SEVENTY-TWO CONSECUTIVE PROPOSITIONS IN TRANSVERSALS. 47 



Theorem XXXI. 



Let B C, A A 7 meet in H And let B C, m n meet in l x \ 



CA, BA 7 „ ml; CA, nl „ rn 1 > . 



AB, CA, „ n) AB, Im „ n x ) 



Then l^m^ is a straight line, as is well known. As it depends entirely on the position of A 7 let 

 it be called (p (A 7 ). Then ; — 



<p (A 7 ) is the polar of A 8 to the imaginary conic, a 2 + /5 2 + 'y 2 =0. 

 (p (B 7 ) „ B 8 „ 



<p (C 7 ) „ c s „ ,, ,, 



Theorem XXXII. 



(p (A 8 ) is the polar of A 7 to the imaginary conic, a 2 4-/3 2 -J- r y 2 =0. 



<P (B 8 ) „ B 7 „ „ 



9 (C 8 ) „ C 7 „ ,, „ 



Theorem XXXIII. 



(p (P x ) is the polar of P 2 to the imaginary conic, a? +j3 2 + r y 2 = 0. 



(p (."2) )> "1 >) )> ;> 



Theorem XXXIY. 



(p - 1 (&! a x ©J is the pole of & 6 9$ 6 @ 6 to the conic, a 2 + /3 2 + 7 2 = 0. 



0-»(a 6 » e e 6 ) „ «!*!©! „ 

 Theorem XXXV. 



(p - 1 (8L U 7 © 7 ) is the pole of & 8 2S 8 ® 8 to the conic, a 2 + /3 2 +7 2 =0. 



0-i(a 8 * 8 © 8 ) „ a 7 ?3 7 e 7 „ 



The principle of reciprocation would introduce here a number of Propositions 

 which it is unnecessary to enunciate. 



Theorem XXXVI. 



Let A A v B 7 C 7 meet in l 2 \ And let B C, m^n 2 meet in a x 



BB 1; C 7 A 7 „ mA; CA, n 2 l 2 „ ft 



CC 1S A 7 B 7 „ nj AB, l 2 m 2 „ 7, 



Then ;• — « 1 /3 1 7 1 is a straight line. 



Theorem XXXVII. 



Let A A 2 , B 8 C 8 meet in l z ~\ And let B C, m 3 n 3 meet in a 2 \ 



BB 2 , C 8 A 8 „ m, \; CA, n 3 l 3 „ #, 



CC 2 , A 8 B 8 „ nj AB, Z 3 m 3 „ 7, J 



Then ; — # 2 /3 2 7 2 is a straight line. 



Theorem XXXVIII. 



Let /8 1 7 2 » |5 2 Ti meet m a 3 1 



Ti a 2' 72 a i » & f' — Then, a 3 6 3 y z is a straight line. 

 «!&, «2^1 » 7 3 J 



