44 
DEMONSTRATION OF PASCALS THEOREM. 
[9 
This is an immediate consequence of the equations 
. . x 3) X i} X 5 , Xg 
= 
. . X 3 , X iy Xg, Xg 
2/s, 2/4, 2/s, 2/s 
2/s, 2/4, 2/s, 2/6 
Z 3 , Z iy Zg , Zg 
rp rp rp rp rp rp 
> *A / 2> '"3) *^4) ^5? 
X\t X 3 , 
2/i, 2/2, 2/s, 2/4, 2/s, 2/e 
2/i, 2/2, • • 
Z l> , ^4, Z 5) Z 6 
¿1, ¿2, • • 
Consider now the points 1, 2, 3, 4, 5, 6, the coordinates of these being respectively 
x lf y 1} z x x 6 , y 6 , z 6 . I represent, for shortness, the equation to the plane passing 
through the origin and the points 1, 2, which may be called the plane 12, in 
the form 
12 x + y 12 y + z 12, = 0 ; 
consequently the symbols 12*, 12 y , 12 z denote respectively y x z 3 — y 2 z x , z x x. 2 — z 2 x u x x y 2 — x. 2 y x , 
and similarly for the planes 13, &c. If now the intersections of 12 and 45, 23 and 56, 
34 and 61 lie in the same plane, we must have, by Lemma (1), the equation 
12*, 
45*, 
23*, 
56*, 
12„ 
45j/, 
23^, 
56y, 
12,, 
45 z , 
23 z , 
56,, 
23*, 
56*, 
34*, 
61 
23„ 
56^, 
34j„ 
61, 
23„ 
56„ 
34 z , 
61 
Multiplying the two sides of this equation 
by the two sides respectively of the equation 
x 6 , x u 
2/e, yi, 
z ii 
x 3 , . . . = 612.345, 
2/2, • • • J 
¿2 , 
x 3 , 
x iy 
^5 
2/3, 
2/4, 
2/5 ! 
^3 , 
¿4, 
Zg 
and observing the equations 
x s 12* + y 6 12 y + z 6 12 z = 612, 
112 = 0, &c.