6 
But, Q being a point in the line BC, Q.M = B. MO + OC, 
PN = A . MN + B . MO + OC ; 
or, if x, y, » denote the co-ordinates of P, 
z = A x + By + c, 
the equation to the plane. 
Cor. The lines in which a plane intersects the co 
ordinate planes are called its traces on those planes ; thus 
AC, BC, are two of the traces of the plane BCA, the third 
being the line in which, if prolonged, it would intersect the 
plane of xy. Hence, we perceive the meaning of the constants 
in the equation to a plane, 
z — Ax + By + c ; 
for A, B, are the tangents of the angles at which the traces on 
zx,yz, are respectively inclined to the axes of x and y pro 
duced in the positive directions ; and c is the portion of the 
axis of z produced in the positive direction, intercepted 
between the plane and the origin. 
It is important to observe, that the foregoing method 
of finding the equation to a plane, applies equally to the case 
where the co-ordinates are oblique, and leads to a result of 
the same form ; so that whether a plane be referred to rect 
angular or oblique axes, its equation may be represented by 
z = Ax + By + c; but in the latter case, A will signify the 
ratio of the sines of the angles at which the trace on zx is 
inclined to the axes of x and z ; and B the ratio of the sines 
of the angles at which the trace on yz is inclined to the axes 
of y and z. Hence, all results which involve no other as 
sumption than z — A x +By + c, for the form of the equation 
to a plane, will be equally true for oblique and rectangular 
co-ordinates. 
8. To investigate the equation to a plane under the form 
X y z 
~ + — + — = 1. 
a b c