18 



and ft 



4. a* -(2 a 2 -e 2 sin 2 <£) J 



(2a 2 -e 2 sin 2 c£) a 

 \ integrating round the curve, the length of the column 



/ i\ 



= 1 = 2 



M 1 +~) dy 



r 



= 2 



2a : 



J 



V \ 4a 4 -(2a 2 -- e 2 sin 2 <f>) 2 } 



r 



e sin <f> d <f> 



= 2 

 J 



r 0i 



2 a 2 d<£ 



V (A a 2 - e 2 sin 2 <£) 

 d<j> 



= 2a 



V ( 1 - — sin 2 </> 

 ^° \ 4« 2 / 



The limits of the integral are from y = o to y = h. 

 When y = o, <£ = o. 



e-h f 



When y — h, cos </> — 



.*. cos <^>j 



I 



f 



h + f 



■h + f 



The value of — is thus expressed in terms of an elliptic 

 2a e 



integral having a conveniently small modulus — , but, as the 



2a 



integral is not this time a complete one, the value of h is not 

 so easily determined as in the previous case. 



The integral reduces to the form in case 1 if f = o. Also 



I 

 if e = f, so that there is no bending, — = o .*. P = o, so that 



2a 



bending begins in this case as soon as any load is put on. 



