TO THE DETERMINATION OF THE EFFICIENCY OF MACHINERY. 9 
insensible, or it may be drawn taking friction into account. In the former 
case it will be represented as in fig. 5a, which is obtained as follows :—Link 1 
may first be drawn through the axis of ¢, for we know that the half links at 
the joints ae, eb must lie normally to the surface of these joints (being friction- 
less), and must therefore lie in one straight line, passing through the centre of 
the pins at ae and eb. For similar reasons we draw 5 and 6 through the axes 
of elements d and f The forces exerted by the links d and ¢ on the element 
a, must be balanced by the force due to the third joint a, therefore the direction 
' of this balancing force must pass through the intersection of 1 and 5, and must 
be normal to the surface of the pin at ab. We therefore are now able to 
draw the link 2. For similar reasons we must draw. the link 4 through the centre _ 
of the pin at cb, and through the intersection of the lines 5 dnd 6. The element 
b is in equilibrium under the action of the four forces acting at the four joints ; 
in other words, the resultant of 2 and 4 must be equal and opposite to the 
resultant of 1 and 6, so that we may complete the dynamic frame by drawing 
the link 3 as. shown; this last link, however, would be equally well placed 
as shown in fig. 50, where it joins the intersection of 2 and 6 with that of 1 
and 4. Both the frames shown in fig. 5a or fig. 50, are kinematically equivalent 
to the actual machine shown in fig. 5 im the following sense:—A given 
small contraction of the element ¢ would, supposing all.the other elements to 
be rigid, produce a definite extension of the element / in the actual machine. 
A small contraction in link 1 equal to that in element e would, supposing 
all the other links of the frame inextensible, produce an extension in link 
6 equal to that produced in / in the machine. We may calculate the 
relation between the stresses in ¢ and_/ by the relative rates of their contraction 
and extension, that is to say by the principle of virtual velocities, or we may 
calculate the relative stresses between links 1 and 6 of the frame by the 
ordinary principles of statics, for instance, by a “reciprocal figure.”* The 
ratio between the stresses in e and f, and that between the stresses in 1 and 6, 
would be identical whichever method were adopted. It need hardly be said 
that the method by virtual velocities would be much thé simpler. It is not 
until we wish to take friction into account that the utility of the dynamic frame 
becomes apparent. In order to take friction into account we have to change 
the form of the frame only in this respect, that the links, instead of being normal 
to the surface of each joint, must be inclined so as to make the angle of repose 
with the normal to that joint, and must be so placed that the reaction due to 
the elasticity of the link—or in other words, the stress in the link—may oppose 
the motion of rotation of the pin in the eye of the link ; in brief, the link must 
make the stated angle with the surface of the joint. In the present example, 
* Vide J. Clerk Maxwell on Peeaorocal Figures, Phil. Ba April 1864; and Fleeming Jenkin, 
Trans. Roy. Soc. Ed. vol. xxv. 1869. ' 
VOL. XXVIII. PART I. Tw 
