2 PROFESSOR FLEEMING JENKIN’S APPLICATION OF GRAPHIC METHODS 
late the efficiency of a machine by taking into account the spaces through which 
these stresses act in equal times, so as to show any loss of energy which may 
arise from deformation, such as may be due to the stretching of ropes. 
W. J. MAcquorn RAnKINE introduced the word efficiency to denote the ratio 
of the useful work done by a machine to the whole work done. He showed that 
the efficiency of a train of mechanism is measured by the continued product 
of the efficiencies of all the successive pieces or combinations, and he gave 
methods by which the efficiency of certain elementary pieces and modes of 
connection could be ascertained. These methods assume that in each case 
the effort is known in magnitude, position, and direction; also that the position 
and direction of the resistance are known. The weight of the piece is taken 
into account, but no mention is made of the resistance due to inertia, which 
could, however, be treated in the same manner as the forces due to weight. 
RANKINE did not carry his explanation of the subject so far as to show how 
to find for any actual complete machine the direction of the successive efforts 
and resistances; nor does he draw attention to the fact that they are inter- 
dependent. We cannot determine the efficiency of a whole machine by calcu- 
lating the efficiency of each part separately without regard to its position in 
the machine, for it is this position which determines the directions of the driving 
and resisting efforts. These directions cannot be found by mere kinematic 
analysis, but depend not only on the form of the neighbouring parts, but also 
on their friction, weight, mass, and rigidity. In many cases the chief difficulty 
in determining the efficiency of a machine consists in determining the direction 
and point of application of the effort and resistance to the motion of each part. 
These directions are found by the dynamic frame. The writer has endeavoured 
to take up this subject where RanxINE left it, and to give a general method by 
which the efficiency of the great majority of actual machines can be practically 
calculated. In doing this, he has found the graphic method convenient. 
Since the method adopted is based on a novel analysis of machines, it has 
been found necessary to begin by some elementary definitions. 
§ 2. Elements of Machines.—All machines consist of parts so joined that any 
change in the force with which one part presses against another will produce 
some change in the force with which the other parts are pressed or held 
together. This condition obviously holds good when the parts are rigid, and it 
constitutes the test whether flexible ties or elastic fluids form part of a given 
machine. | 
A loose coil of rope cannot by this definition form part of a machine, but a 
rope is part of a machine, whenever a change in the tension of the rope changes 
the pressures or tensions between other parts of that machine. Similarly, steam 
or water forms part of a given machine whenever a change in the pressure of 
that steam or water changes the force exerted between other parts of that 
