306 Mr. A'. G. M. Michell on the Elastic Stability of 

 therefore 



d*~ [r ' } fr&V 7& 



and 



dar \ Pi/3 2 7 p 2 fax 



Of which the solution is 



ClU 11 • £ . Tl «K 



-r- = D sin - + -ej cos - , 

 ax c c 



where D and E are arbitrary constants, or 



= A cos- + Bsin- +0. . . . . (8) 

 c c 



If the beam is to be deformed symmetrically on the two 

 sides of the origin, B must be zero. If also, as previously 

 assumed, 6 is zero when x is equal to + Z, 



C= —A cos-, 

 c 



so that 



0=All— cos- jcos -, 



72/3 



Also since y and 6 are both zero when x is equal to /, --— 



is also zero, at that point, and the condition of critical sta- 

 bility is 



1 n 



cos - =0, or 



c 

 - = —t- , where n is any integer and 



c 2 ~ r A/% A- 



In this case the number of loops in the beam between the 

 points +1 and — lis odd. 



If 6 is zero at the origin, in equation (8) 



C=— A, and 0=Bsin^ 



c 



so that 6 is also zero at +1 and — I. 



— =mr is the condition of critical stability. 



