THE ABSOLUTE SYSTEM 19/11 



so, by (3), °f t = F,(x ; 0) (* = 1, . . . , n) 



which proves the theorem, since F\(x ; 0) contains t only in 

 x v . . . , x n and not in any other form, either explicit or 

 implicit. 



Example 1 .* The absolute system of S. 19/10, treated in this 

 way, yields the differential equations 



dx x 



~dt 



dx 2 



~dt 



The second system may not be treated in this way as it is not 

 absolute and the group property does not hold. 

 Corollary : 



— #2 



Ji\Xi, . . . , X n ) = 



■Fi(x v . . . , x n ; t)\ (i = 1, . . . , n) 



dt 



(2) Given the differential equations, they may be written 

 dxi =fi{x v . . . , x n ).dt (i = 1, . . . , n) 



and this shows that a given set of values of x v . . . , x n , i.e. 

 a given state of the system, specifies completely what change 

 dxi will occur in each variable xi during the next time-interval 

 dt. By integration this defines the line of behaviour from that 

 state. The system is therefore absolute. 

 Example 2 : By integrating 



dx 1 



~dt 



dx 2 



dt 

 the group equations of the example of S. 19/10 are regained. 



Example 3 : The equations of the homeostat may be obtained 

 thus : — If xi is the angle of deviation of the ith. magnet from 

 its central position, the forces acting on xi are the momentum, 

 proportional to xi, the friction, also proportional to xi, and the 

 four currents in the coil, proportional to x v x 2 , x 3 and # 4 . If 

 linearity is assumed, and if all four units are constructionally 

 identical, we have 



j t (mxi) = — kxi + l(p — q)(a il x 1 + . . . + a^x^ 



(i = 1, 2, 3, 4) 

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