19/12 THE STATE-DETERMINED SYSTEM 



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 = l(p — q)/m and j = k/m, the equations may be written 

 dxjdt — x { \ {• 



dxjdt = &(««*i + . . . + fl, 4 a 4 ) -jx t J {l = l > 2 ' 3 ' 4) 



which shows the 8-variable system to be state-determined and 

 linear. 

 They may also be written 



dati 



dt ~ Xi 



dx t k (Up — q). , , , 



> (i = 1, 2, 3, 4) 



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

 ±i tends rapidly towards 



k q (gfl^l + • • • + a i*Zi) 



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

 towards which i t is tending. In the limit, 



^ = i t = ^^(-Bft + • • • + a iiXl ) (i = 1, 2, 3, 4) 



Change the time-scale by t = ^ t, and 



dxjdr = a n x x + . . . + a u x 4 (i == 1, 2, 3, 4) 



showing the system x l9 x 2 , x 3 , x± to be state-determined and linear. 

 The a's are now the values set by the input controls of Figure 



8/2/3. 



19/12. The theorems of the preceding sections show that the 

 following properties are equivalent, in that the possession of any 

 one implies the possession of the remainder. 



(1) The system is state-determined. 



(2) From any point of the field departs only one line of be- 



haviour. 



(3) The lines of behaviour are specifiable by equations of form: 



dxi/dt ='f i (x l , . . . , x n ) (i = 1, . . . , n) 



247 



