HARMFUL EFFECTS OF CROWDING UPON GROWTH 115 



conditions one would expect to find growth retardation with increase 

 in numbers to be much greater than if volume relations alone were 

 the responsible factor. 



Inspection of his experimental results in comparison with a simple 

 formula built on the assumption that the growth would be inversely 

 proportional to the group stimulation, 



'^'{^ 



i) 



in which y represents size, x stands for the number of animals in a 

 given space, and ^ is a constant, shows that the influence of the 

 stimulation is not on this order but is approximated by taking an 

 exponential value of x, namely x^^'. The equation then becomes 



X3/2 



x{x—i) 



o. 



y = k 



Vx 



Values calculated from this formula fit fairly well with Bilski's ob- 

 servations on the effect upon the growth of differing numbers of 

 tadpoles in jars of equal size; the observations of Semper on the 

 growth of snails in relatively small vessels; and, according to Bilski, 

 with the observations of HofTbauer on growth in carp. Another for- 

 mula derived by a continuation of the same reasoning better fits 

 Semper's results with snails in larger volumes. 



Bilski recognizes the general significance of his results and believes 

 that such diverse phenomena as the reported dependence of size of 

 mammals upon available land, and other similar relationships, in- 

 cluding even a correlation between the size of children and available 

 space, nmy depend upon an application of this principle. Farr (1843, 

 1875) worked out an equation essentially similar to that of Bilski to 

 describe the relation between death-rate and the density of human 

 populations. Brownlee (1915, 1920) finds that Farr's law fits a wide 



