On the Composition and Decomposidon of Foi'ces. 331 



two particular resultants (22), and each of these being 

 equal to the sum of those which compose it (25), it fol- 

 lows that the general resultant is ecjuai to the excess of the 

 sum of the forces which act one way, above the sum of 

 the forces which act the contrary way. 



2S. Cor. 3. If the forces P, g, R,'S remain parallel 



among themselves, and without changing in quantity, tak6 

 ?4)other direction, and become/?, q^ r, .9, . . . ., the resultant 

 t of the two first will still pass through E and be equal to 

 their sum p + ^. Likewise thtr resultant v of the three 

 forces p, q^ r passes through the point F, and is equal to 

 p + g-^r, Ii) the same manner the resultant r of the fo^f 

 forces p, q, ^, s will pass through the point G and will be 

 eqtial fo the isum p -{-q + r-^s. Hence the general resultarit 

 of all the forces pj y, r, s . . will alwa^'S pass through the 

 same point as the genera(ttesaltint of the first foi*cfes:PiQ', 

 H, S, &c.... •■-;>•' .*'" '■' • , 



We see then that when the qiiantities and the points of 

 application of parallel forces remain the same, the resultant 

 of these forces always passes through the same point what* 

 ever may be their direction, and the qo-lntiiy of this re- 

 sultant is always equal to their surfj.* * '' '" ■ ' 



The point through which the resultant of parallel forces 

 always passes, whatever may be their direction^ is called 

 the centre of parallel fen' ccs. 



It is easy to perceive that if the points of application 

 A, B, C, D, . . . of the parallel forces P, O, R, S . . . .are in 

 the same plane, the centre of these forces is in the same 

 plane; for this plane contains the right line AB, and con- 

 sequently the point E in this right line, which is the centre 

 of the forces P, Q ; it contains als'o the right line EC, and 

 of course it contain? the centre F of the forces P, Q, R: 

 it also contains the right hue FD, and consequently it con- 

 tains the centre G of the forces P, Q, R, S ; atnd so on. 



J f the points of application are in the same right line, 

 we can demonstrate in the san:e manner that the centre of 

 parallel forces is in the same right line. 



THEOREM. 



29. Two forces applied to the satne body cannot have 

 a resultant, unless their directions concur in the same 

 jpornt, and are contained in the same plane. 



Demonstration. When the directions of two forces 

 i3o not concur in the same point, they cannot be considered 

 as destined to inove a single point; therefore a single force 

 cannot produce the same eilect, and consequently they have 

 iio resultant. thkokiim. 



