THE BILATERALITY OF APPENDAGES. 301 



But even these effects are only of secondary importance, since 

 it is clear that any change in the volume of the lateral appendage 

 beyond the area of its insertion can have no reference whatever to 

 the primary construction of the system, although it may tend to 

 produce all the normal appearances of sliding-growth. Much less, 

 therefore, can similar effects, produced by changes in the volume 

 of axillary shoots, etc., as in the case of the Pine-cone, and 

 aggregated inflorescences {Helianthus), have any bearing whatever 

 on the subject : such changes can only be included as tertiary 

 factors ; and one wonders more and more at the curious standpoint 

 which has sought to find in these tertiary effects a basis for 

 the Dachstuhl theory. On the other hand, the possibility stiU 

 remains that in the theoretical and mathematical construction of 

 asymmetrical phyllotaxis systems there may still exist a primary 

 cause for these displacement movements ; and as a matter of fact, 

 the necessity for such readjustment of the free portion of the 

 appendage follows directly from the mathematical consideration of 

 the log. spiral theory.* 



The discussion of this readjustment, which implies a slight 

 rotation of the primordia about their centres of construction, and 

 is also only applicable to cases of asymmetrical construction, may 

 therefore be postponed until the equation to the quasi-circle 

 primordium has been deduced : at present it will be sufficient to 

 consider the geometrical consequences of adding the compensation 

 for such " sliding-growth " to the theoretical diagrams. 



The spiral construction diagrams, as expressed by intersecting 

 curves giving rhomboid meshes, present a good working idea 

 of a typical foliage-bud in which the members have, in consequence 

 of mutual growth-pressures, filled all the room available to each 

 primordium, and thus assume an obliquely rhomboidal section 

 {cf. Pinus, fig. 88 ; Araucaria, fig. 41). A section of such a bud 

 at the level of the growing-point includes the insertion-areas of 

 the members close to the apex ; but away from this the members 

 will be cut at some distance above their bases, owing to the 

 character of the dome-shaped apex. The plane diagram, therefore, 

 represents insertion-areas only, and these alone are now under 

 * Gf. Mathematical Notes. 



