CHAPTER 24 



Constancy and Independence 



24/1. The relation of variable to variable has been treated by 

 observing the behaviour of the whole system. But what of their 

 effects on one another ? Thus, if a variable changes in value, can 

 we distribute the cause of this change among the other variables ? 

 In general, it is not possible to divide the effect into parts, 

 with so much caused by this variable and so much caused by that. 

 Only when there are special simplicities is such a division possible. 

 In general, the change of a variable results from the activity of 

 the whole system, and cannot be subdivided quantitatively. 

 Thus, if dx/dt = sin x + xe y , and x = \ and y = 2, then in the 

 next 0-01 unit of time x will increase by 0-042, but this quantity 

 cannot be divided into two parts, one due to x and one to y. 



24/2. But a relationship which can be treated in detail is that of 

 ' independence '. By the principle of S. 2/8 it must be defined in 

 terms of observable behaviour. 



Given an absolute system and two lines of behaviour from two 

 initial states which differ only in their values of x® (the difference 

 being A#°), the variable x k is independent of Xj if x k 's behaviour is 

 identical on the two lines. Analytically, x k is independent of Xj 

 in the conditions given if 



F k (4, . . . , x% . . . ; t) = F k (xl . . . , ^ + AflJ, . . . ; t) (1) 

 as a function of t. In other words, x k is independent of Xj 

 if XfcS behaviour is invariant whemthe initial state is changed 

 by A4 



This narrow definition provides the basis for further develop- 

 ment. In practical application, the identity (1) may hold over all 

 values of Aa?° (within some finite range, perhaps) ; and may also 

 hold for all initial states of x k (within some finite range, perhaps). 

 In such cases the test whether x k is independent of Xj is whether 



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