EXPERIMENT STATION BULLETINS. 447 



acid, therefore the last amount of acid formed would amount to 2'^-'^axy. 

 There is no means of ascertaininj:^ the correctness of either probabil- 

 ity'. In fact, bacteria do not divide all at the same moment; this is 

 assumed only for the convenience of computation. To determine the 

 full values, the calculus is necessary. 



It has been assumed above that the original number of bacteria a 

 produces the total amount of acid (i.ri) before they multiply. Corres- 

 pondiug to this, it is also assumed in the further computation that 

 the final nund)er h has been formed just at the moment, the determi- 

 nation is made. These two assumptions will counterbalance one an- 

 other to some extent. The final amount of acid is therefore 2"V/a?7/. 



The total acid, *S', which is determined by titration, is the sum of 

 all these figures. 



S = axy -f 2axy + 2^axy + +2'^-hixy. 



= axy (1+ 2 + 2^ 4 2' + +2'^-i) 



= axy (2"-l) according to the sum formula of geometrical 

 progression. 



t{h - a) log 2 



X is the amount of acid formed by the average cell in one hour, i. e. 

 the fermenting capacity. aS^ is the acid formed in the time t, a and 6 are 

 the initial and fiual number of bacteria. In computing, care has to be 

 taken to reduce both acid and bacterial numbers to the same unit of 

 medium, either 1 cc. or 100 cc. or 1,000 cc. 



After tills manuscript had been finished, an attempt at computing the 

 fermenting capacity of the single cell by Burchard was found in Archiv 

 f. Hygiene Vol. 36, p. 283 (1899). Burchard, working with urea bacteria, 

 considered the geometrical mean of the number of bacteria in the begin- 

 ning and at the end of his experiments as the average of the acting cells. 

 The geometrical mean gives the number of cells in the middle of the 

 experiment, but this is not the average of all acting cells. By his method, 

 he obtains figures which are much larger than those obtained by the 

 above formula. It is easily proven, by Burchard's own data, that they 

 are wrong. Burchard computes from his data that the fermenting capac- 

 ity is 0.00,000,003 mg = 300 x 10^ "^ mg. urea per cell and hour; the 

 rate of reproduction is 6.3 hours, the final number of bacteria is 42,- 

 720,000 per cc, the amount of urea decomposed 1.78 mg. per cc. The 

 urea decomposed by the last generation is 21,360,000 X 6.3 X 0.00,000,003 



