Theorems in the Geometry of Coordinates. $55 
which is the equation of the line passing through the origin, 
and parallel to the line whose equation is 
Oh 
i a B al 1. 
Turorem II.—If the equations of two planes be repre- 
sented by 
fi and aoe See, 
Mes #1 By val 
then will the sum of these equations, viz. 
(ity Geghe (tsb 
& my p By eS. SBS: . 
be the equation of the plane passing through the line of in- 
tersection of these planes, and through the three points of in- 
tersection of the diagonals of the three quadrilaterals formed 
by the intersection of the given planes with the axes of co- 
ordinates. 
Let OX, OY, OZ be the oblique axes of coordinates, and 
let P, Q, R be any three points in 
the axes of w, y, z respectively, and 
P’, Q’, R! any other three points in 
the same axes. Draw the traces 
PQ, QR, RP, P’Q'!, Q' R', R! P! 
forming with the coordinate axes 
the three quadrilaterals P Q Q’ P’, 
QRR Q, PRR P’, and let I, H, 
G be the points of intersection of 
the diagonals of these quadrilate- 
rals respectively. Then if we put 
OP=4 |0Q=6 |OR=y¥ 
OPaa|O@=6,|OR’=y, ~ 
we shall have the equations of the several planes as below. 
(PQR) .. ctptoe srcotea tle GED 
(?QR) .. Sih See eta fl dh plunge 
(POUR. eras i tts Ne Ala e! 
(PQ R’) tat «| sulle) 
(P'Q'R’) caper et - (5.) 
(2 QR) as Fue tc ~ 2 + «) (6) 
