21/7 PARAMETERS 



(2) In the canonical equations, f k {x^ . . . , x n ) is identically 



zero. 



(3) In the equations of S. 19/7, F k (x\, . . . , a>°; f) = #j>. 



(Some region of the phase-space is assumed given.) 



In a state-determined system, the variables other than the step- 

 and null-functions will be referred to as main variables. 



Theorem: In a state- determined system, the subsystem of the 

 main -variables forms a state -determined system provided no step- 

 function changes from its initial value. 



Suppose x v .... x k are null- and step-functions and the main- 

 variables are Xk+i, . . . , x n . The canonical equations of the 

 whole system are 



dxjdt — 



dxje/dt = 

 dx k +i/dt =f k+1 (xi, . . . , X* xt+i, . . . , x n )\ 



dxjdt =f n (x l9 . . . , x k , x k+ i, . . . , x n ) 



The first k equations can be integrated at once to give x x = x® y 

 . . . , Xk = x%. Substituting these in the remaining equations 

 we get: 



dxt+i/dt =fk+i(x%, . . . , x\, x k +i, . . . , x n )) 



dxjdt = f n (x° v . . . , x° k , Xk+i, . . . , x n )j 



The terms x\, . . . , a?£ are now constants, not effectively functions 

 of t at all. The equations are therefore in canonical form; so the 

 system is state-determined over any interval not containing a 

 change in x®, . . . , x° k . 



Usually the selection of variables to form a state-determined 

 system is determined by the real, natural relationships existing 

 in the real ' machine ', and the observer has no power to alter 

 them without making alterations in the c machine ' itself. The 

 theorem, however, shows that without affecting whether it is 

 state-determined the observer may take null-functions into the 

 system or remove them from it as he pleases. 



It also follows that the statements : ' parameter a was held con- 

 stant at a ', and ' the system was re-defined to include a, which, 



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