363 
and also the series of frequencies, now agree admirably 
with the empirical data (reduced numbers) as far as the 
point x — 17. From here the theoretical ©'s approach 
the final value 1.000; whereas the empirical ®'s increase 
more rapidly to the final value 1.070, and diverge more 
and more from the former, since this is the region where 
we meet the non-typical leaflets. Their number in each 
interval is expressed by the difference between the ordi- 
nates of the frequency curves. 
These results justify the following conclusion: 
About 192 — 931/;, of the bracts in question belonged 
to a type governed fully by the Law of the Geometric 
Mean. The remaining 7°, (in this case about 241) were 
non-typical, transitional leaflets, some inserted lower and 
the rest higher, but all of greater length. 
J have also measured the breadth of these same bracts, 
and I should like to have treated this series in an analogous 
manner. But in this respect the lower transitional leaflets 
were extreme variates on the plus side, being shaped like 
normal bracts but more robust; on the contraryÿ the more 
linear-shaped higher leaflets were extreme variates on the 
minus side. Î have ascertained the fact that, in compa- 
rison with a logarithmic curve, the empirical distribution 
shows ,a tail“ on either side. Although [I suppose that 
the middle part would follow a logarithmic distribution, 
J have not tried to find the theoretical curve, as it would 
never afford a sfriking instance. 
And this alone Î intended to show: the existence of 
striking cases where it is almost certain that organs have 
been governed by the Law of the Geometric Mean. 
It is my personal conviction, that Logarithmic distrib- 
ution represents a more frequent case in nature than 
even the Normal curve. Among 145 frequency series 
from my own material no less than 111 showed positive 
