PROFESSOR DICKSON ON SOME ABNORMAL CONES OF PINUS PINASTER. 517 



This derivation is highly improbable, and, indeed, quite inadmissible « 

 although it is interesting to note that in Cone III. (see diagram in Plate XXII- 



— of the ordinary system is actually derived from the system 



1, 4, 5, 9, &c. Failing this derivation, we may ask if it is possible for the 

 ordinary simple system to be derived from the bijugate by abortion of one-half 

 of the scales % At first sight this derivation seems, and perhaps is, much more 

 improbable than the first one. In the very remarkable cone, however, described 

 above, and figured in Plate XIX. fig. 3, a process of abortion of this kind, arrested 

 half way, seems to have occurred. In looking at the lower part of this cone ? 

 it is quite apparent that had the large and small scales been developed equally 

 we should have had a bijugate arrangement. It is just conceivable, however 

 that these small scales may be the result of a " superfcetation " (to borrow MM. 

 Bravais' very obstetrical term) ; but this does not seem at all probable. The 

 derivation by abortion would, of course, be represented thus, — 



2 4 6 10 16 



12 3 5 8 



(b.) The system 1, 3, 4, 7, 11, &c, is derivable from the ordinary bijugate 

 only by way of the simple spiral of the ordinary system, and this may be done 

 by either of the methods indicated under the last head. 



(c.) The ordinary trijugate, as already indicated, is directly derivable from 



the bijugate, thus, — 



2 4 6 10 



3 6 9 



(d.) The system 1, 4, 5, 9, &c, is also directly derivable from the bijugate 

 of the ordinary system, thus, — 



2 4 6 10 

 14 5 9 



In looking at the two foregoing schemes of derivations, it is evident that the 

 great difficulty lies in the absence of any probable derivation, either of the 

 ordinary bijugate from the ordinary simple spiral, or conversely, of the ordinary 

 simple spiral from the bijugate. As to the other systems, it is evident that 1, 

 3, 4, 7, 11, comes most naturally from the simple ordinary spiral ; while the 

 ordinary trijugate and the system 1, 4, 5, 9, 14, are most readily derived from 

 the ordinary bijugate. At one time I was inclined to accept the bijugate as the 

 fundamental form ;* but the more I reflect on the matter, the more I am con- 



* This view I published (under reserve) in the abstract of this communication in the Society's 

 Proceedings, vol. vii. p. 453. 



VOL. XXVI. PART II. 6 T 



