THE MACHINE WITH INPUT 4/12 



Ex. 2 : A machine with input a, has the transformation 



( x' = y — az 

 T:iy= Iz 

 i^z' = .Y + « 



What machine (as transformation) resuhs if its input « is coupled to its 

 output z, by a = — z? 



Ex. 3: (Continued.) Will this second machine behave differently from the 

 first one when the first has a. held permanently at — 1 ? 



Ex. 4: A machine has, among its inputs, a photoelectric cell; among its outputs 

 a lamp of variable brightness. In Condition 1 there is no connexion from 

 lamp to cell, either electrical or optical. In Condition 2 a mirror is placed 

 so that variations in the lamp's brightness cause variations in the cell's 

 potential (i.e. so that the machine can "see itself"). Would you expect 

 the behaviours in Conditions 1 and 2 to difter? (Hint: compare with Ex. 3.) 



INDEPENDENCE WITHIN A WHOLE 



4/12. In the last few sections the concept of one machine or part 

 or variable "having an effect on" another machine or part or variable 

 has been used repeatedly. It must now be made precise, for it is 

 of profound importance. What does it mean in terms of actual 

 operations on a given machine? The process is as follows. 



Suppose we are testing whether part or variable / has an immediate 

 effect on part or variable j. Roughly, we let the system show its 

 behaviour, and we notice whether the behaviour of party is changed 

 when part /'s value is changed. If part /'s behaviour is just the same, 

 whatever /'s value, then we say, in general, that / has no effect ony. 



To be more precise, we pick on some one state S (of the whole 

 system) first. With / at some value we notice the transition that 

 occurs in part j (ignoring those of other variables). We compare 

 this transition with those that occur when states 5*1, ^'2, etc.^ — other 

 than S — are used, in which 5*1, S^, etc. differ from 5" only in the value 

 of the i-th component. If Si, S2, etc., give the same transition in 

 part j as S, then we say that / has no immediate effect on j, and vice 

 versa. ("Immediate" effect because we are considering y's values 

 over only one step of time.) 



Next consider what the concept means in a transformation. 

 Suppose its elements are vectors with four components {u,x,y,z), 

 and that the third line of the canonical equations reads 



y' = 2uy — z. 



This tells us that if y is at some value now, the particular value it 

 will be at at the next step will depend on what values 11 and z have, 



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