GROWTH 283 



(4) Again, we can take it that — decreases simply in inverse propor- 



w at 



tion to the lapse of time 



1 dw k 

 w dt t' 

 whence w = bt^. 



This is the 'parabolic' or 'double-log' curve; it should give a straight line 

 when log weight is plotted against log time, whereas the 'exponential' or 



'single log' formula lu = be^\ which holds when is a constant, gives 



w dt 



a straight line when log iv is plotted against time. The double-log formula 

 fits better with the weights of such growing systems as embryos, at least 

 in the early stages, but like the single-log expression, it has no upper limit 

 as time increases and can therefore obviously only hold for part of the 

 life-history of most animals, which reach a final adult size. (It is possible 

 that certain animals, including fish, continue growing indefinitely.) 



All the formulae have been applied by various authors to the actual 

 data derived by weigliing embryos during their growth. Such observed 

 growth curves are generally roughly sigmoid in shape, but they may give 

 evidence of a number of cycles of growth, so that applying any one type 

 of formula one may have to invoke a set of different constants for each 

 cycle. Even so, it has never been possible to show that any one of the above 

 formulae fits the facts so exactly that it must represent the actual situation 

 and the others can be excluded. There are many snags in fitting theoretical 

 curves to the actual observations. Li the first place, many growth curves 

 have been derived by weighing a sample population at each of a series of 

 ages, taking the average of each age group, and joining the points together 

 to give the overall curve. It is difficult to do anything else if we are inter- 

 ested, for instance, in the foetal growth of a mammal. But there will, of 

 course, be a certain variation in the stage of development reached by 

 individuals of the same temporal age, and if there were any sudden spurts 

 or slowings-down of growth these might be obscured by taking averages. 

 For instance, there is usually a sudden spurt in the growth of a boy at the 

 time of puberty. In some boys this occurs rather earlier, in some rather 

 later. If one derives a growth curve by weighing groups of boys at various 

 ages, the spurt becomes distributed among a number of different groups 

 and its existence concealed. 



Even when a growth curve can be obtained by weighing a single indi- 

 vidual at various times during its life, the curve is bound to suffer from a 



