. The principles of projective geometry applied to the straight line and conic . CE,BK „\/KK - -;^ ( - VA) (- sJY) = -1, .•. AS\ CF, BL „VAA . The correlative theorem is as follows : Every fair of involution pencils in the same plane are in perspective: if both thepencils have real or both imaginary double rays, the perspective is real: if one hasreal and the other imaginary double rays, the perspective is imaginary. Let the centres of the involutions be B and C and let BA and CA be the rayscorresponding to BC in the two involutions. Let a pair of rays of the involution. 104 Principles of Proj
. The principles of projective geometry applied to the straight line and conic . CE,BK „\/KK - -;^ ( - VA) (- sJY) = -1, .•. AS\ CF, BL „VAA . The correlative theorem is as follows : Every fair of involution pencils in the same plane are in perspective: if both thepencils have real or both imaginary double rays, the perspective is real: if one hasreal and the other imaginary double rays, the perspective is imaginary. Let the centres of the involutions be B and C and let BA and CA be the rayscorresponding to BC in the two involutions. Let a pair of rays of the involution. 104 Principles of Projective Geometry + , and \/KK centre C, meet AB in C■^ and C^ and a pair in the involution centre B meet -4CinB\ and ^2- With the notation of the correlative theorem draw two lines AS andAS such that the ratios of S and *S, the points where they meet BC, are ^ j^=. Let (7(7i and CC. meet ASin R and T. Join BR andiJ^ to meet JC in /i/ and B.^. Since ., CYj, Z?5i are concurrent, F .Ci6i= -1, 1 and since AT, (7(%, BB, sire concurrent, /f/i = 1, \/KK .-. h^ . Therefore the lines BB{ and 5^2 are conjugate rays of the involution whosecentre is B. Hence the involutions are in perspective with AS as axis. Similarlythey are in perspective with AS && axis. 61. Analytical formulae connected with an involution. If the notation of Art. 14 be used and B and C are taken as the points ofreference and x-yX.^,; x^Xi ; Xr^x^, the ratios of ^11^2 ; -^3^4 ; ^lo-^c, ■re given by +^hx +6 =0, aV j^2hx +h =0, ax- + ^hx + h = 0,the following results are obtained : (1) The condition that BC; ^1^42; ^3^4 should be three pairs of conjugatepoints of an involution is X^ = X^Xi. (2) The condition that ^11^2 ■> ^3-^4 ! -^^b^G should be three pairs of conjugatepoints of an involution is 1 1 1 Xi-\-X2 Xz + Xi Xr, + XQX\X2 XiXi Xr^Xa Involution 105 (3) The condition (1) that BC; A^; A^Ai should be three pairs ofconjugate points of an involution may also be written ha — h
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