19/11 DESIGN FOR A BRAIN 



The canonical equations of an absolute system 



19/11. Theorem : That a system x v . . . , x n should be absolute 

 it is necessary and sufficient that the #'s, as functions of t, should 

 satisfy differential equations 

 dx 1 



i — fl\ x l> - • • t x n) 



~rr — JnK^n • • • » x n) 



(i) 



where the /'s are single-valued, but not necessarily continuous, 

 functions of their arguments ; in other words, the fluxions of 

 the set x v . . . , x n can be specified as functions of that set and 

 of no other functions of the time, explicit or implicit. 



(The equations will be written sometimes as shown, sometimes 

 as dxi/dt =fi(x v . . . , #») [i == 1, . . . , n) . (2) 



and sometimes abbreviated to x = f(x), where each letter repre- 

 sents the whole set, when the context indicates the meaning 

 sufficiently.) 



(1) Start the absolute system at x\, . . . , a% at time t = 

 and let it change to x v . . . , x n at time t, and then on to 

 x x + das l9 . . . , x n + dx n at time t + dt. Also start it at 

 SB l9 . . . , x n at time t = and let time dt elapse. By the group 

 property (S. 19/10) the final states must be the same. Using 

 the same notation as S. 19/10, and starting from a£, Xi changes 

 to Fi(x° ; t + dt) and starting at x% it gets to Fi(x ; dt). 

 Therefore 



Fi(x° ; t + dt) = Fi{x ; dt) (i = 1, . . . , n). 



Expand by Taylor's theorem and write ^rFi(a ; b) as F'i{a ; b). 



Then 



Fi{x° ; t) + dt.Fl(x° ; t) = Fi{x ; 0) + dt.F'lx ; 0) 



{% = 1, . . . , n) 



But both Fi(x° ; /) and Fi(x ; 0) equal xi. 



Therefore F^x ; t) = F^x ; 0) {i = 1, . . . , n) . (3) 



But Xi = Fi(x° ; t) \i = 1 n) 



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