512 PROFESSOR DICKSON ON SOME ABNORMAL CONES OF PINITS PINASTER. 



ance itself. To do this we have only to look at the upper portion of Cones II. 

 and III. In each of these a change is ushered in by the partial fusion of two 

 adjacent scales. These scales may be viewed either as consecutive members of 

 one secondary spiral, or as adjacent members of two parallel ones ; as, however 

 their relative position is strictly defined by reference to the spiral in which they 

 are consecutive members, which it cannot be by simple reference to any two 

 parallel spirals in which they occur, the former description is that which must 

 be adopted. In these cases it is impossible to doubt that it is the coalescence or 

 fusion of two consecutive scales in one of the secondary spirals ivhich leads to or 

 causes the general disturbance of the arrangement. When such a disturbance 

 takes place, a convergence of certain secondary spirals becomes prominently 

 visible ; but this convergence is not at all more real, although more apparent, 

 than the convergence of other secondary spirals whose component members do 

 not happen to be in contact. It will be noted, moreover, that this disturbance 

 affects the numbers of all the secondary spirals, excepting only those among 

 which the fusion of consecutive scales occurs ; for example, in Cone II. consecu- 

 tive scales in one of the spirals by 4 have coalesced, and it is these spirals by 4 

 alone which run continuously throughout the two systems without diminution in 

 number. That a similar explanation legitimately applies to all cases of " con- 

 vergence," even where the duplex nature of what I would term the scale of 

 convergence is not demonstrable, I think few will be inclined to doubt. 



At this point, I may now conveniently refer to the method to be adopted in 

 numbering the scales on the cones exhibiting " convergence." From the 

 circumstance of the " scale of convergence" resulting from the fusion of two 

 adjacent scales which are usually at a considerable interval from each other on 

 the generating or fundamental spiral, it is evident that the disturbance conse- 

 quent thereon must, at the very least, extend to all the scales which would 

 have been included between the numbers of the two scales which have 

 coalesced, if, indeed, it does not involve a region of the cone extending both 

 above and below the level of the scale of convergence. At first sight it might 

 appear to be the simplest method to consider the scale of convergence as the 

 point of passage from the one system to that succeeding it — as at once the last 

 term of the lower system, and the first of the upper. Here we might reckon 

 the scales in the lower system either up to the lower, or up to the higher, of 

 the two components of the scale of convergence. In either way, however, we 

 should encounter a difficulty; as in the former case a considerable number of 

 scales would escape the reckoning, while in the latter the double enumeration 

 of a considerable number of scales would be involved. The disadvantage of 

 double enumeration also attaches to a method suggested to me by a friend, 

 viz., to number the scales from below the level of the convergence, according to 

 the spiral of the lower system, as far up as one can go above it, and from above 



