( 167 ) 
and one species even occurs in the Bahamas. The two extremes of 
the different routes have therefore at last met. 
The species of the genus usually show definite characters, allow- 
ing their exact separation. Intermediate forms are nevertheless 
abundant, and have frequently been referred to hybridization. There 
is, however, no evidence whether they are hybrids or merely chance 
variations. In general the change from the normal is in but one or 
two characters, such as the shape, pubescence or serration of the 
leaves or the shape of the involucral scales, and it is seldom diffi- 
cult to determine the specific identity of the variants. Within the 
species there is often great variability, and extreme forms have fre- 
apie been regarded as distinct. Thus there is an unbroken series 
of forms between typical V. etgantea and V. oligantha, or be- 
tween V. ¢xtertor and V. Baldwini, and in each case the two must 
be reduced to one single species. 
The number of flowers in each head has frequently been used as 
a basis for separating species and has been found quite constant 
within certain limits. The number follows closely the phyllotactic 
series 8, 13, 21, 34 and 55. The series may be constructed by 
adding each two numbers to form the next higher. An intermedi- 
ate series may be constructed by omitting one number and adding 
each alternate two. This series includes the numbers 11, 18, 29 and 
47. In practice the ripe achenes are counted, pulling them from 
the involucre by forceps, and the number, if none is abortive, is 
invariably found to correspond closely with some of the series. The 
lowest numbers, 8, 11, and 13, are usually followed exactly; the 
upper ones may vary one or two in either direction. In the de- 
scriptions the variations have seldom been taken into account, and 
only the typical number is indicated, thus Sin V. Lettermannt, 13 
in V. comcinna and 21 in V. fasciculata. 
Ky To THE NORTH AMERICAN SPECIES 
Some or all of the ial scales constricted below a membranous green 
leaflike append Section BACCHAROIDES. 
Involucral scales cere or scarious,*but without a conspicuous green 
terminal appendage. 
chenes terete, without ribs, or faintly ribbed. Section TEPHRODES. 
Achenes ribbed or furrowed. Section LEPIDAPLOA. 
Section BACCHAROIDES 
One species in North America. 1. V. anthelmintica, 
