CONSTANCY AND INDEPENDENCE 24/10 



24/9. From the /'s of the canonical equations (24/3(1) ) form 

 the differential matrix [f] by inserting, in the (kj)-th position 

 (at the intersection of the A:-th row and the ;'-th column) an or 

 R according as dfk/dxj is, or is not, zero (in the region of phase- 

 space considered). Then the square, cube, etc., of [/] will contain 

 in the (A^')-th position an element which is zero or non-zero as the 

 second, third, etc., tests of 24/4(1) are or are not zero. If now 

 these powers are summed, to S : 



« = m + [/? + • • • + 1/]- 1 , . . (i) 



a zero element in S at the (Jcj)-th position means that all the 

 terms of the series were zero, and therefore that Xk is independent 



Of Xj. 



The same independence will make zero the element at the {kj)-t\\ 

 position in the matrix whose (^)-th element is zero or non-zero 

 as dFn/dJl is or is not zero. This integral matrix, [F], must 

 therefore satisfy 



[*]=« (2) 



24/10. The restriction is now added that the behaviour of each 

 variable xt is to depend on its own starting-point. (Physical 

 systems not conforming to this restriction are, so far as I am 

 aware, rare and peculiar.) The principal diagonal of [F] will 

 then be found to have all its elements non-zero. In such a case, 

 [F] is not altered if we add to it the matrix / of S. 24/8, and we may 

 sum up as follows : 



If a dynamic system is specified by 



— l =f i {x 1 , . . . , x n ) (i = 1, . . . , n) 



and if [/] is an jRO-matrix where each (ftj)-th element is or R 



as ~- is or is not zero respectively (in some region within which 



CXj 



the nullity does not change), and if [F] is an .RO-matrix where 

 each (kj)-th element is or J? as Xk is or is not independent of Xj 

 respectively in the same region, and if each x's behaviour depends 

 on its own starting point, then 



m = [/] + [/]« + • • ■ + [n n - 1 ■ ■ a) 



This equation gives the independencies when the differential 

 matrix is given; for x k is or is not independent of x 5 as the 



245 r 



