THE PHYLLOTAXIS OF HELIANTHUS TUBEROSUS. 663 



ii. A transition from opposite leaves into the f occurred in about 16 per cent of those 

 stems commencing at the base with decussate leaves. It can apparently only be effected 

 some previous approximation to the " tricussate " grouping of leaves. The process 

 will be best understood by referring to the diagram (fig. 5), which illustrates the actual 

 position as observed in nature ; in which it will be noticed that, from the 1st to the 9th 

 leaf, the spiral is to the left, and that, in consequence of the 8th and 9th leaves becoming 

 confluent, the latter is ranged over the first, while the 10th becomes nearly opposit* 

 and is over the second. From this point (the 10th leaf) the spiral turns to the right, 

 and, by the 14th and 15th, the 17th and 18th, becoming confluent, the 17th leaf is ovei 

 the 10th, and thus the f- arrangement is commenced, and henceforth continued uninter- 

 ruptedly into the undeveloped terminal bud. 



A more usual way, however, of passing from opposite leaves into the '} arrangement 

 is by throwing out an extra leaf at right angles to the diameter passing through them 

 (fi"\ 6), and by converging the first pair towards the opposite side, and in this maimer 

 exhibiting a tendency to produce a whorl of three leaves, and consequently four in any 

 single coil commencing with the position of some leaf. A point to be observed in this 

 method is, that if the confluent pair be exactly on the same level, either one may be 

 taken as the second leaf of a cycle commencing with the first isolated leaf. Thus the 

 8th or 9th leaf will be immediately over the 1st, according as the spiral is chosen to one 

 side or the other, corresponding to the divergences f or f respectively. The leaves of 

 each confluent pair generally become isolated at last, one being elevated above the other, 



so that a true spiral is ultimately secured. 



It must be observed that the same angular distance between the two confluent leaves 

 answers for two kinds of spiral, viz. f and f . This is of course, strictly speaking, hnp<^- 

 sible, as the angular divergence for * is 102° 51' 77"+, and for f, 135°; but this condi- 



&""**■ *"»^*b 



nly transitional. When the leaves become isolated the proper angular div 



2 

 7 



is subsequently secured, and is generally , 



iii. The next fraction of the secondary series is ft. This was only occasionally 

 represented, with a commencement of decussate leaves at the base of the stem, by a 

 previous transition into the f spiral-and by the "vertical" line passing through the 

 1st, 8th, and 15th leaves being itself really inclined, and so forming a secondary spiral, 

 in consequence of which the 12th leaf falls approximately over the 1st. ^ ^ 



iv. In like manner A was occasionally reached by a passage through ft and ?. 



t 



HI. Transitions from the Tricussate arrangement into Spirals of the Primary series. 



i. Three verticillate leaves were never resolved directly into the I arrangement ; y. 



in certain stages of change it occasionally happened that a 4th leaf would tall nearly over 



another selected as the 1st. Similarly no case appeared of a direct transi ion mo the 

 ! arrangement. Indeed it appears that tricussate verticils are more readily resolvable 

 into spirals of the secondary series, just as decussate leaves are most readily convert.hi, 



into those of the primarv. . . + i„„ 



ii- The diagram (fig. 7) will illustrate the manner of transition from tricussate leaves 

 into the | divergence, a change which occurred in about 30 per cent. In this case the 



