310 Prof. K. Pearson. Mathematical Contributions [Jan. 20, 



will suffice to demonstrate this heterogeneity. Certain individuals have 

 the normal number of about 8*5 branches to this whorl, but about a fifth 

 of the total number of individuals on\j develop about half this normal 

 number of branches. To illustrate this I have in Diagram 2 plotted 

 the mean number of branches to the whorl, and fitted these means 

 with a parabola of the third order,* using only whorls 2 to 10. The 

 equation to this parabola is 



y = 9-451,443 -0-549,4302;?-0-179,9881x 2 -0-008,4206a- 3 , 



the origin being at the 6th whorl, and y giving the mean number of 

 branches for x whorls from the 6th. We have the following results : — 



Observed number of 



Position of whorl. branches. Calculated. 



1 7-718 8-752 



2~~ 9-327 9-308 



3 9-718 9-707 



4 9-827 9-898 



5 9-746 9-829 



6 9-564 9-451 



7 8-891 8-714 



8 7-491 7-565 



9 5-736 5-956 

 10 3-964 3-835 



A much worse fit was obtained by striking a cubical parabola through 

 all ten points. 



It will be seen that the excellency of fit fully justifies the use of 

 this curve. But that there is a large deviation from the observed 

 mean of the 1st whorl, when we calculate its value from the curve 

 thus obtained. Somewhat reluctantly, therefore, I felt compelled to 

 omit the consideration of the 1st whorl from my investigations. Had 

 I possessed a sufficient number of specimens I should have separated 

 my material into two classes, those plants with normal 1st whorl and 

 those with abnormal 1st whorl. But with my available material I 

 should have had considerably less than 100 individuals to deal with, and 

 accordingly I settled to take nine homologous parts only, namely, the 

 2nd to the 10th whorls, in which the differentiation appears to be solely 

 due to position on the plant. Above the 10th whorl, the phenomenon 

 of forking obscures the determination of branches to the whorl, while 

 below the 2nd whorl the full or partial development of branches to 

 the whorl seems to be determined by the local lower vegetation 

 round the stem. 



Taking Table VIII, I found for the mean of the means 8*2515 

 branches, and for the standard deviation of the means o- M , o- M 2 = 

 * By the method indicated in ' Bionietrika,' vol. 2, p. 11. 



