CHAP. H.] DIFFERENT POINTS OF APPLICATION. 17 



this plane and hold the forces in equilibrium f The direction 

 of action of the resultant is given at once from the force poly- 

 gon [Art. 5 (&)]. It must act in a direction from 2 to 0, and 

 must be equal to 2 0, taken to the scale of force. ISTow at any 

 point in the line of direction of P 1? as for instance 1, let us 

 suppose the component given by C to act. What is then the 

 resultant of P t and CO? A glance at the force polygon gives 

 us 1 C, because this line closes the polygon made by C 0, 1 

 and 1C. At 1 then, the three forces S (parallel and equal to 

 C 0) Si (parallel and equal to 1 C) and P x are in equilibrium, 

 and there is no tendency of the point 1 to move. But 1 C or 



51 may be considered as acting in the plane at any point in its 

 line of direction ; therefore at 2 its intersection with P 2 pro- 

 longed. Suppose at 2, S 2 or 2 C to act. We see at once from 

 the force polygon that 2 C, C 1 and P 2 are in equilibrium. 

 There is therefore no tendency of the point 2 to move, and the 

 two forces P x P 2 are then in equilibrium with C 0, 1 C, C 1 

 and 2 C. J3ut since the resultant of C and 2 C or of S and 



5 2 is also the resultant of the forces, and since it must there- 

 fore act through the point of intersection of S and S 2 ; we 

 have only to prolong these lines to intersection b. Through 

 this point the resultants R^ must pass and acting downwards 

 (from to 2) as indicated in the Fig., it replaces P! P 2 . Act- 

 ing upwards it would hold them in equilibrium. We thus 

 easily find the point 2 in the plane at which 2 C or S 2 must 

 be applied, when C or S acts at 1, and S S 2 are thus found 

 in proper relative position. The position, intensity, and direc- 

 tion of the resultant are thus completely determined. 



Had we taken any other point than 1, as the point of applica- 

 tion of C 0, we should have found a different corresponding 

 point for application of 2 C, but in any case the prolongations 

 of 2 C and C would intersect upon the line a b, prolonged if 

 necessary. The same holds true for any position of the " pole " 

 C. This construction is evidently general whatever the posi- 

 tion or whatever the number of the forces. We may thus 

 obtain any number of points along the line a b ; that is, the 

 resultant also, may act at any point in its line of direction. 



[NOTE. That b is a point in the resultant of PX andP z can 

 be proved in a method purely geometrical. In the two " com- 

 plete quadrilaterals " 1 2 C and 1 b 2 a, the jive pairs of 

 corresponding sides 1 and a 1, 1 2 and a 2, 2 C and b 2, C 

 2 



