350]
ON THE CLASSIFICATION OF CUBIC CURVES.
385
And the corresponding positions of the critic centres on the ellipse are
A 2 , (harmonic point), the other two centres imaginary,
if 2 , real centre, the other two centres imaginary,
C" 12 , a twofold centre, G' 3 a one-with-twofold centre,
D 1? (asymptote point), D 2 : D ;j ,
G"E 13 , twofold centre, G"E. 2 one-with-twofold centre,
A, real centre, the other two centres imaginary,
A», harmonic point; the other two centres imaginary.
95. And for the inclinations < and > tan -1 2, the only difference is that the
positions are AC.l)EG"A', viz.
A, at infinity,
M, between A and G",
C, touching upper branch of envelope,
D, through asymptote point D x ,
E, through crunode,
G", touching upper branch of envelope,
A, between C" and A',
A', at infinity,
and that instead of the points G"E 13 and G" E. 2 we have separately the points G" 13 , C" 2
and E x , E 3 , E., as shown in the figure.
96. For the better understanding of the figure it is to be observed that the
points Do and D. lie on the line x — ^: this depends on the theorem No. 81 of the
Memoir on Involution, viz. of the critic centres which belong to a satellite line
through the vertex (0, 0, 1), one is the vertex itself, the other two lie on the line
x+y — z — 0; or making the transformation (x + iy), ^ (x — iy), z for x, y, z and
writing x + z = 1, of the critic centres which pass through the vertex (0, 0, 1) (the
asymptote point), one is this point itself, the other two lie on the line x — z = 0, that is
x = L
97. Again it is to be observed that the centre E 3 lies on the line x = 0, and the
centres E ly E 2 on the circle (indicated by a dotted line in the figure) («+|) 2 + y 2 = f :
this depends on the theorem Nos. 73 and 74 of the Memoir on Involution, viz. of the
critic centres for satellite lines through the node (acnode) (1, 1, —4), one lies on the
line x + y- 0, and the other two lie on the conic z (x + y + z) — 4>xy = 0 ] making the
substitution
C. V.
2 (x + yi), h (x - yi), Z for X, y, z,
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