18 
PROFESSOR T. A. HEARSON ON THE KINEMATICS OF MACHINES. 
and T to lie in the same straight line away from one another, the four links then 
forming a triangle, will require that 
t + Z — <V X .(a). 
Also in order that they may be able to overlie one another, again forming a triangle 
with the two other links, the difference of t and z must not be less than the difference 
of v and x, 
t^z=>v^x .(/3). 
Therefore Law II. may be expressed as follows :— 
In order that two consecutive links may he capable of turning relatively to one another, 
it is necessary that the sum of the lengths of those two links shoidd not he greater, nor 
the difference let>s, than that of the other two links. 
If these two criteria of complete turning motion be applied to each of two joints 
which are opposite one another, for example, the joint s between T and Z, and the 
joint w between V and X, it will follow that for complete turning at those joints, 
either t = x and z = v .( 1 ), 
or t — v and z — x .( 2 ). 
The equalities ( 1 ) are also those which will permit of turning motions at the joints 
u and y, and if t does not exceed z, the equalities ( 2 ) will permit of turning motion 
at the joint u. 
Thus, for four O motions, opposite links must be of equal length. 
For three O motions and one CJ, the two links adjacent to the U must be equal to 
one another, and longer than the other two links, which must also be equal to one 
another. 
It thus happens that in order that there may be O motions at two joints opposite 
one another, it will be necessary to have an O motion at at least one other joint, so 
that a combination consisting of two O motions alternating with two U’s is not 
possible. 
There being 0 motions at the two consecutive joints s and u, then, if t added to the 
length of one of the other links equals the sum of the other two, and, if added to 
each of the others the sum is less, or if t added to each one of the three gives a sum 
less than the sum of the two remaining, then, in either case, there can be TT motions 
only at both w and y, but in the former case it will be possible for two complete revo- 
lutions to occur at s and u contemporaneously with one complete to and fro swing at 
each of w and y. 
If the proportions are such as to satisfy neither set of conditions previously 
mentioned, then it will be possible for the motions at all the four joints to be U 
motions only. 
