34 PROFESSOR FLEEMING JENKIN’S APPLICATION OF GRAPHIC METHODS 
between links at the ends of link 4. The problem is rather simpler than that 
described in the last paragraph, masmuch as link 5 of the dynamic frame is not 
loaded. 
We will now consider how this figure is modified by the introduction of 
friction. Beginning with link 1a im fig. 380, we can draw line 5 making the stated 
angle with the surface of the guide bars, and lines L, L,, which are identical 
with the lines of the same name in fig. 37a. We must, however, subdivide 
L, in a new ratio, for it is clear that the line of bearing pressure 2a will pass 
over the friction circle, and so that the trial point where it intersects line 5 
will be some distance to the right ; 28 must pass under its friction circle, and 
the point which 28 must pass through on the left must obviously be near the 
bottom of its friction circle. Draw the line 2, joining two trial points, refer 
L, to the two ends of link 2, or, in other words, subdivide L,, fig. 38), in the ratio 
in which L, subdivides link 2; from the point of subdivision in fig. 384, draw 
2 parallel to 2 in fig. 38a. The intersection of 2 and 5 gives the point 0. The 
piece c is held in equilibrium by four forces L., 6a, 28, and the reaction from the 
bearing: the intersection of 28 with 6 (fig. 38a) gives one point through which 
the loaded link 3 must pass, and our trial point for the other end must obviously 
be a little to the left of the friction circle at the main bearing, where the tangent 
38 cuts line 1, and this tangent 38 must be a little less steep than the line 38 in 
fig. 37b. The load L, is now to be subdivided between the two ends of the loaded 
link 3, the two components being /, and /’, ; the polygon of the forces in equilibrium 
at the upper right hand end of 3 can now be drawn in fig. 380, these are 28, /., 
6a and 3; the two former are known and the directions of the two latter; the 
polygon can therefore be drawn with the lines arranged in the order named, and 
thus the magnitude of 6a can be determined. The problem is now solved, but 
if. we wish to complete our drawing of the frame, we must rearrange the last 
drawn polygon, so that the forces in the reciprocal figure come in their natural 
order as shown by the full lines in fig. 380, then finishing L, we can, as in 
figures 37a and 376 complete the reciprocal figure and frame without diffi- 
culty. If we have chosen our trial points well, the lines of the frame will be 
tangent to the friction circles. If they cut these or do not touch them, we must 
correct our choice of the trial points until the desired figure is found. As the 
friction circles are generally very small, a single trial is generally sufficient for 
a draughtsman who has mastered the theory. The result is really remarkable. 
When the loads have been determined a reciprocal figure of nine lines enables 
us to ascertain the true relation between effort and resistance in a horizontal 
direct acting steam engine, taking into account the weight, inertia, and friction 
of every part of the simple train of joints and elements. 
§ 30. Loaded Dynamic Frame when neither e nor f are attached to the extremities 
of other members of the Machine.—This case presents some geometrical peculiari- 
