450 PHYLLOTAXIS OF CONES. 
whole circumference which intervenes between any two consecu- 
tive leaves of the same spiral whorl.’ Stretch a wire or band with 
marks or appendages so as to be alternate, two-ranked as are the 
leaves in the elm; then by giving the band a twist, it brings the 
marks three-ranked, like the sedges; still farther torsion brings 
them five-ranked, like the leaves of a cherry tree; still more twist 
and they stand as the scales of the American larch, which is ex- 
pressed by the fraction three-eighths. 
e -= common series of fractions found in alternate leaves 
is 3, 4, 2, 8, #5. o%, 42, 24, 34, etc. The relations of these sev- 
eral numerators and denominators have been repeatedly shown by 
various authors. 
After the first two fractions, each succeeding one may be made 
by adding both of the previous numerators for its numerator and 
both of the previous denominators for its denominator. Each 
denominator is the same as the second succeeding numerator. 
Also, taking the orders of secondary spirals nearest the vertical 
line, on each side, right and left, the number of parallel spirals of 
the lower order of these two will give the numerator; and this 
number, added to the number of parallel spirals of the higher 
order will give the denominator.”—Henfrey. Also “the number of 
the parallel secondary spirals is the same as the common difference 
of the numbers on the leaves that compose them.”—Gray. These 
relations enable us to number easily each scale of any cone, oF 
count the spirals each way, and then determine with accuracy the 
fraction expressing its Phyllotaxis. Balfour and Gray in their 
text books say the Phyllotaxis is uniform in the same species, and 
that one direction or the other generally prevails in each species, 
and that both directions are sometimes met with in different cones 
of the same tree. Several other text books make the same asser- 
tions. Most authors on this subject with which I am familiar say 
there are only rare cases of other series of a P. Duchartre 
mentions two other series, viz : 4, 3, 2, 3,, v5) sf) ete.) p b $ 14? 
zz, etc., and observes that-the same relation exists in different 
fractions of each series as exist in the fractions of the more COM- 
‘mon series. 
Mr. Hubert Airy recently read a paper before the Royal So- 
England, an abstract of which is given in “ Nature” for 
Rini 6, 1873. After mentioning some experiments which show 
the intimate relations of different fractions of the common series, 
