365] INTERSECTIONS OE A PENCIL OF FOUR LINES BY A PENCIL OF TWO LINES. 485 
Q; and the four lines are aa', bb', cc', dd, meeting suppose in P; then the twelve 
points are 
a'd.c'b, ad'.cb', a'c .d'b, ac' .db' lying in a line through P, 
a'b.d'c, ab' .dc', dd.b'c, ad'.be „ „ , 
a'c .b'd, ac' .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 ^ ^ (or any other arrangement having the 
diagonals ab. cd), and thence writing down the four expressions 
a'c, a c, a'c, ac' 
d b, 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’, c'), the points ab'. a'b, ac . a'c, be'. b'c 
lie in a line through Q; and similarly the points ab'. a'b, ad'. ad, 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, c, d), (b', c', d'), 
the line in question also passes through cd'. c'd; that is, the six points ab'. a'b, 
ac'. a'c, ad'. a'd, be'. b'c, bd'. b'd, cd'. c'd lie in a line through Q, which is in fact the 
before-mentioned 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 respectively, the triangles Mad, Mbb', 
Ncc, Ndd are in perspective. Hence, considering the two triangles Mad, Ndd (or, if 
we please, the complementary set Mbb', Ncc'), the corresponding sides are 
Ma, Nd meeting in ab'. dc', 
Md, Nd' „ a'b. dc, 
ad, dd „ P ; 
that is, the points ab'. dc', a'b. dc lie in a line through P. 
Similarly ad. dd and be'. b'c lie in a line through Q; or, calling these points H, I 
respectively, the triangles Had, Hdd, Ibb', lec' are in perspective; and considering 
the combination Hdd, Ibb' (or, if we please, the complementary set Had, led), the 
corresponding sides are 
Ha , lb meeting in ad . bd, 
Hd, lb' „ a'd.cb', 
aa' , bb' „ P ; 
that is, the points a'd .c'b, ad'.cb' lie in a line through P.