THE DETERMINATE MACHINE 3/10 



3/9. ''Unsolvable'" equations. The exercises to S.3/6 will have 

 shown beyond question that if a closed and single-valued trans- 

 formation is given, and also an initial state, then the trajectory from 

 that state is both determined (i.e. single-valued) and can be found 

 by computation. For if the initial state is x and the transformation 

 T, then the successive values (the trajectory) of x is the series 



X, T{x), T2(x), T^ix), T\x), and so on. 



This process, of deducing a trajectory when given a transforma- 

 tion and an initial state, is, mathematically, called "integrating" the 

 transformation. (The word is used especially when the transforma- 

 tion is a set of differential equations, as in S.3/7; the process is then 

 also called "solving" the equations.) 



If the reader has worked all through S.3/6, he is probably already 

 satisfied that, given a transformation and an initial state, he can 

 always obtain the trajectory. He will not therefore be discouraged 

 if he hears certain differential equations referred to as "non- 

 integrable" or "unsolvable". These words have a purely technical 

 meaning, and mean only that the trajectory cannot be obtained 

 if one is restricted to certain defined mathematical operations. 

 Tustin's Mechanism of Economic Systems shows clearly how the 

 economist may want to study systems and equations that are of the 

 type called "unsolvable"; and he shows how the economist can, in 

 practice, get what he wants. 



3/10. Phase space. When the components of a vector are numerical 

 variables, the transformation can be shown in geometric form; and 

 this form sometimes shows certain properties far more clearly and 

 obviously than the algebraic forms that have been considered so far. 

 As example of the method, consider the transformation 



y' = ^x + iy 



of Ex. 3/6/7. If we take axes x and y, we can represent each 

 particular vector, such as (8,4), by the point whose x-co-ordinate 

 is 8 and whose j-co-ordinate is 4. The state of the system is thus 

 represented initially by the point P in Fig. 3/10/1 (I). 



The transformation changes the vector to (6,6), and thus changes 

 the system's state to P'. The movement is, of course, none other 

 than the change drawn in the kinematic graph of S.2/17, now drawn 

 in a plane with rectangular axes which contain numerical scales. 

 This two-dimensional space, in which the operands and transforms 

 can be represented by points, is called the phase-space of the system. 

 (The "button and string" freedom of S.2/17 is no longer possible.) 



37 



