James Clerk Maxwell (1831-1879)

4

                                 MR CLERK MAXWELL ON

Professor FLEEMING JENKIN, in a paper recently published by the Society,
has fully explained the application of the method to the most important cases
occurring in practice.

In the present paper I propose, first, to consider plane diagrams of frames
and of forces in an elementary way, as a practical method of solving questions
about the stresses in actual frameworks, without the use of long calculations.

I shall then discuss the subject in a theoretical point of view, and give a
method of defining reciprocal diagrams analytically, which is applicable to
figures either of two or of three dimensions.

Lastly, I shall extend the method to the investigation of the state of stress
in a continuous body, and shall point out the nature of the function of stress
first discovered by the Astronomer Royal for stresses in two dimensions, extend-
ing the use of such functions to stresses in three dimensions.

                                  On Reciprocal Plane Rectilinear Figures.

  Definition.—Two plane rectilinear figures are reciprocal when they consist
of an equal number of straight lines, so that corresponding lines in the two
figures are at right angles, and corresponding lines which meet in a point in
the one figure form a closed polygon in the other.

Note.—It is often convenient to turn one of the figures round in its own
plane 90°. Corresponding lines are then parallel to each other, and this is
sometimes more convenient in comparing the diagrams by the eye.

Since every polygon in the one figure has three or more sides, every point in
the other figure must have three or more lines meeting in it. Since every line
in the one figure has two, and only two, extremities, every line in the other figure
must be a side of two, and only two, polygons. If either of these figures be taken
to represent the pieces of a frame, the other will represent a system of forces
such that, these forces being applied as tensions or pressures along the correspond-
ing pieces of the frame, every point of the frame will be in equilibrium.

The simplest example is that of a triangular frame without weight, ABC,
jointed at the angles, and acted on by three forces, P, Q, R, applied at the
angles. The directions of these three forces must meet in a point, if the frame
is in equilibrium. We shall denote the lines of the figure by capital letters,
and those of the reciprocal figure by the corresponding small letters; we shall
denote points by the lines which meet in them, and polygons by the lines which
bound them.

Here, then, are three lines, A, B, C, forming a triangle, and three other
lines, P, Q, R, drawn from the angles and meeting in a point. Of these forces
let that along P be given. Draw the first line p of the reciprocal diagram
parallel to P, and of a length representing, on any convenient scale, the force
along P. The forces along P, Q, R are in equilibrium, therefore, if from one