of Edinburgh, Session 1870-71. 393 
obvious that to get at P we must count y leaves along AQ, and v 
LL W/ 
along QP. If, however, t- ^ -, count v leaves along aq, and y 
v n 
along qp. P, or p, thus found is the next leaf of the fundamental 
spiral to A or a ; the next is derived from it by a second applica¬ 
tion of the same process, and so on. 
There is no necessity for restricting the development, as given 
above, to once round the cone. Suppose we go several times round 
and that A, a , a, &c., are successive positions of the same leaf. The 
processes given above may be employed, and the results will be of 
the same nature. But this extension enables us to obtain (more 
and more approximately, sometimes accurately) a right angle aAo, 
where o is a leaf reached after several turns of the fundamental 
spiral. This indicates that the leaves may be grouped (approxi¬ 
mately or accurately) in lines parallel to the axis of the stem or 
cone. When this can be done accurately, it is easy to see that 
(since one of n , m , is greater, and the other less, than the number 
v y 
of leaves in one turn of the fundamental spiral) the difference of 
azimuth of two successive leaves of that spiral must be expressible 
in the form 
0 ry + si/ 
A 7T —-; 
rm + sn 
where s and r are necessarily very small positive integers in all the 
ordinary cases of phyllotaxis, since they are the numbers of leaves 
in AK, Re, respectively, which are portions of the spirals on which 
or parallel to which, m and n were measured. 
The fraction 
ry -f sv 
rm + sn 
has been called the divergence of the fundamental spiral. Of its 
constituents the numbers m, n, r , s are at once given by inspection 
of any cone or stem, and (from m and n) y and v are easily 
calculated. 
To extend this investigation to the cases in which the divergence 
is altered by torsion of the cone, it is merely necessary to notice 
that such a process alters only r and s. It produces, in fact, a 
simple shear in the developed figure. 
