19/21 DESIGN FOR A BRAIN 



equations can be written down, and their integration predicts the 

 behaviour of the whole system. The other method is to study 

 the behaviour of the whole system empirically. From this 

 knowledge the group equations are obtained : differentiation of 

 the functions then gives the canonical equations and thus the 

 relations between the parts. 



Sometimes systems that are known to be isolated and complete 

 are treated by some method not identical with that used here. 

 In those cases some manipulation may be necessary to convert 

 the other form into ours. Some of the possible manipulations 

 will be shown in the next few sections. 



19/21. Systems can sometimes be described better after a change 

 of co-ordinates. This means changing from the original variables 

 x v . . . , x n to a new set y v . . . , y m equal in number to the 

 old and related in some way 



y% = </>i(x l9 ...,#„) (i = 1, . . . , n) 



If we think of the variables as being represented by dials, the 

 change means changing to a new set of dials each of which 

 indicates some function of the old. It is easily shown that such 

 a change of co-ordinates does not change the absoluteness. 



19/22. In the ' homeostat ' example of S. 19/11 a derivative 

 was treated as an independent variable. I have found this 

 treatment to be generally advantageous : it leads to no difficulty 

 or inconsistency, and gives a beautiful uniformity of method. 



For example, if we have the equations of an absolute system 

 we can write them as 



& —M®v . . . , as*) = (» = i, . . . , n) 



treating them as n equations in 2n algebraically independent 

 variables x v . . . , x n , x v . . . , x n . Now differentiate all the 

 equations q times, getting (q + l)n equations with (q + 2)n 

 variables and derivatives. We can then select n of these vari- 

 ables arbitrarily, and noticing that we also want the next higher 

 derivatives of these ?i, we can eliminate the other qn variables, 

 using up qn equations. If the variables selected were z l9 . . . , 0» 

 we now have n equations, in 2n variables, of type 



&i(z v . . . , z„, z v . . . , i n ) = (f = 1, . . . , n) 



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