506] 
ON THE MECHANICAL DESCRIPTION OF A CUBIC CURVE. 
149 
We then have a sin 0 — b sin <£ ; and moreover the length AI being 
and therefore IC = c — A cos (0 + </>), we have 
whence also 
or we have 
CF _c-h cos (6 + </>) 
sin (f) 
x = QG =dsm(f) ; 
xy = d{c — h cos {6 + </>)} ; 
that is 
or rationalising and reducing, this is 
x 2 y 2 
2bh 
ad 
a?y — 2cdxy + -12 — cA + A 2 ^1 + j & 2 + d 2 (c 2 — A 2 ) = 0, 
a quartic curve with two dps. 
In the particular case a = b, the relation between 0. <£ is simply 0 = 
should become unicursal. 
Writing in the equation ^ = 1, the equation takes the form 
[xy - d (c + A)} = 0 ; 
the second factor is extraneous, and the curve is the hyperbola 
x (y — tvj — d (c — A) = 0, 
as at once appears from the foregoing irrational form of the equation. 
In the particular case A = c, the equation contains the factor x, and 
it becomes 
K^“a<U , ) _2rf! ' + <; ( 1 + a) X = °’ 
viz. we have here a cubic curve with three real asymptotes meeting in 
is also the centre of the curve. 
If simultaneously a = b and A = c, then the equation is 
( 2c \ 
y — x) (xy — 2cd) = 0, 
2c 
the actual locus being in this case the line y — x = 0. 
A cos (0 + (j>) r 
(f) ; the curve 
omitting this 
point which