Section 7. On the motion of fluids and resistance of projectilert. Two systems of similar particles similarly placed, with given ratios between their densities, and beginning to move similarly in proportional times, will continuo to do so, if there be no contact of particles except at instants of reflaxion, and if there be no attracting forces of particles on one another but such as are as the diameters of corresponding particles inversely and the square of the velocities directly. 2 Con (33) And finite parts of these systems are resisted in a ratio compounded of the duplicate ratio of the velocities and diameters, and the ratio of the densities. 6 Cor. (34) The circular end of a cylinder encounters twice as much resistance as a sphere moving with the same velocity. Scholium, containing among other things the construction (without demonstration) of the solid of least resistance, which shows that Newton must have carried his fluxions (before 1687) far enough to solve some problems at least in what is now called the calculus of variations. (35) A medium consisting of equal particles at equal distance., to find the resistance it offers to a sphere. 7 Cor. and Schol. (36) To find the motion of water issuing through an orifice at the bottom of a cylinder. 10 Cor. Lemma 4. (37i Resistance to a cylinder moving in a nowelantio fluid. 3 Cor. and Schol. Lemma 5, 6, 7. A cylinder, sphere, or spheroid, of the same circular section, pieced in a cylindrical canal of running water, or moving equally In It, is equally urged or resisted. Schol. (3S) Resistance to a globe in a non-elastio fluid. 4 Cor. (39) The same when the globe is in a cylindrical canal. Schol. (40) The same, showing how to find the resistance experimentally. Scholium, con
taining accounts of fourteen experiments.
Section 8. On motion propagated through a fluid. (41) Proseure is not propagated through a fluid in right lines, unless when the particles lie in right lines. Cor. (42) Every motion propagated through a fluid diverge. from the direct path. (43) Every tremulous body excites in an elastic medium pulses in every direction ; but in a non elastio medium, a circular niotion. Cur. (44) 3 Cor. (45) Oscillation of water in a bent tube. (46) The velocity of waves. 2 Cur. (47) In pulses, the motion of the particles follow the law of an oscillating pendulum.* Cor. (48) The velocities of pulses in different media are as the square roots of the elastic force directly and the density inversely. 3 Cor. (49) Given the density and elasticity, to find the velocity. 2 Cor. (50) To find the length of the pulses. Scholium.
Section 9. On the circular motion of fluids. (Refutation of Des Cartel's vortices on the hypothesis that the resistance which arises from the want of lubricity of the parts of is fluid, is proportional to the velocity with which the parts of the fluid are separated.) (51) If a cylinder of infinite length revolve about its axis in is uniform and infinite fluid, and create a vortical motion, the periodic times of the particles of fluid are as their distances from the axis. 6 Cur. (52) But if the revolving body be a sphere, these periodic times are as the squares of the distances from the centre. 11 Cur. and Schol. (53) A body revolving in a vortex so as to return to Its place, must be of the same density as the parts of the vortex, and move in the Male trimmer. 2 Cor. and Scholium, completing the refutation above mentioned.