336 On the Composition and Decomposition of Forces, 



rcction. as the force R. \Vc find the quantities of the 

 forces P, O, by means of these proportions, 



AD:AB::R:P 



AD : AC : : K : g. 



PROBLEM. 



39. To determine the resultant of any nuniber of forced 

 P, Q, R, S, &c. whose directions, comprised or not com- 

 prised in the same plane, concur in the same point A. 



Demonstration. Upon the given directions lav off, 

 from the point A, the lines AB, AC, AD, AE . . . .'pro- 

 portional to the forces P, Q, R, S respectively ; and 



considering, at first, any two of these forces as P, O, and 

 completing the parallelogram ABFC, the diagonal Kv will 

 represent the quantity and direction of the particular re- 

 sultant T of these forces (36). 



Instead of the forces P and Q we may now take their 

 resultant T, and considering the two forces T, R, repre- 

 sented by AF and AD, complete the parallelogram AP'GD, 

 and the diagonal AG will represent the quantity and di- 

 rection of the resultant V of the forces T, R, or of the 

 three forces P, Q, R. Again, instead of the three forces 

 P, Q, R, we may take their resultant V, and considering 

 the two forces V, S, complete the parallelogram AGHE, 

 the diagonal of which All will represent the quantity and 

 direction of the resultant X of the two forces V, S,'or of 

 the four P, O, R, S. 



In the same manner we may proceed to determine the 

 resultant of any number of forces whatever, and the last 

 resultant will be the general resultant of the system. 



40. Cor* If all the forces P, O, R, S . . . . are applied to 

 the point of concourse A of their directions, to produce an 

 equilibrium, we must first find the quantity and direction 

 of their resultant, and then apply to the point A a force 

 equal and directly opposite to it. But if the forces are 

 applied to other points of their directions, invariably con- 

 nected among themselves; to produce an equilibrium we 

 must apply to any point of the direction of their resultant 

 a force which shall be equal and directly opposite to that 

 of their resultant, provided that the point of application of 

 this force be also connected in an invariable manner to 

 those of the forces P, Q, R, S . . . . 



PROBLEM. 



41. To determine the resultai?t of any number of forces 



i 



