60 
In this case also the projections of the generating lines are 
tangents to the principal sections of the surface. 
Thus (changing the sign of x in both equations) 
— x = mz + h will be a tangent to z~ = l'at, 
if z 2 + I'mz + I'h = 0 be a perfect square, 
nfl' 
or 41' h = l’ 2 m 2 . or h = ; 
4 
f ml\ 
— x = m I z + —i 
is the equation to a tangent; which, measuring x in the 
positive direction, coincides with the equation to the projection 
of the generating lines on zx. 
78. We shall terminate this Section with demonstrating 
the following general and important property of surfaces of 
the second order. 
If two surfaces of the second order have a plane section 
in common, their other curve of intersection, if it exist, will 
also be a plane curve. 
Let the equations to the two surfaces be 
Ax 2 + By 2 + Cz 2 + %A'yz + 2B 1 zoc + 2 C'ooy + 2 A' so + 2 B"y 
+ 2C"z + D = 0, 
aa? 2 + hy 2 + cz 2 + Za'yz + 2b'zx + 2c xy + 2 ax + 2 h"y 
+ 2c" z + d = 0, 
and suppose them to have a common section in the plane 
of xy ; then making z = 0, the curves represented by the 
equations 
Ax 2 + By 2 + 2 C' xy + 2 A" x + 2 B" y + D = 0, 
ax 2 + by 2 + 2c xy + 2a"x + 2b"y + d = 0,