APPENDIX 405 



fermentation by a mathematical calculation, provided 

 that the increase in cells is known. 



After an attempt by Burchard (1899) which was based 

 on a wrong mathematical principle, Rahn (1911) devel- 

 oped a formula in the following way: 



If we assume that the single cell up to the time where 

 it is completely divided produces the amount of products 

 s, then the amount of products in the successive genera- 

 tions will be 



as 2as 4as Sas . . . a2'^s 



It is arbitrary to state which shall be the last member 

 of this progression, whether 2"* or 2""^ Rahn decided 

 that 2""^ would be more accurate to assume. The total 

 amount of the products formed, S is the sum of all these 

 amounts. 



S = as + 2as -\- 4as + 2^as + • • • + 2"-ias 



Applying the sum formula of geometrical progressions, 

 we get 



S = as(2" - 1) = s(a2" - a) 



and since 2"a = b, we get 



S = s{b - a) 



or 



S 



s = 



b — a 



In this formula, s is the amount of products formed 

 by one cell in one generation, i.e., in the time g. Ordi- 

 narily, the fermenting capacity per hour will be more 



