372 
ON THE CLASSIFICATION OF CUBIC CURVES. 
[350 
i 
i 1 
59. The equation of the harmonic conic corresponding to the satellite line \x + gy + vz=0, 
is 
It is a hyperbola passing through the angles of the triangle, and through the harmonic 
point (1, 1, 1) or (i, ^). Observing that the points in question are the intersections 
of the two rectangular hyperbolas (pairs of lines) x(y — z) = 0, y(z—x) = 0, it follows 
that the harmonic conic is a rectangular hyperbola. 
60. The coordinates of the centre are f 2 , g 2 , h 2 and the centre is consequently a 
point on the circle which is the twofold centre locus. 
The asymptotes of course meet in the' centre, and they again meet the circle in 
two points which are the intersections of the circle with the line fx + gy+ hz = 0. 
61. The Harmonic Conic is the same for the satellite lines which have a given 
direction, and we may to determine it take a satellite line which touches the envelope. 
If the constants a, /3, y satisfy the condition a. + /3 4- 7 = 0; then the equation of a 
of contact with the envelope are as a 4 : /3 4 : y 4 ; the coordinates of the twofold centre 
as a 2 : /3 2 : y 2 ; the coordinates of the one with twofold centre as 
a 2 (/3 - 7) : /3 2 (7 - a) : y 2 (a - /3). 
The values of f g, h are as a 3 (/3 3 — 7 3 ) : /3 3 (7 3 — a 3 ) : 7 3 (a 3 — /3 3 ), or what is the same 
thing, as a 3 (/3 — 7) : ¡3 3 (7 — a) : 7 3 (a — ¡3), or as a 2 (/3 2 — y 2 ) : /3 2 (7 s — a 2 ) : y 2 (a 2 — /3 2 ) : the 
last-mentioned values show that the line fx + gy + hz = 0 passes through the harmonic 
point (1, 1, 1) or (1, i, f), and also through 
the point (a 2 , /3 2 , y 2 ) which is the twofold centre. 
62. On account of the symmetry of the figure in regard to the three asymptotes, 
it is sufficient to construct the harmonic conic for a direction of the satellite line 
inclined to the base at an angle not > 30°, and this is what is accordingly done: it 
may however be remarked that for the limiting inclination = 0°, that is, when the 
satellite line is parallel to the base, the harmonic conic becomes a pair of right lines, 
the base and perpendicular; but that for the other limiting inclination = 30°, that is, 
when the satellite line is perpendicular to one of the legs of the triangle, the 
harmonic conic is still a proper hyperbola, and is situate symmetrically in regard to 
the leg in question; the two limiting cases will be readily understood by means of 
the general case shown in the figure.