734] 
ON THE KINEMATICS OF A PLANE. 
105 
from the expressions of x, y we eliminate t, we obtain a relation between (x, y), 
which is that of the rolled-on curve in the fixed plane. 
The system may be written 
or, if we take 0 as the independent variable, 
x x = a' sin 0 — ß' cos 0, x = a — ß', 
y-L = a! cos 6 + ß' sin 6, y = ß + a'. 
To find the variations of /, we have 
xi = a" sin 6 — ß" cos 6 + a' cos 6 + ß' sin 6, = a" sin 6 — ß" cos 6 + y x , 
yi = a" cos 6 + ß" sin 0 — a sin 6 + ß' cos 0, = a!’ cos 0 + ß" sin 0 — x 1} 
y' =/3' + a", 
x' = a -ß". 
Hence 
Xi = X cos 0 + y' sill 0, or X = xi COS 0 — yi sin 0, 
yi = — x’ sin 0 -F y' COS 0, y' = xi sin 0 + yi cos 0, 
values which give x'- + y' 2 = xi 2 + yi 2 , which equation expresses that the motion is in 
fact a rolling one. 
Imagine the two curves, and the initial relative position given; say the two 
points A, Aj (fig. 2) were originally in contact, then the arcs A I, A X I are equal, and, 
calling each of these s, and X, Y, X u Y 1 the coordinates of I in regard to the two 
Fig. 2. 
V 
x 
0 
sets of axes respectively, we have X, Y, X 1} Y 1 given functions of s, such that 
X' 2 + Y' 2 = 1, Xi 2 + Yi* = 1, the accents now denoting differentiation in regard to s. 
We have, from the figure, 
C. XI. 
14