8 JOURNAL OF THE 



It is well known that the laws of statics alone suffice to 

 determine the stresses in any truss, whose pieces are free 

 to move at the joints, when the number of bars is just that 

 necessary to strictly determine the form. When there are 

 superfluous bars or continuous members without free play 

 at the joints, the theory of elasticity must be used to give 

 the additional equations, which, added to those furnished 

 by the ordinary laws of statics alone, give as many equa- 

 tions as unknown stresses, from which the latter are ob- 

 tained by elimination. The theory of ''least work '' offers 

 a direct solution of such problems. 



It may be observed, if a truss is subjected to such con- 

 ditions^ that more than two joints are fixed in position, 

 that there may be more bars than are strictly necessary to 

 define the form, even when m = 2 n — 3. It is always easy 

 in such cases to ascertain the number of "superfluous 

 bars" by supposing the truss built out from two joints 

 taken as fixed, apex by apex, towards the other fixed joints. 



The number of bars just sufficient to fix the position of 

 each apex, other than the fixed ones, is easily seen; all 

 other bars are superfluous to this end and must be so treated 

 when applying the method of least work. 



4. Derivative of the Work of Deformation with respect 

 to an external force. Deflection. Consider a truss of in- 

 variable form, without superfluous bars, and let a force 

 unity act in the direction of and along the line of action of 

 any external force P. Then when all the original exter- 

 nal forces, such as P, are removed and we have only the 

 force unity acting on the truss, with the corresponding re- 

 actions, if any, call // the stress in any bar due to the force 

 unity in question. Also call the length of this bar <?, its 

 cross section zv and modulus e, and let us conceive it as 



a s 



elongating an amount a 1 = > «i" the exact amount 



e w 



