THE ABSOLUTE SYSTEM 19/24 



These have only to be solved for z v . . . , z n in terms of 

 z l9 . . . , z n and the equations are in canonical form. So the 

 new system is also absolute. 



This transformation implies that in an absolute system we can 

 avoid direct reference to some of the variables provided we use 

 derivatives of the remaining variables to replace them. 



Example : x l = x x — x 



Xn i)X 1 ~\~ Xaj 



can be changed to omit direct reference to x 2 by using x x as a 

 new independent variable. It is easily converted to 



dx x 



~dt ~~ 

 dx. 



'1 



which is in canonical form in the variables x x and x v 



19/23. Systems which are isolated but in which effects are 

 transmitted from one variable to another with some finite delay 

 may be rendered absolute by adding derivatives as variables. 

 Thus, if the effect of x x takes 2 units of time to reach a? 2 , while 

 x 2 's effect takes 1 unit of time to reach x v and if we write x(t) 

 to show the functional dependence, 



then ^=/i(«i(ft «#-»» 



dx 2 (t) 

 dt 



= f 2 {x 1 (t - 1), x 2 (t)}. 



This is not in canonical form ; but by expanding x x (t — 1) and 

 x 2 (t — 2) in Taylor's series and then adding to the system as 

 many derivatives as are necessary to give the accuracy required, 

 we can obtain an absolute system which resembles it as closely 

 as we please. 



19/24. If a variable depends on some accumulative effect so 



that, say, x t =f\\ <j>{cc 2 )dt>, then if we put <f>{x 2 )dt = y, we get 



213 P 



