14. PROFESSOR FLEEMING JENKIN’S APPLICATION OF GRAPHIC METHODS 
we shall find reason to consider machines of class 2 rather as half machines 
than complete simple machines. When couples are admitted in place of 
driving and resisting links, the dynamic frame necessarily includes two 
stiff bars, between which the couple or couples act. We then have three 
cases :—1. The resisting and driving couples may act between the same pair of 
stiff bars. This gives a dynamic frame of merely two bars, with the two pairs 
of links by which the couples are exerted. 2. A driving or resisting couple 
between two bars may be combined with a resisting or driving link between 
the same bars. 3. A resisting couple or a driving couple may, in the quadri- 
lateral of figs. 8 and 9, be substituted for any link, the couple being exerted 
between two bars replacing two links, which, together with the link removed, 
form a triangle. When we examine the usual combinations of elementary parts 
forming actual machines, we shall in all cases find that these combinations may 
be represented by a dynamic frame of one of the classes described. 
§ 12. Efficiency of Elements.—The relation between the energy exerted and 
the useful work done in a machine is affected by a loss of energy in transmis- 
sion through the elements, as well as by a loss in transmission past the joints. 
At each joint we may say that only a certain fraction of the energy received is 
transmitted, the remainder being wasted in overcoming useless friction ; the 
fraction transmitted is the measure of the efficiency of the joint (RANKINE). 
Let this fraction be called J. Similarly, let the ratio between the energy 
received and that transmitted by each element be called e, then the efficiency 
of the whole machine consisting of a linear train of joints and elements will be 
the product J,J,J,.... X@,@,¢,.... This formula is, however, of little 
practical use, because the values of J,J,, &c., are materially influenced by the 
directions of the forces at each joint, and these cannot be assumed for one joint 
independently of the others. In other words, the values of J are not inde- 
pendent of one another. ‘The value of the product J,J,J;, . ... . &¢., can only 
be found by solving a large number of troublesome simultaneous equations, or 
by means of the dynamic frame. With respect to ¢¢,¢,.... we must dis- 
tinguish between two cases. 1. Those in which the element is in equilibrium 
under the external forces independently of any progressive change in its own 
form. 2. Those in which the element is not in equilibrium under the forces 
applied at the joints. As an example of the first class, I may take a straight 
rope used to transmit power. Although the rope stretches, yet the whole pull 
at one end is transmitted to the other end, but there is aloss of work, because 
the distance traversed by the driving end is greater than that traversed by the 
following end. In all cases of this kind the values of e are not only inde- 
pendent of one another, but do not affect the values of J. They do not alter 
the relation between effect and resistance, and their aggregate effect is easily 
taken into account; e for each element is a constant fraction, which can be 
