A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

BOOK VI.

133

(Y. A'B'C'C); and hence the anharmonic ratio of
(Y. ABCC') equal the anharmonic ratio of the pencil
(Y. A'B'C'C).
Cor. l. —If the point A moves towards 0, the point
A' will move towards infinity.
Cor. 2.-The foregoing Demonstration will hold if
some of the pairs of conjugate points be on the production of OX in the negative direction; that is, to
the left of OY, while others are to the right, or in the
positive direction.
Cor. 3.-If the points A, B, A, &c., be on one side of 0,
say to the right, their corresponding points A', B', C', &c.,
may lie on the other side; that is, to the left. In this
case the As AYA', BYB', CYC', &c., are all right-angled
at Y; and the general Proposition holds for this case
also, namely, The anharmonic ratio of any four points is
equal to the anharmonic ratio of their four conjugates.
Cor. 4.-The anharmonic ratio of any four collinear
points is equal to the anharmonic ratio of the four
points which are inverse to them, with respect to any
circle whose centre is in the line of collinearity.
DEF. — When two systems of three points each, such as
A, B, C; A', B', C', are collinear, and are so related that
the anharmonic ratio of any four, which are not two couples of conjugate points, is equal to the anharmonic ratio
of their four conjugates, the six points are said to be in
involution. The point 0 conjugate to the point at infinity is called the centre of the involution. Again, if
we take two points D, D', one at each side of 0, such that
OD2 = OD'2 = k2, it is evident that each of these points is
its own conjugate. Hence they have been called, by
TOWNSEND and CHASLES, the double points of the involution. From these Definitions the following Propositions are evident:(1). Any pair of homologous points, such as A, A', are
harmonic conjugates to the double points D, D'.
(2). Three pairs of points which have a commonpair of
harmonic conjugates form a system in involution.