150 ON THE MECHANICAL DESCRIPTION OF A CUBIC CURVE. [506 
Writing h = c, and for greater convenience h — c = d = 1; also to fix the ideas 
supposing b < a, and writing - = k, = sin X, then we have 
(X 
sin 0 = k sin </>, 
X = sin (j), 
1-cos (0 + <ft). 
J sin (f) 
that is 
xy — 1 —VI — ic 2 V1 — k 2 x 2 4- kx 2 , 
giving the rationalised equation 
x (y 2 — 2kx 2 ) — 2?/ + 4# = 0 ; 
the angle <£ may be anything whatever, but 6 varies between the limits ± X, the 
simultaneous values of these angles and of the coordinates being 
0 = 
0 
0 = 0 
x = 0 
y = o 
<f>= 
90° 
0 = x 
x = 1 
y = 1 + sin X 
</>= 
180° 
0 = 0 
x = 0 
y = ±° 0 
</> = 
270° 
0 — — X 
X = — 1 
y = — (1 + sin X) 
0= 
360° 
0 = 0 
x = 0 
y = 0; 
and it thus appears that the mechanism gives the continuous branch which belongs 
to the asymptote x = 0 of the cubic curve ; the other two branches belong to 
x = sin (f>, y = -f ,~h ^, which would require a slight alteration in the arrange 
ment of the mechanism. 
I remark that if AH, HI had been unequal, then writing Z HI A = this would 
be connected with 6 + </> by an equation of the form 
sin (6 + — m sin 
and the coordinates x, y would be rational functions of the sines and cosines of 
d, (f>, the deficiency is in this case >1.