12 JOURNAL OF THE 



w'lich, from what precedes, is equal to the total relative 

 displacement of the apices, where P and P' are applied. 

 Hence, in any case, to find the relative displacement of 

 two apices, between which two equal and opposed forces, 

 P and F\ act, we have only to take the total derivative of 

 F with respect to one of the forces P, so that it is not nec- 

 essar}' to designate the two opposed forces by different 

 letters. 



5. Demonstration of the Theorem of Least Work. 



Let us suppose that we have a truss of any kind, with 

 superfluous bars numbered n, n -h i, . . . , whilst the 

 (n — i) necessary bars (system N) are numbered consecu- 

 tively, I, 2, ... , (n— i). 



Let, 



Xj, X2, . . . Xii_i = stresses in bars i, 2, . . . , (n — i) 

 of a frame supposed to consist of necessary bars alone 

 (system N) subjected to the actual loading. 



Ui, u,, .... u„_i = stresses in bars i, 2, . . , (n — i) of 

 system N alone, by forces unity acting towards each other 

 from either end of the original position of superfluous bar 

 //, all the superfluous bars being removed. 



Vi, V2, . . . v„_, = stresses in bars 1,2,..., (n — i) of 

 system N alone, caused by forces unity acting towards 

 each other from the apices of superfluous bar (n i- i), all 

 the superfluous bars being removed. 



Similarly we proceed for other superfluous bars, if any. 

 The stresses X, //, ?', . . . , can all be found by the laws of 

 statics alone. Now designating the length, cross section 

 and modulus of elasticity of any bar, by <7, zc and ^, re- 

 spectively, with the same subscript as the number of the 

 bar, we have the total elastic work of deformation of all the 

 bars, including the superfluous bars, expressed by 



G= V2 y I — I ^- ^4 1- 14 ^- . . . , 



--m 



e„w„ e„ + iW„4- 



