VII RELATIVE GROWTH 227 



an insect extremity), an equation of the same form cannot apply to the organ as 

 a whole because a sum 



Vx -^Vi ^Vi + • • • = ^1^"" + ^z^"' + ^3^''' + • • • (74) 



is not of the form bx^. Hence the allometric equation can only be considered a 

 first useful approximation (Reeve and Huxley, 1945). This objection does not apply 

 to the size dependence of physiological processes and similar phenomena. 



Although understood to be in the nature of an approximation, the principle of 

 allometry can be given a rationale and interpretation from different viewpoints. 



1. Physiologically, the mechanism underlying allometric growth becomes clear 

 if the allometric equation (7.3) is written in a slightly different form: 



±=a-^.-^ {7-5) 



d^ d^ X 



That is, the part y receives from the increase of the total system (d^r/di) a share 

 which is proportional to its ratio to the total system [yjx) . The allometry coefficient, 

 a, is a distribution coefficient indicating the capacity of j to appropriate a certain 

 share of the total increase. 



Allometric growth therefore is an expression of the competition of parts of the 

 living organism for available resources. Although particular allometry constants 

 can hardly be explained in this way, physiological competition is well documented 

 by observations and experimental facts (p. 23off'.). 



2. Functionally, allometry can be conceived of as being an expression of the 

 principle that the organism must remain functioning and self-maintaining in 

 spite of variations of absolute size: principle of biological similarity (Glinther and 

 Guerra, 1955; Guerra and Giinther, 1957). 



It is a well-known engineering principle that any machine (in the broadest sense) requires 

 changes in proportion of component parts or in rates of processes in order to remain 

 functional, when enlarged from a small-scale model to actual size. Any magnitude Q_ of 

 the machine can be expressed, in the cm-g-sec system, as 



Q. = UM^T^ (7-6) 



(L = length, M = mass, T = time). If q, I, m, t are the corresponding measures in the 

 model, and Ljl = X, A//m = \x, Tjt = t, the general reduction coefficient is: 



y = X^JJlM (7.7) 



and (l = q-X (7-8) 



Since dimensionally [j. = A^ d {d = density) and x = X *, equation (7.7) can be written 

 (for constant d) as: 



y = X^^3y^<5/2 (7.9) 



This is known as Newton's reduction coefficient for mechanical similarities. For electro- 

 dynamics, another relation between X and t applies, namely t = X, so that the electro- 

 dynamic reduction coefficient is 



y' =-XP-iY-6 (7.10) 



Similarly, any biological function (y) will be related to that of a unit system, e.g. i kg 

 by W = X^ ( 14^ = weight of the actual system) and 



J, = ^I4^i(^*3y + <5') (7.1,) 



Literature p. 253 



