450 
Proceedings of the Royal Society 
convergence of secondary spirals, i.e ., by abortion of one, or possibly 
coalescence of two, resulting in diminution of number. For 
example, after referring to the possible derivation of an arrangement 
with 5 and 7 secondary spirals (series ^ | , ?I , &c.), from an 
Z O 7 1 Z 
ordinary one with 5 and 8, by abortion of one of the spirals by 8, 
they add that “the series 1, 4, 5, 9 . . . does not admit of 
explanation by the way of abortion, and that one can deduce it 
from the ordinary series only by supposing a superfoetation or 
addition of a new spiral among the secondary spirals by 8.” 
“This hypothesis,” they continue, “appears to us altogether 
improbable, since in the face of an immense number of instances 
where two spirals converge into one, we cannot on the other hand 
cite one (apart from rectiserial stems) where one spiral diverges into 
two similar and parallel ones.”* 
The two cones of Pinus Pinaster which form the immediate 
subject of Dr Dickson’s paper, and for which he is indebted to the 
kindness of R. Smyth, Esq., Emyvale, Co. Monaghan, Ireland, are 
interesting cases of convergence of spirals. These, together with a 
few other cases already noted by Dr Dickson, seem to throw some 
additional light upon this question of the origin of variations in 
the spiral arrangements in a given plant, where not unfrequently 
spirals belonging to several distinct systems occur. 
In the first of the cones received from Mr Smyth, there is at the 
base a right-handed spiral (series | | > &c 0 
with the secondary spirals 9 S, 14 D, 23 S. A little above the base, 
however, two of the 9 spirals to the left run into one, leaving, from 
that point up to about the middle of the cone, an arrangement of 
secondary spirals 8 S, 14 D, 22 S = a left-handed bijugate of the 
1 1 2 3 5 5 
series - , , &c., with divergence — About the 
o 4 7 11 18 lo x 2 
middle of the cone two of the 14 spirals to the right run into 
one, leaving, from thence to the top of the cone, an arrangement 
13 
of secondary spirals 8 S, 13 D, 21 S = a left-handed — spiral of the 
.112 3. 
ordinary series &c. 
* L. c. pp. 104, 105. 
