ELISHA MITCHELL SCIENTIFIC SOCIETY. 1 5 



the necessary bars and one siiperjiuous bar at a time be a 

 minimum; or preferably^ that tJie worlz of all tJie bars be a 

 minimum^ provided zve asszime the fiction^ that the stresses 

 of the superfluous bars are entirely independent of each 

 other. 



It is this which constitutes the method of " least work." 



When there is but one superfluous bar, the true stresses 

 correspond exactly to a minimum of elastic work, but for 

 a greater number of superfluous bars this is not necessarily 

 true, since the stresses in the superfluous bars are functions 

 of each other and not independent, as we assume in form- 

 ing eqs. (7). This consideration has not been pointed out 

 by any previous author, as far as the writer knows. 



The theorems of " deflection" and " least work" have 

 now both been proved by aid of the method of virtual 

 velocities, which, it is seen, is especially adapted to the 

 object in view, as it leads easily and unmistakably to the 

 theorems, and leaves, no doubt, whatsoever as to the exact 

 interpretation of results. 



The theorems are easily extended to solid beams, com- 

 posed of molecules, resisting any change of distance apart 

 by forces varying directly as the changes of distance, ac- 

 cording to the law of elasticity first assumed; for such 

 bodies can be treated, therefore, as articulated systems, 

 whence the above theorems directly apply, the unknown 

 stresses between certain molecules taking the place of the 

 stresses in the superfluous bars of the preceding demonstra- 

 tions. The theorems are therefore perfectly general and 

 apply to solid beams, articulated structures, or combina- 

 tions of the two, including structures having certain mem- 

 bers continuous over certain apices ; but it would take us 

 too far in this article to give the most convenient methods 

 of dealing with such composite structures, which may be 

 found, however, partly in the article by the writer in the 

 April, 1 89 1, Transactions Am. Soc. C. E. , and very fully 

 in Castigliano's very exhaustive treatise before mentioned. 



