112 
138. Any diametral plane of a surface having a center 
is parallel to the tangent plane applied at the extremity of 
the diameter to which it is conjugate. 
Taking the center for origin, the equation to the surface 
will be 
ax* + by 2 + cz 2 + 2a'yz + 2ft'zx + 2c xy + d = 0; 
therefore the equation to the tangent plane at a point ocyz is 
(Cor. l, Art. 107), 
{a x + h'z + cy) (x' - x) + {by + a z + ex) {y - y) 
+ (cz + a!y + b’x) {z' — z) = 0, 
or {ax + b' z + cy) x + {by + az + ex) y + {cz + ay + b' x)z 
+ d — 0 ; 
and if it be applied at the extremity of the diameter whose 
equations are x = mz, y = nz, the equation becomes 
d 
{am + cn + b')x + {bn + cm + a) y + (c+ h'm + an)z + - = 0, 
and therefore (Art. 18) represents a plane parallel to the 
diametral plane which is conjugate to the diameter x - mz, 
y = nz, the equation to which is (putting a" = b" = c" = 0, 
in the equation Art. 137) 
{am + c n + b') x + {bn + cm + a) y + (c 4 b'm + an)z'= 0. 
This result might have been foreseen ; because the straight 
lines in which a diametral plane and a tangent plane at the 
extremity of the conjugate diameter are cut by any plane 
through that diameter, must, by the nature of lines of the 
second order, be parallel to one another. 
139. We shall now proceed to the reduction of the 
general equation of the second degree 
ax” + by 2 + cz 2 +2n'yz+2b'zx+2c'xy+2a"x+2b"y+ 2c"z+d = 0,