22 

 2 a 2 



.'. i = y 2 — 2 (e — c) y + 2a 2 (since p = o when y — o) 



(l + p 2 ) 2 



also p = o when y = h 



h 

 .\ h 2 -2(e-c)h = o .*. e-c = — 



2 

 2 a 2 



.". 1 = y 2 -hy + 2a 2 



(l + p 2 ) a 

 This is a differential equation of the same form as in 

 Case 2. Dealing with it in precisely the same manner, we 

 obtain the result 



d<f> 



1 = 2 



V ' A 2 



1 sin 2 6 



J n 16a 2 



4 a 





J 



V , h 2 



1 - sin 2 6 



16a 2 



the limits of the integral in this case taking a more simple 

 form. 



I 

 Thus — = K , where iT is the complete elliptic integral 

 4a h 



of the first kind with k = — . 



4a 



Z * 



If h = o f or there is no deflection, — = — 



4a 2 

 AEIir 2 



i.e., P = is the least value of P which can produce 



I 2 



any deflection, and we see that the least possible value of 



— is — . 

 4a 2 



The bending moment at the top and bottom of the 



h 

 co\umn = P f-M = P (e-h)-P c = P — . 



2 



The bending moment at the centre of the column 

 h 

 = P(e-h) = P— also. 

 2 



