403] 
129 
403. 
ON PASCAL’S THEOREM. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. ix. (1868), 
pp. 348—353.] 
I consider the following question: to find a point such that its polar plane in 
regard to a given system of three planes is the same as its polar plane in regard to 
another given system of three planes. 
The equations of any six planes whatever may be taken to be X = 0, Y = 0, Z — 0, 
U=0, V — 0, IF = 0, where 
X+ Y+ Z+ U+ V+ W = 0, 
aX + bY + cZ+fU+gV +hW =0, 
and so also any quantities X, Y, Z, U, V, W satisfying these relations may be regarded 
as the coordinates of a point in space; we pass to the ordinary system of quadriplanar 
coordinates by merely substituting for V, W their values as linear functions of 
X, Y, Z, U. 
This being so, the equations of the given systems of three planes may be taken 
to be 
XYZ= 0, UVW = 0, 
and if we take for the coordinates of the required point (x, y, z, u, v, w), where 
x + y+ z+ u+ v+ w — 0, 
ax + by + cz + fu + gv + hw = 0, 
then the equations of the two polar planes are 
X Y Z A 
—I 1— — 0, 
x y z 
U V W 
—I 1— — o, 
u v w 
C. VI. 
17