42 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



to suit special requirements; while (4) secondary packing would 

 ensue as a result of unequal growth pressures, and (5) the same 

 tendency towards perfection of symmetrical construction of a radial 

 system would result in the substitution of secondary circular -vortex 

 construction for the primitive spiral system. 



Although much of these deductions may appear at first s^ht 

 fanciful, it is evident that if such generalizations can be successfully 

 applied to the special case of leaf-distribution, the original con- 

 ception becomes much strengthened. Nor are the hypotheses here 

 put forward more imaginative than that of the paraboloid apex of 

 Sachs which remains incapable of proof, or his construction for the 

 apical cell of Pteris which does not satisfy the evidence of his 

 own drawings. 



The proposition is then that the genetic spiral is a logarithmic 

 spiral, homologous with the line of current flow in a spiral-vortex ; 

 and thai in such a system the action of orthogonal forces will he 

 mapped out "by other orthogonally intersecting log. spirals, — the 

 " parastichies." 



In comparing such a leaf -producing spiral-vortex with that of 

 the cell-producing vortex of Pteris, the differences will be due to 

 the absence of the disturbing secondary phenomena. Thus, there 

 is clearly no apical construction to be impressed on the series of 

 members, and the members again are wholly free from one another ; 

 they are so far at liberty to assume as far as possible the form of 

 spheres, but when formed in close contact they will exert lateral 

 pressures on each other which, when they cease to be orthogonally 

 distributed, will induce slipping. The theoretical spiral-vortex of 

 phyllotaxis will thus have one disturbing factor only, namely the 

 pressures due to any unequal individual growth of the component 

 members. 



Gkowth. 



So far, the mass of protoplasm constituting the growing apex has 

 been considered as a constant mass of fluid presenting radiating and 

 circular or spiral- vortex phenomena, as it appears, that is to say, at 

 any given moment of observation. 



