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JOURNAL OF THE ROYAL HORTICULTURAL SOCIETY. 



This secondary series will be seen, on comparing it with the primary, 

 to differ in commencing with the fractions ^, ^, &c. in place of ^, ^, &c. ; 

 but afterwards each successive fraction may be written down as in the 

 primary series by simply adding the two successive numerators and 

 denominators respectively. 



If, now, we project on a plane a cycle of any one of the spiral 

 arrangements represented by a fraction of this secondary series, as in 

 the case of |, we shall find that a complete circumference will invariably 

 contain four leaves instead of three. And, moreover, the angular diver- 

 gence of any leaf from the next in succession will be found in a similar 

 manner to be that fractional part of 360°. Similarly, just as all angular 

 divergences of the leaves of the primary series lie between 120° and 180° 

 inclusively, all those of the leaves of the secondary series lie between 90° 

 and 120°, the limiting point being at an angular distance from the first 

 leaf of 99° 30' 6" + . Lastly, it must be observed that the fractions 

 of the secondary series are the successive convergents of the continued 

 fraction : 



i 



3 + 1 

 1 + 1 



1 + &c 



In a manner analogous to the above, we might construct a tertiary 

 series, commencing with the fractions \, i, and which would then appear 

 as follows : J, -\, f, / T , &c. Such a series, moreover, does exist 



in nature, as well as others, in Lycopodium\ e.g. the fractions f, J, T - r 

 corresponding to the series ^, J, &c. ; \, &c. ; }, J, &c. respectively. 

 Though these series are rarely to be met with now, it is interesting to 

 find that of the trees of the Coal period several of the family allied to our 

 existing Lycopodium, or Club-moss, illustrated them. Indeed, I have 

 found f on Araucaria imbricata on one branch, but the usual arrange- 

 ment belongs to the usual series, -J, J, &c. This conifer is a living 

 representative of a very ancient type. Having, then, before us three 

 analogous series, it is obvious that we might construct any number of 

 such series, and finally all would be represented by the algebraical forms, 

 where a is any number : — 



11 2 3 5 



&c. 



a a + 1 2a + 1 3a + 2 6a + 3 



These fractions being the successive convergents of the continued 

 fraction 



i 



a + 1 

 1 +1 



1 + See. 



In all the preceding investigations I have supposed the space between 

 any two successive leaves on the stem to have been sufficiently developed 

 to enable me to trace an imaginary spiral line through the leaves. But 

 it sometimes happens that such spaces, called internodes, are so short, or 

 are practically wanting, that the leaves become crowded together, so that 



