600. If in a range consisting of four points, A, B, C, D, we take A and B, called conjugate points, as the extremities of a sect, this is divided internally or externally by C; and distinguishing the "step" AC from CA as of opposite "sense," so hat AC CA, the ratio AC/BC is never the same for two positions of C. The like is true of the positive or negative number AD/BD.

=

The ratio [AC/BC]/[AD/BD] is called the cross ratio of the range, and is written [ÅBCD].

601. Four elements may be arranged in twenty-four different ways:

[ÅВCD], [BÅDC], [ĊDAB], [DĊBA],
[ABDC], [BACD], [DCAB], [CDBA],
[ACBD], [CADB], [BDAC], [DBCA],
[ACDB], [ĊABD], [DBAC], [BDCA],
[ADBC], [DACB], [BCAD], [ĊBDA],
[ADCB], [DÅBC], [ĊBAD], [BCDA];

but four cross ratios in each of these six rows are equal, as may be readily proved by writing out any two in a row.

602. If in a cross ratio the two points belonging to one of the two groups be interchanged, the cross ratio changes to its reciprocal.

[Proved by writing out their values.]

Thus the ratio in the second row is reciprocal to that in the first, fourth to third, sixth to fifth.

603. If in a cross ratio the two middle letters be interchanged, the cross ratio changes to its complement.

For we have, taking account of sense or sign, BC + CA+ AB = 0;

.. BC. AD + CA. AD + AB. AD = 0;

.. BC. AD + CA. [BD+AB]+AB. [CD — CA] ='o; :. BC. AD + CA. BD + AB.CD = 0;

.. 1+[CA.BD]/[BC.AD]+[AB.CD]/[BC.AD] = 0; .. 1 — [AB. CD]/[CB. AD] = [AC.BD]/[BC. AD];

... I

604. By 603 [ADCB] == 1 — [ACDB] = [by 602] I —

1/[ÅĊBD] = [by 603] I Thus if the cross_ratio

1/[1 − (ABCD).

[ÀBCD] = λ, then the six cross

ratios derivable from these four co-straight points are ^,,

λε

605. Theorem. If S be a point without the range ABCD, and if through C a straight be drawn parallel to SD, meeting SA, SB in G, H, respectively, then GC/HC = [ÅBCD].

Proof. GC/SD = CA/DA. SD/HC = DB/CB.

GC SD CA DB СА

DA

.. GC/HC = [CDAB] = [ÅBCD].

the

606. If two transversals meet straights of the pencil S [abcd] in A, B, C, D and in A', B', C', D', then [ABCD] [À'B'C'D'].

=

DB

Proof. Through C and C' draw GH and A G'H' || to SD. Then GC/HC = G'C' /H'C'.

607. The cross ratio of the pencil S [abcd] means the cross ratio of the four points ABCD on any transversal, and is written S [ABCD].

608. If two ranges or pencils have equal cross ratios they are said to be equi-cross.

609. Mutually equiangular pencils are equi-cross.

B

B

a

с

B

C

D

FIG. 259.

610,. The joins of corre sponding points of two equicross ranges which have two corresponding points coincident are concurrent.

FIG. 260.

610'. The crosses of corresponding straights of two equi-cross pencils which have two corresponding straights coincident are co-straight. Proof. Let the join of the two crosses B and C cut the common straight in A, and cut d in D. Then is D also on d', since by hypothesis d' cuts ABC in a point D' such that [ABCD] = [ÅBCD].

612'. Ranges whose points

611. Corollary. Equi-cross ranges or pencils are projective. 612,. Pencils whose straights pass through four fixed points on a circle, and whose vertices lie on the circle, are equi-cross.

Proof. The pencils are mu

tually equiangular.

613, [Pascal.] In a cyclic hexagon the crosses of opposite sides are co-straight.

lie on four fixed tangents to a circle and whose bearers are tangent to the circle are equi

Proof. By 612,, the pencils B. ACDE, and F. ACDE are equi-cross; .. [RHDE] = [QCDL]; .. by (610,) RQ, HC, EL

are concurrent.

614. If the figure formed by joining the six concyclic points. by consecutive sects in any order be called a hexagram, there are 60, and Pascal holds for each.

615. Pascal holds for six points, three co-straight and also the other three.

INVOLUTION.

616. If a system of pairs of co-straight points AA', BB', CC', etc., be so situated with regard to a point O on the same straight that OA. OA' OB. OB' = OC. OC', etc., they are said to be in involution. The point O is called the center, and AA', BB', CC', etc., are called conjugate points of the involution. The points E, F, situated on the range, on opposite sides of C, such that OE' OF OA. OA' are called the double points of the involution.

=

If straights be drawn from a point S outside the range to A, A', B, B', C, C', etc., they form a pencil in involution, and SE, SF are called the double straights of the pencil.

[Observing sense or sign, the double points and double straights are real only when conjugate points of the involution. are on the same side of the center.]