7 
x 
a 
y z 
+ T + “ = 
b c 
Let AC (fig. 4) in the plane of zx be the straight line to 
which the generating straight line is always parallel, BC in 
the plane of yz that along which it moves; G a point in the 
generating line DE in any position, ON=x, NH=y, HG = z, 
its co-ordinates. 
Draw DF, FE parallel to OC, OA, and let OA, OB, OC, 
which are called the intercepts of the co-ordinate axes be re 
spectively denoted by a, b, c. 
Then since G is a point in DE, 
but 
Z X 
DF + FE = 1; 
0) 
DF y 
— = 1 -f 
c b 
FE 
a 
y 
. , . DF 
therefore, multiplying the first term of (l) by ——, the second 
FE 
term by , and the second member by the equal quantity 
1 
y 
6’ 
and transposing, we get 
z at y 
cab 
a relation holding between the co-ordinates of any point in the 
generating line in any position, and consequently the equation 
to the plane generated, and expressed by the intercepts of the 
co-ordinate axes produced in the positive directions to meet the 
plane. 
Cor. It is easily seen that this result might have been 
deduced from Art, 7. For if we take z = Ax + By + c to be 
the equation to the plane ABC (fig. 4), and in it make z = 0, 
y - 0, x = a for the point A, we get 0 = A a + c; and making 
» = 0, x = 0, y = b for the point B, we get 0 = Bb + c ; there 
fore, substituting for A and B their values, the equation to the 
plane becomes, as above,