21/6 



PARAMETERS 



may be many-one, for while one value of the vector (a v a 2 , . . .) 

 will indicate one and only one field, several such vectors may 

 indicate the same field. Thus the relation is a mapping of 

 Bourbaki's type. The possibilities are sufficiently indicated in 

 Fig. 21/2/1; in the upper line P, each dot represents one value 

 of the vector of parameter- values (a v o 2 , . . .); in the lower line 

 F, each dot represents one field. Notice that (1) every vector 

 value indicates a field; (2) no vector value indicates more than 

 one field; (3) a field may be indicated by more than one vector 

 value; (4) some fields may be unindicatcd. 



21/3. If the a's can take m combinations of values then m fields 

 are possible. The m fields will often be distinct, but the possibility 

 is not excluded that the m may include repetitions, and thus not 

 be all different. 



21/4. If a parameter changes continuously (i.e. by steps that 

 may be as small as we please), then it will often happen that the 

 corresponding changes in the field will be small; but nothing here 

 excludes the possibility that an arbitrarily small change in a 

 parameter may give an arbitrarily large change in the field. 

 Thus the fields will often be, but need not necessarily be, a con- 

 tinuous function of the parameters. 



21/5. If a parameter affects immediately only certain variables, 

 it will appear only in the corresponding /'s. Thus the canonical 

 representation (of a machine with input a) 



dxjdt =f 1 (x 1 , x 2 ; 

 dxjdt =f 2 (x lf x 2 ) 



corresponds to a diagram of immediate effects 



21/6. Change of parameters can represent every alteration which 

 can be made on a state-determined system, and therefore on any 

 physical or biological 4 machine '. It includes every possibility of 

 experimental interference. Thus if a set of variables that are 

 joined to form the system x =f{x) are changed in their relations 



263 



