The general problem is to determine the deflection of the point _J_ when the bridge carries both dead and moving load over the entire span, as shown in Fig. 29. The general plan of procedure is first to find the stresses due to this combined load in every member of the truss, so that the corresponding lengthening or shortening is at once shown. The effect of this lengthening and shortening for any single member _BC_ in producing deflection at _J_ is then determined; the sum of all such effects for every member of the truss is next taken, and that sum is the deflection sought. In this case the vertical deflection will be found, because that is the deflection generally desired in connection with bridge structures, but precisely the same method and essentially the same formulæ are used to find the deflection in any direction whatever. The following notation will be employed: Let _w_ = deflection in inches at any panel-point or joint of the truss; ” _P_ = any arbitrary load or weight supposed to be hung at the point where the deflection is desired and acting as if gradually applied. This may be taken as unity; ” _Z_ = stress produced in any member of truss by _P_; ” _S_ = stress produced in any member of truss by the combined dead and moving loads; Let _l_ = length in inches of any member of the truss in which _Z_ or _S_ is found; ” _A_ = area of cross-section of same member in square inches; ” _E_ = coefficient of elasticity. _S_ or _Z_ may be either tension or compression, and the formulæ will be so expressed that tension will be made positive and compression negative. The change of length of the chord member _BC_ produced by a stress gradually increasing from zero to _S_ is _S_ ----_l_. _AE_ If it be supposed that _BC_ is a spring of such stiffness that it will be compressed by the gradual application of _Z_ exactly as much as the shortening of the actual member by the stress _S_, the deflection of the point 4 with the weight _P_ hung from it, and due to that compression alone, will be precisely the same as that due to the actual shortening of _BC_ by the combined dead and moving loads. It is known by one of the elementary principles of mechanics that, since _P_ acts along the direction of the vertical deflection _w_, the work performed by the weight _P_ over that deflection is equal to the work performed by _Z_ over the change of length _l_. Hence _l l Sl_ --- _Pw_ = --- _Z_ ----, or 2 2 _AE_ _Z Sl_ _w_ = --- ----. (42) _P AE_