THE PHYLLOTAXTS OF HELIANTHUS TUBEROSUS. (J59 



only; but the 14th, after two coils, appearing in the same vertical line, there is a return to the 

 divergence f. The 11th and 12th, as also the 14th and 15th, leaves eoli.re respectively. At 

 the 18th leaf the spiral ceases. Then followed two whorls of three leaves each, as shown b\ 

 the dots. A new spiral to the left next commences [from 1 (bis)] ; and as the 24th is the first 

 leaf which is found to fall as nearly vertical as possible over any one preceding (viz. 13th), and 

 as there are three coils from 13th to the 24th, the divergence ^ is established, and then* < 

 forward continued upwards without further interruption. 

 Fig. 9. This diagram illustrates the limiting positions of leaves of spirals corresponding to divergenoea 



of the different series . 



If 0° be the position of any leaf taken as the commencement of any spiral arrangement <>t tin 

 primary series, then the second leaf of that spiral will fall between 120° and 180° inclusively. 

 Similarly any second leaf of a spiral of the secondary series falls between 90° and 120 



inclusively. 



And of the tertiary series any second leaf will fall between 72° and 90°. In a like mainu i 

 " limiting arcs/' as they may be called, can be found for all higher series, which, it will b< 

 observed, decrease in size as the No. of the series increases. 



The "limiting angle" to which all divergences of the primary series continuously approx- 

 imate as they ascend that series is 137° 30' 28". The same for the secondary series i 

 99° 30' 6". The same for the tertiarv series is 77° 57' 19". 



