Zimmerman—Instant Angular and Linear Velocities. 515 
centrodes of A and C, or that links A and O are in form the 
same as that of the rolling centrodes moving one upon the other 
without slip. 
The case of relative motion of links A and O in this mechan¬ 
ism is simple; but if we attempt to see the relative motion be¬ 
tween the two links B and D—the exceedingly complex cen¬ 
trodes make it quite out of the question to follow the rolling 
motion as such. The centrodes pass to infinity plus and minusy 
and do not show outwardly the angular motion. The various 
possible ways of passing the dead centers produce various curves 
between the points lettered c' and e', also Id and n\ of space 
centrede bd; and in the body centrode bd, between the points 
d and e, also, l and m. 
When three bodies move in the same plane, as A; B and D, 
each moves with respect to the 1 other about some point, and the 
three centers or centres ab, ad and bd must lie in the same 
straight line. 
The point bd, is the point of contact of the two centrodes of 
B and D, or as was stated above, the centrodes may be considered 
expanded links, point bd may be said to be the contact point, or 
common point of the two links B and D. As a point in each link 
it must of necessity travel about the center of rotation of that 
link with respect to the stationary body.. Also, since bd is a 
common point to the two links, bd must be a point, when con¬ 
sidered in each, which has the same velocity about either center 
of rotation. In general, then:—the directional motion and 
velocity of points in the centrodes which come in contact, are 
the same. 
During the motion of any mechanism, two facts, first:—that 
the directional motion of the contact points of the centrodes 
which roll together, is perpendicular to the; line of centers, and 
secondly, that their velocity is the same, as each of these points 
of the centrodes comes in contact at the centro, enable us to 
make this deduction:— 
“We may replace the complex centrodes by a simple pair of 
centrodes in the form of circles, which circles roll together 
without slip and have the property of expanding and contract¬ 
ing, each independently, opposite to one another, or together 
according to the same law as the contact points of the regular 
centrodes.” 
The above named circles will always be in contact at the 
