PROFESSOR DICKSON ON SOME ABNORMAL CONES OF PINUS PINASTER. 513 



that level, according to the spiral of the upper system, as far down as 

 one can go below it. The method now to be described, and which I have 

 employed in numbering the outline figures of the cones in Plate XX. figs. 1-7, 

 and in the construction of the plans or diagrams of the same in Plates XXI. 

 and XXII., although to a certain extent artificial and arbitrary, yet has 

 the advantage of reducing to a minimum the number of doubtful or ambiguous 

 scales. My procedure is as follows : — The arrangements above and below a 

 given convergence are noted ; then, in order to ascertain how the secondary 

 spirals of two such arrangements would most naturally fit or run into one 

 another, constructions of the two arrangements are made in such a way that 

 the last term of the lower system coincides with the first (*. e., No. 0) of the 

 upper ; or if either or both of the systems happen to be conjugate, one of the 

 last terms, or of the first, or of each, as the case may be, is to be placed at the 

 common point of the two systems. Further, the constructions are made so 

 that those secondary spirals which correspond in number and direction 

 in the two systems shall be continuous ; these spirals being, as above indi- 

 cated, those among which the fusion of consecutive scales has, or is 



presumed to have, occurred. To take an example. Supposing we have 



2 7 



to join or fit together a -=■ spiral below, and a — spiral above, as in Plate 



XXII. fig. 2 ; a construction is made of the lower spiral, and its last term is 

 taken as the starting-point of the one above, which is thence constructed in 

 such a way that the lines of its secondary spirals by 9 are continuous with the 

 9 in the arrangement below. The lines of the secondary spirals by 8 in the 

 lower, and of those by 7 in the upper system, are now drawn (these being the 

 lines among which the convergence is most apparent, and which, of course, 

 have approximately the same direction), when the next proceeding is to join the 

 8 below to the 7 above, which is effected as follows : — the lower extremities of 

 the seven upper lines are joined to the upper extremities of those seven of the 

 eight below, which lie nearest them in the same direction. The upper extremity 

 of the eighth or remaining lower line is then joined to the nearest (in the same 

 direction) of the lower extremities of the seven above, when it becomes 

 apparent that, in the construction, it is this last point which is the point of 

 convergence, and, moreover, that this point corresponds to No. 6 of the upper 

 system. The cone under examination is now taken, and the "scale of 

 convergence" marked as No. 6 of the upper spiral. This upper spiral is now 

 reckoned back to No. 0, up to which point, on the other hand, the scales of the 

 lower spiral are continuously reckoned. When this has been done, it will be 

 found, in the case before us, that one scale has been excluded from the 

 enumeration ; and, on turning to the construction, it will be seen that this 

 scale occupies a position corresponding to a point where one of the junction 



VOL. XXVI. PART II. 6 S 



