SYSTEMS IN INVOLUTION.

332°. If A, A', B, B' are four points on a common axis, whereof A and A', as also B and B', are correspondents, a point O can always be found upon the axis such that

OA. OA'=OB. OB'.

This point O is evidently the centre of the circle to which A and A', and also B and B', are pairs of inverse points, and is consequently found by 257°.

Now, let P, P' be a pair of variable conjugate points which so move as to preserve the relation

OP. OP=OA.OA'=OB. OB'.

Then P and P' by their varying positions on the axis determine a double system of points C, C', D, D', E, E', etc., conjugates in pairs, so that

OA.OA'=OC. OC'=OD.OD'=OE.OE' = etc.

Such a system of points is said to be in involution, and O is called the centre of the involution.

When both constituents of any one conjugate pair lie upon the same side of the centre, the two constituents of every conjugate pair lie upon the same side of the centre, since the product must have the same sign in every case.

With such a disposition of the points the circle to which conjugates are inverse points is real and cuts the axis in two points F and F'. At these points vari

F

B

B

i

able conjugates meet and become coincident. Hence the points F, F' are the double points or foci of the system.

From Art. 311°, 1, FF' is divided harmonically by every pair of conjugate points, so that

FAF'A', FBF'B', etc., are all harmonic ranges. When the constituents of any pair of conjugate points lie upon opposite sides of the centre, the foci are imaginary.

333°. Let A, A, B, B', C, C' be six points in involution, and

let O be the centre.

Draw any line OPQ through O,

and take P and Q so that

OP.OQ=OA. OA',

and join PA, PB, PC, and PC',

and also QA', QB', QC', and QC. Then,

OA. OA'=OP. OQ,

A B C C' B'

... A, P, Q, A' are concyclic. .. 4¿OPA=2OA'Q. Similarly, B, P, Q, B' are concyclic, and LOPB=LOB'Q, etc.

Similarly,

LAPB=LA'QB'.

¿BPC=¿B'QC',

LCPC'=LC'QC, etc.

Hence the pencils P(ABCC') and Q(A'B'C'C) are equianhar

monic, or

Hence also

{ABCC'} = {A'B'C'C}.

{ABB'C}

{ABB'C} = {A'B'BC'}, {AA'BC} = {A'AB'C'}. And any one of these relations expresses the condition that the six points symbolized may be in involution.

334°. As involution is only a species of homography, the relations constantly existing between homographic ranges and their corresponding pencils, hold also for ranges and pencils in involution. Hence

I. Every range in involution determines a pencil in involution at every vertex, and conversely.

2. If a range in involution be projected rectilinearly through any point on a circle it determines a system in involution on the circle, and conversely.

Ex. The three pairs of opposite connectors of any four points cut any line in a six-point involution.

A, B, C, D are the four points,

and P, P' the line cut by the six connectors CD, DA, AC, CB, BD,

RR

= B{Q'P'RR'} = {P'Q'R'R},

T

B

(302°)

{PQRR'} = {P'Q'R'R},

and the six points are in involution.

Cor. I. The centre O of the involution is the radical centre of any three circles through PP', QQ', and RR'; and the three circles on the three segments PP'', QQ', and RR' as diameters are co-axal.

When the order of PQR is opposite that of P'Q'R' as in the figure, and the centre O lies outside the points, the co-axal circles are of the .p.-species, and when the two triads of points have the same order, the co-axal circles are of the c.p.-species.

Cor. 2. Considering ABC as a triangle and AD, BD, three lines through its vertices at D, we have—

CD

The three sides of any triangle and three concurrent lines through the vertices cut any transversal in a six-point involution.

EXERCISES.

1. A circle and an inscribed quadrangle cut any line through them in involution.

2. The circles of a co-axal system cut any line through them in involution.

3. Any three concurrent chords intersect the circle in six points forming a system in involution.

4. The circles of a co-axal system cut any other circle in involution.

5. Any four circles through a common point have their six

radical axes forming a pencil in involution.