21/3 DESIGN FOR A BRAIN 



(3) In the group equations, F k (x®, . . . , a?J ; t) = x^. 

 (Some region of the phase-space is assumed given.) 

 Since we usually consider absolute systems, we shall usually 

 require the parameters to be held constant. Since null-functions 

 also remain constant, the properties of the two will often be 

 similar. (A fundamental distinction by definition is that para- 

 meters are outside, while null-functions may be inside, the given 

 system.) 



21/3. In an absolute system, the variables other than the step- 

 and null-functions will be referred to as main variables. 



21/4. Theorem : In an absolute system, the system of the main- 

 variables forms an absolute subsystem provided no step-function 

 changes from its initial value. 



Suppose x l9 . . . , Xjt are null- and step-functions and the main- 

 variables are Xk+u . . . , x n . The canonical equations of the 

 whole system are 



dxjdt = 



dxjc/dt — 

 dxk+i/dt = fk+i(x v . . . , x k , x k+ i, . . . , x n ) 



dXn/dt =f n (x v . . . , X*, X k+1 , . . . , X n ) 



The first k equations can be integrated at once to give x x = x\, 

 . . ., Xk = x Q k . Substituting these in the remaining equations 

 we get : 



dx k+ i/dt = fk+i{x\, . . ., x% x k +i f . . ., x n T\ 



ClX n /dt — Jn\pC\i • • • 5 #jfc> Xk+1, • • •» x n ) J 



The terms x^, . . ., x^ are now constants, not effectively functions 

 of t at all. The equations are in canonical form, so the system is 

 absolute over any interval not containing a change in a?J, . . . , x^. 

 Usually the selection of variables to form an absolute system 

 is rigorously determined by the real, natural relationships existing 

 in the real ' machine ', and the observer has no power to alter them 

 without making alterations in the ' machine ' itself. The theorem, 

 however, shows that without affecting the absoluteness we may take 



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