34 MOMENT OF ROTATION OF FORCES. [CIIAP. IV. 



The above principle is due to Oulmann, and will be referred 

 to hereafter as Culmanrfs principle. 



37. Application of the above to Equilibrium Polygou. 



Let P M be a number of forces given in position as repre- 

 sented in Fig. 19 (a) PI. 5. By forming the force polygon Fig. 

 19 (b), choosing a pole C, and drawing S S x , S 2 , etc., we form 

 the equilibrium polygon abed ef, Fig. 19 (a). 



The resultant of the forces P w acts in the position and direc- 

 tion given in the Fig. Kow, as we have seen in Art. 22, 

 regarding the broken line a b c d e as a system of strings, we 

 may produce equilibrium by joining any two points as a andy 

 by a line, and applying at a and f the forces S and S 4 . Let us 

 suppose this line a f perpendicular to the direction of the 

 resultant. Since we can suppose the broken line or polygon 

 fastened at any two points we please, this is allowable, and 

 does not aft'ect the generality of our conclusion. 



Then the compression in the line a f is given by H, the 

 " pole distance" or the distance of the pole C from the result- 

 ant in the force polygon. We have therefore at a the force 

 H and V t = H acting as indicated by the arrows. At a then 

 Y! acting up, H and S acting away from a, are in equilibrium, 

 or Y! is decomposed into H and S , as shown by the force 

 polygon. 



According to Culmanrfs principle then, the moment of V x 

 with reference to any point, as m or o, is equal to H x o m. 

 Therefore H being known, the ordinates between a f and S 

 are proportional to the moment of V x at any point. Vj acting 

 upwards gives positive rotation (left to right) with respect 

 to m. 



At the point b, P! may be replaced by a force K parallel to 

 R and a force K 1 along S x [see force polygon]. This we see 

 at once from the force polygon where K and K 1 make a 

 closed polygon with P x , and taken as acting from to K and 

 K to 1, replace P x . But the'se two forces are in equilibrium 

 with S t and S , or 1 C and C [see force polygon], and since 

 K 1 and I K balance each other, all the forces acting at b may 

 be replaced by S , K and K C. We have then at b the force 

 K resolved into components in the directions S and S^ 



By Culmann's principle, therefore, the moment of O K 

 about any point as m, is proportional to the ordinate n m, and 

 since K acts downward this moment is negative. Hence the 



