260 THE RATE OF GROWTH [ch. 



But if the substance produced exercise a catalytic effect, then the 

 velocity will vary not only as above but will also increase as x 

 increases: the equation becomes 



V = -T- ^k'x(a — x), 



Cut 



which is the elementary equation of autoca.talysis. Integrating, 



at a — X 



In our growth-problem it is sometimes found convenient to choose 

 for our epoch, t', the time when growth is half-completed, as the 

 chemist takes the time at which his reaction is half-way through; 

 and we may then write (with a changed constant) 



This is the physico-chemical formula which Reed and Holland 

 apply to the growing sunflower-stem — a simple case*. For a we 

 take the maximum height attained, viz. 254-5 cm. ; for t\ the epoch 

 when one-half of that height was reached, viz. (by interpolation) 

 about 34-2 days. Taking an observation at random, say that for 

 the 56th day, when the stem was 228-3 cm. high, we have 



K in this case is found to be 0-043, and the mean of all such 

 determinations t is not far difierent. 



Applying this formula to successive epochs, we get a calculated 

 curve in close agreement with the observed one; and by well- 

 known statistical methods we confirm, and measure, its "closeness 

 of fit." But jtist as the chemist must vary and develop his funda- 

 mental formula to suit the course of more and more comphcated 

 reactions, so the biologist finds that only the simplest of his curves 



* H. S. Reed and R. H. Holland, The growth-rate of an annual plant, Helianthus, 

 Proc. Nat. Acad, of Sci. (Washington), v, p. 135, 1919; cf. Lotka, op. cit., p. 74, 

 A sifnilar case is that of a gourd, recorded by A. P. Anderson, Bull. Survey, 

 Minnesota, 1895, and analysed by T. B. Robertson, ibid. pp. 72-75. 



t Better determined, especially in more complex cases, by the method of least 

 squares. 



