18 PROFESSOR FLEEMING JENKIN’S APPLICATION OF GRAPHIC METHODS 
link 4 is shifted. When the ropes are long, their efficiency must also be taken 
into account, when our object is to compare energy exerted with work done. 
When we simply wish to compare effort and resistance, the loss of energy due 
to the stretching of the rope may be neglected. Inasmuch as the axes of 
elements ¢ and / are assumed to lie in parallel planes, perpendicular to the axis 
of a, the forces in the elements e¢ and / (unless parallel) give rise to an 
injurious couple on the bearings, which, except when these are very far apart 
relatively to the distance between the planes of ¢ and\/, ee diminishes the 
efficiency of the machine. 
§ 15. Inclined Plane.—The idea involved in problems on the “ inclined plane ” 
is that one element, sliding on another with a plane joint between them, shall 
be employed to maintain equilibrium between forces applied to the sliding 
element in a plane perpendicular to the joint. We may embody this idea in a 
simple complete machine, as shown in fig. 13, where 0 is a fixed element, ¢ 
a driving element jointed to } and a, the sliding piece having a plane joint 
with b ; / the resisting element jointed with / and a: the axes of e and fare ina 
plane perpendicular to the joint a). We have here a self-strained combination — 
fulfilling all the required conditions. The dynamic frame with friction is shown 
in fig. 13¢. Links 1 and 4 are drawn tangent to the friction circles, and link 3 
is drawn from their intersection A, making the stated angle with the plane 
joint ab. The bar 2 may be drawn anywhere, but is conveniently shown 
parallel to the joint ab. When link 1 coincides with link 3 or makes a greater 
angle with the joint ad than link 3 does, the mechanism will not work. | 
§16. The Hanging Pulley.—The hanging pulley becomes a complete simple 
machine, when the driving element and resisting element are attached to a com- 
mon element, as shown in fig. 14; } isthe fixed element, e the rope by which the 
effort is exerted, a the pulley, 7 the element on which useful work is done. Links. 
1 and 4 are drawn for the dynamic frame in two parts as shown; the moment 
of the couple dividing the parts being that required to bend and unbend the 
rope, and its force the force exerted on the rope. The pulley will take up a 
position in which the link 3 drawn tangent to its two friction circles cuts the 
intersection of 1 and 4 at A. 
It is curious to observe that while Professor REvLEAvx has very properly 
rejected the lever, inclined plane, hanging pulley, and wheel and axle, as 
elements of kinematic analysis, nevertheless these so-called mechanical powers 
do furnish the characteristic features of four simple machines of class 2 in which 
the number of elements is restricted to four. We shall find that the wedge 
is a characteristic feature in a simple machine with six dynamic links of class 1- 
These considerations show that dynamical and not kinematical reasoning 
guided the mechanicians who selected the so-called “ powers.” 
817. Example of a complete Machine having a Dynamic Frame of Six Links.— 
