BOSCOVICH'S THEORY. 



759 



Natural 

 Philoso- 

 phy. 



Pr.ATE 

 LXV. 



Fig 2. 



Boscovich'j All this while, we have supposed the points A and 

 theory of 5 to t, e 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, and 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 c a at right angles 

 to DC, and making the perpendicular dh equal Ida, 

 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, layoffs A 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 



Plate 

 I XV. 



rig. 3. 



Pl.ATI 



I. XV. 



: is. 1. 



excess in attraction. The curve will pass through Boscovich'' 

 the point D, and the directions will be changed as il >eol 7, of 

 in the former case. If a perpendicular be drawn p^i- 

 through B, it will Le an asymptote to the curve on ph Y . 

 either side, since the repulsion of B will prevent ab- * y--*/ 

 solute contact. There may be several limits or in- 

 tersections, either between A and B, 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 Plat* 

 line, their mutual action will be 0, if the three dis- LXV. 

 tances AD, DB, AB, be each the distances of limits. Fi S- 5 ' 

 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. 



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 cither extreme point, if the 

 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 ADB, 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 two forces AQ, BT be perpendicular to AB, Fig. 5r 

 or parallel to one another, the third force CF mill 

 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 CKzzVB, the 

 triangle CIK is equal and similar to BTV or TBS ; 

 and, therefore CI=BT, and IK=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 reciprocal 

 ratio of the distances AD, DB,from the right line 

 CD, drawn through C in the direction of the forces. 



