350] 
ON THE CLASSIFICATION OF CUBIC CUKVES. 
371 
and they may be spoken of simply as the perpendiculars, but it is to be borne in 
mind that the former is the proper construction of these lines. 
55. The equation of the envelope is 
\/x + \/y + \/ z = 0, 
or, what is the same thing, 
x* + y* + up — 4 (yz 3 + y 3 z + zaf + z 3 x + xy 3 + afy) 
+ 6 (y 3 z 3 + z'-x 3 + x 3 y 3 ) — 124 (x 3 yz + xy 3 z + xyz 3 ) = 0. 
The curve consists as shown in the figure of a trigonoid branch inscribed in the 
triangle and of three acnodes outside the triangle. 
56. The side x = 0 touches the curve in the point (0, 1, 1) or (0, |), which 
is its intersection with the perpendicular y — z = 0; the side x — 0 has with the curve 
at the point in question a four-pointic intersection. The last-mentioned line y — z= 0 
meets the curve in the point (—4, 1, 1) or (2, — which is one of the acnodes, 
and therefore a point of twofold intersection; then again in the point (16, 1, 1) or 
(f, Yg) which may be considered as a vertex of the trigonoid branch, and finally 
in the before-mentioned point (0, 1, 1) or (0, £), which is the point of contact with 
the side x — 0. 
57. The equation of the twofold centre locus is 
\Zx + *Jy+*Jz = 0, 
or in a rational form 
x 3 + y 3 + z 3 — 2 yz — 2 zx — Ixy — 0, 
which is in the case of an equilateral triangle, a circle inscribed in the triangle and 
touching the sides at their midpoints respectively. The circle is shown in the figure. 
58. The equation of the one-with-twofold centre locus is 
— (— x + y + z) {x — y + z) (x + y — z) + xyz = 0, 
or, what is the same thing, 
xr i + y 3 + z 3 — (yz 3 + y 3 z + zx 3 + z 3 x + xy 3 4- x 3 y) + Sxyz = 0. 
It is a cubic having the harmonic point (1, 1, 1) or (i, i, t), for an acnode, touching 
the sides of the triangle externally at their midpoints respectively, and having the 
three asymptotes 
5x — 4<y — 4z = 0, — 4>x + 5y — 4<z = 0, 4# — 4y + 5z = 0, 
or, what is the same thing, a? = f, y = %, 2 = the form of the curve is shown in the 
figure. 
47—2