114 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



whorls, we have yet to determine whether the whole effect is to be 

 explained in this way. To test this we have to ansv/er the following 

 question: Is there a sensible difference between the means and varia- 

 bilities of (a) whorls at the nodes where branches originate, and (6) an 

 equally large sample of similarly situated whorls taken without refer- 

 ence to the presence of branches? An approximate answer to this 

 question can be obtained by the use of our growth equation (i) and the 

 data provided in table 58. If in the equation 



Y = 7.9520 + 1.3608 log (x — 0.8015) 



we substitute for x the values for the mean position of "branch origin" 

 whorls in the different series given in table 58, and solve for Y, we shall 

 get the probable mean leaf- number in a group of primary-branch whorls 

 situated in the same average position on the axis as are the whorls 

 where secondary branches originate. But in the distributions for 

 "branch-origin" whorls all parts of the plant have been included, and 

 not merely primary branches. Consequently the predicted means from 

 the equation will not be strictly comparable with the observed means in 

 table 55. It is to be expected, however, that if the means for whorls 

 where branches originate were calculated for primary branches alone 

 they would not differ greatly from the values given in table 55. This 

 seems probable from the fact that the two other portions of the plant 

 besides primary branches which contribute most largely to the table 55 

 means are the main stem and secondary branches. But since, as has 

 been shown above (p. 31), main-stem whorls have a relatively high leaf- 

 number, while on the other hand secondary-branch whorls have a low 

 leaf-number (cf. p. 41), owing to the high proportion of proximal whorls, 

 it may fairly be supposed that the effect of these two portions of the 

 plant will about balance each other in the means given in table 55. 

 That this supposition is in fact a reasonable one is shown by the results 

 which follow. 



In table 59 are given in parallel columns the observed mean leaf- 

 number in whorls at nodes where branches originate and the pre- 

 dicted leaf-number from equation I, where x takes successively the 

 mean values given in table 58 for Series I, II, III, and IV. 



The agreement between observation and prediction is closer than 

 probably would have been expected. The table shows that at the outside 

 not more than 0.2 leaf in the excess of the means for "branch-origin" 

 whorls over "whorls in general" can be due to the combined effect of 

 (a) any hypothetical influence of the presence of a branch at the node 

 to which a whorl belongs, and (6) the inclusion in our observed means 

 of main-stem and secondary-branch whorls. In Series I the predicted 



