
32 PROFESSOR FLEEMING JENKIN’S APPLICATION OF GRAPHIC METHODS 
were enabled to draw the octagonal polygon described in the last paragraph 
depended on our knowledge of the four points which determined the position 
of four angles of the polygon, or one point on each of the eight lines. When 
we try to ascertain the actual lines of bearing pressure, taking into account 
the friction of the machine in motion, we find that the conditions determining 
their direction are more complex, since now we do not know any fixed point in 
any line. The conditions are, however, only changed to this extent, that the 
lines of bearing pressure must make the stated angle with the joints, instead 
of being normal to the surfaces of those joints. By trial, an octagon is easily 
found fulfilling this condition as well as the general condition of being in equi- 
librium under the forces applied at the joints. The manner of proceeding 
which seems most easy is to draw, first the polygon without friction, and then 
to sketch a modified polygon having its sides tangent to the friction circles, or 
making the stated angle with the joints. The sides of this trial polygon 
intersect at certain points which may be called trial points. When the friction 
circles are not very large, it is easy, by the exercise of a little judgment, to draw 
the trial polygon so as to make these trial points agree very closely with the 
true points of intersection, even at the first attempt. Then, referring the loads 
to the trial points, we draw a new polygon and reciprocal figure, figs. 36 and 
36a, as for the frame without friction. If this second polygon has sides which 
make the stated angle with the joints, the problem is solved. Otherwise, a 
second selection of corrected trial points must be made, and a third trial poly- 
gon drawn : it will seldom if ever be necessary to make a third trial. We thus 
get a dynamic frame which truly represents the directions of the forces at every 
joint in the actual machine, and this frame will be called the complete dynamic 
Srame of the machine, or the loaded dynamic frame with friction. The resist- 
ance which can be overcome by a given effort 1 in the driving link, is shown by 
the line 6 in the reciprocal figure 36a, which has been used as an auxiliary in 
drawing the loaded dynamic frame, and this resistance will be the actual 
resistance which could be overcome by the given effort in the given machine, 
under the given conditions as to speed, friction, mass, and weight. 
§ 29. Application of the Method to an ordinary Horizontal Single-Acting 
Steam Engine.—In fig. 37, let the lines 6 and ¢ represent the centre lines of the 
connecting rod and crank of an engine, while the line @ represents the direction 
of the motion of the piston. Let the line f represent the direction and position 
of the resistance overcome ; and let this resistance be represented as in previous 
examples by a stress between d and c. Let the lines L,, L,, L,, and L, 
represent the loads on the elements a, b, c, and d of this engine, where d is the 
frame, it being remembered that these loads must be such as would balance one 
another; in other words, L,, the resultant reaction due to the foundation, must 
be the equilibrant of the three others; L, is the weight of the balanced fly-wheel 
