ON THE KINEMATICS OF A PLANE. 
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through 0 draw at right angles to this a line meeting the same circle in B, then, 
as before, the points A and B move along the fixed lines OA 0 , OB 0 ; or as regards 
the relative motion, taking A, B as fixed points, we have the originally fixed plane 
now moving in such wise that the two lines OA 0 , OB 0 thereof (at right angles to 
each other) pass always through the points A and B respectively, and the curve is 
that described by the point S on the line OA ; the point 0 describes the circle on 
the diameter AB (the small circle), equation r x = 2c cos B x ; and OQ having a given 
constant value = y, we have for the curve described by the point S the foregoing 
equation r x = 2c cos 6 1 — y; or writing y = —f that is, taking S on the other side of 
0 at a distance OS = /, the equation is r x = 2c cos 0 X 4- /; viz. this is a nodal Cartesian 
or Limaçon, the origin being an acnode or a crunode according as f> or < 2c ; and 
if /= 2c, then we have the cuspidal curve or cardioid tç = 2c (1 + cos tfj), = 4ccos 2 £0 1 . 
The general conclusion is that the centre 0 of the large circle describes on the 
moving plane a small circle (centre O x ), and that every other point of the fixed plane 
describes on the moving plane a Limaçon having for its node a point of the small 
circle, and being, in fact, the curve obtained by measuring off along the radius vector 
of the small circle from its extremity a constant distance. 
Considering in connexion with the point, coordinates (x 1} y x ), (x, y), a second 
point, coordinates (Zj, Y x ), (X, Y), in regard to the two sets of axes respectively, 
we have 
x — (c + x x )cos 6 — y-i sin 0, X = (c + X x ) cos d—Y 1 sin 0, 
y = (c — x x ) sin 0 — y x cos 6, Y = (c — Xj) sin 0 — Y x cos 0 ; 
from the first two equations we have 
cos 0 : sin 0 : 1 = x (c - x x ) + yy x : xy x + y{c + x x ) : c- - x{- - y? ; 
and substituting these values in the second set, we find 
X : Y : 1 
= x {c 2 •+■ c (X x — x x ) — X x x x — Y x y^\ + y { c (y x — Y x ) + y 1 X 1 — x x F x } 
: x { c (3/1 — F].) — y x X x + x x Fi} + y {c 2 — c (X x — x x ) — X x x x — Y x y^ 
: c 2 - a?! 2 - 2/1 2 ; 
or the points (x, y), (X, F), considered as each of them moving on the fixed plane, 
are homographically related to each other. 
To find the curve enveloped on the fixed plane by a given curve of the moving 
plane, we have only in the equation f(x 1} y 1 ) = 0 of the curve in the moving plane 
to substitute for x 1} y x their values in terms of x, y, 0, and then considering 0 as 
a variable parameter, to find the envelope of the curve represented by this equation. 
And, similarly, we find the curve enveloped on the moving plane by a given curve 
of the fixed plane.