160 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



of buds and similar arrested systems, of which the Piae-cone affords 

 a good illustration. 



In the first-mentioned case the resultant curves will be spirals 

 of Archimedes or helices; in the latter, modified logarithmic 

 spirals, which may be conveniently described as "retardation 

 spirals." 



In the former case, again, the visible result will be the straighten- 

 ing out of the " orthostichy lines," and the spires of a spired system 

 may thus become ultimately quite straight, such an effect being 

 well marked in typical Cacti whose seedlings have obvious spiral 

 ridges. The consideration of such growth-forms as these also 

 illustrates the fact that the final effect is due not only to the 

 assumption of equal volume in the members themselves, but also 

 to the attainment of equal length in the secondary zones of elonga- 

 tion which constitute the internodes. So long, therefore, as the 

 internodes are growing, the same appearance as that presented by 

 a gradated series of members wiU be maintained, even when these 

 members are practically equal. The spires of Uii^horbia higlandu- 

 losa (fig. 77), and Sedirni reflexum (fig. 76), thus continue to be 

 well marked after the leaves have reached the adult condition 

 owing to continued growth in the main axis. In the limit, the 

 rows become much straighter, but usually only after the fall of 

 the leaves. The "orthostichy" lines thus appear to become 

 straighter and straighter, as growth slows down in successive 

 members and internodes ; but they will always be spiral lines so 

 long as growth continues throughout the whole system. 



Spiral " orthostichy lines " and " spires " are thus usually more 

 obvious in buds and bud-sections than in adult structures, as 

 originally noted by Bonnet; while if the whole system stops 

 growth simultaneously these spiral orthostichy lines or spires 

 become fixed, and the resulting structure has the appearance of a 

 permanent bud. 



In such a construction the secondary phenomenon of dorsiven- 

 trality produces very striking results. Thus the fact that a leaf 

 increases tangentially to a greater degree than in radial extent may 

 be regarded as due to a diminished radial rate of growth. With- 

 out going into further detail at present with regard to such a 



