104 
ON THE KINEMATICS OF A PLANE. 
[734 
a, ¡3, the coordinates of 0 X in regard to Oxy; and by 0, the inclination of 0 x x x to 
Ox. And denoting by x, y, x x , y x the coordinates of a point P in regard to the two 
sets of axes respectively, then 
x = a + x x cos 0 — y 1 sin 0, 
y = /3 + x 1 sin 0 + y x cos 0; 
or, what is the same thing, 
x x = (x — a) cos 0 + (y — (3) sin 0, 
y x = — (x — a) sin 0 + (y — /3) cos 0; 
or, as these last equations may be written, 
x x = ctj -h x cos (— 0) — y sin (— 0), 
yi = /31 + x sin (— 0) + y cos (— 0), 
where a x , /3 X , = — a. cos 0 — /Ssin 0, a sin 0 — /3 cos 0, are the coordinates of 0 referred to 
the axes 0 1 x 1 y 1 , and — 0 is the inclination of Ox to 0 x x x . 
When the motion is given, a, /3, 0 are given functions of a single variable 
parameter, say of t*; or, if we please, a, /3 are given functions of 0. 
The velocities of a given point (x, y) are determined by the equations 
x = a' — (x x sin 0 + yi cos 0) 0', 
y' = ¡3’ + (#! cos 0 — y-i sin 0) 0'; 
that is, 
x’ - d = - (y - /3) 0', 
y'~f3' = (x - a) 0’; 
or, as these equations may also be written, 
— (,x' — a!) sin 0 + (y' — ¡3') cos 0 = x 1 0', 
— {x — a) cos 0 — (y' — /3') sin 0 = y x & 
Hence if x =0, y = 0, we have 
x x 0' — a sin 0 — ¡3' cos 0, or a = (y — ¡3) 0', 
yj 0' = ot! cos 0 + (3' sin 0, — ¡3' = (x — a.) 0', 
which equations determine in terms of t, x x and y x the coordinates in regard to the 
axes O x x x y x , and x and y the coordinates in regard to the axes Oxy, of I, the centre 
of instantaneous rotation. 
If from the expressions of x x , y x we eliminate t, we obtain an equation between 
( x i, 2/i)> which is that of the rollrng curve in the moveable plane; and, similarly, if 
* t may be regarded as denoting the time, and then the derived functions of x, y in regard to t will 
denote velocities ; and, to simplify the expression of the theorems, it is convenient to do this.