PROFESSOR T. A. HEARSON ON THE KINEMATICS OF MACHINES. 
19 
For the motion at the joint of a quadrilateral, which, in the movement, causes a 
continually diminishing angle, it will be convenient to use a small o, to distinguish it 
from the motion which causes an increasing angle, which will be represented by a 
large O. 
In the combination of four O’s, the two large O’s may alternate with the two 
small o’s, in which case the machine movement may be represented by a formula 
written thus ' ^ or conven i ence hi printing the letters may be put in line. 
The links may be inserted thus —0—o—0—o—, and the end links imagined to be 
joined up, or the joining up may actually be exhibited thus 0—o—0—o; or, in many 
cases, it will be sufficient to leave the links themselves to the imagination, and 
show only the sequence of motions thus OoOo. This movement is that which is 
obtained by the mechanism known as “ parallel cranks,” which is employed in loco¬ 
motive engines to connect together two pairs of driving wheels by means of an 
outside coupling rod. 
If the two angles which increase are in sequence, as would be expressed by OOoo 
or OooO, or ooOO, this movement is that belonging to so-called “ anti-parallel 
cranks.” 
If two complete rotations take place at one joint in the time occupied by one at 
another, or in the time taken by a complete swing to and fro, it will be convenient to 
express that fact by a doubled 0, thus, or by O 3 . 
Thus the combination of three O’s and one U may be written thus, o0 3 oU, or 
when reversed thus, Oo 3 OU. Considerations of symmetry indicate that the angular 
velocity at the two single O’s must be equal, and therefore, on account of the 
changing magnitude of the angle at the U joint, it will be alternately greater and 
less than half the angular velocity at the fourth joint. 
As previously described, we may have the movement 0 3 o 3 UU, and also OoUU. 
Where a U motion occurs, as for example between X and Z, fig. 1, the construc¬ 
tion of the links may be modified to the form shown in fig. 2, and yet precisely the 
same relative motion of all the links will be retained. 
For the link X, of fig. 1, a block X, in fig. 2, is substituted, which swings about a 
centre y, which is the centre of curvature of the circular slot way formed in the 
link Z. X may be regarded as being guided by a portion of the surface of a large 
pin, instead of by the whole of the surface of a smaller one, the centre of the pins 
occupying the same relative position in each construction. 
It is easy to conceive the centre of curvature y to be removed to a distance and 
the slotway to become less curved, until, in the limit when y is at an infinite distance, 
the slotway will be straight, and a sliding motion between X and Z will be 
substituted for a swinging motion. The resulting movement may be conceived to 
be that due to a four-bar mechanism, in which X and Z are infinitely long. 
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