22/14 THE EFFECTS OF CONSTANCY 



in the canonical representation, Xj occurs effectively in /*. (The 

 range of /'s arguments is assumed to be specified.) 



22/12. The diagram of ultimate effects can also be shown to have 

 the property that an arrow goes from Xj to x* if and only if, in 

 the equations of S. 19/7, x9 occurs effectively in Fk (over some 

 specified range). 



(These matters were discussed more fully in the First Edition, 

 but need not be repeated at length.) 



22/13. It is worth noticing that, given n arbitrary points, a 

 diagram of immediate effects can be drawn by the arbitrary 

 placing of any number of arrows. That of the ultimate effects 

 cannot, however, be so drawn ; for an arrow from p to q and one 

 from q to r imply an arrow from p to r. Thus, while diagrams 

 of immediate effects are, in general, unrestricted, those of ultimate 

 effects must be transitive. 



22/14. The thesis of S. 12/10 can now be treated rigorously. 

 Figure 12/10/1 is given to be the diagram of immediate effects, 

 and the whole is assumed to be isolated and state-determined. 

 (For compactness below, the subscript A will be used to mean 

 ' any variable in the ^4-set'; and similarly for B and C.) Then 

 the canonical representation of the whole must be of the form 



xa =/a(xa, xb) 1 



xb=/b(xa, x b , xc)\ . . (1) 



XC =fc(XB, X C ) J 



with xc not in/,4, and xa not in/c. The two parts of the theorem 

 can now be proved. 



(1) Suppose the 2?'s are null-functions (over some specified 

 range). They are therefore constant. Write their values col- 

 lectively as p. The topmost line of (1) then becomes 



XA =fA(XA, ft) 



which shows that the system composed of the variables xa is 

 state-determined (so long as (} is constant). Further, the integrals 

 Fa of these equations cannot contain x°, so the system A is inde- 

 pendent of the system C. 



A similar proof will show that C is state-determined and 



279 



