POSITION REGRESSION— PRIMARY BRANCHES. 71 



ber of leaves per whorl increases with each successive whorl, and in such 

 a luay that not only does the absolute increment diminish, but also the 

 rate of increase diminishes, as the ordinal number of the whorl meas- 

 ured from a fixed point increases. Or, put more briefly, the rate of 

 increase in leaf-number at any given point as we go out on the branch 

 varies inversely as the number of whorls which separate the given point 

 from a fixed point, the a. of our equation. 



As a consequence of the fact that the rate of increase in leaf-number 

 constantly diminishes as we go out on the branch, it is evident that the 

 actual number of leaves per whorl observed on any branch becomes 

 practically almost constant after the first 10 to 15 whorls. Even though 

 there be a tendency to increase, the rate is so slow that in discrete variates 

 such as we have here it would only be detected w^ith very large numbers, 

 while with ordinary numbers likely to be met in practice it would appear 

 that there was a constant number of leaves. To show how slow the rate 

 of change is we may determine at what position on primary branches 

 the mean number of leaves would be 11, that is, one more than what 

 we find as the highest mean number on the plants here dealt with. 

 Put in another way, we may determine how many whorls would have to 

 be successively formed in accordance with the law of growth which 

 holds up to the 30th whorl before the mean number of leaves per whorl 

 would become 11. To do this we have merely to substitute 11 for Yin our 

 equation and solve for x. Doing this we have 



log {x - 0. 8015) = ^-^5^5 = 2 . 23977 



whence 



a; =177.5 



or in round numbers, the mean number of leaves per whorl will not 

 become 11 until the 177th or 178th whorl is reached! But the facts pre- 

 sented in the last section show that there is comparatively little diver- 

 gence in the number of whorls to the branch (i.e., the length of the 

 branch) in plants collected from different localities, provided they are 

 taken at roughly the same period of the growing season. All show 

 about the same number of whorls to the branch, and this number is very 

 far short of anything like 100, even. There is no evidence whatever 

 from our material that Ceratophyllum in a state of nature ever attains 

 to such size as to have 175 whorls, or anything approaching that number, 

 on primary branches. Our material includes several large plants, so 

 that it can not be maintained that we are dealing with smaller-sized 

 individuals than usually occur. In particular, plant 2 of Series IV 

 was one of the largest Ceratophyllum plants I have ever seen. Yet, as 



