122 
Proceedings of the Royal Society of Edinburgh. [Sess. 
O x and 0 2 are fixed points ; Q 1 and Q 2 are two variable points on two 
linear loci such that Q X Q 2 passes through O r The angles OjQjR and 0-,Q 2 P 
are constant, and R is a point on a line l 3 . If R0 2 P is also an angle of 
constant magnitude, the locus of P is, according to Maclaurin, a quartic 
curve with a double point at 0 2 . But if, when Q 2 P passes through 0 2 , 0 2 P 
coincides with it, then the locus degenerates into a cubic devoid of a double 
point. 
In reality the curve is, in general, a unicursal quartic having three double 
points, while the cubic is also unicursal and therefore possesses a double 
point. 
Dem. 
Take the origin of co-ordinates at 0 2 . 
Let 0 2 P have the equation 
y — mx = 0 ( 1 ) 
and 0 1 Q 1 Q 2 have equation 
L x + th 2 == 0 . . . (2) 
Then QjR has an equation of the form 
xA 2 (t) + yB,(t) + C 2 (t) = 0 ( 3 ) 
The line 0 2 R which is in 1 — 1 correspondence with 0 2 P cuts Q X R on l 3 . 
TTpdpp 
( 4 ) 
( 1 ) may .’. be written 
y*&)-xf &)= 0 (?) 
Also the equation to Q 2 P is of the form 
x¥ 2 (t) + yQ 2 (t) + K 2 (t) = 0 ( 6 ) 
On solving (5) and (6) for x and y we obtain the unicursal equations to 
a unicursal quartic, which possesses a double point at 0 2 , it is true, but also 
possesses other two double points in general. 
Suppose, however, when Q 2 P passes through 0 2 that 0 2 P coincides with 
it, then the curve is a unicursal cubic, in which 0 2 is an ordinary point, 
but the curve necessarily has a double point elsewhere in virtue of its 
unicursality. 
Dem. 
Let a be the value of t for which Q 2 P passes through 0 2 . 
Let R 2 (0 in (6) = (£ — a)(t — /3), and let 
m o =M a )l ( p2( a ) = - 
(7) 
