106 
ON THE KINEMATICS OF A PLANE. 
[734 
or, what is the same thing, 
say 
sin 0, cos0 = Y'X' - Y'X, X'X' + TIV; 
and then, as before, 
x = a + x x cos 6 — y x sin 6, 
y = /3 + Xx sin 6 + y x cos 0; 
or, what is the same thing, 
x — X = cos 6 (x x — Xx) — sin 6 (y x — Yx), 
y — Y = sin 0 (ocx — Xx) + cos 0 (yx — Yx), 
where X, Y, X 1} Y 1} and therefore also 0, denote given functions of s. The formulae 
will be of a like form if X, F, X 1} Y x are given functions of a parameter t. 
A well known but very interesting case is when two points of the moving plane 
describe right lines on the fixed plane. This may be discussed geometrically as 
follows: Suppose that we have the points A, G (fig. 3) describing the lines OA 0 , 
OC a , which meet in 0; through A, G, 0 describe a circle, centre Ox, and with centre 
Fig. 3. 
V 
B 0 * 
0 and radius =200 x , describe a circle touching the first circle in a point /; and suppose 
that A 0 , Go denote points on the second circle. Then it is at once seen that, considering 
the first or small circle as belonging to the moving plane, and the second or large 
circle as belonging to the fixed plane, the motion is in fact the rolling motion of 
the small upon the large circle; and, moreover, that each point of the small circle 
describes a right line, which is a diameter of the large circle. In fact, the angle 
IOxG at the centre is the double of the angle I0G at the circumference; that is,