=107. Statically Determinate Trusses.=—The bridge structures which have been treated require but the simplest analysis, based only on statical equations of equilibrium of forces acting in one plane, i.e., the plane of the truss. It is known from the science of mechanics that the number of those equations is at most but three for any system of forces or loads, viz., two equations of forces and one of moments. This may be simply illustrated by the system of forces _F₁_, _F₂_, etc., in Fig. 27. Let each force be resolved into its vertical and horizontal components _V_ and _H_. Also let _l₁_, _l₂_, etc. (not shown in the figure), be the normals or lever-arms dropped from any point _A_ on the lines of action of the forces _F₁_, _F₂_, etc., so that the moments of the forces about that point will be _F₁l₁_, _F₂l₂_, etc. The conditions of purely statical equilibrium are expressed by the three general equations _H₁ + H₂_ + etc. = _F₁_ cos _a₁_ + _F₂_ cos _a₂_ + etc. = 0; (35) _V₁ + V₂_ + etc. = _F₁_ sin _a₁_ + _F₂_ sin _a₂_ + etc. = 0; (36) _Fl = F₁l₁ + F₂l₂_ + etc. = 0. (37) If all the forces except three are known, obviously those three can be found by the three preceding equations; but if more than three are unknown, those three equations are not sufficient to find them. Other equations must be available or the unknown forces cannot be found. In modern methods of stress determinations those other needed equations express known elastic relations or values, such as deflections or the work performed in stressing the different members of structures under loads. A few fundamental equations of these methods will be given. In Figs. 19, 20, and 21 let the truss be cut or divided by the imaginary sections _QS_. Each section cuts but three members, and as the loads and reactions are known, the stresses in the cut members will yield but three unknown forces, which may be found by the three equations of equilibrium (35), (36), (37). If more than three members are cut, however, as in the section _TV_ of Figs. 22 and 23, making more than three unknown equations to be found, other equations than the three of statical equilibrium must be available. Hence the general principle that _if it is possible to cut not more than three members by a section through the truss, it is statically determinate_, but _if it is not possible to cut less than four or more, the stresses are statically indeterminate_.