507] 
CERTAIN QUARTIC CURVES BY A MODIFIED OVAL CHUCK. 
153 
Let the coordinates of the fixed point P, referred to axes through A, the first of 
them perpendicular to, and the second coincident with, AB, be 6, c; let the distance 
AB be = a; and let 0 denote the angle BAO: then, if x, y are the coordinates of 
P referred to the origin 0 and axes Ox, Oy, we have 
x + a cos 0 = b sin 0 + c cos 0, 
y = — b cos 0 + c sin 0, 
which, if a be constant, gives a quadric equation, or the curve is an ellipse; and, in 
particular, if 6 = 0, that is if the point P is on the line AB, then we have 
¿c = (c — a) cos 0, y = c sin 0, 
or the curve is 
x 1 if 2 
P y- = 1. 
(c — a) 2 c 2 
But if a is a given function of 0, then the equation is still found by eliminating 
0 between the two equations for x and y. In particular, if the distance AB is given 
as the perpendicular upon the tangent of a circle, as shown in the figure, then if k 
be the radius AO of this circle, and X the inclination of AO to Ax (k and X being 
taken to be constants), we have 
a — k cos (0 + X), 
and the equations are 
x = b sin 0 + {c — k cos (0 + A.)} cos 0, 
y — — 6 cos 0 + c sin 0. 
The elimination is nearly the same as if 6 were = 0; viz. we may determine y, a in 
such wise that 
6 sin 0 + c cos 0 = y cos (0 + a), = y cos <p suppose, 
— 6 cos 0 + c sin 0 = y sin (0 + a), = y sin $ ; 
C. VIII.