19/11 DESIGN FOR A BRAIN 



where p and q are the potentials at the ends of the trough, I 

 depends on the valve, k depends on the friction at the vane, 

 and m depends on the moment of inertia of the magnet. If 



h = * t j = _ then the equations can be written 



m m 



dxi _ . 

 ~dt~ Xi 



dx~ 

 — = h(a il x 1 + . . . + a i4 oj 4 ) — jxi 



(i = 1, 2, 3, 4) 



which shows the 8 -variable system to be absolute. 

 They may also be written 



dxi _ . 



dt ~ m\ k {ChlXl + * * * + a ^ l *> 



Let m — > 0. dxi/dt becomes very large, but not dxi/dt. 

 So xi tends rapidly towards 



k ^ Xl + • • • + ^4^4) 



while the CD's, changing slowly, cannot alter rapidly the value 

 towards which xi is tending. In the limit, 



^ = £i = fcZ-%^ + . . . + aii x t ) (i = 1, 2, 3, 4) 



Change the time-scale by r = , 1 ; 



— = a il x 1 + . . . + ai^ (i = 1, 2, 3, 4) 



showing the system x v . . . , x± to be absolute and linear. The 

 a's are now the values set by the hand-controls of Figure 8/8/3. 



19/12. That a system should be absolute, it is necessary and 

 sufficient that at no point of the field should a line of behaviour 



208 



