514 PROFESSOR DICKSON ON SOME ABNORMAL CONES OF PINUS PINASTER. 



lines between the upper seven and lower eight spirals happens to intersect one 

 of the lines by 9 running continuously through the two systems in the other 

 direction. In the case before us, there is only one such intersection ; but, as 

 we shall presently see, there may be two or more such points in similar 

 associations of other spiral systems, which, in the same way, will be found to 

 correspond in position to scales excluded from the enumeration ; or, again, 

 there may be no such intersections, in which case every scale in the cone can 

 be included in the continuous enumerations. Again, it will be found that the 

 different transitions differ in the value in the upper system to be attached to 

 the scale of convergence. To illustrate the above, I may refer to the diagrams 

 in Plates XXI. and XXII., with which the outline figures of the corresponding 

 cones on Plate XX. may be compared. In doing so, it will only be necessary 

 for me to indicate the spirals by their systems, — thus, 1, 2, 3, 5, 8, &c, or 1, 3, 4, 

 7, 11, &c, and so on, — the particular term of its series to which a given spiral 

 belongs being quite immaterial as regards the present question. The unnumbered 

 scales on the outline figures of the cones, and the intersections corresponding 

 thereto in the diagrams, I have marked with asterisks. In Plate XXI. fig. 1, 

 we have the system 1, 4, 5, 9, 14, &c, passing into the bijugate 2, 6, 8, 14, &c, 

 with three intersections corresponding to three unnumbered scales on the cone, 

 and No. 2' of the upper system as the point or scale of convergence ; also, this 

 same bijugate 2, 4, 6, 8, 14, &c, passing into 1, 2, 3, 5, 8, 13, &c, with one 

 unnumbered scale, and No. 6 of the new system as the scale of convergence. In 

 Plate XXI. fig. 2, the system 1, 3, 4, 7, 11, &c, passes into the bijugate 

 2, 4, 6, 10, &c, with no unnumbered scales, and No. 1' of the new system as 

 the scale of convergence. In Plate XXII. fig. 1, we have the trijugate 3, 6, 9, 

 &c, passing into 1, 4, 5, 9, &c, with two unnumbered scales, and No. 4 of the 

 new system as the scale of convergence ; then 1, 4, 5, 9, &c, passing into 

 1, 2, 3, 5, 8, &c, with no unnumbered scales, and No. 4 as the scale of 

 convergence ; and, lastly, 1, 2, 3, 5, 8, &c, passing into 1, 2, 5, 7, &c, with one 

 unnumbered scale, and No. 3 as the scale of convergence. In Plate XXII. 

 fig. 2, we have 1, 8, 9, 17, &c, passing, as above mentioned, into 1, 2, 7, 9, 

 16, &c, with one unnumbered scale, and No. 6 as the scale of convergence ; 

 and 1, 2, 7, 9, 16, &c, passing into the trijugate 3, 6, 9, 15, &c, with two 

 unnumbered scales, and No. 1 as the scale of convergence. * 



With regard to the second point that I proposed to consider, — viz., as to what 

 constitutes affinity of different spiral systems as regards their possible or actual 

 derivation one from another ; in other words, upon what the aptitude of 

 different spirals to pass into one another depends, — it might, a priori, have been 



* It might, perhaps, he possible for a mathematician to furnish a formula, whereb}', from two 

 spiral systems given, to deduce the number of ambiguous scales, and the value in the upper system of 

 the scale of convergence, thus saving the trouble of a preliminary geometrical construction. 



