COJS^STANT PHYLLOTAXIS. 103 



Thus a lateral axis which might normally lay down (8 + 13) curves 

 along which the lateral members would be built owing to a bulk- 

 ratio of (5 : 1), would if impoverished reduce the number of curves 

 in correlation with the lowering of the bulk-ratio. 



Thus (8-1-13) would reduce simply by dropping out one short 

 curve to (8 + 12) ; but the difference in the bulk-ratio thus implied 

 would be very small.* On the other hand, by dropping out one 

 of the long curves, i.e., 8 to 7, it would be necessary to take for the 

 short curves the nearest whole number which gives the same ratio 

 and working-angle; this would be 11, since 8 : 13 : : 7 : 11'375. 



The log. spiral construction diagram shows that by so doing the 

 bulk-ratio is lowered to (4'2) : 1. 



In the same way, if the long curves were dropped to 6, the 

 nearest number in the required ratio is 10, since 8 : 13 : : 6 : 9'75 ; 

 and such a change would be correlated with a fall of the bulk-ratio 

 to (3'8) : 1. The next stage of reduction would give as in the 

 normal case (5 + 8) with the bulk ratio 3 : 1. 



The systems- (7 -F 11) and (6-1-10) therefore represent approxi- 

 mations to a bulk-ratio (4 : 1) intermediate between (5-|-8) and 

 (8-1-13), and maybe expected to occur together with these con- 

 structions : thus (7-f-ll) was noted among plants of Monanthes 

 polyphylla as equally common with (5 -)- 8). In the same way (6 -f 

 10) occurs commonly in Pinus {Pinea, pumUio), Podocarpusjaponica, 

 etc., also as an enlargement of a (5 -f- 8) system rather than as an im- 



* The case of such a construction as (8-H2) would be of special interest, in 

 that, while it presents no difficulty from the point of view of the loss of one curve 

 from an (8 + 13) system, the practical result is the immediate formation of a 

 tetrajugate system, since (8 -I- 12) = 4 (2-1-3), and such a system would work out as 

 four concurrent genetic spirals producing four members simultaneously instead of 

 one at a time. The occurrence of such a construction among normal specimens 

 thus either involves a fundamental change in the building-mechanism, or else it 

 implies that the genetic spiral is now comparatively unimportant and only 

 secondary to the curves of the parastichies. 



Among a batch of several hundred cones of Pinus austriaca, one such cone was 

 found in which a short curve was lost below the widest diameter of the closed 

 cone ; farther up, as in the general case, other curves including long ones were 

 also lost, but the effect of the first loss was, in the case of the dry expanded cone, 

 to render the parastichy system (8-1-12) as seen on the base, over a considerable 

 area. 



