ON THE CALCULATION OF THE EQUILIBRIUM AND STIFFNESS OF FRAMES. 601 



COR. If the other pieces of the frame are extensible, the complete value 

 of the extension in DE due to a tension F in BC is 



where 2(epq) means the sum of the products of epq, which are to be found 

 for each piece in the same way as they were found for A. 



The extension of the line BC due to a tension F in BC itself will be 



2 ) may therefore be called the resultant extensibility along BC. 



PROBLEM II. A tension F is applied between B and C ; to find the 

 extension between D and E when the frame is not /simply stiff, but has 

 additional pieces R, S, T, &c. whose elasticities are known. 



Let p and q, as before, be the tensions in the piece A due to unit 

 tensions in BC and DE, and let r, s, t, &c. be the tensions in A due to 

 unit tension in R, S, T, &c. ; also let R, S, T be the tensions of R, S, T, 

 and p, a; T their extensibilities. Then the tension A 



= Fp+Rr+Ss + Tt + &c.; 

 the extension of A 



the extension of R 



= - Ft(epr) - RZer 1 - SZers - TZert + &c. = Rp ; 

 extension of S 



= - F1(eps) - R2(ers) - S^es* - T1(est) = So- ; 

 extension of T 



= - F2(ept) - R2(ert) - S2(est) - TZ(e?} = Tr ; 

 also extension of DE 



= - F2(epq) - Rl(eqr) - S2(eqs) - T2(egt) = x, 



the extension required. Here we have as many equations to determine R, S, T, 

 &c. as there are of these unknown quantities, and by the last equation we 

 determine x the extension of DE from F the tension in BC. 

 Thus, if there is only one additional connexion R, we find 



and 



VOL. I. 76 



