JOURNAL 



OF THE 



WASHINGTON ACx\DEMY OF SCIENCES 



Vol. 11 June 19, 1921 No. 13 



MATHEMATICS AND BACTERIOLOGY.— 0« the dilution method 

 of counting bacteria} P. V. WELLS and W. F. WElls. (Com- 

 municated by S. W. Stratton.) 

 The dilution method of counting the number of bacteria in water 

 is an example of the use of a geometric scale when the variations are 

 so large that an arithmetic scale is cumbersome. We shall investigate 

 briefly the theoretical basis of this method, with a view to the stand- 

 ardization of the experimental procedure, and shall show that its re- 

 sults have a remarkably simple interpretation. 



FUNDAMENTAL THEOREM 



Consider a "universe" containing .4 cc. of water, and B bacteria. If 



a sample of a cc. is examined, the chances of finding n, n — l,....2, 

 1, bacteria, respectively, in the sample are given by the terms of 

 the hypergeometric series 



B{B - 1)...{B - n-^ 1 ) 

 A^(A^- l)...iN -n-\-l) 



r __W__ 7i(n^-^ W{W-1) 1 



[^+''B-n+l~^ 2! {B-n + l)iB-n + 2)^--\ ^^^ 

 where the water is conceived as composed of W "particles," giving 

 the total "population" N = B + W, and the sample contains m parti- 

 cles. Each particle, whether bacterium or water, is assumed to have 

 an equal chance of being sampled. The general term, for the proba- 

 bility of finding C bacteria in the sample is 

 ^ ^(^-1) .... (B-n-fl) 



^ A^(A^ -1)....{N -n+ 1) 

 r njn - 1) .... {n - C -f 1) W{W - 1) .... {W - n + C + 1 )1 . 

 L a {B-n + l){B -n-{-2)....{B-C)l ^ 



But since the population N is arbitrary, this result is of little use as it 

 stands. 



Taking the population N as infinite, and n as extremely large, but 

 small compared with A^, and placing X^B/A as the number of bac- 



' Received April 29, 1921. 



265 



