1901-2.] Prof. Chrystal on Theory of Miller's Trisector. 
13 
We thus get the branch AGE, approaching the asymptote C F, 
whose equation is 
x + y — 3/^/2 . 
The same result may he obtained very simply by seeking the 
intersections of the sextic with the variable circle, 
x 2 + y 2 — r 2 (5). 
If we combine this equation with (3), we get 
(x 2 -y 2 ) 2 = (3r 2 ~4:) 2 lr 2 (6). 
Since (5) and (6) have only eight finite intersections, we see that 
four of the intersections of the sextic with the circle must be at 
infinity. It appears, therefore, that the sextic touches any circle 
at each of the circular points at infinity. 
The rest of the finite intersections are given by the equations 
x 2 + y 2 = r 2 x 2 - y 2 = ± (3r 2 -4 )jr . 
We have therefore the following parametric representation of points 
on the sextic trisectrix : 
x 2 = (r - l)(r + 2) 2 /-2r , y 2 = (r + l)(r - 2) 2 /2r ) 
xfi==(r+ l)(r - 2) 2 /2r , y 2 = (r- l)(r + 2) 2 /2r j" ’ 
Since the second of these is derivable from the first by merely 
changing the sign of r f we see that the sextic is completely repre- 
sented by 
where r may have any real value numerically greater than unity. 
The formulae for the branch AGE are obviously 
where r ranges from + 1 to + oo . 
If we now mirror the branch A G E in the axes of x and y and 
in the octant lines successively, so as to get the eight congruent 
branches indicated by the form of the equation, we get the curve 
drawn in fig. 2, where 0 A = 1, OB = 2, 0 C = 3/^/2, 0 G = 2/^/3. 
It will be seen that the curve has two pairs of parallel asymp- 
totes, corresponding to two double points at infinity. 
There are also eight finite double points : two on each of the 
