On the Parallelogram of Forces. 307 
Mat, then coss20—0, .. =c0e but when 2—O, we evidently 
agen cP 
have 4 ue ...k- = _, which gives k=1; hence r=z cos. 4, (6), 
2 2 2 
and by changing 4 into sn (as in (2),) v into y, we have y=zcos. 
(5 - ), or y=z sin. 4, (7); by adding the squares of (6) and (7), 
we have (since cos.2d+sin.?6=1,) 2?-+-y? =z”, (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 compo- 
nents x and y. . 
Suppose now, that the directions of @ 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 
x, we have x+y cos. a= the components resolved in the direction of 
x, but z cos. 6 = the resultant resolved in the same direction, .°. 
x+y cos. a=zcos. 4; and by resolving the components, and the re- 
sultant in a direction perpendicular to that of x, we have y sin. a= 
zsin.é; by adding the squares of these equations we have x?+- 
Qxy cos.a+y?=z?: 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 =r cos. a, y=r cos. b, z=rcos.c; whose squares, when ad- 
ded, give (since cos.2a + cos.2b-+cos.*c=1,) v2? +y? +22? =r? ; 
hence the resultant is represented in direction and quantity by the di- 
agonal of the rectangular parallelopiped, whose adjacent sides denote 
the components @, y, 2. 
Let us now suppose that aay 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 a, y, z, 
through M; and let 7, 7’, &c. denote the forces, a, 6, c, the angles 
which the direction of r makes with the directions of 2, y, z respect- 
ively, and let a’, 6’, c’ denote the corresponding angles for 7’, and so 
on; let R denote the resultant, A, B, C severally, the angles which 
its direction makes with the directions of «, y, z. ‘Then by resolv- 
Vou. XXVI.—No. 2. 40 
