40 



ANY NUMBER OF FORCES IN EQUILIBRIUM. Art. 34 



Fig. 25. 



be the equilibrant of R t and P 2 ; it must, therefore, pass through 



C also. Draw the two force triangles 123 and 453 (with R as a 



common side), in the order in which the lines are numbered; the 



intersections determine the magnitudes of P 2 , E^ and R 2 . We 



thus have a force quadrilateral 1 , the diagonal line representing 



two equal and opposite forces. 



This procedure evidently does not apply when the forces are 



parallel. For a method of solution which will give .Bj and R 2 



and the direction of R lt when P^ and P 2 are fully known, see 



Art. 46, Fig. 39. 



34. Any Number of Forces in Equilibrium. Fig. 25 shows 

 a body acted upon by the force P at m. If at 

 any point n, two equal and opposite forces par- 

 allel to the line of P and each equal to P, be in- 

 troduced they can not change the effect. P is 

 then equivalent to a force P parallel to it acting 

 at any point n and a couple whose moment is Pa. 

 It follows that any force may be considered as 

 acting parallel to itself through any point if a 



couple is introduced whose moment is equal to the product of 



the force into the dis- 



tance through which 



it has been supposed 



to be moved. This prin r- 



ciple is of much prac- 



tical importance. 



Fig. 26 shows a 



number of non-con- 



current forces which 



have been treated in 



the above manner ; 



each one has been 



shifted to a common 



point 0, so that there 



results a system of 



concurrent forces and 



a system of couples. The system of couples may be reduced to a 



.single couple (30) whose moment is 



Pa--Pa etc. 



26. 



'This is the method of Culmann, the founder of graphic statics. 



