18 
Proceedings of Royal Society of Edinburgh. [sess. 
angle A O X (2 0), then B is a point on the curve in question. And 
if we take X 0 Y as axes, we have 
x = a (cos 2 0 + cos 4 6) = 2 a cos 0 cos 3 0 , | 
y == a (sin 2# + sin 40) = 2a cos 6 sin 30 j ' ’ 
from which we get at once the equation 
(x 2 + y 2 ) 2 - 3a 2 (x 2 + y 2 ) -2a s x = Q .... (10). 
The curve is therefore a bicircular quartic. 
This quartic may be readily identified with a well known curve 
by means of its characteristic property. If we take a point O, such 
that Ofi = OF = a, join OB, and draw OD parallel to AB, it is 
seen that 0 A B D is a rhombus. We therefore have OD = DB=a. 
Hence the locus of D is a circle whose centre is O ; and it follows 
that the locus of B is a particular variety of Pascal’s Limagon. 
If we put y> = 2 0, and take O X as prime radius, the polar equa- 
tion of the curve is 
r = a (2cos <£ + 1) (11) ; 
and the Cartesian equation referred to XOY' is 
(x 2 + y 2 - 2ax) 2 — a 2 (x 2 + y 2 ) . . • . (12). 
The origin is obviously a real node, whose tangents are 
y — ± J3x. 
Prom (10) it is obvious that the lines y±ix = 0 are imaginary 
cuspidal tangents at the circular points at infinity. The quartic is 
therefore trinodal, bicuspidal, and also unicursal. Its Pluckerian 
numbers are n = 4, 8=1, k = 2 ; m = 4, r = 1, i = 2. 
If we regard the curve as a Cartesian oval, it is easily seen that 
O is the triple focus ; and that the other three real foci are 
H(OH = Ja) and O, the latter counting twice. It is thus readily 
found that we have the property 
20B + 4HB = 3 a (13). 
It will, in fact, be easily verified that 
2 J{x 2 + y 2 ) + i J{(x - f) 2 + ?/ 2 } = 3« . . . (14) 
rationalises into the equation (12). 
Taking a hint from Kempe’s trisecting link-work, already re- 
ferred to, we can construct a seven-bar link- work which will enable 
us to draw the Quartic Trisectrix, or, indeed, a Limagon of any 
