ELISHA MITCHELL SCIENTIFIC SOCIETY. Ii 



plus, the displacement is in the direction of the force; when 

 minus, in a contrary direction. When two equal forces, 

 directed both toward or both from each other, along the 

 same line (as in the case of the horizontal thrusts of an 

 arch hinged at the abutments), are designated by the same 

 letter P, if we call P and P^ the two forces and F the work 



d F d F 

 of deformation of the truss, then and give the ac- 



d P dP^ 



tual displacements of P and P\ along the directions of the 

 forces; both minus or both plus, according as the motion 

 is opposed to the direction of the force or with it; so that 

 d F d F 



r gives the total relative displacements of P and 



d P d Pi 



F\ In case we can regard the apex, at which either force 



d F 



as P^ acts, as fixed, then represents, as usual, the dis- 



d P 

 placement of one apex with respect to the other. 



If a truss has siiperjiiious members, we can suppose them 

 removed and that two opposed forces act at either end of 

 each bar, each equal to the final stress in the member and 

 acting in the same direction. Then if we designate bv P 

 and P\ the forces replacing the action of any one bar, at 

 either end, upon the apices, then if F represents the work 

 d F d F 



of the necessarv bars, \ gives the total relative 



d P d P^ 

 displacement of the apices. Xow^ as w^e can regard P* = P 

 as a function of P, the total derivative of F with respect to 

 d F d F d P^ d P^ 



P is \ ; but since P = P\ = i, there- 



d P d P^ d P d P 



fore the total derivative of F ivith respect to Pis equal to 

 d F d F 



+ 



d P d P^ 



