VARIABILITY OF SUCCESSIVELY FORMED WHORLS. 



95 



a positive correlation between the first few successive whorls, diminish- 

 ing the farther out we go. The material for investigating the matter 

 in these cases is, however, so meager that it is not worth while to discuss 

 it in detail. 



Table 48.— Coefficients of correlation from table ^7. 



THE VARIABILITY OF SUCCESSIVELY FORMED WHORLS.— THE SECOND 

 LAW OF GROWTH IN CERATOPHYLLUM. 



We come now to the consideration of a matter of very considerable 

 interest and importance in connection with the general laws of morpho- 

 genesis in Ceratophyllum. It is as to whether whorls in different 

 positions on the plant show equal degrees of variability. We have seen 

 that the mean leaf-number of successively formed whorls changes in a 

 regular and orderly way. We have now the further problem: Does the 

 variability of whorls successively formed similarly show a tendency to 

 orderly change, and if so, what law does this change follow? 



In discussing the subject we may adopt the notation and methods 

 used by Pearson (:05) in his memoir on "Skew Correlation." If we let 

 o■,^. denote the standard deviation of an a--array of a character B, and o-y 

 the total variability of the same character, then (Pearson, loc. cit., 

 p. 10) : "A curve in which the ratio of o-», to the standard deviation o-^ 

 is plotted to X may be termed a scedastic curve. " 



Further, Pearson says (p. 22) : 



I must remind the reader, however, that the form of the regression line does not 

 in any way limit the nature of the distribution of the array about its mean; the varia- 

 bility of an array, i. e., the standard deviation of an array, having for its mean value 

 ay^i — yj^^ may or may not be the same for all arrays. If it is the same, or all arrays 

 are equally scattered about their means, I shall speak of the system as a homoscedastic 

 system, otherwise it is a heteroscedastic system. 



For every array of the correlation tables for position and leaf -number 

 given above (pp. 59 and 60) I have calculated the ratio -i^, with the re- 



