506] 
147 
506. 
ON THE MECHANICAL DESCRIPTION OF A CUBIC CURVE. 
[From the Proceedings of the London Mathematical Society, vol. iv. (1871—1873), 
pp. 175—178. Read November 14, 1872.] 
If the coordinates x, y of a point on a curve are rational functions of sin cos</>, 
vr — k 2 sin 2 <£, the curve has the deficiency 1, and conversely in any curve of deficiency 1 
the coordinates x, y can be thus expressed in terms of the parameter </>. Hence 
writing sin 0 — k sin </>, the coordinates will be rational functions of sin </>, cos cf>, cos 6, 
or say of sin (f), cos cf), sin 0, cos 0; and for the mechanical representation of the relation 
k sin cf) — sin 0, we require only a rod 0A rotating about the fixed point 0, and con 
nected with it by a pin at A, a rod AB, the other extremity of which, B, moves in 
a fixed line Ox. The curve most readily obtained by such an arrangement is that 
described by a point G rigidly connected with the rod AB; this is however a quartic 
curve (with two dps., since its deficiency is =1). I first considered the cubic curve 
xy —1 — V(1 — x 2 ) (1 — k 2 x 2 ), 
or say 
xy — 1 = — V(1 — x 2 ) (1 — k 2 x 2 ); 
writing herein x = sin <£, and as before k sin 0 = sin 0, we have then y sin <£ = 1 — cos 0 cos <f>; 
which values may be written 
X = Sin (f), 
y = 
1 — cos (0 + <f>) 
sin (f) 
sin 0. 
I found, however, that this was not the cubic curve most easily constructed; and I 
ultimately devised a mechanical arrangement consisting of 
1. Rod OH, and connected with it by a pin at H, rod HI (*). 
1 There was a mechanical convenience in this, but observe that producing OH to meet IP in T, the 
single straight rod OHT might have been made use of. 
19—2