S26 On the Composition and Decomposition of Forces, 



6lbers the contrary; after having found the smli of all those 

 which act one way, and also the smn of all those which 

 act the contrary way, the resuhant of all these forces is 

 equal to the difference of these sums, and is directed the 

 same way as the greater. 



14. Therefore, to produce an equilibrium with all thestf 

 forces, we must apply to the same point, and in the di-» 

 rection of the less sum, a force equal to the difference of 

 the sums. For this force will be equal and directly oppo- 

 site tatheir resultant. 



THEOREM. 



15. If to the extrt^tnities off dn inflexible right line AB, 

 (fig. 3) two equal forces P, O, be applied, both of which 

 ^ct the same way, and the directi'orts of which AP, BO, are 

 parallel to each other : 



1. The direction of the resultant R of these forces is pa- 

 rallel to AP, BO, and passes through the middle of AB. 



2. The resultant is equal to thelipm P+Q of .jhese twd 

 forces. ^..;^) h.in «£-n.>^ .•^•:.r...ii '^ 



Demonstration of Tf^HE iftks^ Part, li^t anothet' 

 inflexible right line DE perpendicular to the direction of 

 the forces be invariably attached to the right line AB, and 

 produce the direction of the forces P, Q, till they meet the 

 tight line DE in D and E; we may consider these forced as 

 applied at D and E. 



Divide DE into two equal parts in C, and on that side to 

 which the forces tend to move this right line place an im- 

 moveable obstacle at C, and the right line ED will be ct 

 rest 3 for, the parts of the line on each side of the obstacle 

 being equal, there is no reason why one of these equal 

 forces should overcome the other; therefore the resultant 

 will he destroyed by the obstacle C, or the resultant will 

 pass through the point C. In the same manner it may b4 

 shown that the direction of the resultant of the two forces 

 passes through I, the middle of any other right line GH 

 parallel to DE. 



Therefore, it passes at the same time through the two 

 points C, T, which are equally distant from the direction 

 of the forces P, Q, of course it is parallel to them, and 

 passes also through the middle of AB. 



Part II. The direction of the two forces P, Q, and that 

 of their resultant R, being parallel, we may consider them 

 as concurring in a point at au infinite distance, and the 

 two forces P, Q, as both applied to this point: now thcx 

 resultant of two force*! apjiiied to the sair.e point is equal 

 to their-sum (10): therefore tlie resultant of the two forces 

 1*. O. is equal to the sum P-f Q of those forces. 

 '^^^^ ^ 16, Cor. 



