72 Mi- Greenhill, On height consistent with stability. [Feb. 7, 



from which the greatest height a jelly in the form of a paraboloid 

 may be made can be inferred. 



IV. Generally, if we have a solid of revolution like a tree, 

 of which the radius of the section of the trunk at the depth x 

 below the top is r, and if W be the weight in pounds of the part of 

 the tree above this section, then if the tree be of such a height as 

 to be slightly bent under its own weight, the differential equation 

 of the central line of the tree is, as before, 



and differentiating with respect to x, 



*-*B("S- i *i;s r -'-"» » 



This differential equation is equivalent to Bessel's differential 

 equation if r and W are proportional to powers of x. 



For suppose r = \x m , W= fix 11 ; then 

 which is reduced to Bessel's differential equation by putting 



l-4??» 



p = x—2~z, and x n ~ im+2 = r 2 . 

 For instance, in I. in = 0, n— 1; in II. m — 1, n = 3; 

 in III. m = i, n = 2. 



The solution of (2),« subject to the condition that p is finite 



or -y- = when x = 0, is as before 

 ax 



l-im n-4»i+2 



p = Ax 2 J 4m- 1 (/C.C 2 ), 



where 



M-4OT+2 



16/i 



ttAV(?i-4»i + 2) 



9\2 ' 



