PARAMETERS 21/2 



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) =yj(a:; 1) 

 (j)(x) = ip(x ; 2) 

 then the two representations are identical. 



As example of its method, the action of S. 8/10, 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 lt x 2 and x 3 were used, and that the 

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

 were (S. 19/11) 



dxjdt = a 11 x 1 + a 12 x 2 + a 13 x 3 ^\ 



dx 2 /dt = a 21 X ± -f- «22^2 + a 22 X 3 f 



dxjdt = a 31 x ± + a 32 x 2 + a 33 x 3 ) 



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 ±1 + a 12 + a 21 + a 22 )x x + {a 13 + a^)^ 

 dxjdt = (a 31 + 032)^ + a 32 x 3 



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



b t were : 



dxjdt = {a lx + b(a 12 + a 21 + a 22 )}x 1 + (1 - b)a 12 x 2 



+ (« is + ^23)^3 

 dxjdt = a 21 x x + a 22 x 2 + o 23 x 3 



dxjdt = (a 31 + ^32)^! + (1 — b)a 32 x 2 + a 33 x 3 _ 



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.) 



21/2. A variable x^ 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% 



(2) In the canonical equations, fk(x lt . . . , x n ) is identically 



zero. 



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