24/3 DESIGN FOR A BRAIN 



r-Q Ffc(aj?, . . . , a?2 » /) = 0. (These relations and notations are 



collected in S. 24/19 for convenience in reference.) 

 Example : In the system of S. 19/10 



X-^ I= X^ -j- XyZ ~i~ t 



x 2 = x% + 2t 

 x 2 is independent of x v but x x is not independent of x 2 . 



24/3. We shall be interested chiefly in the independencies intro- 

 duced when particular variables become constant : when they are 

 part-functions, for instance. Such constancies are most naturally 

 expressed in the canonical equations, for here are specified the 

 properties of the parts before assembly (S. 19/19). We there- 

 fore need a method of deducing the independence from the 

 canonical equations, preferably without an explicit integration. 

 Such a method is developed below in S. 24/3 to 10. (The method 

 recently developed by Riguet, however, promises to be much 

 better.) 



Given an absolute system 



-^ =f i (x 1 , . . . , x n ) (t = 1, . . - , n) . (1) 



it is required to find whether or not x^ is independent of Xj, some 

 region of values being assumed. The region must not include 

 changes of values of step-functions or of activations of part- 

 functions ; for the derivatives required below may not exist, 

 and the independencies may change. 



If the functions fi are expandable by Taylor's series around the 

 point X®, . . . , x„, we may write their integrals symbolically 

 (S. 19/27) as 



Fi(4, . . . , xl ; t) = <**x\ {i = 1, . . . , n) . (2) 



where X is the operator 



/K . . . , a£)£g + . . . +f n {xl . . . , fl©gjg. 



(The zero superscripts will now be dropped as unnecessary.) 



pi 



Expanding the exponential, and operating on (2) with =— , 

 the test whether x k is independent of Xj becomes whether 



sf— <*- 1. *...•> • • (3) 



242 



