1881.] Mr Greenhill, On height consistent with stability. 73 



and the greatest height h to which the tree can grow without 

 flexure is given by 



n-4m+2 



c = ich 2 , 

 or h = (-■] r ^ I " rr2 , 



where c is the least positive root of the equation 



J im-l (x) = 0. 

 n-4»t+2 



By assigning different values to m and n according to the 

 growth of the tree, and to E, /x, and X according to the elasticity 

 and density of the wood, the greatest straight vertical growth of a 

 tree can be inferred. The application of these formulae to the case 

 of the large trees of California would be interesting, but in the 

 absence of the numerical data required, I am unable to carry this 

 out. 



This paper was written for Dr Asa Gray, Professor of Botany 

 in Harvard University, Cambridge, Mass., and was to have been read 

 at the meeting of the American Association last year, but arrived 

 too late. 



A pine tree, as described in Sproat's British Columbia (1875), 

 is said to have grown in one straight tapering stem to a height of 

 221 feet, and to have measured only 20 inches in diameter at the 

 base. 



Considered as an example of Article II., and neglecting the 

 weight of the branches, a diameter of 20 inches at the base would 

 allow of a vertical stable growth of about 300 feet. 



Perhaps the best assumptions to make for our purpose as to 

 the growth of a tree are, (i) to assume a uniformly tapering ti'unk 

 as a central column, and (ii) to adopt Ruskin's assumption (Modern 

 Painters), that the sectional area of the branches of a tree, made 

 by any horizontal plane, is constant. 



This is equivalent to putting m = 1 and n = 1 in equation (2), 

 and then the solution depends on the least root of the associated 

 Bessel's function of the order — 3. 



Generally, for a homogeneous body, n= 2m + 1, and the dia- 

 meter at the base must increase as the § power of the height, 

 which accounts for the slender proportions of young trees, compared 

 with the stunted appearance of very large trees. 



