78 



VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



this fact will be pointed out later, when certain other results are in 

 hand. 



(6) There is the same uniformity in the correlations for the different 

 series as was observed in the case of the primary branches. 



(c) It is obvious from the values of r and v that the regressions are 

 not linear. 



The precise character of the regression of the leaf-number on posi- 

 tion is given in table 39. In this we have for each of the first four series 

 and the combined Series I, II, and III, the mean number of leaves for 

 each position on the branch. 



It is at once apparent that the secondaries start with a smaller mean 

 number of leaves to the whorl than do primaries, but that the increase in 

 the first few whorls from the origin is more rapid than in primary 

 branches. Unfortunately in the case of secondary branches the number 

 of observations on which the means are based gets small very quickly, 

 so that the results begin to be irregular even before we reach the 10th 

 whorl. The figures in the table show these points, but they are brought 

 out more clearly in the diagrams in fig. 13, which show graphically the 

 regression lines for Series I, II, III, and IV. 



Table 40. — Comparison of observed and calculated mean leaf-num,ber for successive 

 whorls of secondary branches. Series I, II, and III combined. 



From these diagrams we see that the general features of the regres- 

 sion are essentially the same in secondary as in primary branches. 

 There is clearly a functional relation of very much the same sort between 

 the number of leaves to the whorl and the order of formation of the 

 whorl. To determine the nature of this functional relation, we may in 

 the light of our experience with primary branches assume at once that 

 it is of the form 



y = A -\r Clog {x — o.) 



and proceed to fit a curve of this type to the data by the method followed 

 in the former case. This was done, using as material the combined 



