TO THE DETERMINATION OF THE EFFICIENCY OF MACHINERY. 30 
ties. Let the elements of the frame be the four lines a, 6, c, d, shown as thick 
black lines in fig. 39, Plate XIT., and let the elements ¢ and/ be joined to these at 
points intermediate between their extremities. Each element is then in equili- 
brium under the action of three forces, and the simple dynamic frame is the 
quadrilateral 2, 3, 4, 5, having its angles on the prolongations XX, and YY, of 
the directions of the stress in ¢ and f, and having its sides so placed as to pass 
through the joints ab, bc, cd, da, denoted by the letters MNPQ. It is not 
quite obvious how this quadrilateral may be drawn. It may be proved 
that all quadrilaterals of which the angles lie in the lines XX, and YY, and of 
which three sides pass through M, N, and P, have their fourth sides so placed 
as to intersect at one point E; the point E can therefore be found by drawing 
two trial quadrilaterals, and this point can then be joined with Q and so give 
the direction of one side of the desired quadrilateral ABCD. Professor Tarr, 
who pointed out this fact to the writer, also showed that the point E might be 
more simply found as follows :—Produce MN until it intersects e prolonged in 
X, join X with P; similarly, produce NP until it intersects / prolonged in Y, 
and join YM; the point E lies in the intersection of XP with YM; the line 
QE gives the direction and position of one side of the quadrilateral ABCD. 
A second quadrilateral has been drawn on the figure for the sake of illus- 
trating the form which it assumes when the fourth point is g, chosen out- 
side the angle XEY. 
When, as in fig. 40, Plate XI., the four members a, 0, ¢ and d are all loaded, 
the problem becomes still more complex. The octagonal equilibrated polygon for 
the four loads and two stresses in ¢ and /, otherwise named 1 and 6, are shown 
in fig. 40, with lettering analogous to that employed for the simpler cases. This 
polygon was formed in a somewhat indirect manner, and it is probable that a sim- 
pler geometric method may be found if the case should arise frequently in practice. 
(1.) The relation between astress in ¢ and one in / was found by a simple frame 
and reciprocal figure, fig. 41; (2.) The same process was repeated for a stress 
in ¢, and a stress L, between the elements a and d, fig. 42; (3.) The process 
was repeated for a stress L, between 6 and d, fig. 43; (4.) The process was 
repeated for a stress L, between ¢ and d, fig. 44; (5.) By addition the stress in 
e was found which was required to overcome the given stresses due to / and to 
the four loads ; (6.) The several loads were referred to the joints ; (7.) Polygons 
of force were drawn for each joint, and by these polygons the inclinations of 
2B, 38, 48, and 58, fig. 40, were found, the intersections of these lines with 
the loads (including ¢ and /) gave the eight angles of the polygon; (8.) The 
reciprocal figure, fig. 40a, was drawn, by which the work was checked, and the 
inclinations of the sides of the polygon verified. 

