ARTICLES 



411 



For the purposes of our present discussion we may, then, 

 write : 



dX 1 



dt 



dXi 

 ~aT 



= F,{X U *.,... X n ) « 

 = F 2 (Xi, X 2 , . . • X H ) 



(5) 



—jf = F»{X\, X 2 , . . . X n ) ) 

 We define C iy C*, . . . C n by the relation 



Q 



Fi(C lf C 2 , . . . C n ) — Fi{C 1} Ci, 

 Fn{C\> C 9 , . . . C„) 



o 



and introduce new variables 



dx. 



so that 



dt 



— Ji\ X l> X %> • • ' X n) 



(6) 



(?) 

 (8) 



We shall assume that the functions / are analytic, and 

 write 



dt ( Q ) 



+ «ai^"i 2 + #/22#2 2 + . . . 



Omitting the discussion of special cases, 1 a general solu- 

 tion of the system (9) is, 



X t = a,ie A '* + a i2 e^' + . . . + a ilx e^' + a n2 e' A »+^" + . . . (10) 



where the exponential coefficients \ u X 2 , ... X,, are the n 

 roots for A, of the equation 



A(X) = 



a 



11 



</ 



■ji 



X a X 2 . . 



«22 A, 



In 



2« 



a 



«i 



a 



n'i ' 



tin 



= o 



(II) 



and the coefficients a are determined in well-known manner. 8 

 Marks of Irreversibility. — The question now arises, Where 

 in our equations and in their solution do we find the earmarks 

 of irreversibility ? 



1 Such as the case when the determinantal equation (11) has multiple roots. 

 This offers no special difficulties, but is here omitted for the sake of brevity. See, 

 for example, Picard, Traits a" Analyse, vol. iii, p. 30. 



3 See, for example, Webster's Dynamics, 1912, pp. 159, 161. 



