$$4 On the Composliion and Dccowposition of Forces* 



the point F of its direction, tends to turn this arc in the 

 eamc manner as the force M (32), and may be substituted 

 for it in order to counterbalance the force P : therefore the 

 two forces P, Q, are also in equilibrio about the fixed point 

 P, and their ;;esultant will be destroyed by the resistance of 

 this point, and consequently the direction of this resuhunt 

 will pass through the point D, 



But we have seen that the resultant of the two forces 

 P, O, passes through the point of concourse A of their 

 dircctioiTS (30) ; therefore this resultant will be directed 

 according to the diagonal AD. 



34. Cor. 1. If from any point D taken in the direction 

 AD oF the resultant of two forces P, Q, we draw the right 

 lines DB, DC, parallel to the directions of these forces, 

 we shall have a parallelogram A BCD, the sides of which 

 AB, AC, are proportional to the forces P, Q; that is, we 

 shail have P : Q : : AB : AC or : : DC : DB. 



For if these sides are not proportional to the forces, their 

 resultant will be directed according to the diagonal of the 

 parallelogram, the sides of which are proportional to these 

 forces (33), and not according to ADj which is contrary 

 to the supposition. 



35. Cor, 2. If from the point D, taken npon the di- 

 rection AD of the resultant of two forces P, Q, we let fall 

 tlie perpendiculars DE, DF, upon the directions of these 

 forces ; these perpendiculars will be reciprocally propor- 

 tional to the forces P, Q. 



For it has been shown above (34) that P : O : : DC : 

 DB j and the triangles DBE, DCF, being similar, give 

 DC :DB : : DFrDE^ 

 therefoFe P : Q : : DF : DE. 



THEOREM. 



36. When the directions, of two forces P, Q, are com-^ 

 prised in the same plane, and concur in a point A (fig. 11); 

 if we take upcn these directions the right lines AB, AC, 

 proportional to these forces, in such a manner that we havt: 



P;Q::AB:AC; 

 and having finished the parallelogram AB DC; the rei- 

 sullant R of these two iorces will be represented in quantity 

 ind direction by the diagonal AD of the parallelogram ; 

 thatift, we shall h^ve ,. 



V > / P : Q : R : ; AB : AC : ADi 

 Df.monstration. We have already seen (33) that the 

 resultant of tb^e tw(i ibrces P, Q, will be directed according 

 to the diagonal AJi) bf the parallelogram; il therefore re- 

 mains. 



