MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 345 



Note VIII. — Sliding-Growth. 



From Note I. it is clear that each primordium is primarily 

 bilaterally symmetrical about two axes, represented respectively 

 by a radius of the system and a circle passing through its centre 

 of construction ; and so long as the primordia are free from 

 adjacent members each will retain this form, except in so far as it 

 becomes affected by secondary alterations in the rate of growth in 

 these respective directions ; and this holds for the case of symmetry 

 and also for the more general case of asymmetrical construction ; 

 that is to say, in an asymmetrical construction in which no 

 contacts are made, no sliding-growth effect takes place, and the 

 leaf-members would be horizontally extended and isophyllous 

 on the adult cylindrical shoot. But as soon as the members 

 make lateral contact, the mathematical conditions undergo a 

 change, as previously noticed, the members become represented 

 by oblique rhombs, obliquely placed and anisophyllous ; the 

 packed leaf of an asymmetrical system thus becomes secondarily 

 an asymmetrical structure, while in a symmetrical system, 

 on the other hand, it still retains its original symmetry in 

 relation to the radial and circular paths of the system : in other 

 words, the free portion of a leaf in a spiral system is free to obey 

 its structural properties as a quasi-circle, but so long as it is packed 

 and makes close contact with adjacent members its growth-form 

 becomes distorted into the form of a quasi-square rhomb. In a 

 spirally constructed bud, therefore, with leaves growing more or 

 less in contact below, but free from one another above, as the 

 expression of a conoid growth-form which was initiated from a 

 point and extended until it reached the product of an adjacent 

 centre, the change from the lower distorted region to the upper 

 symmetrical part, when this takes place throughout the whole 

 system, will convey the impression that a slight twist has taken 

 place in the members, as the tangential diagonal of the oblique 

 rhomb changes to a true circular path. It is this appearance of a 

 tendency to a readjustment on the part of the free portion of 

 the appendage which gives the primary tendency to slip in the 

 bud, and the phenomenon provisionally included under the term 



z* 



