317 
606] QUARTIC CURVE AS FUNCTIONS OF A PARAMETER, 
and consequently 
. a?d — 3a6c + 2 b 3 
viz. cos cf) is given as a rational function of the coordinates (A, fx); there is, as before, 
the trisection; and we then have 
ax + b = 2 (c/x — b) cos ^</>, y = Xx, 
giving x and y as functions of X, fx, <f>; that is, ultimately, as functions of X. I have 
not succeeded in obtaining in a good geometrical form the relation between the point 
(x, y) on the given quartic and the point (X, ¡x) on the nodal quartic. 
Reverting to the expression of tan <jf>, it may be remarked that a = 0 gives the 
values of X which correspond to the four points at infinity on the given quartic 
curve ; axl 2 + 4ac 3 + 4b 3 d — 6abcd — 3b~c 2 = 0, the values corresponding to the ten tangents 
from the origin; and a~d — 3abc + 2b 3 = 0, the values corresponding to the nine lines 
through the origin, which are each such that the origin is the centre of gravity of 
the other three points on the line. 
I take the opportunity of mentioning a mechanical construction of the Cartesian. 
The equation r' = — A cos 6 —N represents a limaçon (which is derivable mechanically 
B 
from the circle r' = — A cos 6), and if we effect the transformation / = r 4—, the new 
r 
B 
curve is r + 7 + ^ cos 6 + N = 0 ; that is, r 2 + r ( A cos 6 + N) + B = 0, which is, in fact, 
the equation of a Cartesian. The assumed transformation r' = r + — can be effected 
immediately by a Peaucellier cell. 
I