1910-11.] The Rate of Multiplication of Micro-organisms. 649 
XLV.— The Rate of Multiplication of Micro-organisms: A Mathe- 
matical Study. By A. G. M ‘Kendrick, Captain and 
M. Kesava Pai, M.D. (Pasteur Institute of Southern India). Com- 
municated by Professor M‘Kendrick. 
(MS. received March 13, 1911. Read June 19, 1911.) 
The problem of the rate of multiplication of micro-organisms is one 
which has often been attacked, hut which has not, to our knowledge, 
been reduced to a simple law. 
If there be an unlimited supply of nutriment, an organism reproduces 
itself by compound interest: in a geometrical progression — i.e. 1, 2, 4, 8, etc. 
That is to say, that the rate of growth under unimpeded conditions is 
proportional to the number present, at any moment, or 
a> %-h- 
In test-tube experiments, however, this simple state of affairs is complicated 
by the fact that the supply of nutriment is limited, and consequently, 
as time goes on, the rate of multiplication falls off. 
Every living organism employs the nutriment which it has absorbed 
for two objects : first, the maintenance of the individual ; and, second, 
its reproduction. As, however, in the case of those micro-organisms with 
which we shall deal, the rate of multiplication is very fast, we may, 
for all practical purposes, consider that the amount of food-stuff utilised 
for their upkeep is negligible, and assume that the whole of it is employed 
in reproduction. 
If we accept this simplifying assumption we may say that organisms 
in a test-tube multiply, by a simple conversion of the available food-stuff, 
into other organisms, and that the rate of multiplication is proportional 
to the concentration of that food-stuff. 
If a be the original concentration of food-stuff, the concentration at 
the time t will be ( a — y ). 
Introducing this factor into equation (1), we have 
( 2 ) 2 t =h y( a ~y)> 
which means that the rate of increase of fast-growing organisms is 
proportional to the number of organisms present , and to the concentration 
of the food-stuff 
