24/15 DESIGN FOR A BRAIN 



found that P and Q must both be zero. So df B /dx A and df B /dx c 

 must be zero over the region. This can be achieved in several 

 ways without fn being zero, i.e. without Xb being constant. Two 

 examples will be given. 



(1) If fn is a constant, then Xb will increase uniformly, i.e. will 

 not be constant, but Xa and xc will still be independent. Without 

 a fourth variable, the linear change is the most which x B can make 

 if the system is to remain absolute. 



(2) If fn is a function of other variables not yet mentioned, y 

 is not restricted to a constant rate of change. Thus if there is a 

 variable u which dominates y we could have a system 



dx 

 ~dt 



x + y 



du _ 

 dt~ 3 



dy 



= sin u 



dt 

 dz 



dt= y+Z j 

 which is clearly absolute. Its solution is : 



x = (x Q + y° + 



sin u° + — cos u )e l — y° 



cos IT 

 3 



10 



sin (w c 



U = u° + 3t, 



y = y Q -J- - COS U 







1 



cos {u° + 3/), 



30 + gjj cos (u<> + 30, 



(2° + y° + — sin u° + — cos u°)e l 



10 



10 



cos U K 



sin (u° 4- St) + — cos (u° + 3f). 



10 v ^ ; ^ 30 v ^ y 



Not even the rate of change of y is constant, yet x and z are 

 independent. 



Physically the conclusions are reasonable. The various con- 

 ditions which make x and z independent all have the effect of 

 lessening or abolishing x's and s's effect on y. The abolition can 

 be done either by making y constant, or by driving y exclusively 

 by some other variable (u). A well-known example of the latter 

 method is the ' jamming ' of a broadcast by the addition of some 



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