198 RELATION OP PHYLLOTAXIS TO MECHANICAL LAWS. 



gencies more or less symmetrically placed on a foliar structure 

 which only resembles a shoot in that the aggregated carpels con- 

 stitute a mass exhibiting radial symmetry. 



II. Higher members of the series 3, ^, 7, etc., and ^, 5, 9, etc. 



Examples of such constructions occur in Dijpsacus and Helianthus, 

 side by side with bijugate representatives, and clearly represent in 

 the latter the expansion-series of seedling variations. 



As pointed out by Bravais {loc. cit., p. 100), great care is required 

 in the case of Dipscuius in which single curves are readily dropped 

 or added in the middle of the inflorescence (figs. 38ai, &), and the 

 ratios derived from the number of parastichies will often vary in 

 the different portions of the head. Many examples are given by 

 Bravais; thus a capitulum presenting (23-1-37) would be a member 

 of the 1, 4, 5 ... . series, but the omission of one curve in either 

 direction, by reducing the system to (22-1-36), would cause it to be 

 included under a bijugate construction of the 1, 3, 4 ... . series. 



In the case of Helianthus the curved systems acquire a greater 

 degree of constancy, and the ratios, with rare exceptions, are per- 

 fectly definite. Thus "Weisse obtained one bijugate and six anoma- 

 lous capitula of the types (18-|-29) and (47-1-76) among 140 plants. 

 Although Weisse's pot-plants were obviously very poorly nourished, 

 the percentage of anomalous capitula was no greater than in plants 

 grown in the open, so that it does not appear that such anomalies 

 are directly induced by bad environment. As previously noted, two 

 bijugate capitula were found among a batch of 15 from one garden, 

 while another batch of 15 plants, grown under the most unfavour- 

 able conditions (B. G. 0., 1900), included three anomalous heads (29 

 -|-47) and (47-1-76), as well as one which could not be counted at 

 all.* From the point of view that variations are initiated in the 

 seedling, these results would not be surprising, and they would seem 

 to imply that the expansion series proceeded normally in spite of 

 bad environment. That these constructions are not merely due to 



* Pot plants were placed in an open bed late in June, and remained without 

 water throughout the whole of a dry hot season. They grew about 3 feet high 

 and produced capitula which only began to expand early in November, when 

 they were all cut down by a hard frost. The remaining twelve were half 

 (34-1-55), the others (55-t-89). 



