dicated by the arrow, fig. 66. The directions and
magnitudes of these two forces are already drawn (fig. 67 a) in a fitting
position to represent part of the polygon of forces at XEFA; beginning with
the upward thrust EX, continuing down XA, and drawing AF parallel to AF in
the frame we complete the polygon by drawing EF parallel to EF in the
frame. The point F is determined by the intersection of the two lines, one
beginning at A, and the other at E. We then have the polygon of forces
EXAF, the reciprocal figure of the lines meeting at that point in the
frame, and representing the forces at the point EXAF; the direction of the
forces on EH and XA being known determines the direction of the forces due
to the elastic reaction of the members AF and EF, showing AF to push as a
strut, while EF is a tie. We have been guided in the selection of the
particular quadrilateral adopted by the rule of arranging the order of the
sides so that the same letters indicate corresponding sides in the diagram
of the frame and its reciprocal. Continuing the construction of the diagram
in the same way, we arrive at fig. 67 d as the complete reciprocal figure
of the frame and forces upon it, and we see that each line in the
reciprocal figure measures the stress on the corresponding member in the
frame, and that the polygon of forces acting at any point, as IJKY, in the
frame is represented by a polygon of the same name in the reciprocal
figure. The direction of the force in each member is easily ascertained by
proceeding in the manner above described. A single known force in a polygon
determines the direction of all the others, as these must all correspond
with arrows pointing the same way round the polygon. Let the arrows be
placed on the frame round each joint, and so as to indicate the direction
of each force on that joint; then when two arrows point to one another on
the same piece, that piece is a tie; when they point from one another the
piece is a strut. It is hardly necessary to say that the forces exerted by
the two ends of any one member must be equal and opposite. This method is
universally applicable where there are no redundant members. The reciprocal
figure for any loaded frame is a complete formula for the stress on every
member of a frame of that particular class with loads on given joints.

[Illustration: FIG. 68]

[Illustration: FIG. 69]

Consider a Warren girder (fig. 68), loaded at the top and bottom joints.
Fig. 69 b is