19/7 THE STATE-DETERMINED SYSTEM 



19/3. A variable is a function of the time. A system of n 

 variables will usually be represented by x l9 x 2 , . . ., x n , or some- 

 times more briefly by x. The case where n = 1 is not excluded. 

 It will be assumed throughout that n is finite; a system with an 

 infinite number of variables (e.g. that of S. 19/17) will be replaced 

 by a system in which i is discontinuous and n finite, and which 

 differs from the original system by some amount that is negligible. 

 Each variable x t is a function of the time t; it will sometimes be 

 written as x^t) for emphasis. It must be single-valued, but need 

 not be continuous. A constant may be regarded as a variable 

 which undergoes zero change. 



19/4. The state of a system at a time t is the set of numerical 

 values of x-^t), . . . , x n (t). Two states (x v . . . , x n ) and 

 (Vv • • • > Vn) are e< I ual if x % = Vi for a11 *• 



19/5. A transition can be specified only after an interval of time, 

 finite and represented by At or infinitesimal and represented by 

 dt, has been specified. It is represented by the pair of states, 

 one at time t and one at the specified time later. 



A line of behaviour is specified by a succession of states and the 

 time-intervals between them. Two lines of behaviour are equal 

 if all the corresponding states and time-intervals along the suc- 

 cession are equal. (So two lines of behaviour that differ only in 

 the absolute times of their origin are equal.) 



19/6. A primary operation is a physical event, not a mathe- 

 matical, requiring a real machine and a real operator or experi- 

 menter. He selects an initial state (x\, . . . , a?J), and then 

 records the transition that occurs as the system changes in 

 accordance with its own internal drives and laws. 



19/7. If, on repeatedly applying primary operations, he finds 

 that all the lines of behaviour that follow an initial state S are 

 equal, and if a similar equality occurs after every other state 

 S',S", . . . , then the system is regular. 



Such a system can be represented by equations of form 

 x x = F x (xl . . . , xl ; t) 



X n — F n \ x \9 ' • • » x n i 



243 



