2. Case of great velocities. Direct determination of the resistance of the air by the aid of the balistic pendulum. Experiments of Hutton, their results. Experiments made at Metz in 1839 and 1840. General expression of the resistance based upon the total of the results obtained, and containing a function of the velocity in three terms. Search after a function in two terms fit to replace in each particular case the general expression. _Seventeenth Lecture._--(17.) Theory of the motion of projectiles in the air. Differential equations of the motion. Hypothesis on the relation of the element of the trajectory to its projection. Calculations based on this hypothesis, and leading to the final equation of the arc of the trajectory. Inclination of the element of the trajectory. Velocity of the projectile at a given point. Duration of the passage. _Eighteenth Lecture._--(18.) Examination of the functions employed in the formulas of the science of projectiles. Formation of the balistic co-efficient, and the series contained in the functions. Relations of the series and the functions to each other. Arithmetical tables designed to give their values. Determination of the relation of an arc of the trajectory to its projection. Error resulting from the introduction of the constant relation in balistic calculations. _Nineteenth Lecture._--(19.) Application of balistic theories to the movement of projectiles thrown at great angles. Analysis of the trajectory, and determination of all the circumstances of the movement. Trajectory of shells considered as a single arc. Solution of several problems involved in this hypothesis. Determination of the range. Velocity corresponding to a given range and angle of projection. Angle of projection corresponding to a known initial velocity and range. Angle of greatest range. Variation of the velocity of the projectile during the whole of its passage. Limit of velocity of projectiles falling vertically in the air. _Twentieth Lecture._--(20.) Application of balistic theories to the motion of projectiles thrown at low angles. Case where the relation of the arc to its projection can be supposed sensibly equal to unity. Problems relative to direct fire; distinction established between the angle of projection and the angle of fire. In ordinary cases in practice the angle of fire is very nearly independent of the height of the object aimed at. Relations between the angle of projection, the angle of elevation of the object aimed at, and the angle of descent. Problems relating to plunging fire. (Ricochet fire.) Determination of the initial velocity and the angle of projection for a projectile which has to pass, firstly, through two given points; secondly, through one given point, the trajectory having at this point a known direction. Case of practical impossibility.