446 STATE BOARD OF AGRICULTURE. 



cation in the period t. The bacteria are assumed to multiply in geom- 

 etrical progression. 



a 2a 2'a 2^a 2'a 2^ a 



The final number of bacteria, h, will then equal 2'iff. From this 

 equation, the number of generations, n. during the time of observation, 

 t. is easily deducted mathematically : 



a 

 r?, =:log^ — , or, if changed to the common Briggs logarithms: 



n log»»2 = log'" — 

 a 



logio 



° a 



n =- 



log>''2 



Let the average rate of reproduction, i. e. the time required for the 

 average cell to double, be y. This rate of reproduction, multiplied by 

 the number of generations, must equal the total time of the experiment. 



y n ^=^ t 



t 



y =" — 



or substituting the above value for n, we obtain 



t log 2 



y = 



log h/a 



If we let X equal the amount of acid formed by one cell in one hour, 

 a single cell will produce during one generation, i. e. during the time 

 y, an amount of acid equal to xy. a cells will i)roduce axy acid in 

 one generation. Then they multiply, and 2a cells are forming acid, 

 which will amount, during this second generation, to 2axy. The division 

 of cells takes place again, and now there are 2'a cells which form alto- 

 gether 2-axy acid. The acid produced by the successive generations is 

 therefore : 



axy, 2axy, 2^axy, 2^axy 



The final figure cannot be stated, unless arbitrarily. 



The final number of cells is 2"a, but it cannot be stated when these 

 bacteria have been formed and whether they have i)roduced any acid 

 and how much. The two extremes are the following: (1) The cells 

 have been formed for some time, and are just ready to divide again, 

 when the determination is made, in this case, the last acid formed 

 would amount to 2"axy; (2) The bacteria just finished their last 

 division when the final count was made, they have had no time to form 



