516 PROFESSOR DICKSON ON SOME ABNORMAL CONES OF PINUS PINASTER. 



The idea which, perhaps, most naturally occurs is that the simple spiral 

 of the ordinary system is the fundamental form ; we may therefore, in the 

 first place, look how the other forms may be derived therefrom. 



(a.) The ordinary bijugate is derivable from the ordinary simple spiral 

 through the intervention of the system 1, 3, 4, 7, 11, &c, thus, — 



1 2 3 5 8 13 



1 3 4 7 11 



2 4 6 10 



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



system, thus, — 



1 2 3 5 8 13 



13 4 7 11* 



(c.) The ordinary trijugate appears to be derivable from the simple spira 

 of the ordinary system only by way of the bijugate, and hence thus, — 



1 2 3 5 8 13 



1 3 4 7 11 



2 4 6 10 



3 6 9 



(d.) The system 1, 4, 5, 9, 14, is similarly derivable from the ordinary 

 system by way of the bijugate, thus, — 



2 3 



5 



8 



13 



21 



1 3 



4 



7 



11 



18 



2 



4 



6 



10 



16 



1 



4 



5 



9 



14 



To the foregoing derivations the most serious objection is, that it seems 

 improbable that the bijugate, which is the commonest of the anomalous forms, 

 should be the result of two convergences ; and still more so, that the not very 

 rare trijugate of the ordinary system, and the also not very rare simple system 

 1, 4, 5, 9, 14, &c, should each be the result of three such transitions. 



In the next place, let us take the ordinary bijugate and see how the other 

 systems may be derived from it. 



(a.) The ordinary simple system is derivable by convergence from the 

 bijugate only, it would seem, by way of the system 1, 4, 5, 9, &c, thus, — 



2 4 6 10 



14 5 9 



12 3 5 8 



* It will be remembered that in Cone III. the system 1, 2, 5, 7, 12, &c, is derived from the 

 ordinary system by coalescence of consecutive scales in one of the secondary spirals by 5. Here, the 

 system 1, 3, 4, 7, lb, &c, -would be derived from the same by coalescence of consecutive scales in one 

 of the secondary spirals by 3. 



