402 G E O M (4.) The point conjugate to the point at infinity is called the " centre " of the involution. livery involution has a centre, unless the point at infinity be a focus, in which case we may say that the centre is at infinity. In a hyperbolic involution the centre is the middle point between the. foci. (5.) The product of the distances of two conjugate points A, A’ from the centre 0 is constant :— Ugh . O-y= C. I’roof.—-Let A, A’, and B, B’ be two pairs of conjugate points, 0 the centre, I the point at infinity, then (A B01) = (A’B’IO), or OA. OA’=-OB . OB’. In order to determine the distances of the foci from the centre, we write F for A and A’ and get or”=c; 0F=:l:t"E. Hence if c is positive OF is real, and has two values, equal and opposite. The involution is hyperbolic. If c=O, OF =0, and the two toci both coincide with the centre. If e is negative, it/c becomes imaginary, and there are no foci. Hence we may write-— In a hyperbolic involution, OA . OA’=lt-'3, In a parabolic involution, OA. ()A’=O, In an elliptic involution, OA . ()A’= — la‘-’. From these expressions it follows that conjugate points A, A’ in a hyperbolic involution lie on the same side of the centre, and in an elliptic involution on opposite sides of the centre, and that in a parabolic involution one coincides with the centre. In the first case, for instance, 0A. 0A’ is positive; hence OA and 0A’ have the same sign. lt also follows that two segments, AA’ and BB’, bctvecn pairs of conjugate points have the following positions:——in a hyperbolic involution they lie either one altogether within or altogether with- out each other; in a parabolic involution they have one point in connnon; and in an elliptic involution they overlap, each being partly within and partly without the other. Proof. —We have UA . ()A’=()B . OB’=lt-‘3 in case of a hyperbolic involution. Let A and B be the points in each pair which are nearer to the centre 0. If now A, A’ and B, B’ lie on the same side of 4), aml if B is nearer to 0 than A, so that OB<()A, then OB’>OA’; hence B’ lies further away from 0 than A’, or the segment AA’ lies within BB’. Aml so on for the other cases. (6.) An involution is determincd— (a) By two pairs of conjugate points. I Ience also (B) By one pair of conjugate points and the centre; (7) By the two foci; (8) By one focus and one pair of conjugate points; (e) By one focus and the centre. (7.) T he condition that A, B, C aml A’, B’, C’ may form an involu- tion may be written in one of the forms- (ABCC’) =(A"13'c'c), or (A BCA’) = (A’B’C’A), or (A BC’A’) = (A’B’CA ), for each expresses that in the two projective rows in which A, B, C and A’, B’, C’ are conjugate points two conjugate elements may be interchanged. 8.) Any three pairs, A. A’, B, B’, C, C’, of conjugate points are connected by the relation — 1‘-V. . 52’ _ A’C 15'». (‘ 1:“ ' I’roof.—We have by (7) (ABC’A’) =(A’B’C‘A), which, when worked out, gives the above relation. The latter is easily remembered by aid of the following rule of writing the first sidc. First write B C A 7.: Ti 'T;’ and then fill up the gaps in numerator and denominator by A’, B’, C’ respectively. . § 7_8. ’1‘_m=:oP.EM.—TIzr sides of anyfour-point arr: cut by any line ‘(Tl _spr points in im:olutz'ou, opposite sirlcs being cut in conjugate pom 8. Let A113-,C,D, (fig. 30) be the four-point. If its sides be cut by the line 11 in the points A, A’, B, B’, C, C’, if furtl-.01-, C11), cuts the ]lll(5 .lBl Ill C2, and “'0 Project the [‘0V AIBICZC to 2} Once fl-Oln J), and once from ('1, we get (A’B’C’C) = (B.C’C). interchanging in the last cross-ratio the letters in each pair we get (A’|l’(,"()) = (A B('C’). Hence by § 77 '_T) the points are in involution. E T It Y The theorem may also be stated thus :- .’I'/u:orem.—Tlic three points in which any line cuts the sides of a triangle and the projections, from any point in the. plane, of the vcrticcs of the triangle on to the same line are six points in involu- tion. Or again— The projections from any point on to any line of the six vcrtices [rnoJ E(‘Tl’F.. c'[ A n' C- A- P 7” Fig 30. of a four-side are six points in involution, the projections of oppo- site vcrtices being conjugate points. This property gives a simple means to construct, by aid of the straight edge only, in an involution of which two pairs of conjugate points are given, to any point its conjugate. § 79. The theory of involution may at once be extendcd from the row to the flat and the axial pcncil—viz., we say that there is an involution in a flat or in an axial pencil if any line cuts the pencil in an involution of points. pairs of conjugate rays or planes ; it has two, one, or no focal raj,»- or planes, but nothing correspomling to a centre. An involution in a flat pencil contains always one, aml in gem-— ral only one, pair of conjugate rays which are perpendicular to one anothcr. For in two projective llat pencils exist always two corre- sponding right angles (3' 40). Each involution in an axial pencil contains in the same manner one pair of eon_jugate planes at right angles to one another. As a rule, there exists but one pair of conjugate lim-s or planes at right angles to each other. But it is possible that there are more, and then there is an infinite number of such pairs. An in- volution in a flat pencil, in which evcry ray is pcrpcndicul-.u' to its conjugate ray, is said to be ci7'cuIm'. That such involution is possible is easily seen thus :——if in two concentric llat pencils each ray on one is made to correspond to that ray on the other which is perpendicular to it, then the two pencils are projectivc, for if we turn the one pencil through a right angle cach ray in cm- coincides with its corresponding ray in the other. But these two projective pencils are in involution. A circular inrolul1'mz has no focal rays, because no my in .1 pencil coincides with the ray perpendicular to it. §80. Tu1:nI:Ii:I.—.Ercry elliplicul iurululiuu {u a real‘ may 7:- consitlcrcrl as a srcfion of a circular involution. I’roQf.——ln an elliptical involution any two seglnents AA’ and BB’ lie partly within partly without each other (fig. 31). 1lLllL‘C two circles described on ’AA and BB’ as diameters will intersect in two points E and IS’. The line ICE’ cuts the base of the involution at a point 0, which, from a well known proposition (Eucl. Ill. 35), has the property that OA.()A’=()B.0B’, for each is equal to OE. OE’. The point 0 is therefore the centre of the invohuion. If we wish to construct to any point C the conjugate point C’, we may draw the circle through Fig. 31. Clili’. This will cut the base in the rcquircd point C’ for UCJIC’ =OA.0A’. But EC‘ and lit!’ are at right angles. Ilcnce the involution which is obtained by joining li or Ii’ to the points in the given involution is circular. This may also be exprcssul thus :— Ercry cl/1'p!icul z'nroIufion has (he property (hat there are two clcfi-nilc points in the plum‘ from 2I:Iu'ch an y luv) conjugate poinls are seen under 0 riylzl angle.
An involution in a pencil consists of