CHAPTER 23 



The Ultrastable System 



23/1. The definition and description already given in S. 8/6 and 

 7 have established the elementary properties of the ultrastable 

 system. A restatement in mathematical form, however, has the 

 advantage of rendering a misunderstanding less likely, and of 

 providing a base for quantitative studies. 



If a system is ultrastable, it is composed of main variables Xi 

 and of step-functions at, so that the whole is absolute : 



-£ =fi(x; a) (i = 1, . . . , n) 



d ^ = gi(x; a) (i = l, 2, . . .) 



The functions gi must be given some form like that of S. 22/2. 

 The system is started with the representative point within the 

 critical surface cf)(x) = 0, contact with which makes the step- 

 functions change value. When they change, the new values are 

 to be random samples from some distribution, assumed given. 



Thus in the homeostat, the equations of the main variables are 

 (S. 19/11) : 



dx' 



-± = a il x 1 + a i2 x 2 + a i3 x 3 + a ti x± (i = 1, 2, 3, 4) 



The a's are step-functions, coming from a distribution of ' rect- 

 angular ' form, lying evenly between — 1 and -f- 1. The critical 



surfaces of the a's are specified approximately by | x \ ± - = 0. 



4 



Each individual step-function a^ depends only on whether Xj 

 crosses the critical surface. 



As the a's change discontinuously, an analytic integration of 

 the differential equations is not, so far as I am aware, possible. 

 But the equations, the description, and the schedule of the 

 uniselector-wirings (the random samples) define uniquely the 

 behaviour of the x's and the a's. So the behaviour could be 



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