It is now evident that in the process of reduction of a coplanar system no change is made at any stage either in the sum of the projections of the forces on any line or in the sum of their moments about any point.

The formal analytical reduction of a system of coplanar forces is as follows.

The assemblage of parallel forces P can be replaced in general by a single force, and the coplanar system of forces Q by another single force.

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The theorem that any coplanar system of forces can be reduced to a force acting through any assigned point, together with a couple, has an important illustration in the theory of the distribution of shearing stress and bending moment in a horizontal beam, or other structure, subject to vertical extraneous forces.

Again, any coplanar system of forces can be replaced by a single force R acting at any assigned point 0, together with a couple G.

Addition of their several projections agreed with the assumption of Buee and Argand for the case of coplanar lines.

Since CE equals BE these directions are equally inclined to, and coplanar with, the normal to the mirror.

But the symbols of ordinary algebra do not necessarily denote numbers; they may, for instance, be interpreted as coplanar points or vectors.