June 19, 1921 wells and wells : counting bacteria 269 



Example 2 

 (20 samples, 11 negative in 10,000 cc, all negative in 100,000 cc.) 



d Per cent Neg. d •\- e 



4 55 « = 3.78 



X = 6 X 10^ = 6,000 bacteria per cc. 



To determine Po with precision, more samples must be taken at 

 each dilution than is practicable for a single specimen of water, but 

 the method is very useful where many bottles of water are collected, 

 or where the samples represent a seasonal distribution. The use of 

 the geometric mean in such cases is justified only by the type of such 

 frequency distributions. This point will be considered in a subse- 

 quent paper. 



METHOD OF BACTERIAL COUNTS 



No information is gained in the "per cent negative" method from the 

 number of colonies found on the positive plates. Returning to the 

 fundamental equation (6) , the probability of finding C bacteria in the 

 diluted sample is 



Pc=-exp.(-2) (6) 



Now the bacterial counts are usually fairly large (OlO). In this case 

 Stirling's formula gives 



C./ = CfV2^exp. (-C) (14) 



Comparing (6) and (14), 



Pc = 



V 



= f-Yexp.(C-.) (15) 



This is the frequency curve of bacterial counts of a given sample (at a 

 fixed dilution). The mode {Mc)^ or most probable value, of C is 

 given by the condition 



^^ = ln ^ - ^ = (when C^Mc) (16) 



PdC C 2C 



Neglecting — compared with In C, involving an error of 2 per cent when 



C= 10, and less for larger values, we have 



Mc = z = a0^ (Oio) (17) 



That is, when q: = /3 = 1, the number of bacteria per cc. (X) is simply 

 the product of the dilution D and the mode Mc of the frequency dis- 



