550 JOURNAL OF FORESTRY 



H . „_ ,[E 



or, 3^ =1.97 ^- 



as in {A). 



For trees growing as cylinders, as do some palms, the differential 

 equation (1) of the neutral axis becomes 



where c^^w-^Er"^, w being the weight per cubic foot and r the radius. 



This equation can be solved only in series form. If we assume p = 

 .v"', then (7) gives 



m(m- l).r'"-'+c.v"'+' = (8) 



(8) shows that (7) has two series solutions, one beginning with x" and 

 the other with x, consecutive powers differing in exponent by 3. It is 



needless to get the second series, for — is to be zero when x is zero. 



If we assume 



x= s"~*rl„;c'"+'" 



n = 



(8) gives 



(w+3w)0n+3«-l).4„.+c.4„_, = 

 .". for m = 0, 



3n(3n— 1) 

 .". the only usable solution of (7) is 



,.^,(l_^^^,3+_^,e___^^,.+etc.) 



-Ao(l 2[3 J'^2.5T43 J 2.5.8 L-^ ■ [ 3 J "^^^""-J 



This series is evidently convergent. If we assume p is zero when x 

 is H, we have for determining the height of the cylindrical column the 



smallest positive root of the following, where Z = — r- 



„ , 1 Z , 1 Z^ 1 z\ ^ 



^=^-2 + 2751^-15-8 ■ir3+^^"- 

 This root lies between 2 and 3. A close approximation is 

 3Z = rH3 = 7.815 

 where c = Aw-:-Er- 



,.v 



E 

 1.95 ^j 



