PROFESSOR T. A. HEARSON ON THE KINEMATICS OF MACHINES. 
17 
fro. Such a motion will be referred to as a swinging motion, and represented by U. 
Which of these two elementary motions occur at any joint will depend on the propor¬ 
tions which the lengths of the links bear to one another. 
There are two geometrical laws which govern the association of these two motions 
in this mechanism, and which also apply to other derived mechanisms. 
Law I. The sum of the four angles of the plane quadrilateral, s u w y of fig. 1, 
is constant. 
Law Ilf has to do with the proportions necessary to admit of the complete rotation 
of one link relatively to the next in sequence. It rests on the established fact that 
one side of a triangle cannot be greater than the sum of the other two. The expres¬ 
sion of this law will be given later. 
In tig. 1 suppose the links V and Z to be capable of turning continuously around 
relatively to T in the direction of the arrow, then, with every revolution, the angle 
at s may be regarded as having been increased by the amount of 2tt, and when, in the 
movement, the angle between the two links V and T is reduced to zero by w 
coinciding with w. z , further movement in the same direction may be regarded as 
causing the angle to become negative. 
Regarded in this way Law I. will still hold good, the sum of the four angles 
estimated from any initial position will be constant. 
It will follow from, this that it is impossible to have one O motion only in the 
mechanism, and the other three U motions, or otherwise the sum of the four angles 
would be increased or diminished by 2tt each revolution. 
The law permits of there being four O motions. At two of the joints the angles 
will have to increase, and at the other two to diminish at the same rate, unless it be 
conceived possible for the rate of turning at one joint to be equal to that of the sum 
of the other three. The latter imagined movement will be found not to satisfy the 
conditions imposed by Law II. 
Also Law I. permits the combination of three O’s and one U if the rate of turning 
at one of the O’s is on the average equal to the sum of the rates at the other two. 
Such a movement will be found to be not precluded by Law II. Further there may 
be two O’s and two U’s. 
The capability of complete rotation at any joint will depend on the possibility of 
tne adjacent links getting into the critical positions to lie along the same straight 
line. 
1st. Away from one another. 
2nd. Overlying one another. 
Suppose Z- to be capable of turning completely around relatively to T, then for Z 
* Frequent reference to fig. 1 may be avoided if it is noticed that the small letters for the joints, and 
the capital letters which denote the intervening links, are in alphabetical order. Small letters t v x z 
printed in italics are used to represent the length of the corresponding links. 
MDCOCXCVJ,—A, D 
