15 
1901-2.] Prof. Chrystal on Theory of Miller’s Trisector. 
This last result is merely a particular case of the general theorem 
that the curve 
x=f(sin mf, cos m-fi, sin m 2 0, cos m 2 0, ), 
y = pr(sin mi Q t cos mf, sin m 2 0, cos m 2 0, ), 
where / and g are rational functions, and m v m 2 , integers 
is a unicursal curve. 
If n denote the degree of the sextic, d the number of its double 
points, k the number of its cusps, m the class, t the number of 
inflexions, r the number of double tangents, we find by means of 
Pliicker’s equations, since n = 6, d— 10, k = 0, that m = 10, t= 12, 
r = 24. 
Of the twelve inflexional tangents only four are real, viz., the 
four asymptotes, each of which meets the curve in four points at 
infinity. 
Since the curve is unicursal, its quadrature can be effected by 
means of elementary transcendents. Its quadrature depends on 
the integral 
I d0j ( 4 sin 2 20 + 9 cos 2 20)/cos 2 2 0 , 
which can be expressed in terms of elliptic transcendents. 
Trisection of an Angle by means of the Ruler , the Compass , 
and a Sextic Trisectrix Template. 
Let a template be constructed, one of whose edges, 0 A (fig. 3), 
is the a-axis of the sextic trisectrix when represented by the 
equation (3), and the other the branch A E of the trisectrix. Then 
we may trisect any given angle UOX (<135°) as follows:— 
Place the template so that its 0 falls on the 0 of the given angle, 
and 0 A on one of the arms, say 0 X. Mark the point P where 
the curved edge of the template meets the other arm O U. With 
O A as radius and 0 as centre, describe a circle ; and with half 
0 P as radius, and the middle point of O P as centre, describe 
