Growth Integration 105 



for, example, where no circulation exists, shows. It is al- 

 most certain, too, that the series occur in many plants that 

 have no sap system such as is assumed by the physiological 

 explanation above indicated. Many of the marine algae 

 come under this head, a striking example of which is the kelp 

 Macrocystis pyrifera of the western coast of both Americas. 

 That the laminae of this plant fall into a beautiful direct 

 gradient series is a fact which can not escape the notice 

 of any one who sees them. The question of whether each 

 streamer of laminae reaches finally and necessarily a limit of 

 growth in which the size series is present is not so certain, 

 but from considerable attention to the question I am almost 

 sure this is the case, although the point needs more study. 

 Another interesting and probably useful course of reason- 

 ing about organic growth attempts to connect the results of 

 growth with autocatalytic chemical action. Although these 

 attempts have not, so far as I am aware, taken special cog- 

 nizance of the natural size series which are occupying us, 

 but have been concerned with the weights or volumes of or- 

 ganisms at various stages of growth, there is little doubt 

 that the phenomena we have been considering are closely 

 connected with those dealt with in these attempts. T. Brails- 

 ford Robertson seems to have devoted more thought to this 

 matter than any one else. The following, taken from the 

 summary of conclusions found in his original paper, pre- 

 sents the most essential parts of his theory: "(1) In any 

 particular cycle of growth of an organism or of a particular 

 tissue or organ of an organism the maximum increase in vol- 

 ume or in weight in a unit of time occurs when the total 

 growth due to the cycle is half completed. (2) Any particu- 



x 



lar cycle of growth obeys the formula log K(t-ti), 



A x 



where x is the amount (in weight or volume) of growth 

 which has been attained in time t, A is the total amount of 

 growth attained during the cycle, K is a constant, and t^ is 



