6 PROBLEMS OF RELATIVE GROWTH 



fractional coefficient. But the value of k has an important 

 meaning. 



It implies that, for the range over which the formula holds 

 the ratio of the relative growth-rate of the organ to the relative 

 growth-rate of the body remains constant, the ratio itself 

 being denoted by the value of k. By relative growth-rate is 

 meant the rate of growth per unit weight, i.e. the actual abso- 

 lute growth-rate at any instant divided by the actual size at 

 that instant. 



This is at once seen by plotting the logarithm of y against 



the logarithm of x. In unit time the increase in the logarithm 



of y is k times the increase in the logarithm of x, which may 



be written : 



d , ■, d , 



g.logy-Jglog* 



dy , ,dx , 



i ly - k Tt lx 



This formula, on which I have had the advantage of con- 

 sulting Professor Levy, of the Imperial College of Science, 

 can be deduced on the basis of very simple assumptions about 

 growth in general. One essential fact about growth is that 

 it is a process of self-multiplication of living substance — i.e. 

 that the rate of growth of an organism growing equally in 

 all its parts is at any moment proportional to the size of the 

 organism. A second fundamental fact about growth is that 

 the rate of self -multiplication slows down with increasing age 

 (size) ; a third is that it is much affected by the external 

 environment, e.g. by temperature and nutrition. The two 

 latter considerations affect all parts of the body equally, so 

 that we may suppose that the growth-rate of any particular 

 organ is proportional simultaneously (a) to a specific constant 

 characteristic of the organ in question, (b) to the size of the 

 organ at any instant, and (c) to a general factor dependent 

 on age and environment which is the same for all parts of 

 the body. 



If y stand for the size of the organ, and x for that of the 

 rest of the body, we shall then have 



— = oaG and -- = #yG, 

 dt at 



where a and /? are the specific constants for the rest of the 

 body and for the organ in question, and G measures the 



