272 APPLIED SCIENCE 



result of considerable practical value. When the relation between 

 the stresses due to effort and resistance has been found, it is easy 

 to calculate 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 effi- 

 ciencies 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 interdependent. 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 direc- 

 tions 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 EANKINE 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 



