26 FOECE8 IN THE SAME PLAtfE. [CHAP. II. 



holds the other forces in equilibrium. But now the equilibrium 

 polygon evidently will not close. On the contrary, the first and 

 last strings will be parallel. This position of the pole should 

 then in general be avoided. For any other position of the pole 

 our rule holds good ; viz., 



If the force polygon closes as also the equilibrium polygon, 

 the forces are in equilibrium. If the equilibrium polygon 

 however does not close, the forces cannot be replaced by a single 

 force but only by a couple. The forces of this couple act in 

 the parallel end lines of the equilibrium polygon, and are given 

 in intensity and direction of action by the line from the pole 

 to the beginning of the force polygon [beginning and end coin- 

 ciding]. 



26. Relation between two equilibrium polygons with 

 different poles. We may deduce an interesting relation be- 

 tween the two equilibrium polygons formed by choosing differ- 

 ent poles, with the same forces and force polygon. 



Thus with the forces P! P 2 P 8 P 4 , we construct the force 

 polygon Fig. 14 (a), PI. 4. Then choose a pole C and draw S M , 

 and thus obtain the corresponding equilibrium polygon S a b c d 

 S 4 Fig. 14 (b). Choose now a second pole C'. Draw SV* and 

 construct the corresponding polygon S'o a! b' c' d' S' 4 . [In our 

 figure c and c' fall accidentally nearly together.] 



Join the two poles by a line CO'. Then any two corre- 

 sponding strings of these two polygons intersect upon the same 

 straight line M N parallel to C C'. Thus S and S' intersect 

 at <7, S'i and S t at &, S' 2 and S 8 at Z, S' 8 and S 3 at n, S' 4 and S 4 at 

 m and all these points g, k, I, n and m, lie in the same 

 straight line M N parallel to the line C C' connecting the 

 poles. 



The proof is as follows.* If we decompose P t into the com- 

 ponents S B! and S' S'i, these components are given in inten- 

 sity and direction by the corresponding lines in the force poly- 

 gon. If we take the two first as acting in opposite directions 

 from the two last, they hold these last in equilibrium. The 

 resultant therefore of any two as S and S' must be equal and 

 opposed to that of the remaining two, S^ and S\, and both re- 

 sultants must lie in the same straight line. This straight line 

 must evidently be the line g k joining the intersections of S S'j 



* Elements der Qraphfochen Statik. Bauschinger. Miincken, 1871. Pp. 

 18-19. 



