44 BULLETIN OF THE 



^ . (1. 1-8 IV. (an outer individ.) gives rise to 8 branches and 34 Individ 

 1 2. 9-12 IV. (an inner " ) " "4 " " 27 " 



Inner. 

 Inner. 

 Outer. 



j3. 

 14. 

 j5. 

 IQ. 



r- 



18. 



According to the rule that inner branches ai'e slightly prolific, we should 

 expect cases numbered 4 and 5 in the above table to contain the fewest 

 branches and individuals; in accordance with the rule that marginal 

 branches even of subfans nre more prolific, we should expect tliem, on 

 the contrary, to contain more branches and individuals than cases num- 

 bered 3, 6, etc. The result is usually a condition intermediate between 

 that of the middle and outer branches, such as is partially realized in 

 case number 5. Case number 4 seems to present an unusual condition, 

 which may be correlated with the fact of its close approximation to 

 number .5. (See Fig. 64, 17-20.) From the consideration of this and 

 other cases, I think this conclusion may fairly be drawn : (6) Of the 

 four proximal individuals from which a fan arises, the outer two will bear 

 the greater number of individuals, the inner tivo the lesser. 



Since from rule 2 median individuals (ancestral branches) occupy the 

 margins of fans (or subfans of any degree) and the lateral branches are 

 intermediate, it follows, as a corollary to rule 5, that, in general, the an- 

 cestral branches are the shorter, the lateral branches the longer ; and, as 

 a corollary to rule 6, that from any axil the ancestral branch will of the 

 two give rise to the greater number of individuals ; the lateral branch, 

 conversely, to the less, other conditions being equal. 



We have deduced the laws of lateral budding on different parts of the 

 circumference. We find also that there is a regular variation in the 

 frequency of lateral budding, dependent upon the distance of the region 

 from the primary individual of the fan. This rule, like any other, is not 

 invariable, whatever the other conditions may be ; but it is more or less 

 dependent upon them. A small and regular fan having seven genera- 

 tions ffives this result. 



