PROFESSOR DICKSON ON SOME ABNORMAL CONES OF PINUS PINASTER. 515 



imagined that resemblance in the divergence of the generating spiral would be 

 the bond of union in cases of transition. It is manifest, however, from the 

 instances I have given above, that although in some of the cases we find a close 



resemblance between the generating divergences of the two spirals between 



13 13 



which a passage occurs, — as, for example, between ^-r and ^r, — yet much more 



frequently do we find the transition occurring between spirals with widely dif- 



2 7 



ferent generating divergences, — for example, between -= and y^ , or between 



8 13 



0= and ^-r . Such facts, coupled with the circumstance that in these transi- 

 tions the direction of the generating spiral is very frequently reversed, and that, 

 not unfrequently, we have a transition from a simple spiral to a conjugate or 

 vice versd, are sufficient to show that the aptitude of spirals of different systems 

 to pass into each other is quite independent of what is ordinarily supposed to 

 be the most essential element of a spiral arrangement, viz., the divergence of 

 the generating spiral. On the other hand, it is to be noted that the corre- 

 spondence in the numbers of the secondary spirals and verticals is always very 

 close between the spirals which pass into one another ; indeed, so much so, 

 that the conclusion seems forced upon us, that here we have the essence of what 

 may be called the genetic affinity in such cases, which may be expressed in 

 the following terms : — that, as regards their production or origination, spirals 

 of different systems are to be considered as allied in proportion to the numerical 

 correspondence of their secondary spirals and verticals. 



I shall now turn to the third and last point to be considered, viz., whether 

 it is possible to conceive of the varying spirals in fir cones, or in other plants, 

 being mediately or immediately derived from some one fundamental arrangement 

 in a given set of cases. This question opens up an interesting, but I am afraid 

 very dangerous, field of speculation. To simplify matters we may confine our 

 attention to a few of the commoner arrangements — arrangements which may be 

 found to prevail over an entire cone. Among such (as I have already indicated 

 at the commencement of this paper) are the simple spirals of the ordinary 



system ; after which the bijugate of the ordinary system and spirals of the 



112 3 5 

 system -, -, -, — , _ , &c, are conspicuous ; and after these the trijugate of the 



112 3 



ordinary system and spirals of the system -, -, -, — , &c, may be mentioned. 



t: O v JLtc 



Have these different forms a common origin ? If they have, it is in the highest 

 degree probable that their descent from that common origin is by way of 

 "convergence." Although admitting the occasional occurrence of divergence of 

 secondary spirals, yet I am ready to agree with MM. Bkavais in considering 

 it as an improbable element in the production or derivation of the commoner 

 forms. 



