386 
ON THE CLASSIFICATION OF CUBIC CURVES. 
[350 
and writing also x + z = 1, we find that, for the satellite lines which pass through the 
crunode (1, 0, 2), the critic centres lie one on the line x=9, the other two on the 
conic 
z (x + z) — x 2 — y 2 = 0, 
that is, on the circle x 2 + y 2 + x - 1 = 0, or (x + |-) 2 + y 2 = f. 
98. The circle in question cuts the twofold centre conic 3x 2 — 4x — y 2 = — 1 at its 
intersections with the line x = 0, viz. in the points x = 0, y = + 1; and it moreover 
— (x — 1) (3# — 2) 
touches the one-with-twofold centre locus y 2 
at a point where this 
9« — 5 
same circle meets the ellipse 3x 2 + xy+ y 2 — 2x— y — Q, which is the harmonic conic 
corresponding to the inclination tan -1 2. In fact, writing down the three equations, 
x 2 + y 2 + x — 1 = 0, 
3x 2 + xy + y 2 — 2x — y = 0, 
Q2-1X30-2) 2 
J 9x — 5 
the first and third equations give 
that is 
or, reducing, 
_ (x- l)(3x~2) 2 = 
9x— 5 
— (x — 1) (3« — 2) 2 + (9x — 5) (x 2 + x — 1) = 0, 
(5x - 3) 2 = 0, 
X — X-, 
that is x = ■§, and then from the first or third equation y 1 ^^, or y=± hence the 
circle touches the one-with-twofold centre locus at the points 
® = x =b y = +l\ 
and by means of the second equation we see that the first of these points, viz. the 
point a? = §, y = — }, is a point of the ellipse or harmonic conic 3x 2 + xy+ y 2 — 2x — y = 0. 
99. I consider the analytical theory of the case where the satellite line is parallel 
to the asymptote ; this is in fact similar to the theory ante Nos. 67—69 ; writing 
^ (x + yi), \ (x — yi), z, \ — fii, \ + yi, v 
in the place of x, y, z, y, v, and putting afterwards y = 0, that is, in the transformed 
equation \x + /j,y + vz = 0 writing /¿ = 0, we find for the satellite line A&‘+ v (1 — x) = 0; 
the equation in 0 (the factor 0 + \ — 0 being disregarded) is 
0 2 -\d- 2\v = 0, 
and 
the 
the corresponding 
equation 
x : 1 — x = 
critic 
2 
e + \ : 
centres lie on the line y — 0, at the distances given by 
1 
0 + v ’ 
0 
0 + \’ 
Z = 1 — X = 
0+\’ 
whence x =