l82 GROWTH PRINCIPLES AND THEORY 2 



in general in different developmental stages. The present considerations are 

 mainly concerned with intraspecific comparison. 



Even in more recent surveys (Brody, 1945; Kleiber, 1947; Krebs, 1950; 

 Lehmann, 1956) homeothermic animals are solely taken into consideration. It is, 

 however, necessary to consider the problem on the broader basis of comparative 

 physiology. Furthermore, the case of mammals is by no means simple, but rather 

 is intricate (p. 218), and many familiar conclusions and explanatory hypotheses 

 fall flat if not only mammals but also poikilothermic vertebrates and invertebrates 

 are taken into consideration. 



Such investigation as carried through in the author's and other laboratories led 

 to the following results : 



1. The surface rule also holds for poikilothermic vertebrates and certain 

 invertebrates. The rule is, therefore, of a wide application; but the explanation 

 given by Rubner is too restricted, for in poikilothermic animals there is no 

 thermoregulation, and thus the latter cannot be the basic factor in the relation 

 between body size and metabolic rate. 



2. On the other hand, there are many classes of animals in which the surface 

 rule does not hold. 



J. Thus there are different metabolic types with respect to the relation between 

 metabolic rate and body size. Three such types can be distinguished, this classi- 

 fication applying, as was emphasized, to intraspecific allometry, that is, to 

 individuals of different sizes or to growing animals within one species. 



The dependence of metabolic rate on body size is a special case of the allometric 

 equation which has a wide range of application in morphological, biochemical, 

 physiological, and evolutionary phenomena (p. 224ff.). The dependence of meta- 

 bolic rate on body size can be expressed by the equation: 



M = hvd^, or ) , ^ 



(5.22) 

 log M = log ^ + oc log w ) 



where M is metabolic rate per unit time, w the body weight, and a and h 

 constants. 



Therefore, if metabolic rate is plotted against body weight logarithmically, a 

 straight line is obtained, the slope of which indicates the constant a (tan a = a.). 

 If a = 2/3 or a = 33°4i', metabolic rate is proportional to surface. If a = i or 

 the slope is 45', metabolic rate is proportional to weight. With i > a > 2/3, an 

 intermediary case obtains. 



If weight-specific values are taken, the equation becomes : 



M , , , , 



— = bw''-^ (5.23) 



w 



Hence weight-specific metabolic rates decrease with increasing weight, and the 

 slope of the logarithmic plot is negative. 



