CONSTANT DIFFERENTIAL GROWTH-RATIOS 7 



general conditions of growth as affected by age and environ- 

 ment, 1 then — = — • 



ax ax 



Thus log y = - log x + log b, where b is a constant : i.e. 



y = bx p/a . 



And fi/a, which can also be written k, is a constant, and is 

 also the ratio of the specific components of the growth-rates 

 of y and x respectively. 2 



I am, of course, aware that the existence of growth-cycles 

 and other facts make it impossible to suppose that the ex- 

 pression for change of growth can be so simple as here set 

 forth. We must suppose that each cycle may have its own 

 general and specific components of the growth-rate — i.e. that 

 a, /5 and y may change comparatively abruptly during the 

 life-cycle, and also it is quite possible that other inherent 

 alterations, such as the gradual increase of viscosity of proto- 

 plasm with age (Ruzicka, 1921), will cause gradual and pro- 

 gressive diminution of the specific constants which would 

 account for the various distortions of the S-shaped curve of 

 growth from the form expected on the simplest assumptions. 

 But I am convinced that some such general method of en- 

 visaging growth is sound ; and it is interesting to find our 

 empirical formula for constant differential growth-ratios 

 deducible from it. (See also Schmalhausen, 1927B, 1930.) 



Exactly the same formula would apply to two sums of 

 money put out at different rates of compound interest, pro- 

 vided that they were not accumulating discontinuously by 

 quarterly or annual interest payments, as in financial fact, 

 but continuously, as in the Compound Interest Law, and as 

 in biological growth, k would here denote the ratio of the 



1 One might expect, from certain experimental data, that the factor 

 G would be a simple function of the defect of the size of the organism 

 at any given time from its final size ; but this would not interfere 

 with the validity of our more general formula. 



2 It may well be that the ' general factor ' is not capable of such 

 a simple formulation. But provided that such a general factor does 

 exist — i.e. that the growth both of organ and of rest-of-body is related 

 to some general law of growth affecting the organism as a whole, the 

 deduction of constant differential growth-ratios remains valid. And 

 that such a relation does exist is shown by the work of Przibram, 

 Harrison and others discussed in Chapter VI. 



