LAWS OF TALL-TREE GROWTH 549 



And if w is the weight per unit volume, W= ^wthjux^ 



where c = jj^ 



(1) is the differential equation of the neutral axis when the tree is 

 tilted slightly out of the vertical, under the primary supposition that 

 p is at all points small as compared with unity. This equation can be 

 solved only in series form. 



If p=x"' , (1) gives: 



m(m-\-3)x"'-^-\-cx'"=0 (2) 



Thus there are two ascending series which will satisfy (1), one begin- 

 ning with x^ and the other with x'^. The latter cannot be used, since 

 we have agreed that p is to be small as compared with unity for all 

 values of x; p would be infinite in this series for x equal zero. 



If J' = 2 /!„%"'+'' then by (2) 



{m-^n){m+n-\-3)A„+cA„-i = (3) 



Form = 0, A« = — -^A„_i (4) 



w(w-t-3) ^ ^ 



(4) gives as a solution of (1), 



= ^o(l-i£^+ji.if-j^ .^+etc.) (5) 



(5) is evidently convergent. 



Assuming p equal zero when x is H, we have as the equation deter- 

 mining the greatest still air-height of a cone, not bending under its own 

 weight, 



„ . 1 cH \ {cHY 1 {cHy , 



This expression has its smallest root between 10 and 11. A close 

 approximation is 



cH=m.2 

 But c = 4'w^3t^E, where t = r-i-H 



•••^= 4/7.65^,. 



I w 



