ANOMALOUS SERIES. 207 



shoots of a dichotomy, (4+4) and (4+5) (c/. fig. 79, 2, 3) ; and that 

 this is not a rare or exceptional occurrence is shown by the fact that 

 identical appearances may be found among the shoots of Cactaceae 

 {cf. special section, Echinopsis). The conclusion appears fully war- 

 ranted, that these apices have impressed on them a set of curves, 

 adjusted to the relative size of the lateral member required, which 

 give an approximately symmetrical construction; any accidental 

 variation in the ratio which involves inequality necessarily pro- 

 duces an effect of spirals, while equality in the number of inter- 

 secting curves implies the subsequent appearance of whorls.* 



Thus in dealing with anomalous constructions, the interpretation 

 of the facts observed in terms of a genetic spiral is only possible 

 when the system remains constant, and even in comparatively 

 simple cases the enumeration of the parastichy ratios may prove to 

 sytematists a simpler method of describing the facts observed.f 



In all cases, in fact, except among the very simplest constructions, 

 the " genetic-spiral " hypothesis becomes somewhat of an incubus ; 

 it is quite useless, but still one does not like to throw it over com- 

 pletely. It is true that all complicated constructions are more 

 simply regarded as systems of intersecting curves, and that once such 

 a system is in working order it appears to act along the curved paths 

 of the parastichies, adding or losing these curves as required ; but 

 in the simplest cases on which the spiral construction for asym- 



* Lycopodium Selago presents a point of great interest in that the terminal 

 growth-centre, which clearly is not expressed in terms of an apical cell on the 

 broad flat apex, definitely bifurcates and two independent growth-centres result, 

 each of which initiates its own curve system, with little regard to the other or 

 to the parent centre. These relations have been investigated by Cramer (Pflan- 

 zenpys. Unters., Nageli mid Cramer, iii. p. 10), and not only may the shoots 

 of the dichotomy give dissimilar systems, either symmetrical or asymmetrical, 

 but in cases of both being asymmetrical the genetic spiral may work out either 

 homodromous or antidromous, and thus in one case antidromous to the parent 

 axis. The suggestion is obvious, therefore, that all such new growth-centres 

 produce their systems quite independently and adjust their own bulk-ratio and 

 symmetrical relations. The new systems may with difficulty be expressed as 

 bifurcations of older paths, so far as these reach round each half ; but in terms 

 of genetic spirals they become still more involved, in that, as already seen, true 

 symmetry is readily attaiued. {Gf. Cramer, Plate XXIX. figs. 9-13.) 



+ Cf. Schumann, Monographia Gactacearwm. 







