406 PHYSIOLOGY OF BACTERIA 



interesting. If we call the amount of products formed 

 per cell per hour, x, we find 



s 



X = - 



9 



Substituting in this equation the formula for the genera- 

 tion time from p. 399, we obtain the final expression for 

 the fermenting capacity of a cell per hour to be 



^ " t{h - a) log 2 



We had made the arbitrary assumption here that the 

 last member of the progression to form products was 

 a2''~^; perhaps, we might as well have assumed that it 

 was 2"". The more correct way to solve the problem 

 is the application of calculus. This was done by 

 Slator (1913) and independently, also by Buchanan and 

 Fulmer (1918). The rate with which the products are 

 formed is proportional to the number of bacteria. 

 Using the same designations as before, the cells increase 

 to the progression a2'^, and the products increase accord- 

 ingly in the ratio sa2''. The sum of all products formed 

 between the numbers a and a2'" is given by the integral 

 of the rate of increase 



S = £'sa2''dn 



between the limits n = and n = n' 

 The solution of the integral gives 



^ ^ sa{2'" - 2Q) ^ s(a2^ - a) 

 In 2 In 2 



^ s{h - a) ^ 0.434s(6 - a ) 

 In 2 log 2 



