In the first case, the value of B^ — 4AC does not be-
come equal to zero, but in the second case it does. For
if B2— 4AC=0, then a—ao and b=!x>.
Removing the terms which have vanished, (1) reduces
to A2/2-i-B£i;y+Ca;2-f R=0, (2)

where R=A52+Ba6+Ca2+D5+Ea+F.
Now it is evident that if (2) is satisfied by any ^os«-
tive values of x and y, it is also satisfied by the same,
values taken negatively.
Hence the new origin of co-ordinates must be at the
centre of the curve.
D2



82 ANALYTICAL GEOMETEY.
CHAPTER m.
ANALYTICAL GEOMETEY OF THEEE DIMENSIONS.
(115.) We have seen that the position of a point on a
plane is determined when its distances from two straight
lines drawn in that plane are known ; in a manner very
similar, we shall now proceed to show that the position
of a point in space will be determined when its distances
from three planes are known.
