In the first case, the three points may retain, to sense, their rectilineal situation, however powerful the force may be which tends to disturb them. If the force be in the direction of the line, it will be sufficient if, for the middle point, the attraction in creases very much with the increase of distance from either extreme ; and for either extreme point, if repulsion decreases very much with the increase of distance from the middle. Should the force be im pressed perpendicularly, as, for example, if the mid dle point be urged in the direction DC, then the forces may be so powerful as at a very small distance to resist any other of the same kind. Should the force constantly urge the point D towards C, and AB to the opposite side, we have a bending or in flexion ; and, in like manner, forces acting in the direction of the line joining the points AI)B, will produce a compression or dilatation. The forces re sisting this may be so powerful as to render this change almost imperceptible, or they may be weak, so as to admit of considerable deviation from the ori ginal situation. In this manner we may have an idea of rigidity, and of flexibility and elasticity.
If the tzvo forces AQ, BT be perpendicular to AB, F or parallel to one another, the third force CF trill also be parallel to them and equal to their sum, but in the contrary direction. For, draw CD parallel to them, and also KI to AB ; and since CK=VB, the triangle C I K is equal and similar to wry or TBS ; and, therefore CI=BT, and I K=BS=AR=QP. Wherefore, if IF be taken =AQ, and KF drawn, the triangle FIK=AQP ; and, therefore, FK is equal and parallel to AP or LC, and CLFK is a parallelo gram, the diameter of which, CF, expresses the force of the point C, is parallel to AQ and BT, and is equal to their sum, but in the contrary direction. Also, since SB: BT, as BD : DC, and QA : AR : : DC : DA ; therefore, by equality, AQ : BT : : BD :DA ; that is, the forces in A and B are in the reeiprocal ratio of the distances AD, DB, front the right line CD, drawn through C in the direction of the forces.
This theorem is general, and applies equally to the mutual action of three points having any position, whether in a right line or not. But its application to unequal masses makes it much more general, and will lead us to the equilibrium of the lever, centres of oscillation, percussion, Ste. • If the three points do not lie in a straight line, they ivill be in equilibrio only when the distances ex. pressing the sides of the triangle correspond to li mits. Let AE, EB, BA, (Fig. 7.) be distances constituting an assemblage of this kind ; and let AE=EB : let FEOH be an ellipse passing through E, with A and B its foci. Let AN, Fig. I., be equal to the semitranss,;erse DO=BE=AE, and let DB be less than the breadth of the next arcs NP, Fig. 7. ; and the arcs NM, NO, Fig. 7., equal and similar. It is plain, that if the point E were moved to C, the attraction of A in CL, and repul sion of B in CM, would compose a force in CI along the tangent, which would return C to E ; since BC would be as much shorter than at first as A was longer ; and to these equal removals from the inter. section, equal ordinates or forces will correspond.
But 'should the 'point • E be brought to 0, the forces of A and B will be equal and oppOsite, and no motion will arise, unless the point be otherwise 'somewhat removed from it, in which case it will re cede still farther, and pass with an accelerated mo tion toward1 E or H. The points E and H, there fore, are exactly similar to the limits of cohesion in the original curve, Fig I. ; the points F and 0 are limits of non-cohesion. On the other hand, if the distance BC was that of a limit of non-cohesion, the less distance CB would produce an attraction CK ; the greater AC, a repulsion ; and the resulting force CG would make the point C pass to O. So that, in that case, F and 0 would be limits of cohesion, E and H of non-cohesion.
The point C, if removed a little from the peri phery of the ellipse, will return towards it ; for the increasing attractions when it passes without the ellipse, and the increasing repulsion when within it, will compose a force, in either case, tending towards the periphery and the limits of cohcsion. This as• semblage of three points may even serve to give us some idea of solidity, for if any thing should stop the motion of the point B, Fig. 7., while, the point A is made to revolve round it, as from A to A' ; the point E will, in like manner, pass from E to E', still preserving the original form of the. triangle.—But, enough of the system of three points.
The system of four or more points would afford us r a much greater variety, were we carefully to examine them. We shall only observe, that if two points be situated in the foci of an ellipse, and two others at the vertices of the conjugate axis, they will form a kind of square or rhombus ; and if on the four angles. of this square, there be conceived a series of points of the same kind, to any height, some idea may be got of the solid rod, in which, if the base be inclined, the whole superstructure will immediately be moved to one side. And the celerity of will depend partly on the magnitude of the connecting forces : for should that be weak, the upper part of the structure will move more slowly, and the rod will be bent like a switch. And four points may be placed out of the same plane, so that they will powerfully preserve ', their position, even by the help of a single. limit of distance sufficiently powerful : for the four points may be arranged as a triangular which will therefore constitute a kind of particle most tenacious of its shape. Of four of these particles, disposed in another. pyramid, a particle of a second order may be farmed, less firm on account of the greater distance of the primary particles composing it, whence the action of external points upon it will be more un equal. In like manner, of these particles others may be formed of a higher order, still less firm; and thus at length we may arrive at those, which, being much greater, arc more moveable and variable, upon which chemical operations depend, and of which the grosser bodies are composed ; so that we`would arrive at the same thing as Newton has proposed in his last optical query respecting his primary and elementary par ticles, which compose other particles of various or ders.