All this while, we have supposed the points A and B to be relatively at rest ; but it must be evident that the variet)' is immense, if we take different positions and distances of these points. Boscovich has enumerated many of the more remarkable cases. It will be sufficient for us to notice a few of the more simple, and those especially which may be referred to in the physical application of the theory.
In the first place, the attraction of C towards A and B, in those greater distances at which the curve of forces sensibly coincides with that of gravity, will always be towards 'D, proportional to the reciprocal of the square of DC, and sensibly double of what corresponds to that distance in Fig. 1. And the case will be the same in masses consisting of any number of points ; the attraction being sensibly the sum of the forces of all the points which constitute these masses.
But in those smaller distances, at which the curve winds about the axis, the actions of the points upon each other will sometimes be attractive, sometimes repulsive, and the forces resulting therefrom will be infinitely diversified. So that, although the force of gravity be universal, and depend only on the mass and the distance, yet those properties, which depend on the action of matter at smaller distances, as the reflection and refraction of light, and the separation of colours ; the impressions on the various fibres, in tasting, hearing, smelling, add feeling ; cohesions, se cretions, nutritions, fermentations, precipitations, ex plosions, and all the phenomena of chemistry ; and a thousand others, however various in their effects, may all be satisfactorily explained on the principles of this theory.
Suppose the point C be placed any where in a line DC, perpendicular to AB; or any where in the line joining them, Fig.3. It is evident that, in the first case, the action of B and A being equal and of the same kind, the access or recess of the point C will be in the line DC ; and the curve expressing the forces acting on C, might be found by drawing B d equal to any abscisses from Fig. 1. ; laying off in it, d e its corresponding ordinate ; drawing e a at right angles to DC, and making the perpendicular dh equal 2da, on any of the sides for repulsion, and on the oppo site side for attraction. The curve will cut the axis
in various points ; it will also pass through the point D, and have a similar branch on the opposite side of AB ; in which, however, the sides expressing attrac tion and repulsion will be reversed. Each intersec tion will be a limit, and the point D will be a limit of cohesion or non-cohesion, according as the arch on either side of it is attractive or repulsive. It will also be a weak limit, for the opposite' forces of A and B will nearly destroy each other, although the points be a small matter out of the straight line.
In the second case, where the point C is taken any where in the line AB, the curve which expresses the law of forces may be thus found. Fig. 4. For any point d, assume two abscisses in Fig. 1., the one equal to A d, the other to d B ; and taking the cor responding ordinates, lay off d h equal to their sum or their difference, according as they are of the same or of different kinds, assuming one side of the axis AB to express excess in repulsion, and the other excess in attraction. The curve will pass through the point D, and the directions will be changed as in the former case. If a perpendicular be drawn through B, it will be an asymptote to the curve on either side, since the repulsion of B will prevent ab solute contact. There may be several limits or in tersections, either between A and II, or beyond them ; and according to the distance at which we suppose A and B to be posited, the attractive force of the one may neutralise the repulsive force of the other, or double its attractive force, and vice versa.
Let the three points A, D, B, be in a straight line, their mutual action will be 0, if the three dis tances AD, DB, AB, be each the distances of limits. l The point D may be attracted by both extremes, repelled by both, or attracted by one and repelled by the other. These cases are, however, vastly dif ferent ; in the first, if D be removed from its place to C, it will return to it again ; in the second it will recede still farther. In the former case we have an instance of cohesion ; in the second of non-cohe sion. In the third case, it is plain that the point D will move away from the repelling end, and approach the attractive.