5 02 Intersections of a Pencil of four Lines by a Pencil of two Lines . 



a'd . c'b, ad' . cb', o!c . d'b, ad . db 1 lying in a line through P, 

 a'b . d'c, ab'. dd, a'd . b'c, ad', be' „ „ 



dc.b'd, ad.bd', a'b. c'd, ab'.cd' „ „ 



where the combinations are most easily formed as follows ; viz., 



for the first four points starting from the arrangement °\. c , (or 



any other arrangement having the diagonals ab .cd), and thence 

 writing down the four expressions 



a'd a c a' c ad 

 db' d'b" d'b> db" 



we read off from these the symbols of the four points ; and the 

 like for the other two sets of four points. 



Now, considering the points [a, b, c) and (a', b', d), the points 

 ab' . a'b, ad . a'c, be' . b'c lie in a line through Q ; and similarly the 

 points ab'.a'bj ad', a'd, bd'.b'd lie in a line through Q ; which 

 lines, inasmuch as they each contain the points Q and ab' .a'b, 

 must be one and the same line ; considering the combinations 

 {b y c, d), (b' } d, d 1 ), the line in question also passes through cd'.dd; 

 that is, the six points ab'.a l b, ad. a'c, ad', a'd, bd .b'c, bd'.b'd, 

 cd' . c'd lie in a line through Q, which is in fact the before-men- 

 tioned first theorem. Hence the points ab' . a'b and cd' . c'd lie 

 in a line through Q ; or, calling these points M and N respect- 

 ively, the triangles Maa' f Mbb', 'Ned, ~Ndd' are in perspective. 

 Hence, considering the two triangles Maa', ~Ndd' (or, if we please, 

 the complementary set Mbb', Ncc'), the corresponding sides are 



M# , Nd meeting in ab' . dd, 

 Ma', ~Nd' „ a'b. d'c, 



ad , dd' „ P ; 



that is, the points ab' . dd, a'b . d'c lie in a line through P. 



Similarly ad' . a'd and be' . b'c lie in a line through Q ; or, call 

 iug these points H, I respectively, the triangles Haa', Hdd', Ibb', 

 led are in perspective ; and considering the combination Wdd', 

 Ibb' (or, if we please, the complementary set Haa', led), the cor- 

 responding sides are 



H«, lb meeting in ad' .bd , 



Ra', lb' „ a'd.cb', 



ad, bb' „ P ; 



that is, the points dd . c'b, ad' . cb' lie in a line through P. 



It remains to be shown that the two lines through P, viz. the 

 line containing ab' .dd and a'b. d'c, and the line containing 

 ad' . bd and dd . cb' , are one and the same line. This will be 





