BiylSHA MITCHBLI^ SClENTIiriC SOCIETY. 46 



the method of virtual velocities, to which it readily lends 

 itself. Castli^liano in his "Systemes Elastiques," a v^ork 

 that deserves to become classic, had already deduced the 

 principle in an entirely different manner and g-iven nu- 

 merous applications to structures in metal and stone. 



The writer has followed essential!}^ his manner of ap- 

 plying- the principle, to the truss next to be examined, 

 thoug-h by regarding* the chord, when acting as a beam, 

 to be "free" under the action of certain forces, there is a 

 g"ain in simplicity. 



I have computed in full, the truss shown in plate 2, 

 an inverted Queen Truss. As it corresponds exactly in 

 theory to the truss shown in figure 1, if we suppose the 

 suspended beams there to rest on or be fastened to, the 

 lower chord, the solution of the one truss will indicate 

 that of the other. 



Reference will be made to the following* principles in 

 the theory of elastic work : 



In a straight bar of length a and cross section iy, sub- 

 jected to a gradually applied stress which reaches its max- 

 imum s, the elastic work of deformation, c being* the co- 

 efficient of elasticity of the bar is, 



1 a 



s^ (1) 



2 ew 



In a beam subjected to vertical external forces, if we 

 call the length of a segment of the beam 2I and the mo- 

 ments of the external forces, at the right end of the seg*- 

 nient M^, at the middle M^ and at the left end Mg, the 

 elastic work of deformation, of the segment considered is 



1 / 

 (M|x4M,2+M%) (2). 



2e:i 3 



E being the modulus of elasticity of the beam. 



The principle of least work requires that the sum of 

 the work due to all members of the structure shall be a 

 minimum, when one unknown stress is required. 



