On tJie Composition und Decomposition of Forces, 32t) 



line AB, a force S equal and directly opposite to the re- 

 sultant of the two forces P, O, in such a manner that S = R 

 = P + Q; the three forces P, Q, S, will be in equilibrio 

 (19), and either of the two forces P, g, may be regarded 

 <is equal and-directly opposite* to the resultant of the other 

 two. Hence the resultant of two forces S, O, of whicli 

 the directions are parallel, and which act contrary ways, is 

 a force p, equal and directly opposite to the force P. Now 

 the force P is equal to the difference of the forces S, Q, 

 and acts in a direction contrary to that of the greater S of 

 these two forces; therefore, 1. The resultant p of the two 

 forces S, Q, is equal to their difference S— y, and it acts 

 the same way as the greater, in a direction parallel to that 

 of these two forces. 



Moreover we have P f Q or S : O : : AB : AC (50). 



2. The distances of the point A of application of this 

 resultant from the two points C, B, are reciprocally pro- 

 portional to the forces S, Q. 



23. Remark, If , the ratio of the two forces S, O, and 

 the length of the right line BC are given in numbers, 

 and we would find the distances of the point A from the 

 points B, C, the preceding proportion cannot be directly 

 applied, because in this proportion we know only the two 

 first terms; but from it we easily deduce this by divisioa 

 of ratios; viz. S-Q : Q : : AB-AC or BC ; AC 



in which we know the three first terms. 



We find the distance AB by this other proportion, 

 S-Q' : S : : AB-AC or BC': AB, 



24. Cor, 4. If the two forces S, Q, the directions of 

 which are parallel and which act contrary wavs, are equal 

 between themselves, 1. their resultimt P, which is equal to 

 S— y, (22) becomes nothing; 2. in the,proportion S— Q ; 

 Q : : BC : AC, tiie second term being indelinitely great 

 with respect to the first, which is nothing, the fourth term 

 AC is also indefinitely great with respect to the third. 

 Therefore, the point A of application of the resultant P is 

 at au indefinite distance from the poini; C. Hence, to 

 produce an equilibrium with the forces S, Q, we must ap- 

 ply to the iuilexible right line a force equal nothing, the 

 direction ot" which passes at an indefinite distance: this is 

 ^ot. absurd, but it cannot be executed. 



We see then that it is impossible, by means of a single 

 ftrce, to produce an equilibrium with two equal forces the 

 directions of which are parallel and which act contrary 

 ways ; but by means of two forces we can produce with * 

 them an equihbrium in an indefinite number of ways. 



PJROBLENf 



