a parabola, ellipse, or hyperbola. The original coefficients (a, b, c, f, g, h) may be such 
as not to give any system of real values for a', b', e, p, a; but when this is so the 
equation (a, b, c, f, g, h) (x, y, l) 2 = 0 does not represent a real curve*; the imaginary 
curve which it represents is, however, regarded as a conic. Disregarding the special 
cases of the pair of lines and the twice repeated line, it thus appears that the only 
real curves represented by the general equation {a, b, c, f g, h)(oc, y, 1) 2 = 0 are the 
parabola, the ellipse, and the hyperbola. The circle is considered as a particular case 
of the ellipse. 
The same result is obtained by transforming the equation (a, b, c,f, g, h)(x, y, l) 2 = 0 
to new axes. If in the first place the origin be unaltered, then the directions of the 
new (rectangular) axes 0x 1 , 0y 1 can be found so that h x (the coefficient of the term 
x 1 2/j) shall be =0; when this is done, then either one of the coefficients of xf, y? is 
= 0, and the curve is then a parabola, or neither of these coefficients is = 0, and the 
curve is then an ellipse or hyperbola, according as the two coefficients are of the 
same sign or of opposite signs. 
19. The curves can be at once traced from their equations:— 
y- = 4mx, for the parabola (fig. 13), 
- + p = 1, for the ellipse (fig. 14) 
x* 
as 
y 
Fig. 13. 
p = 1, for the hyperbola (fig. 15) ; 
Fig. 14. 
Fig. 15. 
* It is proper to remark that, when (a, b, c, /, g, h) (x, y, 1) 2 =0 does represent a real curve, there are, 
in fact, four systems of values of a’, b\ e, p, a, two real, the other two imaginary ; we have thus two real 
equations and two imaginary equations, each of them of the form (x - a') 2 +(y-b') 2 =e 2 (x cosa + y cos -p) 2 , 
representing each of them one and the same real curve. This is consistent with the assertion of the text 
that the real curve is in every case represented by a real equation of this form. 
71—2