CHAPTER 21 



Parameters 



21/1. With canonical equations 



— * = f i (x 1 , . . . , x n ) {i = 1, . . . , ri), 



the form of the field is determined by the functional forms ft 

 regarded as functions of x v . . . , x n . If parameters a v a 2 , . . . 

 are taken into consideration, the system will be specified by 

 equations 



-j— = Ji\&ii . . . , x n ; fit 1? & 2 , • • •/ [t = x 9 , m , 9 71). 



If the parameters are constant, the #'s continue to form an absolute 

 system. If the a's can take m combinations of values, then the 

 oj's form m different absolute systems, and will show m different 

 fields. If a parameter can change continuously (in value, not in 

 time), no limit can be put to the number of different fields which 

 can arise. 



If a parameter affects only certain variables directly, it will 

 appear only in the corresponding /'s. Thus, if it affects only 

 x x directly, so that the diagram of immediate effects is 



(X * X -i ^ Xn) 



then a will appear only in f x : 



dxj&t =f 1 (x 1 , x 2 ; a) 

 dxjdt =f 2 {x lf x 2 ). 



But it will in general appear in all the F's of the integrals (S. 19/10). 

 The subject is developed further in Chapter 24. 



Change of parameters can represent every alteration which can 

 be made on an absolute system, and therefore on any physical 

 or biological ' machine '. It includes every possibility of experi- 

 mental interference. Thus if a set of variables that are joined 

 to form the system x = f(x) are changed in their relations so 

 that they form the system x = <j>(x), then the change can equally 



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