1903.] 



to the Theory of Evolution. 



311 



3,938,354. Further, if o- be the standard deviation of the frequency 

 distribution of branches, as found from the bottom row of Table VIII, 

 we have 



o- 2 = 6-721,083. 

 Hence for use in formula (x) we have, since m = 9, 



9 °" 2 : = 2-415,280, — L - 9 -^H> = 0-176,911 (xft, 



<t 2 - <r M 2 m - 1 cr 2 - cr M 2 



Table IX gives the uncorrected homotyposis for the nine whorls 

 treated as simple homotypes. From this I find, 



p = 0-131,258 . (xii). 



Substituting (xi) and (xii) in (x), we find for the homotyposis of the 

 number of branches in the whorls in Equisetum arvense, when corrected 

 for differentiation due to position, 



R = 0-4939. 



This result it must be admitted is extremely satisfactory, and indi- 

 cates how it is quite possible to correct a result like (xii) by allowing 

 for the differentiation of the homologous parts due to serial position.* 



I hope before long to publish other results dealing with homotyposis 

 in serial parts, where the differentiation has every variety of intensity. 

 I think they will suffice to show that differentiation is not a subtle 

 and evasive quality beyond the appreciation of the naturalist who is 

 provided with the training requisite for modern biometric research. 



(9.) The values of R as given by (ix) and (x) may be illustrated 

 from the actual numbers for Equisetum arvense. We have seen in the 

 footnote, p. 304, that 



V = °V°"- 



This in our case gives 



7] = 0-76549. 



But by direct calculation on Table VIII, using whorls 2 to 10, 

 Dr. Lee finds r pc — -0-64616. Hence with the notation of the 

 footnote referred to 



a = 0-7632 o-, 2 M = 0-4104 o-. 



* The value obtained for the crude homotyposis of the members of the whorls 

 in Asperula odorata in my first memoir was p = 01733 (' Phil. Trans.,' A, vol. 197, 

 p. 326). I have little doubt that when we are able next summer to calculate the 

 correction for differentiation in position of whorl, we shall find E for woodruff in 

 good accordance with other homotypic results. My remarks about it were : " In 

 counting the members on the whorls I soon found evidences of differentiation in 

 position, the whorls towards the top of the spray having, as a rule, fewer members 

 than those lower down " {loc. cit., p. 325). Unfortunately I have not kept my 

 records of position. 



