66, 2nd ed. 1875), made great
use of diagrams of forces, some of which, however, are not
reciprocal. Maurice Levy in his _Statique graphique_ (1874) has
treated the whole subject in an elementary but copious manner, and R.
H. Bow, in his _The Economics of Construction in Relation to Framed
Structures_ (1873), materially simplified the process of drawing a
diagram of stress reciprocal to a given frame acted on by a system of
equilibrating external forces.

[Illustration: FIG. 1.--Diagram of Configuration.]

Instead of lettering the joints of the frame, as is usually done, or
the links of the frame, as was the custom of Clerk Maxwell, Bow places
a letter in each of the polygonal areas enclosed by the links of the
frame, and also in each of the divisions of surrounding space as
separated by the lines of action of the external forces. When one link
of the frame crosses another, the point of apparent intersection of
the links is treated as if it were a real joint, and the stresses of
each of the intersecting links are represented twice in the diagram of
stress, as the opposite sides of the parallelogram which corresponds
to the point of intersection.

This method is followed in the lettering of the diagram of
configuration (fig. 1), and the diagram of stress (fig. 2) of the
linkwork which Professor Sylvester has called a quadruplane.

In fig. 1 the real joints are distinguished from the places where one
link appears to cross another by the little circles O, P, Q, R, S, T,
V. The four links RSTV form a "contraparallelogram" in which RS = TV
and RV = ST. The triangles ROS, RPV, TQS are similar to each other. A
fourth triangle (TNV), not drawn in the figure, would complete the
quadruplane. The four points O, P, N, Q form a parallelogram whose
angle POQ is constant and equal to [pi] - SOR. The product of the
distances OP and OQ is constant. The linkwork may be fixed at O. If
any figure is traced by P, Q will trace the inverse figure, but turned
round O through the constant angle POQ. In the diagram forces Pp, Qq
are balanced by the force Co at the fixed point. The forces Pp and Qq
are necessarily inversely as OP and OQ, and make equal angles with
those lines.

[Illustration: Fig. 2.--Diagram of Stress.]

Every closed area formed by the links or the external forces in the
diagram of configuration is marked by a letter which corresponds to a
point o