74 COM'IlloN FOB A SINGLE RESULTANT. 



In Art. 7-J we have r 



to a force R and a couple '/. It (} vanish there remains a 

 single force; and if // vanish, a sin-le c..uj.lr. If neit 

 nor O vanish the forces may reduce to a single force 

 to shew when this is possible. 



To jiit'l tic cr>n<litiu nm<>n'j the forces that they may 1m 

 single resultant. 



Any system of forces can be reduced to a single fore. //and 

 iple G- if thru the forces can be reduced to a single 

 result ' follows that G, R, and N are in equilibrium. 



If R and 8 do not form a couple, they can be r< du ,-<l to a 

 couple ft' and a force R ; therefore L' must balance the emiplc 

 compounded of G and G'. This is impossible by Art. i<>. 

 Ib-nee 7/ and N must form a couple, and this couple must 

 have its plane coincident with that of G, or parallel t that 

 of G, in order that it may balance G. Therefore that the 

 forces may have a single resultant, the direction <>f // mu-t be 

 parallel to the plane of G, or coincident with it ; that is, must 

 be at n't/ It t 'iityles to the axis of G. Hence, usin_r the aol 

 of Art 72, 



cos a cos X 4- cos I cos fi -f cos c cos v = 0, 

 therefore L2X+ J/2 Y+ N^Z= 0. 



80. Conversely, if L* X + ATS F+ STZZ= 0, and ^X, 1 ) ', 

 > not all vanish, the forces can be reduced to a single 



For the plane of the couple G may be made to contain the 

 force It, and the couple may be supposed to have each of its 



forces =R and its arm consequently = . ; the couple may 



then be turned round in its own plane until the force at one 

 end of its arm balances the resultant force R, and there re- 

 mains R at the other end of the arm. 



81. When the forces are reducible to a single resultant, to 

 find the equations to the strai'<jl,t I in, in /'/,/,-/, // acts. 



Let L, M, N denote the moments of the forces round the 

 co-ordinate axes; L', M' t N' the moments of the forces round 



