STEP-FUNCTIONS 22/5 



system. Thus, whether a Post Office-type relay opens or shuts 

 depends only on the two variables : the current in the coil, and 

 whether the relay is already open or shut. 



Such relays and critical states occur in the homeostat. When 

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

 square, cube, or tesseract respectively in the phase-space around 

 the origin. The critical states will fill the space outside this sur- 

 face. 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 

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

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

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

 was often absolute with a field like A of Figure 22/2/1, and often 

 absolute 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 intermittent change by some other, 

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



22/5. The argument can, however, be reversed : if we find that 

 a subsystem has r fields we can deduce, subject to certain restric- 

 tions, that the other variables must include step-functions. 



Theorem : If, within an absolute system x v . . . , x n , x p , . . . , x s , 

 the subsystem x l9 . . . , x n is absolute within each of r fields 

 (which persist for a finite time and interchange instantaneously) 

 and is not independent of x Pi . . . , x s ; then one or more of 

 Xp, . . . , x s must be step-functions. 



Consider the whole system first while one field persists. Take 

 a generic initial state x\, . . . , x% x^, . . . , x° s and allow time t x 

 to elapse ; suppose the representative point moves to a?i, . . . , x n , 



x ' v x s , where each x' is not necessarily different from x°. 



Let further time t 2 elapse, the point moving on to x'u • • • > #n> 

 x'p, . . . , x". Now consider the line of behaviour that follows 

 the initial state x[, . . . , x n , x° p , x q , . . . , x' s , differing from the 



235 



