THE ABSOLUTE SYSTEM 



19/10 



they should form a finite continuous (Lie) group of order one 

 with t as parameter. 



(1) The system is assumed absolute. Let the initial state of 

 the variables be x°, where the single symbol represents all n, and 

 let time t' elapse so that x° changes to x' . With x' as initial 



x»— 



Figure 19/10/1. 



state let time t" elapse so that x' changes to x". As the system 

 is absolute, the same line of behaviour will be followed if the 

 system starts at x° and goes on for time f -f- t". So 



x- = Fi{xi . . . , a£ ; t") = Fi(x° v . . . ,' x\ ; t' + t") 



(i = 1, . . 



n) 



but 



Xi 



Fi{x° l9 



,o . 



n 



(* = i, 



., n) 



giving 



FilF^x"; n • .» Fn(af>; t'); t"} 



= Fi{x° v . . . , x Q n ; t' + n (i = 1, . . . , n) 



for all values of x°, f and t" over some given region. The equation 

 is known to be one way of defining a one-parameter finite con- 

 tinuous group. 



(2) The group property is not, however, sufficient to ensure 

 absoluteness. Thus consider x = (1 -f- t)x° ; the times do not 

 combine by addition, which has just been shown to be necessary. 



Example : The system with lines of behaviour given by 



Xl = x\ + x\t -f Z 2 1 



Xo — Xo -f- "I J 



is absolute, but the system with lines given by 



x x = x\ + x\t + V 



is not. 



'2 — 2 ~" • 



205 



