THE ABSOLUTE SYSTEM 19/15 



bifurcate. The statement can be verified from the definition or 

 from the theorem of S. 19/11. The statement does not prevent 

 lines of behaviour from running together. 



19/13. The theorems of the previous four sections show that 

 the following properties, collected for convenience, in a system 

 x lt . . . , x n , are all equivalent in that the possession of any one 

 of them implies the others : 



(1) From any point in the field departs only one line of 



behaviour (S. 19/8) ; 



(2) the system is state-determined (S. 19/9) ; 



(3) the system has lines of behaviour whose equations specify 



a finite continuous group of order one ; 



(4) the system has lines of behaviour specified by differential 



equations of form 



-^ =fi(xv • • • > ®n) (i = 1, • • • , n) 



where the right-hand side contains no functions of t 

 except those whose fluxions are given on the left. 



19/14. From the experimental point of view the simplest test 

 for absoluteness is to see whether the lines of behaviour are 

 state-determined. An example has been given in S. 2/15. It 

 will be noticed that experimentally one cannot prove a system 

 to be absolute — one can only say that the evidence does not 

 disprove the possibility. On the other hand, one value may be 

 sufficient to prove that the system is not absolute. 



19/15. A simple example of a system which is regular but not 

 absolute is given by the following apparatus. A table top is 

 altered so that instead of being flat, it undulates irregularly but 

 gently like a putting-green (Figure 19/15/1). Looking down on 

 it from above, we can mark across it a rectangular grid of lines 

 to act as co-ordinates. If we place a ball at any point and then 

 release it, the ball will roll, and by marking its position at, say, 

 every one-tenth second we can determine the lines of behaviour 

 of the two-variable system provided by the two co-ordinates. 



If the table is well made, the lines of behaviour will be accur- 

 ately reproducible and the system will be regular. Yet the 

 experimenter, if he knew nothing of forces, gravity, or momenta, 



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