6 — 2 
9] 
43 
9. 
DEMONSTRATION OF PASCAL’S THEOREM. 
[From the Cambridge Mathematical Journal, vol. iv. (1843), pp. 18—20.] 
Lemma 1. Let U = Ax + By + Cz = 0 be the equation to a plane passing through 
a given point taken for the origin, and consider the planes 
ffi = 0, U 2 = 0, £73=0, £7 4 = 0, U 5 = 0, U 6 = 0; 
the condition which expresses that the intersections of the planes (1) and (2), (3) and 
(4), (5) and (6) lie in the same plane, may be written down under the form 
0. 
-Ац A 2 , A 3 , A 4 
Вг. B 2) B„ B 4 
Сг, C 2 , C 3 , C 4 . . 
• • A 3 , A 4 , A 6 , A e 
• . в, , B 4 , B„ Be 
! • . C 3 , C 4 , C 5 , Ce 
Lemma 2. Representing the determinants 
I a-л, y x , z 4 &c. 
Ж 2) У%> 
Х 3> Уз, Z 3 
by the abbreviated notation 123, &c.; the following equation is identically true : 
345 . 126 - 346.125 + 356.124 - 450.123 = 0.