312 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



in the primary construction, but slightly overlapping owing to the 

 secondary sliding-growth effect. The presence of such contact- 

 relations therefore implies a modification of a (2 + 3) log. spiral 

 system. 



Similarly, for the (3 + 5) system (figs. 28, 109), five members 

 overlap by secondary sliding, although in the original condition 

 two would be half covered and the other three would just touch : 



Fig. 108. — Sempervivum calcaratum, Hort. Apex of one-year-old shoot, (2 + 3). 



eight members again form an investment everywhere two members 

 deep. In the same manner, by using higher ratios, any spiral 

 constructions will point to the generalisation that the number of 

 the shorter curves is given by the number of members sufficing to 

 form a single investment of the axis, in the case of an unmodified 

 system, or overlapping slightly to form a single closed cycle in the 

 more usual case of flattened members. 



It will be at once noticed that such a (3 + 5) system, for example, 



