STET-F UNCTIONS 



7/8 



omitted from a system, for its omission leaves the remainder still 

 producing predictable behaviour. 



Systems containing step-functions 



7/8. Suppose that we have a system with three variables, 

 A, B, S ; that it has been tested and found absolute ; that A 

 and B are full-functions ; and that S is a step-function. (Vari- 

 ables A and B, as in S. 21/3, will be referred to as main variables.) 

 The phase-space of this system will resemble that of Figure 7/8/1 

 (a possible field has been sketched in). The phase-space no longer 

 fills all three dimensions, but as S can take only discrete values, 

 here assumed for simplicity to be a pair, the phase-space is 

 restricted to two planes normal to S, each plane corresponding to a 

 particular value of S. A and B being full-functions, the represen- 

 tative point will move on curves in each plane, describing a line of 

 behaviour such as that drawn more heavily in the Figure. When 



Figure 7/8/1 : Field of an absolute system of three variables, of which 

 S is a step-function. The states from C to C are the critical states of 

 the step-function. 



the line of behaviour meets the row of critical states at C — C, S 

 jumps to its other value, and the representative point continues 

 along the heavily marked line in the upper plane. In such a field 

 the movement of the representative point is everywhere state- 

 determined, for the number of lines from any point never exceeds 

 one. 



If, still dealing with the same real c machine ', we ignore S, 

 and repeatedly form the field of the system composed of A and B, 

 S being free to take sometimes one value and sometimes the other, 

 we shall find that we get sometimes a field like I in Figure 7/8/2, 

 and sometimes a field like II, the one or the other appearing ac- 

 cording to the value that S happens to have at the time. 



87 



