GENERAL SUMMARY. 133 



GENERAL SUMMARY. 



This paper deals with a biometrical analysis of intra-individual 

 variability and differentiation in Ceratophyllum. The characters prin- 

 cipally dealt with are (a) the number of leaves in the whorl; (6) the 

 position of the whorl on the plant; (c) the size of the various divisions 

 of the plant; and (d) the position of branches. Some of the chief results 

 as to fact may be summarized as follows: 



(1) Dealing with the intra-individual variation in leaf-number per 

 whorl it is found that the whorls borne on the different axial divisions 

 of the plant (main stem., primary, secondary, etc., branches) are dis- 

 tinctly differentiated in respect to both type and variability. 



(2) The mean number of leaves per whorl is highest in the whorls 

 on the most central division of the plant (the main stem) and decreases 

 regularly as we pass to more peripheral divisions. 



(3) The whorls on the main stem are the least variable in leaf-number, 

 and the variation increases regularly in the more peripheral divisions, 

 till a maximum is reached in secondary-branch whorls. The variation 

 then tends to diminish in the whorls on higher-order branches. 



(4) More than half of the total number of whorls are borne on pri- 

 mary branches. Of the remaining whorls somewhat more are borne on 

 secondary branches than on the main stem. Tertiary and quaternary 

 branches bear relatively few whorls. 



(5) Primary branches are absolutely and relatively more variable in 

 size than are secondaries. 



(6) There is a relatively high degree of correlation between the 

 number of leaves in the whorl and its position on any axis of the plant. 



(7) The degree of this correlation is lowest in the most central 

 division of the plant (main stem) and increases as we pass to the more 

 peripheral divisions. 



(8) The regression of leaf-number on position is not linear, but log- 

 arithmic. 



(9) This leads to what we have called the "first law of growth" in 

 Ceratophyllum, which may be stated as follows: On any axial division 

 of the plant the mean number of leaves per whorl increases with each 

 successive whorl in such a way that both the absolute increment and 

 the rate of increase diminish as the distance (in units of nodes) of the 

 whorl from a fixed point increases. 



(10) Branches which show an excess (or defect) from the average 

 in the number of leaves in the first whorl tend to have the succeeding 

 whorls greater (or less) than their respective averages. This tendency 

 diminishes as we go distad on the plant. 



