68 EQUii.inuir.M <>; m 



/ //// r,,,,'1itions of equ ///A/- iu m of a system of 

 parallel forces acting on a rigid body 



A system of parall can always lc n-duccd to a 



single force and a couple. Since these cannot balance, and 

 neither <>t' tin-in singly can maintain equilibrium, they must. 

 both vanish. That is, 



^/' = 0, and = 0; 

 the latter requires that 



= 0, and 



1 1 nee a system of parallel forces acting on </ //</,'/ IntJt/ will 

 be in equilibrium if the sum of the forces vanishes, awl <ti*<> tin: 

 sum of the moments vanishes with respect to two planes at riyht 

 angles to each other and parallel to the forces. 



Conversely, if the forces are in equilibrium the sum of 

 the forces will vanish, and also the sum of the moments with 

 respect to any two planes at right angles to each other and 

 parallel to the forces. 



71. AVhen 2P=0, the forces reduce to a couple of which 

 the moment is G. When SPis not =0, the forces can always 

 be reduced to a single force; this has already appeared in 

 Art. 66, and may also be shewn thus. The forces will reduce 

 to a resultant U acting at the point (x, y), parallel to the 

 original forces, provided a force E acting at this point will 

 with the given forces maintain equilibrium. The 

 and sufficient conditions for this air, by Art. 70, 



Hence J* = 2P, *'=, 'J = 



These results agree with those of Art. 66. 



7_. To find the resultants of ai< . r offerees a 



on a rigid body in any directions. 



Let the forces be referred to three rectangular axes Ox, Oy, 

 Oz ; and suppose P, , P f , P,,... the forces ; let x lt y lt z t be the 

 co-ordinates of the point of application of P l ; let x^ y y , z 

 be the co-ordinates of the point of application of P f ; and 

 so on. 



