BOSCOVICH'S THEORY; 



163 



Natural 

 Philoso- 

 phy- 



Plat* 

 LXV. 



Swcovich's angle, lying in the internal angles ; or all tend away 

 from the triangle lying in the vertical opposite an- 

 gles : But in ihe second case, with respect to the 

 line joining the two masses, they tend towards the 

 opposite parts from that towards which the force of 

 the first mass is directed. 



Another and more elegant theorem, relating to the 

 directions, is, that the directions of all the three com- 

 pound forces, if produced both ways, will pass through 

 the same point ; and if the point be within the triangle, 

 they tend directly all to it, or all from it ; but if with- 

 out the triangle, two tend directly towards it, and 

 the third from it, or the reverse. 



That all three pass through one point, is thu9 de- 

 monstrated: In any figure, from 13 to 18, which ex- 

 hibits all the different cases above mentioned, let the 



i >g. 13-18. f orce f q j, e tnai w hich enters the triangle, and let 

 the other two HA, QB meet in D. The force be- 

 longing to C is also directed by D. Let CV, Cd be 

 the composing forces, and having drawn CD, let VT 

 be parallel to CA, meeting CD in T : if it be shown 

 that it is equal to Cd, the thing is proved, since, by 

 drawing d'T, we hare d V a parallelogram. Its equa- 

 lity will be seen by considering the ratio of CV to 

 Cd, as compounded of the five ratios CV : BP, 

 BP : PQ; PQ, or BR : AI ; AI or HC : Cd. The 

 first, by calling A, B, C masses, of which these are 

 the centres of gravity, is, from the equality of action 

 and reaction, the ratio B : C. The second sin. PQB 

 or ABD, to sin. PBQ or CBD ; the third A : B ; 

 the fourth sine HAC or CAD, to sine GHA or 

 BAD ; the fifth C : A. The three ratios of the 

 masses compose the ratio BxAxC : BxCxA, 

 a ratio of equality. There remains the ratio sin. 

 ABDx sin.ACD.tosin.CBDx sin. BAD. For sin. 

 ABD and sin. BAD, put AD and BD proportion- 

 al to them ; and v for sin. CAD and sin. CBD, put 



sin.ACDxCD .sin.BCDxCD 



. .p. and ^=r equal to these 



by trigonometry ; and we have the ratio sin. ACD X 

 CD : sin. BCD X CD, that is, sin. ACD or CTV 

 (equal to it sine VI, CA are parallel,) to sin. BCD 

 or VCT, or which is the same, the ratio of CV : VT. 

 Therefore CV : Cd :: CV : VT, or Crf=VT; and 

 'hcrefore CVTD a parallelogram. Q. E. D. 



Cor. Should two of the forces be parallel, the third 

 must also be parallel, and the middle one has the op- 

 posite direction of the other two. 



Cor. If the directions of two forces be given, the 

 third may be found, being drawn through their point 

 of concourse. 



Let us next compare the magnitudes of the forces 

 there immediately occurs this theorem : The acce- 

 lerating forces of any two masses are, in the ratio, 

 compounded of the direct ratio of the sines of the 

 angles, which the line, joining their centres, makes 

 with the lines joining the same centres with the cen- 

 tre of the third, the inverse ratio of the sines of the 

 angles, which the directions of the forces make with 

 the same lines joining them to the third, and the in- 

 verse ratio of the masses. 



For BQ is to AH as BQ : BR, and BR : AI and 

 AI : AH. The first ratio is that of the sines QRB, 

 or CBA, to the sine BQR, or PBQ, or CBD ; the 



ond as A : B ; the third sin. IHA, or HAG, or 



CAD, to the sin. HIA, or CAB : these ratios, 

 changing the order of antecedents and consequents, 

 are the ratios of sin. CBA : sin CAB, which is the 

 first direct ratio ; sin. CAD : sin. CBD, which is the 

 second or inverse ratio, and of the mass A to Bj 

 which is the third and inverse ratio. The demon- 

 stration is the same, if BQ or AH be compared ; and 

 in this demonstration the angles, or their supplements, 

 having the same sines, may be taken indiscriminately. 



From this proposition, a number of elegant corol- 

 laries are derived ; but as they cannot easily be 

 abridged, we refer our learned readers to the work of 

 the author. We shall only observe, that the properties 

 of the lever, and of the equilibrium of forces acting in 

 the same plane, are derived with facility, independent 

 of the usual, but unphilosophical, supposition of in- 

 flexible connecting lines, destitute of all force but 

 cohesion. With equal ease, he derives the proper- 

 ties of the centres of oscillation, conversion, and per- 

 cussion. But ere we take leave of this part of the 

 subject, we cannot refrain from offering to the atten- 

 tion of the reader, the solution of the following pro- 

 blem, respecting the equilibrium of two masses con- 

 nected by two other points, since all that relates to 

 momentum and equilibrium in the lever is compre- 

 hended in it. 



Let there be any number of points of matter in A, 

 which call A, and any number in D, which call D. 

 Let all these points be solicited in the directions AZ, 

 DX, parallel to the given straight line CF, however 

 different may be the forces. Let there be in C and 

 B two points, which mutually act on each other, and 

 on the points situated in A, B, and by these actions, 

 ^hinder all action of the forces in A and B, and all mo- 

 tion of the point B ; the motion of C being prevent- 

 ed by the contrary action of some fulcrum upon which 

 it acts, according to the direction compounded of all 

 the forces it has. Required the ratio of the sum of 

 the forces at A and D must have to this, that the 

 equilibrium may exist, and likewise the magnitude 

 and direction of the force exerted on the fulcrum at 



c. 



Let AZ, DX express the parallel forces of all the 

 points in A and D. That these may be opposed, there 

 should be equal and contrary forces at these points, 

 viz. AG, and DK. These must arise from the ac- 

 tions of the points C and B, according to the right 

 lines AC and AB on A, and on D according to DC 

 and DB. Having drawn GI, GH parallel to BA, 

 AC, it is plain that the force AG must be com- 

 posed of A I and AH, of which the first repels any 

 point in A from C, and the second attracts it to B. 

 On account therefore of the equality of action and 

 reaction, the point C will be repelled from A, and B 

 will be attracted : in like manner C will be repelled 

 from D, and B attracted. The point C therefore 

 has two forces, one in the direction AC, and equal 

 to IA drawn into A ; the other equal to DM into 

 D, and in the direction CD : in like manner, B is af- 

 fected by the two attractions HA x A, and LD X D 

 The force resulting at B ought to be equal, and op- 

 posite to the resulting force at C. It has therefore 

 the direction BC, when the point C is within the an- 

 gle ABC, and the reverse when without it : and to 

 produce the equivalent reaction in CB, we must give 



Boscovich j 

 theory of 

 Natural 

 Philoso- 

 phy- 



i y 9 



Plate 

 LXV. 



Fig. 13-18. 



Equilibri- 

 um. 



Plate 

 LXV. 



Fig. 19. 



