On the Parallelogram of Forces. 30? 



p 



&3= -, then cos. k5=0, .'. =x0 ; but when # = 0, we evidently 



P P P 



have 4=-, .'•&-=-, which gives k = l; hence cc=z cos. 6, (6), 



2 2 2 



P 



and by changing 6 into - — i, (as in (2),) a? into y, we have y = z cos. 



2 



P \ 



- — 6 j , or y — z sin. d, (7) ; by adding the squares of (6) and (7), 



we have (since cos. 2 + sin. 2 d = l,) <z 2 -fy 2 =2 2 , (8); hence it is ev- 

 ident that the resultant is represented in direction and quantity by the 

 diagonal of the rectangle whose adjacents sides denote the corapo- 

 nents x and y. 



Suppose now, that the directions of x and y include any angle a: 

 let z denote their resultant, 6 the angle which its direction makes 

 with that of x ; then by resolving y in the direction of x, and adding 

 Xj we have x-\-y cos. a= the components resolved in the direction of 

 #, but z cos. 6 = the resultant resolved in the same direction, .'. 

 #4-y cos. a=zcos. ; and by resolving the components, and the re- 

 sultant in a direction perpendicular to that of x, we have y sin. a 

 zsln.d; by adding the squares of these equations we have x* -f- 

 2xy cos. a-{-y 2 =zz 2 : hence the resultant is represented in direction 

 ' and quantity by the diagonal of the parallelogram whose adjacent 

 sides denote the components x and y. 



Again, let three forces x, y, z be applied to M, in such a manner 

 that the direction of each is at right angles to the directions of the 

 other two : let r denote their resultant, whose direction makes the 

 angles a, 6, c, with the directions of #, y, z severally ; then we shall 

 have #=rcos. a, y=r cos. 6, z=r cos. c; whose squares, when ad- 

 ded, give (since cos. 2 a + cos. 2 6+cos. 2 c~l,) x 2 +y 2 4-z 2 =r 2 ; 

 hence the resultant is represented in direction and quantity by the di- 

 agonal of the rectangular parallelopiped, whose adjacent sides denote 



the components #, y, z. 



Let us now suppose that any number of forces acting in any direc- 

 tions, are applied to M, to determine the quantity and direction of 

 their resultant. Draw any three rectangular axes denoted by x 9 y, z, 

 through M ; and let r, r% he. denote the forces, a, b. c, the angles 

 which the direction of r makes with the directions of x, y, z respect- 

 ively, and let a\ b\ c r denote the corresponding angles for r', and so 

 on ; let R denote the resultant, A, B, C severally, the angles which 

 its direction makes with the directions of x, y, z. Then by resolv- 



Vol. XXVL— No. 2. 40 



