290 APPLIED SCIENCE 



friction ; the fraction transmitted is the measure of the efficiency 

 of the joint (RANKINE). Let this fraction be called /. 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^ 2 J 3 . . . . x e l e z e 3 .... This formula is, 

 however, of little practical use, because the values of JjJg, etc., 

 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 / are not inde- 

 pendent of one another. The value of the product J\J Z J 3 , . 

 . . etc., can only be found by solving a large number of trouble- 

 some simultaneous equations, or by means of the dynamic frame. 

 With respect to e^e^e^ .... we must distinguish 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. Al- 

 though the rope stretches, yet the whole pull at one end is trans- 

 mitted to the other end, but there is a loss 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 independent of one another, bat do not 

 affect the values of /. They do not alter the relation between 

 effort and resistance, and their aggregate effect is easily taken 

 into account ; e for each element is a constant fraction, which 

 can be 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 

 eje 2 e 3 , etc., 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, an,d may, 

 therefore, generally be neglected. Where, however, the lost 

 work is done against a non-reciprocating force, as in stretching 



