MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 335 



are increased by the gradual transition to apparent spirals of 



Archimedes as the members attain a uniform bulk ; nor could any 



13 

 approximation to the eye of a — orthostichy line be ever taken 



as an indication of a (3 + 5) apical construction. 

 Similarly, other orthostichies may be tabulated : — 



(m^ + vP') f or ( 1 + 2) system = 5 

 ( 2+ 3) „ = 13 

 ( 3+ 5) „ = 34 

 ( 5+ 8) „ = 89 

 ( 8 + 13) „ = 233 

 (34 + 55) „ =4181 etc. 



The first case (1 + 2) is of interest, in that it should be the 

 phyllotaxis of Pandanus and Gyperus, which it obviously is not 

 {ef. figs. 51, 59&). This has been suggested as due, as in the case of 

 Sempervivum, to the change in the spirals consequent on the rapid 

 attainment of approximately equal radial depth (fig. 51). The 

 last case being that of the capitulum of Helianthus taken as a 

 type (fig. 15), in which the system was only carried for between 

 6-700 members before it broke down ; so that even if this type 

 of formula were retained in the descriptive account of phyllotaxis, 

 it becomes quite useless in all high series. 



Note III. — The Form of the " Ovoid" Curve. 



From the equation of Note I., the curve for any given system 

 may be plotted out. Five such curves, those for the 



asymmetrical (3 + 5), 

 asymmetrical (2 + 3), 

 symmetrical (2 + 2), 

 asymmetrical (1 + 2), 

 symmetrical (1 + 1), 



are represented in fig. 111. 



It will be noticed that the form of the (3 + 5) quasi-circle 

 scarcely differs to the eye from a circle, and this approximation 

 is shown by the dotted line. The curve is, however, slightly 



