GENERAL KINETICS OF MATERIAL TRANSFORMATIONS. 



139 



In the expansion of the right-hand member of (6) by Taylor's 

 theorem the absolute term must be zero, in view of (5), hence 



<h:i 



-—■ = an Xx + an a\ + 

 at 



— - = Oil a-i + a'^2 Xi + 

 at 





(7) 



-^ = Onl Xi + a„2 ^2 + . . . + a„„ Xn+ ... 



at 

 A general solution of this system (7) is ^ 



xi = ai e ^1' + ai e ^2' + . ,. + «;; e ^-' ] 



+ ai'2e(^i+^=)'+ ... 

 X2 = «'(c^i'+«'^e^2i4_ ... 



where Xi, X2, . . .X„ are the n roots ^° of the equation of n"* degree 



fl)l — X a]2 Ol3 . ■ . a;r, 



Cf-'l rt22~X fl-13 . . . fl^'n 



A (X) = 



(8) 



Onl 



an2 (Inn — X 



(9) 



5 See Picard, Traite d' Analyse, 1908, v. 3, p. 14; Forsyth, Theory of Differ- 

 ential Equations, 1900, v. 3, pp. 2, 8, 9; Konigsberger, Lehrbuch der Theorie 

 der Linearen Differentialgleichungen, 1889, p. 283. 



5'^ The quantities X in (8), (9) are subject to certain restrictions. See Picard, 

 loc. cit., pp. 9, 10; 17, IS. The method of determination of certain of the 

 constants a in (8) also breaks down in the special case that two or more of the 

 roots of (9) are equal, or differ only by an integral factor. 



We shall not here discuss the case of multiple roots of equations (9), which 



presents no particular difficulty (see Konigsberger loc. cit.; Lotka, Zeitschr. 



f. Phys. Chemie, 1912, v. 80, p. 161); nor the case in which the series in the 



right hand members of (7) contain no terms of the first degree. This case 



does not appear to admit of general treatment. Certain special cases have 



been treated by Picard, Poincare, Dulac, Bendixon, Jordan. (See Dulac, 



Jl. Ec. Polytechn. IX cahier 1904; Bui. Soc. Math. 1906; Bendixon, Acta 



Math. 24; Jordan, Jl. Math. 1906). It should be noted that if the series (8) 



begin with the terms of second degree, for example, a change in sign only of 



. . d.v 

 all the variables x leaves the velocities ~- unaltered, for small displacements. 



Such a state of affairs does not correspond to the nature of the physical sys- 

 tems and processes here under discussion. This special case which presents 

 certain mathematical difficulties, is therefore also a case of no practical interest 

 for our present discussion. 



