654 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
The following figures illustrate this maximum value : — 
Vo- 
a after fourteen hours. 
22,800 
147,000,000 
32,150 
151,000,000 
100,550 
167,000,000 
From a practical point of view this is of extreme importance, as 
obviously error in the number inoculated has little or no effect on the 
total number attained to. Provided that the number planted be com- 
paratively small, it is unnecessary to enumerate each flask, in the prepara- 
tion of vaccines in large quantities. Indeed, with a suitable and accurately 
based system (concentration of broth, etc., being kept constant) vaccines 
could be prepared month after month at standard strength without any 
enumerations being made. B. coli, in the quantities of broth which we 
have used, reaches a maximum in from twelve to fifteen hours. 
When t — 0 
dy 
and yP (the initial rate of increase) = by Q (a — y Q ) , 
dt 
'i.e. 
- ^ y ° = 6(«- y 0 ). 
As in our experiments a is measured in hundred millions and y 0 rarely 
exceeds a hundred thousand, we may consider 
d}og_y _ 0 = ah 
dt 
That is to say, with log y as ordinate and t as abscissa the slope of the 
curve at the commencement is equal to ab. This value can be obtained from 
observed results, and its value substituted in equation (3). The values of 
ab so obtained are entered on the Table of Numbers given above. 
[Note. — ab is measured in logarithms to base e, whereas in the accom- 
panying tables logarithms to base 10 are employed. Log y to base 10 
multiplied by 2-3026 = log y to base e .] 
Since the constant a denotes the original concentration of food-stuff, and 
b depends on the ability of the organism to acquire its food (modified by such 
accelerating or retarding influences as temperature, degree of alkalinity, 
presence of medicaments, etc.), it might have been hoped that in the relation 
