In treating of the system of three points, the sub ject, if generally stated, is reducible to the two fol lowing problems ; viz. 1. Given the position and distances of these points, to find the forces acting on any one composed of the forces by which it is urged by the others, ..the common law of these forces being given by the first figure ; and, 2. Given the law, to find the motions of these points, each of them being projected with given velocities and directions from given places.
The first problem may be solved with comparative • either geometrically or analytically, by means of the curve of forces. The second, if it be requi „site to define the curves described in every case, either by construction or calculation, exceeds, al though the number of points be only three, the powers of the methods yet known ; and is, in fact, no other than that celebrated problem of three bodies, so much sought after by the most celebrated mathe maticians of our time, and to which, only in some par ticular cases, and with the greatest limitations, they have been able to give any solution.
It may be remarked, that if the three points be A, B, and C, Fig. C., and if the distance of any two of them AB, be bisected in D, DC joined, and one third of it be taken as DE, however thepoints be moved by any projection and their mutual forces, the point E will either be at rest, or move uniform ly in a straight line. This depends on the properties of the centre of gravity. Therefore, if the points be left to themselves, C will approach to E, and D will likewise, with half of the velocity of C ; or else they will recede, or move sidewise ; but still pre serving their relative position and distances with re- E spect to E.
As to their tl ir mutual forces. Let there be assumed in Fig. 1., abscisses in the axis, equal to the straight p lines AC, BC, Fig. 2.; and taking out the correspond- • ing ordinates, set off CL if the ordinate to AC 1 tractive, CN if it be repulsive • and, in like manner, set off for BC, CK, or CM.' ; completing the proper parallelogram, its diagonal CF or CH, CI or CG, will exhibit the direction and magnitude of the resulting force, according as the composing forces are both attractive or both repulsive ; or one attrac tive and the other repulsive.
Now, if the point C be supposed to be found al ways in some indefinite line DE, the resulting force may be found for any number of points in that line ; and these ordinates being set off at right angles to the line DE, a' curve drawn through their vertices would express 'the force of the points A and B, at any point in the direction DEC.
Every new direction would require its particular curve ; and the force acting on C, at any point in the same plane, could only be expressed geometrical ly by the perpendicular distance from that plane to a curve superficies.
But it would be more satisfactory to express, not only the magnitude, but the direction of the result ing force. For which purpose, draw FO at right angles to CD, meeting it in O. One curve may ex press the amount CO of the force, in the direction DEC, for every given distance ; and another the va lue of the perpendicular FO ; taking the ordinate on either side of its line of abscisses, according as the action was towards B or towards A.
The force resulting from the action of any number of points disposed in the same superficies, may, in like manner, be expressed by the perpendicular dis tance at any situation, from a plane to a curve super ficies. If there be any of the points, in such a sys tem, situated out of the plane, the force cannot be expressed geometrically in this way, since solidity is the limit of geometric composition. But we arc sur prised to find a mathematician of Boscovich's emi nence say, that geometry is altogether incapable of expressing the law in that ease ; although it may be done by an algebraic equation with four indetermi nate quantities. The locus ad superflciem is, indeed insufficient. Neither is it necessary to express an equation of three indeterminates. A geometrieal construction is possible for the expression of any al gebraic formula. Each of them implies a process to be performed. And the geometric locus differs as completely from an algebraic equation of two or three variable quantities, as a table of logarithms from a formula for finding them. In this case, the geometric construction for any number of points is obvious. It is merely a continuation of that com position of forces, by which the action of two was discovered. It can, indeed, only become definite by supposing all the points given in position ; we may then find the amount of the force for that position. But the algebraic equation can do no more, since it can only be applied to use by finding an arithmetical value of any of its roots.