394 Proceedings of the Royal Society 
Added, March 20 th, 1871, in consequence of some remarks made by 
Professor Dickson at the Meeting of that date. 
It is obvious that if the same leaf, 0, be reached from A by m 
steps of a right-handed, and n of a left-handed, spiral (such that n 
of the former and m of the latter contain, severally, all the leaves), 
another common leaf can be reached by m - n steps of the right- 
handed spiral, and n steps of a new left-handed one (these spirals 
possessing the same property of severally containing, in groups of 
n and m — n respectively, all the leaves). This process may be 
carried on, when m and n are prime to one another, until we have 
steps represented by 1 and 1, in which case we obviously arrive at 
the leaf of the fundamental spiral next to A. It is better, how- 
ever, to carry the process only the length of steps 1 and t, where t 
is determined by the condition that 1 and t + 1 would give spirals 
both right-handed or both left-handed. 
Now, in the majority of cases of fir-cones, it seems that we have 
t, found in this way, = 2, i.e., there are less than three leaves in a 
single turn of the fundamental spiral. It is of course obvious that 
there can never be less than two, and the case of exactly two 
corresponds to the simplest of all possible arrangements, that in 
which the leaves are placed alternately on opposite sides of the 
stem. Fir-cones, therefore, give in general the arrangement next 
to this in order of simplicity. Hence, for such cones, and for all 
other leaf arrangements which are based on the same elementary 
condition, the values of m and n for the most conspicuous spirals 
must be of the forms 
2 , 3 , 5 , 8 , Ac., 
1, 2, 3, 5, A-c. 
These simple considerations explain completely the so-called 
mysterious appearance of terms of the recurring series 1, 2, 3, 5, 
8, 13, &c., &c. The other natural series, usually but misleadingly 
represented by convergents to an infinitely extended continued 
fraction, are easily explained as above by taking t = 3, 4, Ac., Ac. 
As a purely mathematical question it is interesting to verify the 
consistency of the statements just made, where the change in t is 
introduced, with those above made as to the effects of torsion in 
altering r and s. But this may easily be supplied by any reader 
who possesses a small knowledge of algebra. 
