TO THE DETERMINATION OF THE EFFICIENCY OF MACHINERY. 3 
machine. This relation between the parts of a machine is dynamic, and is. 
more general than the kinematic relation which exists between the rigid or 
inextensible parts of a machine. A definite position or motion of one part of 
a machine is not in every case accompanied by a definite motion or position of 
another part, irrespective of the forces in action, neither does the same relative 
position of all the parts of a machine necessarily imply the same forces between 
the parts, but a change in the force exerted between any two parts always 
implies a change in the forces between the other parts. - 
The word ¢ement will in this paper be used to designate the continuous 
parts of machines. Each solid element is a part which is continuous in 
respect that no portion can slide upon or break away from another portion in 
immediate contact with it. Any rigid bar or structure in a machine is an element, 
as, for instance, the connecting rod of an engine, the cylinder and bed-plate, 
the crank and fly-wheel. A belt between two pulleys, or a rope on an axle, when 
these form parts of machines, are separate elements. A slack belt or a slack 
rope forms no part of a machine, since it does not fulfil the first condition of 
altering the force at one end when the tension is altered at the other. A con- 
tinuous fluid forming part of a machine will be called a fluid element. A 
portion of fluid is continuous when a change of pressure at one place is trans- 
mitted by the fluid so as to change the pressures throughout the element. 
Thus the steam in the cylinder of a steam engine is a fluid element, and, 
strictly speaking, this element comprises the steam in the pipes and in the 
boiler, as well as the water in the boiler. Generally, we need only consider 
the steam enclosed in the cylinder cut off and bounded by the slide valve. 
Wherever discontinuity of motion occurs, a solid element will be considered as 
terminated or bounded. Thus two rigid bars joined by a flexible tie will be 
treated as three distinct elements. The flexibility of the tie replaces the 
sliding motion at a joint which would otherwise be required. The discontinuity 
of motion here insisted upon as indicating the surface of separation between 
elements is the property by means of which we shall be enabled to determine 
the relative forces with which separate elements press on one another. Two 
elements are treated as separate when at any part they are discontinuous, and 
also in certain limiting cases when the relative motion at the surfaces of 
contact is infinitely small. | 
The elements of a machine as now defined correspond. closely with Professor 
REULEAUX’ kinematic links. All Professor REuLEAUx’ links are elements, 
but the definition now given of a dynamic element would embrace certain parts 
of machines which can hardly be called kinematic links, as, for instance, the 
steam and water mentioned above. Each element of a machine will in what 
follows be designated by a single small italic letter. 
§ 3. Joints.—The name of joints will be given to the surfaces of separation 
