148 LOTKA. 



= V variables Fi, Y^,--- Y^. The process can in this case be regarded 

 as the resultant of (n — m) single transformations proceeding simul- 

 taneously, and the state of the system, and in particular the velocities 



— - are fulh' defined when we are given the values of the parameters 

 at 



P; the initial values Ai, A2,. . .Am of the variables A'' (masses of the 



several components) ; and the values of the variables Fi, Yo,... Y^ 



which tell us how far each reaction has proceeded. 



Consecutive Transformations. 



• 



A special case arises if the (n — m) "single" transformations whose 

 resultant is the actual transformation can be so arranged in order, 

 that the velocity of the first depends solely on Fi, that of the second 

 on both Fi and Yo, in general that of the j"' on Fi F2. . . Fy. This is 

 the case of what might be termed inirely consccnthe reactions. 



The system of equations (7) in this case takes the form : 



dt 



dy 

 dt 



-J- = fl21 Ih + «22 2/2 + . • . ^ (33) 



-4^ = a,i iji + (h2 2/2 + • ■ ■ + «„;. y, + • 

 dt 



The solution of (33) is here also given by (8) ; however, it takes on 

 a simplified form owing to certain special properties of the exponential 

 constants X and the coefficients a, as follows: 



A. Exponential Constants. 



1. It will be seen that in (33), as compared with (7), every coeffi- 

 cient Urs for which s > r is zero. In consequence of this A (X) reduces 

 to the dexter diagonal, and hence 



Xf = an 



2. Inasmuch as the coefficients a are essentially real, it follows 

 that the X's also are all real. If therefore equilibrium is established. 



