19/25 DESIGN FOR A BRAIN 



the equivalent form 



^?- etc 



dt — ' ' etC * 



which is in canonical form. 



19/25. If a variable depends on velocity effects so that, for 

 instance 



dx 1 _ f dx 2 \ 



dt - Jl \df * v x v 



-jj£ = J2\ X V X 2) 



dx 

 then if we substitute for -=-* in/i(. . .) we get the canonical form 



—jr = fxifiPufitl* x v X 2) 



2 S" I \ 



~fa —J2\ X V X 2) 



19/26. If one variable changes either instantaneously or fast 

 enough to be so considered without serious error, then its value 

 can be given as a function of those of the other variables ; and 

 it can therefore be eliminated from the system. 



19/27. Explicit solutions of the canonical equations 



dxi/dt =fi{x v . . . , x n ) (i — 1,- . . . , n) 



will seldom be needed in our discussion, but some methods will 

 be given as they will be required for the examples. 



(1) A simple symbolic solution, giving the first few terms of 

 x\ as a power series in t, is given by 



an = <* x x\ (t = 1 n) . . (1) 



where X is the operator 



/i«. ■ • • . Ogjg + • • • +/ " ( '* • • • > xl) h n ■ (2) 



and e' x = l+tX+£x*+~X» + . . . . (3) 



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