RISING PHYLLOTAXIS. 129 



Cactaceae. As an extreme case a specimen of Cephalocereus senilis 

 may be cited, in which new curves were added singly, and without 

 rule, at intervals of about 700 leaves. 



While, however, each normal expanding system presents the 

 appearance of growing out of its predecessor so that the curve 

 ratio remains and will thus remain practically constant, the point 

 of view of bulk-ratio becomes lost. An expanding type may 

 represent, as in Helianthus, a fairly constant bulk-ratio affected by 

 an expanding axis, and this as a special case may be separated from 

 types in which the bulk-ratio is the only variable quantity. That 

 such may oceiir is shown by the rising phyllotaxis of such inflor- 

 escences as those of Dipsacus, in which very small florets occur 

 almost immediately after large " decussate " foliage leaves on an 

 axis which does not continuously dilate ; and in the same way the 

 members of a flower may be laid down with a varying bulk-ratio 

 on either a constant or a variable axis. In such case, if the change 

 of ratio is sufficiently large, it is evident that given a constant 

 genetic-spiral, the direction of the parastichies may remain un- 

 affected as in the preceding example of normal expansion. 



The possibility is, however, not eliminated that the change in 

 bulk-ratio may not be correlated with the previous system, and 

 that a- definite break in the phyllotaxis will thus be produced. 

 With the same genetic spiral, that is to say, the bulk-ratio may 

 be so independently affected that the parastichies will reverse and 

 the system show more obvious distortion. 



Such systems may be included under the, term Discontinuous 

 Phyllotaxis, and will be characterised by a reversal of the contact- 

 parastichies. 



Two examples of such phenomena may be considered. 



I. Gyperus alternifolius. The strap-shaped foliage leaves are pro- 

 duced in a (1-1-2) system giving three spiral series (spires) 

 (fig. 59&), which owing to the approximate equality in 

 radial depth of the developing members become approxi- 

 mate spirals of Archimedes, without necessarily implying 

 any torsion phenomena; but beyond these biologically 

 specialised members, small unmodified scale leaves subtend 

 the spikelets of the terminal cluster (fig. 51). These 



