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

For his speculations on sets had already familiarized him with the idea that multiplication might in certain cases not be commutative; so that, as the last term in the above product is made up of the two separate terms ijyz' and jizy', the term would vanish of itself when the factorlines are **coplanar** provided ij = - ji, for it would then assume the form ij(yz' - zy').

**Coplanar** substitution in four hydrogen atoms would involve the pushing apart of the iodine atoms in four horizontal directions.

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

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

By successive applications of (ii) any such **coplanar** system can in general be ~educed to a single resultani acting in a definite line.

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.

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

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.

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.

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.