R. E. BUCHANAN 49 



LAG PHASE 



During this phase the average rate of growth per cell is increasing to the maxi- 

 mum characteristic of the succeeding (logarithmic) phase. Some authors do not dif- 

 ferentiate between this phase and the preceding, terming the two together the "lag 

 phase." 



It is evident that during this period the number of bacteria present is a function 

 of the time; and several attempts, both empirical and theoretical, have been made to 

 formulate the mathematical relationships. In the analysis of certain data Ledingham 

 and Penfold' found that a graph of the logarithms of the logarithms of the numbers 

 of bacteria and the logarithms of the time is a straight line. This leads to the formu- 

 lation: 



h=Beki' (i) 



in which 



J = Number of bacteria after time i 



5 = Initial number of bacteria 



k and 5 = Constants which require evaluation for each 



particular set of experiments 

 e = Base of natural logarithms 



These authors found the value of s to vary from 1.56 to 2.7. 



LOGARITHMIC PHASE 



During this phase the generation time is a constant, as is also the rate of growth 

 per cell. This phase is the one most susceptible to simple mathematical analysis, and 

 is of major importance in the study of the effect of environment upon bacteria. 



Methods for estimating the generation time and number of generations during 

 this phase were apparently first developed by Buchner, Longard, and Riedlin.^ If it 

 be assumed that the cells are multiplying regularly by binary fission, and 



jB = Initial number of bacteria 



6= Number of bacteria after time t 



«= Number of generations in time / 



g=Length of one generation, i.e., time required for 

 the bacteria to double in numbers 



then, 



b=B2» (2) 



and, since « = - , ^ • 



^ b=Bis (3) 



^^log^-log5 (^) 



log 2 



/ log 2 



log ^— log B 



(5) 



' Ledingham, J. C. G., and Penfold, W. J.: /. Hyg., 14, 242. 1914. 



* Buchner, H., Longard, K., and Riedlin, G.: Centralbl.f. BaklerioL, 2, i. 1887. 



