758 



BOSCOVICH'S THEORY. 



Boscovich's 

 theory of 

 Natural 

 Philoso- 

 phy. 



System of 



three 



points. 



Plat* 

 LXV. 



Jig- 2. 



departed, and they will continue therefore to oscil- 

 late in this way for an indefinite length of time. 



Car. 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 infinitum. 



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. 



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 

 ty 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 

 facility, 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. ., 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 the points 

 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- Boscovich's 

 spect to E. theory of 



As to their mutual forces. Let there be assumed p, ' l , u,a 

 in Fig. 1., abscisses in the axis, equal to the straight p i, v . 

 Line's AC, BC, Fig. 2. ; and taking out the correspond- -1-^^ 

 ing ordinates, set off CL if the ordinate to AC be at- 1'i-ate 

 tractive, CN if it be repulsive ; and, in like manner, IXV ' 

 set off for BC, CK, or CM. Then, 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 ase ; although it may be 

 done by an algebraic equation with four indetermi- 

 nate quantities. The locus ad svperficiem is indeed 

 insufficient. Neither is it necessary to express an 

 equation of three indeterminates. A geometrical 

 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 or any of its roots. 



