DESIGN FOR A BRAIN 21/7 



so that they form the system x = <j>(x), then the change can equally 

 well be represented as a change in the single system x = ip(x; a). 

 For if a can take two values, 1 and 2 say, and if 



f{x) = y>(x; 1) 

 <j>{x) = y)(x; 2) 



then the two representations are identical. 



As example of its method, the action of S. 8/11, where the two 

 front magnets of the Homeostat were joined by a light glass fibre 

 and so forced to move from side to side together, will be shown 

 so that the joining and releasing are equivalent in the canonical 

 equations to a single parameter taking one of two values. 



Suppose that units x v x 2 and x z were used, and that the magnets 

 of 1 and 2 were joined. Before joining, the equations were 

 (S. 19/11) 



dxjdt = a n x x + a 12 x 2 + a 13 aO 



2/ — ttoi^i ~~ | Wo? 2 i™ ^S3*»l i 



dX 3 /at = Cl 31 X 1 -f~ a 32<%2 ~T~ a 33 X 3J 



After joining, x 2 can be ignored as a variable since x x and x 2 are 

 effectively only a single variable. But x 2 s output still affects the 

 others, and its force still acts on the fibre. The equations there- 

 fore become 



dxjdt = (a n + a 12 + a 21 + a 22 )x x + (a 13 + a 23 )x, 

 dxjdt = (a 31 + a Z2 )x x + a 3 3 a \ 



It is easy to verify that if the full equations, including the para- 

 meter 6, were: 



dx 1 /dt = {a n + b(a 12 + a 21 + a 22 )}x 1 + (1 — b)a 12 x 2 



+ (a 13 + ba 23 )x 3 



dX 2 /dt = ^21^1 I ^22*^2 l ^23^3 



dxjdt = (a 31 + ba^Xj^ + (1 — b)a Z2 x 2 + a 33 x, 



then the joining and releasing are identical in their effects with 

 giving b the values 1 and respectively. (These equations are 

 sufficient but not, of course, necessary.) 



21/7. A variable Xk behaves as a null-function if it has the 

 following properties, which are easily shown to be necessary and 

 sufficient for each other: 



(1) As a function of the time, it remains at its initial value x Q k - 



264 



