108
ON THE KINEMATICS OF A PLANE.
[734
viz. the curve on the first plane is an ellipse, the semi-axes of which are ± (c + #1),
± (c — Xx), each taken positively; if x? + y? — c 2 , viz. if P be on the circle having AB
for its diameter, then y x 2 = (c + x x ) (c — Xj), and we have
y + x = — (c — x 1 ) i^sin 6 cos 6) -T-y 1 (sin 6 — cos $ j f — — ( c _x x ) -T- y x ,
viz. as mentioned above, the curve on the fixed plane is a right line.
In the general case, we have
x(c- Xx) + yyx = (c 2 - x? - y-c) cos 6,
x Vi + y (c + Xx) = (c 2 - Xx 2 - 2/! 2 ) sin 0,
and thence
{x (c - Xx) + yyx} 2 + [xyx +y(c + Xx)} 2 = (c 2 - Xx 2 - y x 2 ) 2
or, what is the same thing,
sc 2 {(c - Xx) 2 + yc} + 4xycyx + y 2 {(c + Xx) 2 4- y x 2 } = (c 2 - Xx 2 - yx 2 ) 2 .
Considering (xx, y0 as given, the curve traced out by P on the fixed plane is
of the second order; it would be easy to verify from the equation that it is an
ellipse, and to obtain for the position and magnitude of the axes the construction
already found geometrically.
The same equation, considering therein {x, y) as constant and {x ly yx) as current
coordinates, gives the curve traced out on the moving plane; the curve is obviously
of the fourth order. Transferring the origin to A, we must in place of write
Xx — Cx; the equation thus becomes
x 2 {(xx - 2c) 2 + 2/i 2 } + 4 cy x xy + y 1 {x 2 + y 2 ) = {x 2 + y 2 - 2 cx^f;
or, what is the same thing,
(Xx 2 + y 2 - 2cxx) 2 - (x 2 + y 2 ) (xx 2 + yx 2 ) + 4cx {xxx - yyx) - 4c 2 x 2 = 0;
and if we suppose herein x = 0, it becomes
{x{- + yx 2 - 2cxx) 2 - y 2 (Xx 2 + yx 2 ) = o ;
or, writing Xx = cos 9x, yx = Vx sin 6x, where 6 X = angle QAB, this is
(Vx — 2c cos 6x) 2 -y 2 = 0,
or say it is
rx = 2c cos 9x — y,
which is the polar equation of the curve described on the moveable plane by the
point S, whose coordinates in respect to Ox and Oy are (0, y).
There is no loss of generality in assuming ¿c = 0. In fact, starting with any point
S whatever of the fixed plane, if we draw OS meeting the small circle in A, and
