the Equilibrium and Stiffness of Frames. 297 



the extension of A 



=e(¥p + Rr+Ss+Tt+ &c); 



the extension of R 



= - F2 (epr) - R2e?' 2 - S2er* - TSert + &c. = Rp ; 

 extension of S 



= -Y2,{eps) -B2{ers) - SXes 2 -TX(est) = Ser; 

 extension of T 



= -FS(grf) -KX(ert) -SZ{est) -T2(^ 2 ) = Tr ; 

 also extension of DE 



= -F2(g?g0 -R2(*gr) - S2(^«) -TS(^) =#, 

 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 



R=-F 



X(epr) 



{ 



and 



If there are two additional connexions R and S, with elasticities 

 p and cr, 



37= -F 



2e(r 2 + p )%e{f + a) - (2(cr*)) 2 



7,{epr]Z{er8]Z{eqs) +^L{eps)?<{eqr)?,{ers) + 2 t (epq)te{r* + p) y Ze(s* + a)\ 

 -S(g>r)2far)Se(* 2 + a) -2{eps)2{eqs)I,e(r* + p)-Z{epq)(Z(ers)y j 



The expressions for the extensibility, when there are many 

 additional pieces, are of course very complicated. 



It will be observed, however, that p and q always enter into 

 the equations in the same way, so that we may establish the fol- 

 lowing general 



Theorem. — The extension in BC, due to unity of tension along 

 DE, is always equal to the extension in DE due to unity of ten- 

 sion in BC. Hence we have the following method of determin- 

 ing the displacement produced at any joint of a frame due to 

 forces applied at other joints. 



1st. Select as many pieces of the frame as are sufficient to 

 render all its points stiff. Call the remaining pieces R, S, T, &c 



2nd. Find the tension on each piece due to unit of tension in 

 the direction of the force proposed to be applied. Call this the 

 value of p for each piece. 



3rd. Find the tension on each piece due to unit of tension in 



