14 JOURNAL OF THE 



bar 71 is a necessar}' consequence of the real chano^es of 

 length of the necessary bars alone, and it can be found as 

 above, without knowing- the changes of length of the 

 superfluous bars beforehand. The increase in distance 

 between the apices at the extremities of superfluous bar ;/, 

 as determined from "system N," must therefore exactly 

 equal the elongation of bar ;/ under the stress s^ when in 



place. 



d F a„ s„ 



d s„ e„ w, 



Ml 



d F a„ s„ 



or, \ = o (6). 



d s„ e„ w„ 



x\ similar expression obtains for each of the supei-fluous 

 bars, so that we always have as many equations as there 

 are superfluous bars. 



Now each equation of the type above (6), can be found 

 by taking the partial derivatives of the expression for G 

 above, successively with respect to Sj, , s„ -|- ,, . . . , treated 

 as independent of each other^ and placing the results sep- 

 arately equal to zero, so that the equations needed will be 

 of the type, 



dG dG 



= o, == o, (7)- 



d s„ d s„ + , 



From these equations we find, by elimination, Sji, 

 s„ -f ,, . . . , and then substituting these values in equa- 

 tions of the form, 



s = X -I- u s„ + V s„ 4- 1 -h . . . , 

 we find all the stresses, s,, S2, . . . s„_i. 



Theorem of Least Work. Therefore, to deter7n{ne the 

 icnknown stresses^ ive express the ivork of deformation of 

 the zvhole system as a function of the stresses in the bars 

 taken as superfluous^ then treatin^^ these stresses as inde- 

 pendent in the differentiation^ ive express that the ivork of 



