Composition and Decomposition of Forces. 33 



59. Since we have before shown that any two forces and 

 their resultant may be represented by the sides and diagonal of 

 a parallelogram, formed upon the directions of these forces, if p, 

 q, be two forces, represented by the lines AB, AD, in which case 

 their resultant g would be represented by AC, any point F being 

 taken in the plane of these three forces without the angle BAD, 

 and without the vertical angle KAL, we have always 



9 xFG=pxFE + qX FH-, 



and when the point F is taken in the angle BAD, or in the vertical 

 'angle KAL, we shall have in like manner 



p X FG = q X FH p X FE. 



60. The product of a force by the distance of its directioa 

 from a fixed point is called the moment of this force. Thus 

 q x FH is the moment of the force q ; and p x FG is the mo- 

 ment of the force p. 



61. As a force is estimated by its quantity of motion, that 

 is, by the product of a determinate mass into the velocity 

 which it is capable of giving to this mass, the moment of any 

 force has for its measure the product of a mass by its velocity, 

 and by the distance of its direction from a fixed point. 



62. If the perpendiculars FH, FG, FE, are considered as 

 lines inflexible and without mass, connected together and fixed 

 to the point F, in such a manner as to admit only of their turning 

 about this point 5 and we suppose that the forces p, q, and their 

 resultant p, are applied at the extremities E, H, G, we shall see 

 that these three forces tend each to turn the system in the same FJ 

 direction about the point F; and that the two forces q, Q, tend p . ' 

 to turn the system in a different direction from that in which the 



the force p tends to turn it. 



We infer, therefore, that the moment of the resultant, taken with 

 respect to any fixed point F, is always equal to the sum or to the differ- 

 ence of the moments of the two components, according as these compo- 

 nents tend to turn the body or system in the same direction, or in op~ 

 posite directions, about this fixed point. 



63. We conclude, moreover, that in general, whatever be the 

 number of forces p, q, r, -or, fyc., and whatever their magnitudes anc2 Fig 2 ^ 



Mech. 5 



