7/18 THE ULTRASTABLE SYSTEM 



Suppose that the curvature of the surface is controlled by a para- 

 meter which makes A rise and B fall. If the ball is resting at A, 

 the parameter's first change will make no difference to the ball's 

 lateral position, for it will continue to rest at A (though with 

 lessened reaction if displaced). As the parameter is changed 

 further, the ball will continue to remain at A until A and B are 

 level. Still the ball will make no movement. But if the para- 

 meter goes on changing and A rises above B, and if gravitation is 

 intense and the ball fast-moving, then the ball will suddenly move 

 to B. And here it will remain, however high A becomes and 

 however low B. So, if the parameter changes steadily, the 

 lateral position of the ball will tend to change in step-function 

 form, approximating more closely as the passage of the ball for 

 a given degree of slope becomes swifter. 



The possibility need not be examined further, for no exact 

 deductions will be drawn from it. The section is intended only 

 to show that step-functions occur not uncommonly when the 

 system under observation contains fast-acting components. The 

 subject will be referred to again in S. 9/8. 



7/18. In any state-determined system, the behaviour of a variable 

 at any instant depends on the values which the variable and the 

 others have at that instant (S. 2/15). If one of the variables 

 behaves as a step-function the rule still applies: whether the 

 variable remains constant or undergoes a change is determined 

 both by the value of the variable and by the values of the other 

 variables. So, given a state-determined system with a step- 

 mechanism* at a particular value, all the states with the step- 

 mechanism at that value can be divided into two classes : those 

 whose occurrence does and those whose occurrence does not lead 

 to a change in the step-mechanism's value. The former are its 

 critical states: should one of them occur, the step-function will 

 change value. The critical state of an electric fuse is the number 

 of amperes which will cause it to blow. The critical state of the 

 1 constant of proportionality ' of an elastic strand is the length 

 at which it breaks. 



An example from physiology is provided by the urinary bladder 



* I am indebted to Dr. J. O. Wisdom for the suggestion that a mechanism 

 showing a step-function as its main characteristic could conveniently be called 

 a ' step-mechanism '. 



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