Dynamical Principle of Least Action. 303 



work done in the structure is that done in the girder, and we 

 must make it a minimum, subject to the statical condition con- 

 necting M 2 w and the central load W, which, taking moments 

 about one end, is found to be 



W 



4p.2c-2w;c 2 -M 2 =0, 



or 



M 2 + 2wc 2 =Wc. 



Differentiating both equations, we have 



(2M 2 -w c 2 ) dM 2 + (f w c 4 -M 2 c 2 ) dw=0, 



dM 2 + 2c*dw=0. 



Therefore, for a minimum, 



2M 2 -wc 2 =§ wc 2 -iM^ 



gM 2 =J^ 2 ;M 2 =l| W ;c 2 ; 

 but 



M 2 + 2wc 2 =Wc, 

 so that 



or, if / be the span (/=4c), 



M 2 =ih^^=ftW. 



These results agree exactly with those obtained by a writer in 

 the ' Civil Engineer and Architects Journal ' for 1860, and 

 differ slightly from those given by Prof. Rankine in his work 

 on ' Applied Mechanics/ To find v } the central depression of 

 the girder, we have, by the law of conservation, 



2V = Wv; 



but if we eliminate M 2 from the expression for U by aid of the 

 equation connecting M.%, W, w, U will then be a homogeneous 

 function of Ww, so that 



OTT dV dV 



2V =dW W+ d^ W '> 

 comparing which expressions for U, we have 

 _dU__^U « 2 _ dV_ 

 V ~ dW ~ dM 2 ' dW ~ c dM 2 '' 



r.v=^{2M 2 -wc*} 



3EI 



c 2 {28 

 3EI I 25'^ 



4096EI' 



— wc* 2 r = 



