68 Mr Greenhill, On height consistent with stability. [Feb. 7, 



the greatest height to which the pole can reach for the vertical 

 position to be stable; if carried up to a greater height, the pole 

 will curve under its own weight if slightly displaced. 



From the expansion otJ n (x) in a series of ascending powers of x, 



T! , flT* fr _ ^ , 3V 3V , ^ 



-IW" '2-*r(§)\ 2. 4 "*" 2.4.4.10 2.4.6.4.10.16 7' 



and we find by trial that c = l'8S, c 3 = 152, and 



«***-.iW=?r\ 





A = T52 ^ =1-26. 



For instance, for a solid cylinder of pine, 

 #=1500000, about; 

 •6 x 625 37-5 



w = 



12 3 ~ 12 3 



and if the diameter of the pole be six inches, a = 3, h* = J a 2 , and 

 therefore h = 89"45 x 12, and the height in feet is 89*45. 



For a steel wire 



#=31000000, 



_ 7-8 x 62-5 _ 487-5 . 

 W ~ 12 3 ~ 12 s ' 



and if the diameter of the wire be one-tenth of an inch, a = ^, 



, 7 , 9 _ .„/ 31000000 \$ 



and A = 1-26 x 12 -7^=-= r^ 



V487 # o x 400/ 



= 6-81 x 12, 



and the height in feet is 6"81. 



If a load be concentrated at the top B, of weight equal to that 

 of a length I of the shaft, then the differential equation of the 

 central line becomes 



EAF^-=wA P (y -y)dx + wAl(OB-y), 

 clx J o 



and differentiating 



M 2 f P 2 -=-w(x + l)p, 

 ax 



the same differential equation as before, with x + I for x. 



