Growth Integration 105 



for cxanipk', wlicre mo circulation exists, sliows. It is al- 

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

 liave no sap system such as is assumed by the ])hvsiolo^ical 

 explanation above indicated. Many of the marine al^ae 

 come under this head, a striking example of which is the kel() 

 M<icrocystis pyrifera of the western coast of hotli Americas. 

 'J1iat the laminje of this plant fall into a beautiful direct 

 gradient series is a fact whicli 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 scries 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 tlie 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 occu})ying 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 ])a})er, ])re- 

 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. ( ^i ) Any pai'tieu- 



X 



lar cycle of growth obeys the fornuila lay -~ K(t-t,), 



A. X 



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



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



growth attained during the cycle. A' is a constant, and ^ is 



