GENERAL KINETICS OF MATERIAL TRANSFORMATIONS. 151 



and the proportion of the y's is, during the last stages of the process, 

 constant. This last conclusion applies, in fact to all cases in which all 

 the roots X are real and negative. 



General Case. 



We have considered certain special cases in which A (X) reduces to 

 its dextei diagonal, so that all the roots X are real. In general some 

 or all of these roots may be complex, so that in general a more or less 

 complicated series of reactions may give rise to oscillatory pheno- 

 mena.^^ 



In conclusion it may be remarked that the treatment set forth above 

 presents an evident analogy to the theory of small oscillations accord- 

 ing to Lagrange. However, in the case of the oscillations of a mechan- 

 ical system if the number of coordinates is n, the general solution 

 contains 2n arbitrary multiplicative constants; whereas in the case 

 here studied the number of such constants is equal to the number of 

 variables, namely n. 



This point of difference is not without physical significance: 



The multiplicative constants are determined by initial conditions. 

 In the case of an oscillating mechanical system it is necessary, in order 

 completely to determine the motion of the system, to know the 

 n initial values of the ?i coordinates Xi, Xo,. . .Xn, and further, the n 



initial values of the velocities -^^ . . . • — -. 



(It (It (It 



On the other hand, in the case of transformations of the type here 

 considered, not only is it sujficicnt to know the initial values of Xi, 

 Xo, . . . Xn, but, when these are given there remains no further freedom 



of choice for the initial values of the velocities - — . We have here a 



dt 



characteristic property of inertia-free or completely damped systems, 



in which velocities are completely determined when the values of the 



generalized coordinates are given, and in which the accelerations 



vanish with the velocities.^' 



16 See Lotka, Zeitschrift f . phys. Chem., 1910, v. 72, p. 508; 1912, v. 80, 

 p. 159; Hirniak ibid. 1910, v. 75, p. 675. 



17 By differentiating (7) we obtain 



d-xi dxi , (1x2 , 



Hence it is evident, provided none of the coefficients a are infinite, that the 



dx 

 second and all higher derivatives of Xi x^. . .x„ vanish with the velocities — . 



Compare Buckingham, Thermodynamics, 1900, p. 3.3 "■'■ 



