STRENGTH OF MATERIALS 



In the triangle OAC, 



'OC* = AC* + A(? -1AO-AD, or 

 jt^f + V-Zb.AD, 



from which c & 2 r 2 



& '^-2=-2 



Hence w(V-S) 



M=M +- - 



Since r = u + a and a = b U Q * 



M= M Q + ^ ( 2 + 2 aw + a* - a* - 2 aw - w 1 ) 



^ 

 nij 



= M + -(u - u)(u 9 + w + 2 a). 



Since w and ^ are both infinitesimal, i/ -f // (or tlie ditference U-i 

 the absolute values of u and w ) is negligible in comparison with 

 Therefore M= M - wa(u - u ), 



and, consequently, the differential equation of the elastic curve becomes 



JET-gy JQ - w(tt - ?/ ). 



The general integral of this differential equation is found to be 

 (70) u = u + ^+C l B 



l^tt 



in which C f 1 and (7 2 are the undetermined constants of integration.! 



This may be verified by substituting the integral in the above d ; 

 ential equation. 



To determine C l and <7 a it is only necessary to make use of tin* 



terminal conditions at A and 2?. At the point A y I = 0, -7 = 0,and 



(// 



u = U Q . Substituting these values in equation (70) and its first d< 

 ative, it is found that 



^ = and C.=-^2- 



irn 



* Throughout this discussion it should be borne in mind that / is a n :itity. 



t See Johnson, Treatise on Ordinary and Partial Differential K<i 1 ed., 



pp. 85-86; also Calculus, p. 440. 



