296 Prof. Maxwell on the Calculation of 



be regarded as a machine whose efficiency is perfect. Hence 

 the following 



Theorem. — If p be the tension of the piece A due to a tension- 

 unity between the points B and C, then an extension-unity 

 taking place in A will bring B and C nearer by a distance^. 



For let X be the tension and x the extension of A, Y the 

 tension and y the extension of the line BC ; then supposing all 

 the other pieces inextensible, no work will be done except in 

 stretching A, or 



But X = pY, therefore y = —px, which was to be proved. 



Problem I. — A tension F is applied between the points B and 

 C of a frame which is simply stiff; to find the extension of the 

 line joining D and E, all the pieces except A being inextensible, 

 the extensibility of A being e. 



Determine the tension in each piece due to unit tension be- 

 tween B and C, and let p be the tension in A due to this cause. 



Determine also the tension in each piece due to unit tension 

 between D and E, and let y be the tension in the piece A due 

 to this cause. 



Then the actual tension of A is Yp, and its extension is e¥p, 

 and the extension of the line DE due to this cause is — Yepq by 

 the last theorem. 



Cor. — If the other pieces of the frame are extensible, the com 

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



-St(epq), 



where ^(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 



-F2(^ 2 ), 



X(e/? 2 ) may therefore be called the resultant extensibility alongBC. 



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 elasti- 

 cities 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, r their extensibilities. Then 

 the tension A 



= F/? + Rr + Ss + T*-f-&c; 



