THE ABSOLUTE SYSTEM 19/28 



It has the important property that any function @(x lf . . . , x n ) 



can be shown as a function of t, if the aj's start from x®, . . . , x„, 



by 0(x v . . . , x n ) = <^<Z>(^, . . . t 3,0) _ (4) 

 (2) If the functions jfi are linear so that 



-5— = ttj^j -f- di2 X 2 l • • • 1" a \n&n ~T "1 



dx n 



dt ' 



— Cl n iX^ -j- ^712^2 1 • • ■ "T" "nn^n 1 ^n 



(5) 



then if the fr's are zero (as can be arranged by a change of 

 origin) the equations may be written in matrix form as 



x = Ax . . . (6) 



where x and x are column vectors and A is the square matrix 

 [ciij]. In matrix notation the solution may be written 



x = e tA x° .... (7) 

 (3) Most convenient for actual solution of the linear form is 

 the recently developed method of the Laplace transform. The 

 standard text-books should be consulted for details. 



19/28. Any comparison of an absolute system with the other 

 types of system treated in mechanics and in thermodynamics 

 must be made with caution. Thus, it should be noticed that the 

 concept of the absolute system makes no reference to energy or 

 its conservation, treating it as irrelevant. It will also be noticed 

 that the absolute system, whatever the ' machine ' providing it, 

 is essentially irreversible. This can be established either by 

 examining the group equations of S. 19/10, the canonical equa- 

 tions of S. 19/11, or, in a particular case, by examining the field 

 of the common pendulum in Figure 2/15/1. 



Reference 



Shannon, C. E. A mathematical theory of communication. Bell System 

 technical Journal, 27, 379-423, 623-56 ; 1948. 



215 



