1881.] Mr Greenhill, On height consistent with stability. 69 



This state of things will be represented in nature by an 

 exogenous tree like the Areca Palm, growing in a cylindrical shaft, 

 and having a cluster of leaves at the top. 



II. Suppose the rod to taper uniformly up to a point, being 

 a right circular cone of semi-vertical angle a; then at the point P 

 the bending moment, supposing r the radius of the cross section 

 of the rod at P, 



I -vEr* ^ = irw J {y - y) r' 2 dx , 

 and differentiating with respect to x, 



ivE s (** I) — "* j/ w - 



Now r = #tan a, and therefore 



£ ttE tan 4 a -rr ( a 4 -4- ] = — irivp tan 2 a I x' 2 dx 



= — ^ttw ta,n*ax 3 2), 

 2 d 2 p . dp 4uu _ 



* s5 +4iC i + 3¥wi^ =0 «■ 



the differential equation of the central line of the conical rod, the 

 flexure under its own weight being small. 



First put p = x -z, then 



n d 2 z dz ( 4w a \ » 



and again, putting x = r 2 , 



, d 2 z dz ( lQw „ „\ „ , a * 



dr ar V3-& tan a / 



Bessel's differential equation, in which 



2_ ^^ w 2 — 



Therefore the solution of (1), subject to the condition that 

 p is finite when x = 0, is 



/> = Ax~*J 3 (kx-), 



