TO THE DETERMINATION OF THE EFFICIENCY OF MACHINERY. 15 
independently determined, and the fraction expressing the efficiency of the 
machine will be the product of two factors ;—first, the efficiency as found by the 
dynamic frame, and secondly, a coefficient obtained by multiplying together all 
the values of ¢,¢,¢,, &c., for the elements concerned. When the energy thus 
employed acts against a reciprocating resistance in the elements, as where it 
bends the beam of an engine, it is without influence on the whole efficiency for 
a complete cycle of operations, such as a whole revolution of the crank. It 
simply alters the relative efficiency at different periods of the stroke, and may, 
therefore, generally be neglected. Where, however, the lost work is done 
against a non-reciprocating force, as in stretching a rope, it cannot be safely 
neglected, and leads to sensible diminution of efficiency, as where power is 
transmitted by belts and pulleys. Coming to the second class of cases, it is 
clear that when a heavy element is being lifted, lowered, accelerated, or retarded, 
it is not in equilibrium under the action of the external forces at the joints 
calculated in the manner hitherto described; we shall, however, hereafter 
include the forces due to these causes in calculating the forces at the joints, 
and there remains only one mode in which a loss of energy occurs in the course 
of its transmission by an element, namely, its dissipation in overcoming an. 
internal couple. ‘This case finds an illustration in the case of a rope wound on 
to a pulley, or unwound from one ; the pull on the rope is not transmitted in a 
direct line, as we have hitherto supposed, but in consequence of the couple 
required to bend or unbend the rope, the line of action is shifted sideways 
through a length m where mis the moment of the couple, and F the force 
transmitted. This translation of the force affects the values of J for all the 
subsequent joints. It can be represented in the dynamic frame by showing 
the line of action of the force shifted parallel to itself in a disadvantageous 
direction. If in the given problem we know the useful resistance, we must, in 
constructing the diagram, shift the force at the driving end; vice versa, if the 
driving effort is known, we must shift the force at the resisting end. The 
fraction expressing the loss of efficiency due to this cause is not, like that due 
to friction or stretching, independent of the magnitude of the forces involved, 
but will, on the contrary, always involve complete inefficiency when the force 
is very small, and implies a gradually increasing efficiency as the force trans- 
mitted increases. Thus, a small force exerted on a stiff rope passing over a 
pulley produces no effect on the further side, because it is insufficient to bend 
the rope. The loss by an internal couple always diminishes the resistance 
which a given driving effort can overcome, whereas the loss of internal work 
done in overcoming a single force has not this effect. The case of the trans- 
mission of power by fluids in pipes will be examined in a subsequent paper, 
after machines composed of solid parts have been analysed. 
