On ilie Composition and Decomposition (f Forces, Sf 7 



1(5. Co^, J .. To prodiTce an equilibrium with the forces 

 <PyQy we have only to apply ro- the point K a third force 

 • etfual to thvir sum, acting- in a contrary direction, but pa- 

 rallel to A^Por BO: for i his third fofce will be equal and 

 directly opposive to their resultant. 



17. Cor, 2. If after having divided an inflexible right 

 line into any number of equal parts, we apply to? all these 

 ■points of division equal forces the directions of which are 

 parallel aniong themselves ; the resijltant of all these forces 

 passes through the middle of the jight line in a direction 

 parallel to that oF the forces, arid is- equal to their sum. 



For all the particular resultants of these forces taken two 

 and two at equal distances from the middle of the right 

 linti, wjllpa'^s through the timld'le, in the same direction, 

 and each resultant will be equal to the two forces which 

 ate its cbmposantS: therefore, the general resultant will 

 pass thi'ough the middle of the line (16), according to the 

 same direcliOri, and will be equal to the su'm of all the paf- 

 ■tt^ttlar re^u'^taht^', or equal to the ^urh'of all the coiiciposants. 



THEOREM. 



18. If to the extremities of an inflexible right line A3 

 (fi'g. 4.) t\Vo unequal forces P, O, be applied, the directions 

 of which AP,. BQ, are parallel and act the same' way : 



1. Tl^e resultant R is equal to their sum, and its directi6W 

 is parallel to that of the forces. 



2. The point of application C, of the resultant R, divides' 

 the right line AB into two parts which are reciprocally 

 proportional to the forces ; or in buch a manner that 



P:g::BC:AC. 



Demonstration OP First Part. For divide the right 

 line AB into two parts in D, directly proportional to the* 

 forces P, O, and we shall have P : Q : : AD : DB. 



Produce the inflexible line AB both ways, and make 

 AE = AD and BF=BD, also conceive that the two forces 

 P and Q are uniformly distributed through every point of* 

 the rigTit line EP\ and act in directions parallel to the 

 direction of the two forces P, Q ; it is evident that the 

 force P will be distributed through the part DE, and that 

 the force Q will be distributed through the part DF. 



Moreover, the force P will be the resultant of all th^ 

 forces distributed through ED, and O that of all the forced 

 distrilHU'cd through the line DF (17); hence the general 

 resultant of all the forces uniformly distributed tlirough 

 the line EF, will pass through C, the middle of EF, and 

 a^ording to a direction parallel to the direction of the' 



% 4 composants> 



