58 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



tion of the problem is theoretically clear. What we want to know are 

 the correlations and regressions between any character of the members 

 of such linear series and their position. In this particular case we have 

 to determine the correlation and regression between the number of 

 leaves in the whorl and position. 



It is evident that the different axial divisions of the plant must be 

 treated separately. Logically the main stem should be considered first, 

 but for reasons which will appear as we go on, it is desirable to take up 

 the primary and secondary branches before the main stem. The cor- 

 relation tables showing for each series the relation between number of 

 leaves in the whorl and position on primary branches follow. In the 

 case of Series V and VI the tables have been cut off at the tenth whorl 

 because the very small number of entries beyond that point did not 

 make it worth while to keep them. In the other four series every 

 possible whorl has been included. Furthermore, the tables for Series V 

 and VI differ from the others in that in these two series all branches 

 (primary, secondary, tertiary, etc.) have been clubbed together. 

 Otherwise the numbers would have been too small to get any results at 

 all for these series. No confusion need result from this procedure, as 

 the whole discussion of positional differentiation will be based on the 

 first four series, the last two (V and VI) being used merely to illustrate 

 and confirm the conclusions reached from the others. The method of 

 designating position has been fully explained and illustrated in an earlier 

 section of the paper (p. 12 and fig. 2, supra) and need not be repeated 

 here in detail. It is merely necessary to recall that the whorls are 

 numbered in order, beginning at the proximal end of the branch; the 

 "first whorl" (1) thus being in the case of a primary branch the whorl 

 nearest the main stem, in the case of a secondary branch the whorl 

 nearest the primary, etc. 



Mere inspection of these tables shows us at once that the character 

 of the whorls in respect to leaf-number changes as we go out on the 

 branch. It is perfectly clear in a general way that the farther a pri- 

 mary-branch whorl is from the main stem, the more leaves it is likely 

 to have. Or, in other words, we see that there is a clear positional 

 differentiation within an axial division of the plant. The character of 

 successively formed parts changes with the order of their formation. 

 We may at once proceed, then, to the analysis of the laws of this change. 

 As a first step it is necessary to know the exact degree of the correlation 

 between leaf-number and position, and to test whether the regression 

 is linear or not. The test for linearity of regression has been given by 

 Pearson ( :05) . It consists in determining for any system of correlated 



