SUMMARY OF SECTION. 91 



the face of facts of this kind it is difficult to understand how anyone 

 can be so firmly convinced of the Allgemeingilltigkeit of the normal or 

 Gaussian law, as some biologists still are [cf. for example, Ranke and 

 Greiner ( :04) ] . Skewness in variation is a very real biological phenom- 

 enon, which may be changed and modified, not only in degree, but in 

 direction, by various biological factors like growth, as, for example, in 

 the present case, or environmental influences,* etc. 



The general features of our first law of growth are very well brought 

 out graphically by the diagram of plate li. Thus we see clearly that 

 for all divisions of the plant the pro'portionate frequency of whorls with 

 high leaf -numbers becomes greater the longer the plant grows, though 

 at a decreasing rate. This of course results in the increase in mean leaf- 

 number which has been discussed in this section of the paper. 



It is also of some interest to see graphically the proportion of whorls 

 which the different axial regions contribute to the plant as a whole. 

 Primary branches manifestly outweigh any other part of the plant. 



With a knowledge of the law of positional differentiation of whorls 

 we are able to interpret many of the facts regarding variation in Cera- 

 tophyllum which were before obscure. These interpretations will be 

 so obvious to anyone who has followed the results so far set forth that 

 it is not worth while to go over all of them in detail. A few cases 

 only need be cited as illustrations. Thus, on page 18 it was seen, when 

 the gross frequency distributions for Series I, II, and III were com- 

 pared, that "Series I has the lowest frequency of whorls with 10 

 leaves, Series II has about 3 per cent more 10-leaved whorls, while 

 Series III has 4 per cent more of such whorls than does Series I." The 

 explanation of this progressive difference is now clear. A reference to 

 page 13 will recall the fact that the dates of collection for Series I, II, 

 and III were respectively July 22, August 18, and August 25. The 

 proportionate frequency of 10-leaved whorls varies directly as the date 

 of collection, which is exactly what would be expected from our law of 

 growth, since as growth goes on there is an ever-increasing tendency 

 towards the addition of whorls with 10 or a higher number of leaves. 



It has appeared at various points throughout the paper that the gross 

 frequency distributions for. Series V and VI were different in their 

 characteristics, both from all the other series considered and from each 

 other. Yet, as has been seen in this section and will be more fully 

 brought out later, these Series V and VI plants follow the same laws 

 of growth differentiation as do the others. The explanation of their 

 apparent abnormality is to be found principally in the fact that in the 



*E. g., see Pearl (:06). 



