NORMAL RATE OF GROWTH 365 



these organisms ceased to grow. The water-holding power of 

 the salt solution, i.e. the exosmotic property of the artificial sea- 

 water, balanced the inwards pull of the protoplasm. 



II. Normal Rate of Growth. 



(a) Weight. Brailsford Robertson has shown that the rate of 

 increase in weight follows a curve characteristic of autocatalytic 

 reactions, i.e. of reactions in which one of the resultant products 

 acts as a catalyst for the whole reaction. A simple example of a 

 reaction of this type may be found in the inversion of an aqueous 

 solution of cane sugar at 100 C. Part of the product of the 

 reaction (glucose and fructose) appears to undergo further decom- 

 position, giving rise to an unknown acid which accelerates the rate 

 of inversion. 



If x denote the amount of invert sugar formed during hydrolysis, 

 x will also be proportional to the amount of acid produced. Now, 

 by the ordinary compound interest law in which a function varies at 

 a rate proportional to itself an exponential function we have : 



Sx 



-jj. = kx(a-x), 



9 



1 ax 



or, on integrating, k = log 



at a-x 

 where k is a constant. 



As the result of a series of experiments on auto-inversion at 

 100 C., the value of k for this reaction has been put =122 x 10~ 6 . 

 With this value we can tell at any time after the inception of the 

 reaction just how much sugar has been inverted. Further, it 

 is obvious, that as the action proceeds, the velocity due to the 

 concentration of the original substance gradually decreases 

 (i.e. the ordinary mass action without the catalyst), while that due 

 to the concentration of the newly formed substance keeps steadily 

 increasing. Hence, there must, at a certain time, be a maximum 

 velocity due to definite concentrations of a and x. In an auto- 

 catalytic reaction this maximum velocity is reached when the 

 concentration of the newly formed substance is half the concen- 

 tration of the original substance, i.e. when x = a/ 2 . 



Do statistical results bear out the statement that growth is an 

 autocatalytic reaction ? In the following table (LXXI.) is given 

 for comparison, the weight of the human body at various ages, 

 as found and as calculated from the assumption that the rah 

 of growth is autocatalytic. 



