332 On the Composition and Decomposition of Forces^ 



THEOREM. 



30. If the directions of two forces P, Q, applied to two 

 points A, B, (fig. 7 and 8) of the same body, are contained 

 in the same plane and concur in a certain point D : 



Istly. The direction of the resultant of these forces will 

 pass through the point of concourse D ; 



sdly. The direction of the resultant is in the same plauc 

 .as the two forces P, Q, which are its composants. 



Demonstration of I. The point D being found in 

 the direction of both the forces, if we suppose that thi* 

 point is connected to the body in an invariable manner, 

 we may consider the two forces P, Q, instead of being ap- 

 plied to the points A, B, as applied to the point D, and 

 that tiiey have no other cflect than a tendency to move 

 this point; therefore, their resultant may also be consi- 

 dered as having no other eifect. Now a single force can- 

 not net upon a isingle point unless it be immediately applied 

 to this point. Therefore the resultant of the two forces 

 \*, Q, may be regarded as appiied to this point. Hence, 

 the direction of this force passes through the point of con- 

 course of its two composants. 



Demonstkatiojs of the 2D. If at the two points of 

 application A, 13, we conceive an inflexible right line to be 

 attached, in an invariable manner, the effect of the two 

 forces PjQ, and consequently that of their resuhant, is evi- 

 dently a tendencv to niove the right line AB. Now a single 

 force cannot move a right line unless it be immediately 

 applied to some point in this line. Therefore, the resultant 

 of the two forces P, Q, may be considered as applied to 

 some point in the right line AB. Hence the direction of 

 this force passes at the same time thrfi)ugh the point D, and 

 also through a point in the right line AB j it is therefore 

 comprised in the plane of the triangle ABD, determined by 

 the directions of the two composants P, g. 



31. Cor. It follows then that if three forces P,Q, R, 

 applied to the same body, are in equilibrio among them- 

 selves, the directions of these three forces concur in the 

 same point D, and arc comprised in the same plane. 



For these three forces being in equilibrio, any one of 

 them is equal and directly opposite to the resultant of the 

 other two; consequently any two of these forces have a 

 resultant; therefore, (29) the directions of these two force* 

 are comprised i^i the same plane, and concur in the sam© 

 point ; and the direction (30) of the resultant of these twa 

 forces, and conse^^uently that of the third force which 



keeps- 



