236 GROWTH PRINCIPLES ANT) THEORY 2 



A theory of growth-in-time of organs must combine the best available expressions for 

 time growth of the entire organism with the allometric equation as the best 

 available expression for the relative growth of organs. This can be done for both 

 cases of (i") unlimited and (2) limited growth. 



1. Unlimited growth (Second Growth Type). If the growth of the organism is 

 exponential : 



X = x^t" (5.38) 



{x ^ l,w), and an organ obeys the allometric equation: 



y = bx"" (7.1) 



(j; = length, weight of the organ), growth-in-time of the organ must follow the 

 equation : 



y = b'd^'' (7-12) 



with b' = bx^"- and c' = ar. That is, if the body as a whole increases exponentially, 

 so also does an allometrically growing organ. This has been found in the hemipter- 

 ous insect, Notonecta (Clark and Hersh, 1939), and in Blatta, (Voy, 1951 ; Figs, 16, 

 17, p. i97f.)- Equation (7.12) is, however, to be considered a first approximation 

 because a sum of exponentials (growth of the individvial parts) is not itself an 

 exponential (growth of the entire body). 



2. Limited growth (First Growth Type). Combining the Bertalanffy equations 

 with the allometric formula, time growth of an organ can be derived as follows: 



Equations (5.29) and (5.28), see p. 191, can be written in normalized form, i.e. equating 

 final size (/*, w*) to unity: 



/ = (i-ce-^O 



w = {i- ce~'^') ^ 



with c = (i-Zq) or (i-"^o'^^)' respectively. Similarly, the allometric equation for an organ 

 y can be written in normalized form: 



J) = x°' 



with b equated to i, i.e. organ size expressed as a multiple of its size when a. = i. Time 

 growth of the organ then follows the equation: 



y = (i-ce-'^')" (7-13) 



with a = a for A- = / and a = 3a for a: = w. This equation describes organ growth without 

 introduction of new constants; that is, the growth of organs is calculable from that of the 

 whole organism by insertion of the allometry constants. The growth curves according to 

 (7.13) have the following properties. Denoting 



- rc-kf] 



the first two derivatives are: 



9' = eke k' > o 

 ■a" = -ck^cr'^' < o 



Hence ior y: 



y = acp" '9 

 y" = a{a-i) 9"-^ 9'- + a9""' 9" = a9"~2 j [a-\) 9'^ + 99"'^ 



with 



(a_i) cp'2 _L ^^" = (a-i)c2/t2e-2A-r _ {i-cfr'<') ck^e-l<' = ck'^e-'<' {ace-'"-i] 



