19/16 THE STATE -DETERMINED SYSTEM 



a system that is state-determined. In my theory I insist on the 

 systems being state-determined because I agree with the experi- 

 menter who, in his practical work, is similarly insistent. 



Transformations of the canonical representation 



19/14. Sometimes systems that are known to be isolated and 

 complete are treated by some method not identical with that used 

 here. In those cases some manipulation may be necessary to 

 convert the other form into ours. Some of the possible manipula- 

 tions will be shown in the next few sections. 



19/15. Systems can sometimes be described better after a change 

 of co-ordinates. This means changing from the original variables 

 x v . . . , x n to a new set y l9 . . . , y n , equal in number to the 

 old and related by single-valued functions <j> { : 



Vi = <f>i( x v .-.,#«) (i = 1, . . . , n) 



If we think of the variables as being represented by dials, the 

 change means changing to a new set of dials each of which indicates 

 some function of the old. If the functions </> t - are unchanging in 

 time (as functions of their arguments), the new system will remain 

 state-determined. 



19/16. In the ' Homeostat ' example of S. 19/11 a fluxion was 

 treated as an independent variable. I have found this treat- 

 ment to be generally advantageous: it leads to no difficulty or 

 inconsistency, and gives a beautiful uniformity of method. 



For example, if we have the equations of a state-determined 

 system we can write them as 



*< -fii^v ...,#») = (i = 1, . . . , n) 



treating them as n equations in 2n algebraically independent 

 variables x l9 . . . , x n , x v . . . , x n . Now differentiate all the 

 equations q times, getting (q -f\l)w equations with (q + 2)n 

 variables and derivatives. We can then select n of these vari- 

 ables arbitrarily, and noticing that we also want the next higher 

 derivatives of these w, we can eliminate the other qn variables, 

 using up qn equations. If the variables selected were z v . . . , z n 

 we now have n equations, in 2n variables, of type 



( (z v . . . , z ni z l9 . . . 3 z n ) = (i =* 1, . . . , n) 



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