P1/W = L1H/HK = sin (α − λ)/cos (θ + λ),
P2/W = L2H/HK = sin (α + λ)/cos (θ − λ).
It appears, moreover, that if θ be varied P will be least when L1H is at right angles to KL1, in which case P1 = W sin (α − λ), corresponding to θ = −λ.
Just as two or more forces can be combined into a single resultant, so a single force may be resolved into components acting in assigned directions. Thus a force can be uniquely resolved into two components acting in two assigned directions in the same plane with it by an inversion of the parallelogram construction of fig. 1. If, as is usually most convenient, the two assigned directions are at right angles, the two components of a force P will be P cos θ, P sin θ, where θ is the inclination of P to the direction of the former component. This leads to formulae for the analytical reduction of a system of coplanar forces acting on a particle. Adopting rectangular axes Ox, Oy, in the plane of the forces, and distinguishing the various forces of the system by suffixes, we can replace the system by two forces X, Y, in the direction of co-ordinate axes; viz.—
X = P1 cos θ1 + P2 cos θ2 + ... = Σ (P cos θ),
Y = P1 sin θ1 + P2 sin θ2 + ... = Σ (P sin θ).
(1)
These two forces X, Y, may be combined into a single resultant R making an angle φ with Ox, provided
X = R cos φ, Y = R sin φ,
(2)
whence