8 PROFESSOR FLEEMING JENKIN’S APPLICATION OF GRAPHIC METHODS 
equilibrium under the action of external forces equal to those in the half links.* 
Thus any one of these forces may be looked upon as the equilibrant of the 
others, and as acting upon them through a series of links which are subject 
only to compression or tension. In fig. 4, plate II., if the member 6 is jointed with 
members a, c,d, and ¢ by four parallel pins, we might replace 6 in any machine 
by the four half links, fig. 4a, 1, 3, 4 and 5, and a complete link 2, joining the 
intersection of 1 and 5 with that of 3 and 4: this last link will be in equilibrium 
under the action of the forces acting on }, as shown in the half links. In a 
more complicated example the link 2 might be replaced by a complete frame, 
or by a rigid plate. Since the forces acting on one element either intersect at 
one point or in a series of points which may be joined by other links, we may 
always designate the point or points in question by the same letter of the 
alphabet as is used to denote the element. The geometrical intersections will 
be marked by capital letters, whereas the elements are marked by italics. 
When there are several geometrical joints for one element, these points will be 
denoted by the same letter, but distinguished from each other by dots suffixed. 
The substitution of links or half links for an actual element may be effected 
even when forces are parallel, if we admit joints at an infinite distance. If 
now, all the elements of a machine in their relative positions, at any one instant 
be removed in succession, and replaced by their equivalent links or half links, 
we shall substitute for the origmal machine a self-strained frame of links such 
that the stress in each link passing through a joint, will be in all respects equal 
to that on the joint, while the stresses in the driving and resisting links will 
represent the effort and useful resistance in the machine. Each half link at a 
joint of one element is necessarily met and completed by the other half link 
in the same line, due to the reaction of the second element. 
The self-strained frame, composed of links as described above, will be 
called the dynamic frame of the machine with its elements in the given 
relative position. 
§ 8. Hxample.—An example will probably assist in showing what is meant by 
the dynamic frame. Let a machine be composed of the six elements, a, 8, c, 
d, e, f, joined as in fig. 5. Let ¢ be the driving element, and / the resisting element. 
We will suppose in this and in the following examples, that the resulting forces 
all lie in one plane, although the figures may not show the split joints necessary 
to ensure this result. For the present, the effect of weight and inertia will be 
neglected. The machine shown is a complete machine ; the element d has two 
joints, the elements a and ¢ have three joints, and the element 6 has four joints. 
The dynamic frame may be drawn assuming the friction at the joints to be 
* In this frame it might be necessary to include at least one stiff bar or frame to meet opposite 
and equal couples. 
