With respect to the points 'which the curve 'meets the axis, the'y arc either points of section, as E, G, I, or of -contact. In' the -former, there is a transition from attraction. to repulsion,' or the con trary, and these by our author are called limits. These limits are of two kinds ; first, where the transi tion, by an increase of distance, is from repulsion to at traction, as is the case at E, R, which are'called limits of cohesion, for in such a situation the points resist all change of position, viz. separation by means of the attractive force which immediately begins to operate, and mutual approach in like manner by the incipient repulsion. But in the limits of. the second kind; as G, L, P where the transition, by an increase of distance, is froni attraction to repulsion, although the points in such situations do not exert any force on each other, yet the smallest change of distance pro duces a very important alteration : for if they be in the least separated, the repulsive force then acting will remove them still farther asunder ; and, on. the 'other hand, if their distance be diminished in the least, they will tend together more and more. Such limits, therefore, are by Boscovich called limits of non-cohe sion.
The limits of cohesion may be powerful or weak, according to the angle at which the curve intersects the axis, or the distance to which it removes from it. t Ny exhibits a limit of the former; c N x of the latter kind. The most powerful kind of limit at first, at least, is, where the curve at cutting the axis has the ordinate for its tangent, as X, Fig. 6. ; and, in like manner, the weakest is, when the axis is the tan gent, as Y Fig. 6., both being points of contrary flexure.
This being premised, we now proceed to the con sideration of some of the combinations of the points of matter, and of their mutual actions on each other. ' If two points be placed at such a distance from each other, as is equal to that of some limit from the beginning of the line of abscisses, as AG, AE, &c. and without any kind of motion, they must evidently remain there at rest, since they have no mutual ac him]. But if the points be placed out of limits of that kind, they will immediately to approach tor recede by equal intervals. The force continuing 'in one direction, will carry them .to the distance of •the nearest limit, which will, of course, be a limit of cohesion: They will arrive at that with an accele rated motion, and the squares of their velocities will be proportional to the area described by the accom panying ordinate. But they will not stop at this li .mit: Having arrived there with a motion continually -accelerated, they-will go on beyond it, and, of course, they will be immediately acted on -by a force directly •opposite. Their motion will therefore be retarded
the velocity be totally extinguished, by the area -under this second branch of the curve becoming -equal to that intercepted between the ordinate at the original place of the point and the limit aforesaid. Should the area of this second segment be small, 'the original motion of the points will go on; they will pass the second limit of non•cohesion, if they rive-at it with the smallest velocity. Beyond that the original motion will be again accelerated by an action of the same kind as at first, and the. points will.pass -another limit of cohesion. A second retarding force 'will now act, and may at length be equal to the ex tinction of the velocity. If that does not take place exactl•at a limit of non-cohesion, which is scarcely possible, the bodies will be returned again with a se - ries of motions just the contrary of the former, and they will arrive at the same position from which they departed, and they will continue therefore to oscil late in this way for an indefinite length of time.
Cor. The velocity will be greatest at the limits of cohesion, and least at the limits of non-cohesion. No velocity of approach can overcome the repulsion expressed by the first or asymptotic arc, ED. But if the points be placed at first within that arc, the repulsive force may, perhaps, be so great as to carry them over all the subsequent arches, and even through that which expresses the law of general gravity; the points would therefore recede ad iVinitunt.
All this would be the case, were these points left entirely to themselves. But if other external forces act on them, the case might be very different ; for these forces may possibly retain the points in limits of cohesion or non.cohesion, or even in situations out of these limits". Should the two points be projected obliquely, with equal and opposite motions, they would revolve in equal curves round the middle point of the line joining them, which curves, if the law of forces were given, might be formed by the inverse problem of central forces. And it may be observed, that if two points be brought towards each other from ever so great a distance, not directly, but with some small obliquity, (and, indeed, direct motion must be hardly possible,) they will not return back, but, from the nature of central forces, will revolve round the middle point of space, always near each other. Although the interval be not cognisable by the senses, this remark will be hereafter of use, when we come to treat of cohesion and of soft bodies.