10 STRENGTH OF MATERIALS 



or, expanding, 



Now for convenience let the constants in this integral be denoted 

 by A, B, and C respectively ; that is to say, let 



Then the general integral becomes 



y = A sin Cx + B cos Cx, 



At the ends and X, where x = and /, y = 0. Substituting these 

 values in the above integral, 



B = 0, and A sin Cl = 0. 



Since A and B cannot both be zero, sin Cl = ; whence 



C7 = sin- I =XTT, 



where X is an arbitrary integer. Now let X take the smallest value 

 possible, namely 1, and substitute for C its value. Tli.-n 



whence 



(48) f 



which is Euler's formula for long columns. 



Under the load P given by this formula the column is in neul 

 equilibrium ; that is to say, the load P is just sufficient to cause it 

 to retain any lateral deflection which may be given to it. Fr this 

 reason P is called the critical load. If the. load is less tlun this 

 critical value, the column is in stable equilibrium, and any lateral 

 deflection will disappear when its cause is removed. If the 

 exceeds this critical value, the column is in unstable equilibrium, and 

 the slightest lateral deflection will rapidly increase until rupture 

 occurs. 



83. Columns with one or both ends fixed. The above deduction 

 of Euler's formula is based on the assumption that the ends of the 





