CONSTANCY AND INDEPENDENCE 24/5 



By expanding ^— A': 



£- X'+'a:* = £& £- X*x t + X ~ X»x k . (4) 



OXj '—' OXj OXp OXj 



Applying the test (3), if the test for /j, = m gives 



5- X m X k = 



OXj 



then for /li = m -j- 1, by using (4) we need only see whether 



?%k**- ■ ■ ■ ^ 



24/4. We now add the hypothesis that the system is linear (S. 

 19/27). The restriction is unimportant as no arguments are used 

 elsewhere which depend on linearity or on non-linearity. Further, 

 in the region near a resting state all systems tend to the linear 

 form (S. 20/4), and this region has our main interest. 

 Starting with ju = 1 the tests 24/3 (5) become 



OXj 



sfi a/, _ 



z 



dx p dxj 



~* — ' dx p dx a dxj 

 etc. 



(i) 



These tests now use only the /'s, as required. They are both 

 necessary and sufficient. They have been shown necessary ; and 

 by merely retracing the argument they are found to be sufficient. 

 Only the first n — 1 tests of (1) above are required, for products 

 which contain more than n — 1 factors must include products 

 already given, in the first n — 1 tests, as zero. 



The tests are, however, clumsy. The simplicity and directness 

 can be improved by using the facts that we need distinguish only 

 between zero and non-zero quantities, and that the sums of (1) 

 above resemble the elements of matrix products. Sections 24/5-10 

 develop this possibility. 



24/5. An R0- matrix has elements which can take only two 

 values : R (non-zero) and (zero). The elements therefore 



243 



