Section 12. On the attractions of spherical bodies. (70) A particle placed inside a spherical shell is in equilibrium. (71) Spherical shells attract as if their whole masses were collected at their centres. (72) The attractions of spheres on points similarly placed with respect to them are as their diameters. 3 Cor. (73) At different internal points of a solid sphere the attractions are as the distances from the centre. Schol. (74) Solid spheres attract as if the whole masses were collected at their centres. 3 Cor. (75) The same of spheres attracting spheres.
4 Cor. (76) The same of spheres consisting of concentric layers of unequal density. 9 Cor. (77) The same is true when the forces of particles to each other are as their distances. (78) With the same law, the same is true of spheres consisting of concentric layers. Cor. and Schol. : Lemma 29. (79); (80) 4 Cor.; (81), 3 Exam.; (82) : these show the method of finding the attraction of any sphere on a point without it, for any law of force. (83) The force being as the inverse nth power of the distance, to find the attraction of a segment of a sphere on a particle at its centre. (84) The same when the par ticle is not in the centre. Sehol.
Section 13. On the attraction of non-spherical bodies. (85) If the attraction of the body on a contiguous particle be much greater than on one at a little distance, the attraction of the molecules of the attracting body diminishes in a higher ratio than the inverse square of the distance. (86) And the hypothesis of the last is a consequence, if the attraction of the molecules diminishes as the inverse cube of the distance, or faster. (87) If two similar bodies of the same material attract two molecules proportional to themselves and similarly placed, the attractions of the molecules on the two bodies will be proportional to their attractions on their similar particles similarly placed. 2 Cor.
(88) If the particles of a body attract a molecule with forces as their distances, the whole attraction of the body will be towards its centre of gravity, and equal to that of a sphere equal to the body, and having its centre in that centre of gravity. Cor. (89) And the same if there be several bodies. Cor. (90) To determine the attraction of a circle on a point in its axis. 3 Cor. (91) To determine the attraction of a solid of revolution on a point in its axis. 3 Con' relating to cylinders and spheroids. (92) Given an attracting body, to find (experimentally) the law of attraction of its particles. (93) If particles attract as the inverse nth power, a solid bouuded by a plane, but indefinitely extended in all directions on one side of that plane, will attract an external particle with a force proportional to the inverse (n— 3)rd power of the distance from that plane. 3 Cor. Sehol.
Section 14. On the motion of particles from one medium into another. (94) If a particle pass through a medium contained between parallel planes, and be in its passage attracted to or repelled from the boundary of the medium it has left with a force depending on the distance from the boundary ; the sine of the angle of emergence is always in a constant ratio to that of incidence. (95) And the velocity before incidence is to that after emergence as the sine of the angle of emergence to that of incidence. (96) And if the velocity must be greater before than after incidence, the angle of incidence may be made so great that the particle shall be reflected, and the angles of incidence and reflexion are equal. Scholium. (97) To give the boun dary separating two media such a form that all particles issuing from one point may be refracted to another. 2 Cor. (98) To form lens which shall have the property mentioned in the last. Scholium.