In equation (1) the symbol ∫ means that the sum of all the small quantities to the right of it is taken, and _I_ stands for that sum which, in the science of mechanics, is called the moment of inertia of the cross-section about its neutral axis. The value of the quantity _I_ may easily be computed for all forms of section. Numerical values belonging to all the usual forms employed in engineering practice are found in extended tables in the handbooks of the large iron and steel companies of the country, so that its use ordinarily involves no computations of its value. Equation (1) may readily be changed into two other forms for convenient practical use. In Fig. 10 _mn_ is supposed to be a very short portion of the centre line of the beam represented by _dl_. Before the beam is bent the section _FG_ is supposed to have the position _MN_ parallel to _PQ_. Also let _u_ be the small amount of stretching or compression (shortening) of a unit’s length of fibre at unit’s distance from the centre line _AB_ of the beam, then will _udl_ and _uzdl_ be the short lines parallel to _GN_ in the triangle _GmN_ shown in the figure. The point _C_ is the centre of curvature of the line _mn_, and _Cn = Cm_ is the radius. The two triangles _Cnm_ and _mNG_ are therefore similar, hence _udl_ _mn_ _dl_ I ------- = ---- = ----- ∴ u = ---. (2) I ρ ρ ρ If the quantity called the coefficient or modulus of elasticity be represented by _E_, then, by the fundamental law of the theory of elasticity in solid bodies, _a = Eu_. (3) As has already been shown, the greatest stresses (intensities) in the section are +_ad_ (tension) and -_ad_₁ (compression). If _K_ represent that greatest intensity of stress, then _K_ _K = ad_, and _a_ = ----. (4) _d_ If the value of _a_ from equation (4) be substituted in equation (1), _KI_ _M_ = -----. (5) _d_ =79. Practical Applications.=—Equation (5) is a formula constantly used in engineering practice. All quantities in the second member are known in any given case. _K_ is prescribed in the specifications, and is known as the “working resistance” in the design of beams and girders. For rolled steel beams in buildings it is frequently taken at 16,000 pounds, i.e., 16,000 pounds per square inch, about one fourth the breaking strength of the steel. In railroad-bridge work it may be found between 10,000 and 12,000 pounds, or approximately one fifth of the breaking strength of the steel. The quantities _I_ and _d_ depend upon the form and dimensions of the cross-section, and are either known or may be determined. The quotient _I ÷ d_ is now known as the “section modulus,” and its numerical values for all forms of rolled beams can be found in published tables. The use of equation (5) is therefore in the highest degree convenient and practicable.