TO THE DETERMINATION OF THE EFFICIENCY OF MACHINERY. rg 
and at that joint by the element removed. The series of forces supplied by 
any one element are necessarily such as would, when balanced by equal and 
_ opposite forces, leave the element in equilibrium. If, therefore, an element has 
only two bearing joints, and is without weight or inertia, the forces it supplies 
must lie in one straight line, and either push or pull, as the member of an actual 
frame does, when under simple tension or compression; the element, in fact, acts 
like one link ofa frame, as shown in fig.2, where the element a might be replaced* 
by the link 1, shown by the line on which arrow-heads are placed. The ideal 
link replacing the actual element. does not necessarily or generally liein the 
geometrical axis of the element. When there are three bearing joints in an 
element having no weight or mass, the three lines of pressure must lie in one 
plane, and either be parallel or intersect in one point. Any one of the forces 
supplied may be looked upon as the equilibrant of the two others. The forces 
which this element supplies might therefore be replaced by three ideal half 
links, each coinciding in position and direction with the line of bearing pressure 
at the three joints, and all connected by an ideal geometrical joint without friction 
at the point of intersection (which, in the case of parallel forces, will be at an 
infinite distance). Thus element 0, fig. 3, may be replaced by the half links. 
1, 2, 3 intersecting at the geometrical joint B, fig. 3a; links 1 and 3 would 
be half links in tension, link 2a half link in compression. The direction of the 
arrows shows the direction of the stress in the links replacing 0. The direc- 
tions of the links 1, 2, and 3 do not coincide with those of the geometrical axes 
of a, c, and d; indeed, these elements may be stiff bars having many other joints ;. 
but if we know that the element d is moving relatively to a, c, and d, as shown 
by the arrow, then one condition determining the directions of links 1, 2, 3 is 
given us, for these directions must make the stated angle with the surfaces of the 
joints da, bc, and bd. In fig. 3a the links 1, 2, and 3 are therefore shown, not 
passing through the centres of the circles at the joints, but passing on that side 
of the centre on which the force represented in the link would resist the motion 
of the pins supposed to be fast on 0. The small arrows, fig. 3a, show the 
direction of rotation of these pins, and these arrows are lettered 4 to indicate 
that they represent the motion of element 6 relatively to a, ¢,and d. This plan 
of indicating the relative motion of the surfaces at joints will be followed in 
future diagrams. It will always be assumed that the pin is ized in the element 
indicated by the letter at the arrow. When there are more than three joints, 
the forces supplied at each joint are such as would be given by a series of half 
links, one for each joint, corresponding with each line of bearing pressure, and 
themselves joined by other links, so as to form a frame which would be in 
* The writer has in this paper ventured to use the verb “to replace” as it is usually employed by 
writers on chemistry, namely as the. translation of the French word “ remplacer.” 
