SEVENTY-TWO CONSECUTIVE PROPOSITIONS IN TRANSVERSALS. 49 



Theorem XLVIII. 



Q X Q 4 and Q 3 Q. intersect in P 3 . 



Theorem XLIX. 



Q 2 Q 5 and Q 4 Q 6 intersect in P 4 . 



Theorem L. 



QjQg and P X P 2 intersect in P 5 . 



Theorem LI. 



QjQ. and Q 3 Q 4 intersect in P 6 . 



Theorem LII. 



P^Qg and P 5 Q 4 intersect in Q 2 . 



V 



Theorem LIII. 



P^, Q X Q 2 , Q 3 Q 6 intersect in a point. 



Theorem LIV. 



Q X Q 3 , QA, Q 2 Qe intersect in a pomt,— say JT. 



Theorem LV. 



Let CLQ, and Q.P. meet in S x \ . And let Q 3 Q 5 and Q 4 Q 6 meet in S 3 \ T h en •— 

 ofc, and Q> 8 " „ S 2 j' QxQ.andQA „ S 4 j 



S^ X is a straight line. 



Theorem LVI. 



S3S4 j? is a straight line. 



Theorem LVII. 



Let P^Q, and P 2 Q 3 meet in R x 1 . And let P X Q 4 and P 2 Q 5 meet in R 3 "1 Then ._ 

 P^andP^ „ R 2 J' P^andP^ „ R 4 J 



R X R 2 Jf is a straight line. 



Theorem LVIII. 



R 3 R 4 JP is a straight line. 

 VOL. XXIV. PART I. ° 



