RISING PHYLLOTAXLS. 133 



keep up with the expansion of the axis. The preceding examples 

 of Gyperihs and Saxifraga seem to show that as in the general case 

 of normal expanding phyllotaxis, the stations of stable equilibrium, 

 which are constituted by ratios in the Fibonacci series, are at- 

 tained as quickly as possible at any given transition. 



Falling Phyllotaxis. 



It now remains to consider to what extent the converse of these 

 generalisations will hold for reduction of the phyllotaxis systems. 

 That the same general principles obtain is shown by the inflor- 

 escence of Bipsacus, in which curves are dropped out with facility 

 equal to their interpolation, and here as in the stems of Cacti no 

 definite law holds, but one is put in or taken out as required to 

 adjust the bulk-ratio. 



Comparison of the capitula ol Compositae which exhibit the 

 perfection of expanding construction, shows that equal accuracy 

 does not obtain in the reducing stages, and that these are rather 

 of the type already included under the term " discontijiuous." So 

 far as can be determined, the reduction takes place more or less 

 irregularly, but on the whole the ratios of the Fibonacci series 

 continue to mark stations of equilibrium, and these when reached 

 may remain constant for a considerable period, though the arrange- 

 ment is not marked equally well in all cases. The alteration in 

 the bulk-ratio is not sufficient to present the optimum transition 

 which would maintain the set of the contact-parastichies unchanged ; 

 but as this rises, reversal of the parastichies takes place, and a 

 consequent distortion of the system is therefore noticed at a greater 

 or less distance from the periphery of the system. 



The reason for this is obvious : the log. spiral construction 

 which goes on for an infinite extent, however suitable for the 

 production of vegetative leaves, will not answer the purpose of the 

 capitulum, all the florets of which are required to produce seed 

 within a relatively short period. 



In the Helianthus capitukim it is true that a uniform succession 

 of flowers in successively younger and smaller stages is maintained 



