ELISHA MITCHELL SCIENTIFIC SOCIETY. 13 



in which the sum - extends to the necessary bars alone, 

 or bars i, 2, . . . (n — i). In this expression it is under- 

 stood that for j-, the actual stress in any bar, we must sub- 

 stitute expressions of the type, 



S = X — U Sn -f V S„ -j- I 4- . . . , 



on supplying the proper subscripts pertaining to the bar 

 considered. 



The last expression follows at once from the principle of 

 " superposition of effects." 



On designating by F the elastic work of the necessary 

 bars alone, we have 



^iB 



the sum including only the necessary bars and s being ex- 

 pressed as a function of Su, s„-|- j, . . , as above. 



We shall next regard the superfluous bars n, n 4- i, . . . , 

 as temporarily removed and replace -their action by two 

 forces for each bar, each equal to the stress in the super- 

 fluous bar and acting towards each other, as all bars are 

 assumed to be in tension until otherwise determined. 



It has been shown above that treating these forces, s^, 

 Sn-|-i, . . . , as external forces and independent of each other, 

 d F 



that represents the increase in distance between 



d s,, 



the apices at the extremities of bar ;/, the minus sign being 

 used, since the two forces s^, s„, replacing the tension of 

 the bar upon the joints at its ends, act in the opposite di- 

 rection to the displacements. Similarly for the other deflec- 

 tions. Again, since Su is supposed to equal the actual 

 stress in bar ;/ in the complete structure under the loading, 



d F 

 it follows that must equal the elongation of the bar 



d Sn 

 ;/ under the stress Sq when all the superfluous bars are in 

 place, since the real change of length of any superfluous 



