POSITION REGRESSION— SECONDARY BRANCHES. 



75 



That is to say, we find that a mathematical function deduced solely from 

 data given by Ceratophyllum plants collected at Carp Lake expresses 

 perfectly the method of growth of Ceratophyllum plants collected at Ann 

 Arbor. This seems to me to amount to a demonstration that our growth 

 curve is an expression of a real and fundamental morphogenetic law 

 in Ceratophyllum and not a mere chance result, or due to any skillful 

 figure juggling. 



By changing the position of the curve slightly in the opposite direc- 

 tion we obtain a very fair graduation of the regression in Series V, 

 though on account of the fact that Series V and VI include spring 

 plants in which we have the results of a part of two seasons' growth, as 

 has been pointed out above (p. 15), the fit in these cases is not so good 

 as in the other series. 



The result shown in fig. 12 enables us to reach a further interesting 

 conclusion. We see, namely, that in our fundamental growth equation 



Y= A + Clog {x — a) 



it is the constant A which is affected by environmental differences. 

 That is, the absolute size of the elements of the developing system given by 

 a Ceratophyllum branch is modified by environmental differences, but the 

 law which describes the proportionate differentiation of the elements is 

 independent of the environmental history of the plant. Thus we are able 

 by statistical analysis to separate clearly and definitely the effects of 

 external environmental factors and internal form-determining factors 

 in this case. The constant A takes different values in populations 

 living in different environments, while the portion Clog {x — «) remains 

 unaltered, or is only very slightly altered, and we may therefore look 

 upon A as the "environmental constant." 



Table 38.— Correlation between leaf-number and position of secondary -branch whorls. 



Series. 



XL. 

 III. 

 IV 



I, II, and III combined. 



0.601 ±0.036 

 .688± .023 

 .692± .027 

 .650± .016 

 .671± .016 



0.712±0.028* 

 .801± .016 

 .816± .018 

 .727± .013 

 .784d= .011 



Table. 



37 

 37 

 37 

 37 

 37 



»As before, the probable error of the correlation ratio V is calculated from the short formula. 



So far we have been dealing with primary-branch whorls in the 

 discussion of the regression of leaf-number on position. It remains to 

 determine in how far the same relations which we have found for 

 primary branches hold for other divisions of the plant. We may turn 

 first to the consideration of whorls on secondary branches. 



