516 Wisconsin Academy of Sciences, Arts, and Letters. 
common point of the two moving links under consideration, and 
will have their centers at the centers of rotation of these links 
with respect to the stationary link. 
To illustrate: In Fig, 1. Take the three links A, B and 
D. Let A be stationary. Then B and D will be the two mov¬ 
ing links. All points in D' revolve about ad with respect to the 
stationary link, all points in B about ab, and all points in B 
and D revolve about the point bd with respect to each other, bd 
being the point common to the two moving links. 
I may substitute then, for the two complex centrodes, the 
two circular centrodes, which will also be in contact at bd and 
will rotate about the centres ad and ab respectively. 
Fully grasping this relation enables one to more readily 
understand this type of problems, and one can therefore read, 
with equal ease, the relative angular velocity ratios of any two 
links in any mechanism, however complicated, however widely 
separated the links may be and whatever link may be held sta¬ 
tionary. 
When the centros do not. fall within the limits of the drawing, 
an application of the more common geometrical solutions is 
employed with greater facility when the above development is 
understood. 
The relative directional rotation of the two links is also 
instantly recognized by noting the position of the contact points 
of the two moving links with respect to their centers of rotation. 
In Fig. 2 as the mechanism moves from position 3 to position 
1, the two rolling circles or centrodes, which are substituted for 
the more complex ones, expand from the two circles in contact 
at bV , through position aaf to IV, and as in rolling circles “The 
angular velocity ratios of the two centrodes (hence of the two 
links) are to each other inversely as the radii of the rolling 
centrodes measured from their common centre to their respective 
centers of rotation.” Ratio V°V: V°b :: Distance between 
points bb r and ab : Distance between points bb' and ab' or 
_ bb'— ad 
~~bb' — ab' 
the sign — meaning “distance between.” 
Likewise since in position l and V the point IV has the same 
velocity whether considered a point in link l or V, “the velocity 
of any two points in any two links may be compared by finding 
