ASYMMETRICAL LEAST-CONCENTRATED TYPE. 157 



succulent forms, in order to space out these leaves to better 

 advantage with regard to light, is clearly out of the question, when 

 the xerophytic structure indicates that such exposure is not desired 

 and is as purposely avoided by assumption of the symmetry as in 

 the parallel decussate type. 



Since (1 + 1) gives a normal symmetrical construction with, one member 

 only at a node, it is difficult to bring these two-spired types into line 

 with the normal asymmetrical series. The deflection of the members 

 is so slight that it appears possible to regard the case as one in which 

 the (1 + 1) generating curves become slightly unequal, and thus produce 

 asymmetry of the form 1 : (1 + a), where a is very small. From this 

 point of view the two-spired Gasteria becomes of greatest interest, in that 

 it appears to present an example of secondary symmetry which is with 

 difficulty maintained from node to node, i.e., the curve does not keep 

 true. 



The three-spired type, familiar in the leafy shoots of Fandanus 

 and Cyperus, is apparently similarly derived from a (1-1-2) system. 

 A transverse section of the foliage-bud of Cyperus altemifolius 

 shows the three spires very clearly (fig. 51), while the course of 

 the genetic-spiral is as clearly marked as in the case of the seg- 

 ments of the apical cell of the Fern-root (fig. 51, left-hand 

 spiral through 1-9). The spires become again obvious when the 

 axillary reproductive axes are developed in ascending series in 

 November-December (fig. 59&). 



The leaves of Cyperus are highly specialised from a biological standpoint. 

 The first formed members on a shoot are wholly sheathing, so that their 

 phyllotaxis cannot be determined in the full-grown buds ; the foliage 

 leaves elongate tangentially and fold in a peculiar manner without 

 increasing in radial depth to any extent after their first formation. As 

 a consequence the curves soon become approximate Archimedean spirals 

 so far as they can be judged by the eye ; but, as previously pointed out, 

 it does not necessarily follow that such spirals of Archimedes imply 

 torsion. The formation of special folded strap-shaped members is a 

 secondary biological phenomenon which almost effectually masks the 

 orthogonal system so far as it is visible at the apex. 



Thus, it is impossible to say from the direction of the spirals whether the 

 three spirals seen are the complementary "spires" of a (1 + 2) system 

 or the three shorter curves of a (2 + 3), since a left-hand genetic spiral 

 would work out these same curves in either case. The interpretation 

 taken, that the (1 + 2) system is adopted, is based on the fact that two 



