280 PHYSIOLOGY OF BACTERIA 



and Nyman (1907). The fundamental difference in 

 the order of death is well shown in the survivor curves 

 of Figs. 27 and 29 as well as in the black blocks represent- 

 ing the number of organisms dying per unit time. 



(c) MATHEMATICAL FORMULATION OF THE ORDER OF DEATH 



The regular and simple order of death makes a general 

 mathematical treatment possible. The initial number 

 of cells shall be called a, and the number still living after 

 an exposure for the time t shall be h. Since the rate of 

 death at any time is proportional to the number of 

 survivors, this can be written in the form 



dt 

 -? = Kdt 







Integrated, this expression changes to 



-\nh = Kt + C 



for ^ = 0, we have h = a, and consequently 



-In a = C 



We substitute this value for the integration constant 

 into the preceding equation and obtain 



— In b = Kt — In a 

 K' t = In a — Inh 



K - 1 log Q^ - ^Qg ^ 

 ^ ~ t 0.434 



This last expression gives the formula for the death 

 rate, K^ for decimal logarithms. If the number of cells 

 dying per unit time is at all times the same percentage 

 of the number of living cells, K must be constant. The 



