In this equation _x_ corresponds to the position of loading for maximum bending, while the sign (?) represents the distance between the concentrations _W_₍ₙʹ₋₁₎ and _W_ₙʹ. This equation has a very formidable appearance, but its composition is simple and it is constantly used in making computations for the design of railroad bridges. The loads _W_₁, _W_₂, _W_₃, etc., represent the actual weights on the driving-axles and other axles of locomotives, tenders, and cars, and the spacings _a_, _b_, _c_, etc., are the actual spacings found between those axles. In other words, these quantities are the actual weights and dimensions of the different portions of moving railroad trains. The computations indicated by equation (28) are not made anew in every instance. Concentrated weights of typical locomotives, tenders, and cars are prescribed by different railroad companies for their different classes of trains, ranging from the heaviest freight traffic to the lightest passenger train. A tabulation is then made from equation (28) for each such typical train, and it is used as frequently as is necessary to design a bridge to carry the prescribed traffic. The tabulations thus made are never changed for a given or prescribed loading. =87. Applications to Rolled Beams.=—It is to be remembered that these last observations do not limit the use of equations (27) and (28) to railroad-bridge trusses only; they are equally applicable to solid and rolled beams and are frequently used in connection with their design. Great quantities of these beams and various rolled steel shapes are used in the construction of large modern city buildings, as well as in railroad and highway bridge structures. The steel frames of the great office buildings, so many of which are seen in New York and Chicago as well as in other cities, which carry the entire weight of the building, are formed wholly of these steel shapes. The so-called handbooks published by steel-producing companies exhibit the various shapes rolled in each mill. These books also give in tabular statements many numerical values of the moment of inertia, the section modulus, and other elements of all these sections, so that the formulæ which have been established in the preceding pages may be applied in practical work with great convenience and little labor. Tables are also given showing the sizes of rolled beams required to sustain the loads named in them. Such tables are formed for practical use, so that, knowing the distance apart of the beams, their span, and the load per square foot which they carry, the required size of beam may be selected without even computation. Such labor-saving tables are quite common at the present time, and they reduce greatly the labor of numerical computations. CHAPTER VIII.