292) MATHEMATICAL NOTES: 
Now we.can: always: assign sucha value to J: m,. that:(2) shall-be: 
at.right angles to (1). For the condition of perpendicularity is. 
Al + Cm Ba+Cl_ 4 
or, [2 rE, 
which, being a quadratic in J: m with its last term negative, has: 
necessarily real roots: (Indeed it shews that there are ¢wo directions, 
at right angles to each other, in which the chords may be drawn, and: 
in fact gives the directions,of the axes of the curve). 
Hence there exists a straight line such that it bisects all chords 
of the curve drawn at right-angles to it; that is, such that the 
curve is symmetrical with regard to it. 
. Now let us take this line for the axis of +; then for any given 
value of x, the equation to the curve must be satisfied by — y ag 
well as + Ys and this requires C = 0. The equation thus reduceg 
to 
Ax? + By?= F 9 
the form of it proving again that the axis of y is also a line of 
symmetry. 
The equation is now reducible to the three known varieties, accord- 
ing to the nature of the intercepts of the axes, namely : 
(Dy rasscceess the ellipse, a ten) 
(2) cooncoese the hyperbola — = — = =+ 1, including two intersecting 
lines = = + == 0, 
(Gy es wholly imaginary, - rs = aha) 
TI. Curves in which the centre is at an infinite distance, and 
Cc? — AB, the equation being 
Av? + By? + 2 Cay +2 De 4+. 2 Ey =F. 
The same. process as before demonstrates the existence of a line. 
with regard to which. the curve is, symmetrical, Taking this, for 
axis of a, we must have 
; C=0,E=0 
But C = 0 requires either A = 0, or, B = 0 
The latter reduces the equation to 
Ax? + 2 D« =F 
