ME FLAKE. 47 



he co-ordinates of the point of application of /,; 

 let x., y, be those of the p plication of P 9 , and ao on. 



Let A l be the point of application of /;. At O suppose 

 two forces n; u opposite directions each equal aad 



parallel to 1\. Draw Op, perpendicular t- 



ice /' acting at A , is equivalent to I\ acting at 

 .v couple of which 0i is the arm and each fore* . 

 whit h tendi to turn the body from the axis of 9 to that of 



Hence the moment of the couple is 



-jr.cosoj. 



The other forces may be similarly replaced. Hence the 

 system is equival forces 



acting at O. 



in -T xiralM to those of the original forces; and tbo 



!cs of which the moments are 



p /_ \ 



x t sin <, y t cos o^j, 



_ in the plane of the forces. It will be (bund that any 

 one of the above expressions for the moments of the couple* 

 is positive or negative, according as that couple tends to tarn 



xly from the axis of x towards that of jr, or in the 

 tion. 



Let he resultant of the forces acting at 0, let be 



the angle which R makes with the axis of x, and if the 

 >f the resultant couple; then (by Art. Si) 



/fcosa-SPcosa; JZsina-SPsina; 



by Art 48) 



G - P(x sin a -y cota). 



