lO JOURNAL OF THE 



This is a known formula, by means of which the defiec- 

 tion of any truss containing only " necessary " bars in the 

 direction of any given external force or supposed force, can 

 be computed. In using it, strict attention must be paid to 

 the signs of // and a\ plus for tension, minus for compres- 

 sion. 



We shall now put this formula in a different shape and 

 from it eventually deduce the theory of least work. 



If we call X the stress (+ for tension, — for compres- 

 sion) in any bar due to all the loads and their correspond- 

 ing. reactions, when P is omitted, we have the stress in any 

 bar, 



s := X + u P; 



whence, taking the derivative, since X is entirely inde- 

 pendent of P, 



ds 



d P 



(a s d s \ d I /a s^ \ 

 - — = ^ - (5) 



e w d P/ d P 2 \e w/ 



in which it is understood that .v must be replaced by X -f 



1 as^ 



u P. Now b\- eq. (3), represents the elastic work 



2 e w 



of one bar, so that in words (5) shows that if ive express 

 the 7vork of defovuiation of the bars as a fiinction of the ex- 

 ternal forces^ its derivative with respect to one of the forces 

 gives the displacement^ in the direction of the force^ of its 

 point of application. 



This is called by Castigliano "the principle of the de- 

 rivative of work," or it may be termed the theorem of 

 deflection. If we call the work of deformation of the sys- 



dF 

 tem, P", it is plain, from the above that when = J p is 



d P 



