SUMMARY OF SECTION. 89 



tertiary branches) . That is to say, a given mean number of leaves per 

 whorl is produced with the formation of the least number of consecu- 

 tive whorls in the case of tertiary branches; more whorls are required 

 to attain the same result on secondaries, more still on primaries, and 

 most of all on the main stem. 



Before going on to other subjects I wish to discuss certain points 

 brought out by the results of this section which could not well be taken 

 up till all the results for positional leaf differentiation were in hand. 

 In the first place, we note that from the correlation tables for the 

 characters, position, and leaf-number given in this section (tables 30-32, 

 37, 42, and 45) it is a simple matter to build up piece by piece the gross 

 frequency distributions for all whorls on the plant, such as are given 

 in an earlier section of the paper (e. g., table 1). In other words, the 

 arrays of the correlation tables form, so to speak, the dissected elements 

 of our earlier gross frequency distributions. It has been possible in the 

 present case to carry out this process of dissection of variation curves 

 with considerably greater completeness than has hitherto usually been 

 the case. The results of this procedure will, I believe, repay careful 

 study from several different points of view. 



In order to bring out more clearly the general features of the analysis 

 of the frequency distributions, I have resorted to the'graphical method, 

 with the result shown in plate ii. 



In plate li the whorls belonging to different divisions of the plant 

 are represented in different colors, main-stem whorls being given by 

 blue, primary-branch whorls by red, secondary-branch whorls by green, 

 and tertiary-branch whorls by yellow. The data are for Series I, II, 

 and III combined. The abscissas give number of leaves per whorl and 

 the ordinates, frequencies. Instead, however, of representing merely 

 the total height of the ordinates, the frequency of each size of leaf is 

 proportionately divided to show where whorls are located on the plant 

 as a whole and also on the individual axes. These proportional divisions 

 are indicated by the horizontal lines, and the areas they include 

 represent the frequency of whorls in the positions indicated by the 

 Arabic numerals. In order to avoid too great complexity in the diagram 

 the whorls have been to some extent grouped. In the case of the main 

 stem each 10 succeeding whorls were grouped together; in the primary 

 branches the 1st to the 10th whorls, inclusive, are given by single whorls, 

 and from that point to the end of the branch in groups of 5; in the 

 secondary branches the first 6 whorls are given singly, while the 

 remainder of the whorls (7th to 13th inclusive) are in a single group; 

 the first two tertiary-branch whorls are given singly, and the remainder 

 (3d to 6th, inclusive) together. The smooth curves are the graphs of 

 the equations given on pages 23 and 24. 



