No. 422.] QUANTITATIVE STUDY OF VARIATION. 147 
the species studied, the coefficient of correlation between them 
ranging from .549 + to .802 +. 
The fact that the correlation between bracts and rays was 
found to be greater in every case than that between rays and 
disk florets gave rise to the question, What is the relation 
existing between bracts and rays? After a careful study of 
the position of the rays with reference to the bracts, it seems 
that the rays are axillary to the bracts, and that typically a 
nearly constant proportion of the bracts in the capitula of 
. a species produce rays in their axils, the rest remaining empty. 
The material studied showed the mean number of empty 
bracts in each head to be as follows: A. sortii Hook. 22.8, 
A. prenanthoides Muhl. 16, A. puniceus L. 7.9, and A. nove- 
angle L. 1.2. 
The number of species here-studied is too small to permit 
the derivation of laws covering so large a genus as Aster, but 
in these four species the degree of imbrication of the involucral 
bracts was apparently in direct proportion to the number of 
empty bracts. In A. shortii Hook., which has so many empty 
bracts, the scars left by their removal occupies the convex 
surface of an inverted cone, the base of which served as the 
receptacle for the comparatively small number of florets; while 
in A. nove-anglig l., where almost every bract has its ray, 
the scars formed a narrow ring about the broad receptacle. 
In the former there was considerable difference in form and 
size between the minute outer bracts and the inner ray-bearing 
bracts, while in the latter all the bracts were very much alike in 
In a number of capitula of A. nove-anglie L. 
size and form. 
Some of the more 
there were found more rays than bracts. 
marked cases of this kind were examined and the rays distinctly 
seen to form a scattering second row within the full outer row 
which is typical of Aster; in other words, some of the disk 
florets developed ligulate corollas. 
The cue several writers that statistical s 
will prove valuable in taxonomic work is not sustained by the 
results set forth in this paper. It is obviously impossible to 
describe a species by means of the variability constante = y 
such manner as to allow the classification of individuals man P 
