4IO 



ON INCREASE IN SIZE 



[PT, III 



proceeded so far that the depressant effect of the products is measur- 

 able, then the previous equation becomes 



— = k^x [a — x) — k^x^. 



Now when the reaction has proceeded half-way to equilibrium, i.e. 

 when X = la, the equation becomes 



X 



or 



log 

 log 



A - X 



A — X 



= Ak{t- tj), 



Amount transformed 



and this is the well-known equation for the S-shaped curve, where 

 X represents the body-weight (not the increment) at time t, A repre- 

 sents the maximum of final weight which the organism is to reach, 

 ^1 the time at which half this 

 maximum body-weight has 

 been attained, and K a con- 

 stant which has to be deter- 

 mined from a known value 

 of a; at a given time t. By 

 differentiating, it may be 

 found that dx/dt is at a maxi- 

 mum when X = \A — in other 

 words, that the rate of increase 

 in weight is greatest when 

 half the autocatalytic curve 

 has been passed through. 

 Robertson proceeded to apply 

 his calculation to the growth 

 of white rats, figures for which 

 had been reported by Donald- 

 son, Dunn, & Watson, with 

 excellent results over part of ^^^' 4^* 



the curve, though not when the age amounted to more than lOO days. 

 At the time it seemed as if this was a most convincing example 

 of the value of the autocatalytic theory, but maturer consideration 

 showed that this lOO days was but a third of the possible life-span, 

 and the comparison of the two curves as made by Lotka, for in- 

 stance, does not look so impressive. 



