THE SECOND Boo; mostly on resisted motion, contains 0 sections and 53 propositions.
Section 1. When the resistance is as the velocity. (1) The motion lost is as the space described. Cor. Lemma 1. (2) When no forces act but the resistance, the velocities at the beginnings of successive equal times are in geometrical progression, and the spaces described as the velocities. Cor. (3) To determine the resisted motion of ascent or descent when the force of gravity acts. 4 Cur. (4) The same when the particle is projected obliquely. 7 Cor. and Scholium, to the effect that the hypothesis is " magis mathematica quam naturalis." Section 2. When the resistance is in the duplicate ratio of the velo city. (5) When no force acts, equal spaces are described in times which are in increasing geometrical progression, the velocities at the beginning of the times being in the inverse geometrical progression.
5 Cor. (6) Equal and homogeneous spheres, acted on by no forces, describe equal spaces in times which are reciprocally as their initial velocities, in which also they lose the same parts of their velocities.
17) Also, in times which are as the first velocities directly and the first resistances they lose the same fractious of their velocities, and describe spun jointly proportional to the timee and the velo cities. S Cur. Lemma 2; which answers to finding the fluxion, (called here momenta) of simple algebraical quantities. 3 Cor. and Scholium. [Frauxtoses.) (S) When a particle descends or ascends by gravity, the whole forces (gravity and reeiertance compouuded) at the beginning of equal successive spaces, are in geometrical progression. 8 Cor. (9) Determination of the proportions of the times of ascent and descent in the last. 7 Cor. (10) The law of resistance being jointly as the density and the square of the velocity, required the law of density so that (gravity acting) a given curve may be described ; as also the law of velocity. 2 Cur., followed by 4 Examp. and Schol., and also by 8 rules.
Section 3. When the resistance is partly as the velocity, partly as its square. (11) No force acting, and times being taken in arithmetical progression, the reciprocals of the velocities, increased by a certain oonstant quantity, will be in geometrical progression. 2 Cor. (12) But if spaces be taken iu arithmetical progression, the velocities increased by a constant quantity will be in geometrical progression. 8 Cor. (13) Gravity acting, the relation between the time and velocity in the ascent or descent is shown. Cor. and Schol. (14) Relation connecting the space described with the preceding. Cor. and Schol.
&clime 4. On spiral motion in a resisting medium. Lemma 3. A property of the equiangular spiral. (15) The density being inversely as the distance from the centre, and the centripetal force inversely as its square, the particle can revolve in an oquiangular spiral. 9 tier. (16) And, other things remaining, the same can be when the force is inversely an any power of the distance. 3 Cor. and Schol. (17) To find
the force and law of resistance, by which a body may move in a given spiral, with a given law of velocity. (IS) Given the law of force in the last, to find the density of the medium.
Section 5. On the density and compression of fluids, and on hydro rtaties. Definition of a fluid. (19) A homogeneous fluid compressed in a close vessel (gravity, &c., apart) is everywhere equally pressed, and at rest.. 7 C4L.es and Cor. (20) If a solid sphere form tho nucleus of a fluid mars bounded by a concentric sphere, and the parts of the fluid gravitate equally to the centre at equal distances, the pressure sus tained by the sphere is the weight of a cylinder which has the super flees for its base and the height of the incumbent fluid for its altitude. 9 Cor. (21) The density being proportional to the compression, and the centripetal force of particles inversely as their distance from the centre, then at distances in geometrical progression, the densities will be also in geometrical progression. Cor. (22) But if the force be inversely as then at distances in harmonical progression the deursities will be in geometrical progression. Cor. and Schol. (23) If the particles of the fluid repel each other, the density is as the com pression (and then only) when the repellent force of two particles is inversely as the distance of their centres. Scholium. (In cousequence of the particles being supposed to repel only their nearest, Newton treats this only as a purely mathematical result.) Section 6. On the resisted motion of pendulums.* (24) The quan tity of matter in pendulums of the same length, is in a ratio compounded of their weights and of the duplicate ratio of their times of oscillation in vacuo. 7 Cor. (25) A pendulum which moves in a medium in which the resistances are as the of the times, and another moving unresisted in a medium of the same specifio gravity, make their cycloidal oscillations in the same times, and deseribe7proportional parts of their arcs together. Cor. (26) Resistance being as the velocity, cycloidal oscillations are isochronous. (27) Resistance being as the (velocity), the difference between the time of cycloidal oscillation in a resisting medium nod a nomreaisting one of the same specific gravity, will be very nearly as the arcs of oscillation. 2 Cor. (28) The resist :times being as the moments of the time, the reaistauce is to the force of gravity as the excess of an arc of descent (cycloidal) over the subsequent arc of ascent to the square of the length of the pendulum. (29) Resistance being as (vclocity)r, to find the resistance at any point of a cycloidal oscillation. 3 Cor. (30) Easier method of exhibiting the difference of an aro of descent and ascent. Cor. (31) If the resistance be altered in a given ratio, the difference of the arcs of anent and descent is altered in the same ratio. Scholium Generale, containing many experimental results.