taking r combinations 

 of values were subject to occasional change by some other, 

 unobserved system, a system might be found to have r fields. 



22/5. The argument can be used to some degree in the converse 

 direction; for the correspondence of S. 21/2 may be used, with 

 caution, conversely; for though the number of fields does not 

 prescribe the number of parameter-values it does prescribe their 

 minimal number with precision. Thus fields that change like the 

 first row of letters in S. 9/13 demand a minimum of 4 parameter- 

 values, while those that change like the second row demand a 

 minimum of 15. 



If the observer should find that one field persists, the minimal 

 number of parameter- values is, of course, one. If the field should 

 change suddenly to a new field, which persists, he may deduce 

 that the parameter- value must have changed (for no single value 

 could give two fields), and that -the minimal number of values 

 over the new persistence is again one. Thus he may legitimately 

 deduce that the minimal variety attributable to the parameter- 

 values is, on the scale of S. 7/13, that of the step-function — the 

 null-function provides too little, and the part-function an un- 

 necessary excess. (Compare S. 9/10-13.) 



275 



