138 
[504 
504. 
ON THE MECHANICAL DESCRIPTION OF CERTAIN SEXTIC 
CURVES. 
[From the Proceedings of the London Mathematical Society, vol. iv. (1871—1873), 
pp. 105—111. Bead April 11, 1872.J 
The curves in question might be taken to be those described by a point C 
rigidly connected with points A and B, each of which describes a circle: but the 
construction is considered under a somewhat more general form. I consider a quadri 
lateral, the sides of which are a, h, c, d, and the inclinations of these to a fixed line 
a, /3, 7, 8. This being so, if a, b, c, d, and one of the angles, say 8, are constant, then 
we have between the three variable angles the relations 
a cos a. + b cos /3 + c cos 7 + d cos 8 = 0, 
a sin a + b sin /3 + c sin 7 + d sin 8 = 0, 
giving rise to a single relation between any two of the variable angles; and we con 
sider a curve such that the coordinates x, y of any point thereof are given linear 
functions of the sines and cosines of the three variable angles, or, what is the same 
thing, of the sines and cosines of any two of these angles. We thus unite together 
what would otherwise be distinct cases; for everything is symmetrical in regard to the 
sides a, b, c and the corresponding variable angles a, ¡3, 7, irrespectively of the order 
of succession of these sides: and we can thus, in the discussion of the curve, employ 
any two at pleasure, say a, (3, of the variable angles, without determining whether the 
sides a, b are contiguous or opposite. 
Eliminating, then, the variable angle 7, we obtain between a, /3 a relation which, 
if we write therein tan \a. = u, tan^/3 = v, takes the form (*\u, l) 2 (v, 1) 2 = 0; viz. either 
of the variables u, v is expressible rationally in terms of the other of them and of the 
root of a quartic function thereof; say v is a rational function of u and \JU. And