12 PROFESSOR FLEEMING JENKIN’S APPLICATION OF GRAPHIC METHODS 
producing exactly equal and opposite stresses between a and b in two parallel 
directions. Fig. 7a shows the dynamic frame for this machine with friction. First 
we draw the links 1 and 1a tangent to the friction circles for the joints ea eb e,a e,b 
The distance between the links 1 and 1a shows the arm of the driving couple as 
diminished by friction ; next we draw links 4 and 5 by the rules already given, 
then, remembering that the force in link 5 at joint ad must produce an equal 
and parallel bearing pressure on the pin at the joint a4 we draw this bearing 
pressure tangent to the friction circle, so as to make the stated angle with the 
_ surface of the joint ; we also draw the -bearing pressure due to element 4 on © 
another part of the same pin, the intersection of these lines of pressure gives the | 
dynamic joint through which the pull of link 6 must be exerted ; this link is now 
drawn, and the diagram completed by the two bars drawn perpendicular to links 
4 and 5 respectively. Let P, be the force of the original couple, and D the 
distance between link 1 and 1a. Let A be the distance between the lines of 
bearing pressure on. element a@; then P;, the stress in link 5 is given by the 
P,D PD 
expression P,=—7-. Similarly P, = —>. The stresses in links 4 and 5 
being known give the stress in link 6 by the simple composition of forces. The 
portion of fig. 7 referring to the material means of producing the two couples, 
does not necessarily belong to the diagram; the couple between @ and 6 
may in certain cases be produced by some other machine, being, in fact, the 
resisting couple of that other machine, and in that case the efficiency of the means 
of producing the couple must be determined by an examination of the first 
or driving machine. This case is, in fact, one case of compound machinery, 
and will be treated hereafter. In what follows, a driving couple may be 
occasionally described as existing between two elements, without reference to 
the mode in which it is applied; a resisting couple may be spoken of in the 
same manner. 
§10. Assumption that the Links of a Frame le tn one Plane.—One object of 
our investigation is to find a means of ascertaining the efficiency of any 
mechanical arrangement—the word efficiency being understood in the sense 
given it by RANKINE—as the ratio of the useful work done in a machine to 
the whole work or energy expended. Now, Rankine (“ Millwork,” §371 A) 
has pointed out certain conditions which must be fulfilled to give the highest 
- efficiency in any design, viz.:—First, that the useful resistance to the motion 
of any element, the effort to move it and the force due to the weight of the part 
must lie nearly in one plane, or else act in directions parallel to one another ; 
and secondly, that the acting parts must not overhang the bearings. Injurious 
couples are introduced if these conditions are not fulfilled. RANKIN®’s condi- , 
tions are, however, generally fulfilled in all important designs of machinery, and ~ 
when this is the case we may, without serious error, assume all the actions to 
