302 Mr. J. H. Cotterill on an Extension of the 



method, although both methods are only approximate, the reason 

 of which will be seen from what follows. Let i x i% be the slopes 

 at the extremities of a beam acted upon by M 2 M 2 at its extremities, 

 and by the uniform load w, then U being, as shown above, an 

 homogeneous quadratic function of M lf M 2 , we have 

 OTT dV ' dV „ dV 



^^m^ + dW^^'' 



but by the law of conservation 



2U = M 1 i 1 + M 2 ? 2 + ww, 

 where u is the area of deflection of the beam ; and comparing 

 these expressions, we see that 



dV __. ^U__. dV_ 

 dM,"* 1 ' t/M 2 ~ ^2, dw~ U; 

 but the ordinary method is founded on the consideration that 

 the beam is horizontal at its extremities, in other words, that 

 i l =0, 4=0 ; so that the two methods lead to the same result by 

 the same equations. And this will be the case in all questions 

 concerning continuous beams ; but the present method enables us 

 to obtain the requisite equations by differentiation of a single 

 function. 



2. A suspension-bridge platform is stiffened by the applica- 

 tion of a girder of uniform section ; find the bending moment at 

 its centre when loaded in the middle. 



Here, to obtain a complete solution, the work done in the 

 whole structure, chains, piers, suspending-rods, and girders, must 

 be estimated in terms of the bending moments at the centre and 

 the points of attachment of the rods, and the tensions of those rods. 

 To simplify, suppose the rods indefinitely many in number (as is 

 usual in suspension-bridge questions), and the tension per unit 

 of length of the girder w ; also suppose the form of the chain, 

 before the load is put on, parabolic, so that w would be uniform 

 if the load produced no distortion, and will actually be sensibly 

 uniform, because the distortion is inconsiderable. These sup- 

 positions being made, the work done in half the girder would be 



where c is the quarter span, M x the bending moment at the ex- 

 tremity of the girder, M 2 at its centre. I shall consider the 

 case in which the girder is attached, but not fixed at its ends, 

 in which case Mj = 0, and the expression becomes 



In the first place, suppose the chains and rods sensibly in- 

 extensible, and the piers sensibly immoveable, then the only 



