SEVENTY-TWO CONSECUTIVE PROPOSITIONS IN TRANSVERSALS. 



Theorem 5. 



41 



A X A 2 is 

 B, B 2 is 

 C 1 C 2 is 



9 2 -h 2 , 

 -<7(W 2 ), 



-Mi-/), 



A 2 -/ 2 , 



/(i-V), 



0(1 -A 2 ) 



-/(1-A 2 ) 



0. Therefore ; — 



AjAjj, BjBg, C X C 2 intersect in a point, — say P 3 , viz.,f—gh, g — hf, h—fg. 



Theorem 6. 



P, is 

 P 2 is 

 P 3 is 



gK ¥, fg- 



f, g> h - 



f-gh, g-hf, h-fg 



= 0. Therefore P l( P 2 , P 3 range in a straight line. 

 P^P, is/(/-A 2 ), g (A 2 -/ 2 ), h (/ 2 -<7 2 ). 



A X B, B C meet in a 2 , viz., 

 BjCCA „ i$ 2 , „ 

 C X A, AB „ e 2 , „ 



0, 1, -/ 

 ■9, °, 1 



1, -h, 



Theorem 7. 



( CjA, B C meet in a 3 viz., 



Hence ; — 



35 2 © 2 

 © 2 ^ 2 



/*, 1, gh 

 ¥, /, i 



i, fg, g 



Also ; — 



^ 3 ®„ meet in A 3 , viz., 



B 3' 



C. 



© 2 a 2 , € 3 a 3 

 a 2 2S 2 , a 3 3S 3 



)5 



'3' " 



Farther ; — 



iS 2 3 



® 2 a 3 



Hence ;- 



Now; — 



3S 2 ® 3 , 23 3 © 2 meet in A 4 , viz., 



® 2 a 3 , e 3 a 2 „ b 4 , 

 a 2 B 3 , a 3 33 2 „ a 



'4, » 



Also^ A X B, CA „ 33, 



( B X C, A B 



©, 



*'3* J '3 



is r #, ^, l, ^ 



is -j 1, h, hf, I ; 

 is Ifg, 1, /, J 



' 1-flW, A(^-l), ^(A 2 -l) 



A(/ 2 -l), 1-A 2 / 2 , /(^ 2 -l) 



<7(/ 2 -i), /(^ 2 -i) s i-/V 



33 3 © 2 



M/ 2 -i), 



0(/ 2 -l), 



f, hf, h I ; 



/. g> fg J 



h(g*-l), g(h?-\) 



p-h\ /(A 2 -l) 



A being 

 A 



1, 0, 



1-gW, h(g*-l), <?(A 2 -1) 



fc 2 -/, M? 2 -i), ?(^ 2 -i) 



A A 3 A 4 is a straight line. 

 B B 3 B 4 „ ,, 

 C C 3 C 4 „ ,, 



0. Therefore ; — 



AA 3 A t is 

 B B 3 B 4 is 

 C C 3 C 4 is 



Theorem 8. 



0, <7(1 -A 2 ), -A(l-V) 



-/(1-A 2 ), 0, A(l-/ 2 ) 



f{\-g% -g(l-f 2 ), 



A A 3 A 4 , B B 3 B 4 , CC 3 C 4 meet in a point P 4 , viz., 



VOL. XXIV. PART I. 



0. Therefore ;- 



l_/2 \_ g 2 l _ h 2 



~c > ' 1 ' 



f 9 h 



M 



