S30 On the Composition and becomposiiion of Forced, 



PROBLEM. 



25. Any number of forces P, Q, R, S, ^c. the directlonf 

 of which arc parallel, and which act the same way, being 

 applied to the points k, B, C,D, hit. given in position, 

 (fig. 6) and connected in an invariable manner; lo detci^- 

 jnine the resultant of all these forces. • '■'-> '■'^* 



SoLUTroN. Take any two of the forces, as P, Q, ai^rj 

 determine (SO) their resuhant T ; ibis resultant \Vdl be 

 equal to P ; O. and its direction will be parallel to tha't of 

 the two forces P, Q, and we find its point of aj^plication 

 E by the lollowing proportion, P-f O : g : : AB : AE. 



Instead now of the two forces P and O we may substitute 

 their resuhant T, and having dmwn the light line EC, we 

 must determine the resultant V of the two forces T, R ; 

 this resultant V will also be that of the three forces P, Q, 

 R; its quantity equal to T-l R or =P + 04-R, and it's 

 point of application F will be found bv this proportion, 

 T + K or P4Q + R : R : : EC : EF. 



Instead of the three forces P, O, R, we may now subst?- 

 tivte their resultant V, and after having drawn the right 

 line FD, we must find the resultant X of the two forces 

 V, S ; this resuhant X will be that of the four forces P. Q, 

 R, S, and its quantity equal P-fOH-R + S ; and we delere- 

 mine upon FD its point of application G, by the propor- 

 tion, V+S or P + O-f R-fS : S : : FD : FG. 



In the satTie manner we may proceed for any numbei- of 

 forces whatever, and the quantity of the last resuhant will- 

 be equal to the sum of all the forces in t^e system. 



26. Cor. 1 . Henccy by supposing that the point G is 

 invariably connected to the points A, B, C, D . . . we shall 

 have an equilibrium with all the forces P, O, R, S . . . by 

 applying to the point G a force, the direction of which is 

 parallel lo that of the original forces, which acts the contra- 

 ry way, and which is equal to their sum P-f Q + Rh-S . . .v 



' 27. Cor. 2. If among the forces P, Q, iC^, . , . the <ili- 

 rections of which arc parallel, some of them act one way 

 and the remainder the contrary way ; we first determine the 

 particular resultant of those which act oneway, and theii 

 the particular resultant of all those which act the contrary 

 way. For all the forces being reduced to two acting in 

 opposite directions, in determining by art. 23 the result- 

 ant of these two forces we have the getieral resultanl> 

 and, consequently, the force which, if applied in a contrary 

 direction, would keep the whole in equilibrio. The ge^ 

 neral resuhant being equal to the difference between th* 



two 



