188 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



The multijugate systems, so far considered, have been either the 

 (2+4) system regarded as a variant of the (2 + 3), or the cases, 

 6/10, 16/26/42, which represent the normal expansion along the 

 lines already indicated for Helianthvs. 



More elaborate systems, divisible by 3, 4, etc., occur chiefly among 

 the Cactaceae and similar growth-forms, as variations of anomalous 

 systems which become divisible by common factors, and these will 

 be noticed under the special heading. 



Among bijugate types, two cases call for special mention; the 

 (6 + 10) of foliage shoots, and the expansion type (10 + 16) which 

 does not represent the normal sequence, but apparently indicates 

 a stoppage at an intermediate stage in the normal Fibonacci expan- 

 sions. 



The (6 + 10) appears to be initiated directly on vegetative shoots, 

 in which it may be regarded as a variant, possibly often local, of 

 the normal ratio (5 + 8). 



Thus Pinus pumilio cone, normal (5 + 8), varied to (6 + 10) (fig. 

 60a). 

 Sedum elegans shoots, vary (5 + 8) and (6 + 10) (fig. 43). 

 Pinus Pinea seedlings vary (5 + 8) and (6 + 10). 

 Podocarpus japonica leading shoots vary (5 + 8) and 

 (6 + 10) (fig. 42). 

 In dealing with Araucaria, it has already been shown that from 

 the standpoint of bulk-ratio, (6 + 10) represents an intermediate 

 stage approximating 4 : 1 (or 3'8 : 1), and is therefore equally 

 possible as an alternative construction with (7 + 11), which 

 approximates 4:1. The conclusion that (6 + 10) may therefore 

 represent an enlargement of a (5 + 8) system, in which the bulk 

 of the axis is increased without the lateral primordia taking their 

 relative share in the increased nutrition, is unavoidable, and the 

 manner in which (6 + 10) is found associated with (5 + 8) in the 

 examples given strongly supports it ; on the other hand it may be 

 regarded with equal probability as the expression of an inherent 



result depends on the extent to which the members more than fill their full arc or 

 fail to do so. In the latter case four straight rows of narrow leaves are observed 

 {Euphorhia Lathyris, 4-ridged Gacti and succulent Euphorbiae) ; in the former 

 with broad or sheathing leaves the rows may be perfectly irregular (Dicmthus). 



