54 lotka: discontinuous evolution 



essary to carry this out in detail here; the expressions thus 

 obtained for h and k are, however, of special interest 



h =--{(« + f) + l/(a- j 8')* + 4a / /3} ( 22 ) 



* = - o" { (a + j8') - l/~(a-/30 2 + 4c?i8} (23) 



From these expressions it will be seen, that the solution becomes 

 oscillatory as soon as 



(a-|S0* + 4a'j8<0 



It is then convenient to write the solution in trigonometric 

 form, as follows 



x = e- pt {A\coaqt+B\fm\qt} +e" 2pt {A' 2 cos2^+£' 2 sin2gH-C" 2 } 

 + e- 3pt {A / 3 cos3gi + J B , 3 sin3gi + (7 / 3Cosg/ + Z) / 3sin^} (24) 



y = e - pt [a\ cos qt + b'i sin qt) + • • • (25) 



where 



1 1 



p=---( a + 0O and g=-l/ -{(a-/30 2 +4a' l 3} ...(26) 



For lack of numerical data we can not apply the solution thus 

 found to a concrete example of the particular case here considered. 

 However, for a certain type of reaction, which the writer has dis- 

 cussed elsewhere, 2 the laws of chemical dynamics lead to a set 

 of equations 



- — = Lx + Ky + xy (I) 



at 



-^ =Lx + xy (ID 



at 



which will be recognized as a special case of (18) (19); their solu- 

 tion is, in point of fact, of the form (20) (21) (24) (25). 



In order to illustrate the character of the function represented 

 by the series (24), a concrete example of such a reaction has been 



»J1. Phys. Chem. 25: 271. 1910. Zeitschr. phys. Chem. 72: 50S. 1910. 

 Only the firsl two terms of the series are given in these publications. 



