e, and each space between any two forces must receive a distinctive
letter; this method of lettering was first proposed by O. Henrici and R. H.
Bow (_Economics of Construction_), and is convenient in applying the theory
of reciprocal figures to the computation of stresses on frames.

34. _Reciprocal Figures._--J. Clerk Maxwell gave (_Phil. Mag. 1864_) the
following definition of reciprocal figures:--"Two plane figures are
reciprocal when they consist of an equal number of lines so that
corresponding lines in the two figures are parallel, and corresponding
lines which converge to a point in one figure form a closed polygon in the
other."

Let a frame (without redundant members), and the external forces which keep
it in equilibrium, be represented by a diagram constituting one of these
two plane figures, then the lines in the other plane figure or the
reciprocal will represent in direction and magnitude the forces between the
joints of the frame, and, consequently, the stress on each member, as will
now be explained.

Reciprocal figures are easily drawn by following definite rules, and afford
therefore a simple method of computing the stresses on members of a frame.

The external forces on a frame or bridge in equilibrium under those forces
may, by a well-known proposition in statics, be represented by a closed
polygon, each side of which is parallel to one force, and represents the
force in magnitude as well as in direction. The sides of the polygon may be
arranged in any order, provided care is taken so to draw them that in
passing round the polygon in one direction this direction may for each side
correspond to the direction of the force which it represents.

[Illustration: FIG. 66.]

This polygon of forces may, by a slight extension of the above definition,
be called the _reciprocal figure_ of the external forces, if the sides are
arranged in the same order as that of the joints on which they act, so that
if the joints and forces be numbered 1, 2, 3, 4, &c., passing round the
outside of the frame in one direction, and returning at last to joint 1,
then in the polygon the side representing the force 2 will be next the side
representing the force 1, and will be followed by the side representing the
force 3, and so forth. [v.04 p.0555] This polygon falls under the
definition of a reciprocal figure given by Clerk Maxwell, if we consider
the frame as a point in equilibrium under the external forces.

Fig. 66 shows a