2d. Horizontal beam placed upon two supports, and subjected to a weight acting at its center, and with a uniform vertical effect. 3d. Beam placed horizontally on two supports, and having two equal weights symmetrically placed with respect to its center. 4th. Beam placed horizontally on two supports, and subjected to a weight acting at any point whatever throughout its length. 5th. Horizontal beam fixed at both its extremities, and subjected to a weight acting at its center with an equal vertical effect. 6th. Horizontal beam placed on three points of support, at unequal distances, and weighted with two weights acting at the middle of the intervals between the supports. 7th. Vertical beam fixed at the foot, and charged with a weight acting at a certain distance from the axis of the beam. 5. Solids of equal resistances. Most suitable form for cast girders. Applications of the formula of equilibrium and squaring to various kinds of carpentry. 6. On polygonal roofs. Conditions respecting them. Arched roofs, pressure, &c. On the stability of walls required to resist the pressure of roofs. SECOND SECTION: ON THE STABILITY OF REVETMENT WALLS AND ARCHES. 7. On the pressure of earth. Explanation of the theory on Coulomb’s system. Investigation of the pressure of earth by analysis. Hypothesis necessary in order to simplify the calculations. General formula of the value of the pressure, &c. Equations of stability and equilibrium under the hypothesis of slipping and rolling. 8. Simplification of the general equations of equilibrium in three particular cases. Determination of the co-efficient of stability in Vauban’s profile. M. Poncelet’s formula for calculating the thickness of revetment walls with perpendicular face. Transformation of the profile of a revetment to another of equal stability. Vauban’s counterforts, &c. 9. Geometrical method for determining the pressure of earth, whatever may be the profile of the wall and of the earth, taking into account the friction of the earth on masonry. Geometrical determination of the amount of the pressure. Proceeding for the determination, by geometry, of the thickness of a revetment wall at the level of the exterior ground. 10. On buttresses. Geometrical determination of the buttressing of earth, and of its momentum. Simplification of the geometrical constructions of the pressure, of the buttressing, and of their momenta under certain hypotheses. 11. Points of application of the pressure and of the buttress. 1st. In the case of a terrace sloping less than the natural slope of the ground. 2d. In the case of the ordinary revetments of fortification. On the stability of the foundations of revetment walls.