350] 
ON THE CLASSIFICATION OF CUBIC CURVES. 379 
We have therefore 
-2/3\/3-^ = -20, a - ~| = - 4, 
giving 
14 _14VS 
^ 3V3 9 
so that the equation of the parabola is 
{sf-(®- J 5 t )' / §) !, = -¥(*-$); 
that of the other parabola is of course obtained by merely changing the sign of V3. 
76. From the foregoing results we may trace the curve, but this may be done 
somewhat more easily by means of polar coordinates, viz. writing x—r cos 0, y = r sin 0, 
and therefore z — 1 — r cos 0, we have 
that is 
v^r.2 cos l 0 + y/ 1 — r cos 0 = 0, 
r (cos 0 + 8 cos 2 \ 0) = 1 ; 
or since 
cos 0=1 — 8 sin 2 ^ 0 cos 2 \ 0, 
we have 
= 1—8 cos 2 \ 0 + 8 cos 4 1 0, 
cos 0 + 8 cos 4 \ 0 = (1 — 4 cos 2 1 0) 2 
= (— 1 — 2 cos^-0) 2 , 
and the equation is 
0 = 0° gives r = a, 
1 
1 (1 + 2 cos | 0) 2 ’ 
0 = 180° gives r= 1, 0 = 240° gives r = oo, 0 = 360° gives r = 1, 
values which agree with the results obtained by rectangular coordinates. 
77. The form is shown in the figure; we see that the curve consists of a lower 
branch without any singularity; and of an upper branch which cuts itself in the 
crunode. 
78. The equation of the twofold centre locus (making in the form vx + \/y + *Jz = Q, 
the foregoing transformation) is 
that is 
V2 i x + yi) + ( x ~ yi) + Vl — x = 0, 
X + ViC 2 + y 2 = 1 — X, 
or 
/l Jx 2 + y 2 = 1 — lx, 
48—2