June 19, 1921 wells and wells: counting bacterl\ • 267 



As a function of z, this equation is a special case of Pearson's Type III 

 frequency curve, and the mode Mz, or most probable value of z is 

 simply Mz = C. 



"per cent negative" method 

 The quantities a and /3 are usually taken as 1 cc.(q! = /3= 1). The 

 probability of a negative result (no bacteria) in the diluted sample is 

 simply the negative exponential 



Po = exp. (— -) (7) 



This is the frequency curve of the negative plates in dilutions D. 

 The mode {Md= ^), or most probable value of the negative dilution 

 D is infinite, as it should be. The fraction of negative plates in dilu- 

 tion D is Pq, and the percentage of negative plates 100 Pq, when the 

 number of samples taken is sufficiently large to overcome the 

 fluctuations of sampling, and when there is no constant error in the 

 experimental procedure. 



In practice the sample (1 cc.) is diluted in the definite dilutions 

 D=10, 100, 1000, etc., cc. of water, and Fo observed. In order to 

 compute X from these results, place in (7) 



E = \n 

 Then 



f-^) = ^ (8) 



\Pj D 



x= d +e (9) 



Where x = log X 



d = \ogD\ (10) 



e == log E ] 



The arithmetic means d and e are therefore related by the simple ex- 

 pression 



x=d+~e (11) 



and the number of bacteria per cc. X is given by the product of the 

 geometric means 



X = GdGe (12) 



Where 



e — log Ge J 

 The "per cent negative" method requires for convenient application 

 a table giving e in terms of Po, as in table 1. 

 It is evident from the magnitudes in the second column of table 1 



