576 GEOMETRY. [790 
quantities p, v; and the identity x? + y? + z? = x 2 + y 2 + z 2 can be at once verified. 
It may be added that the transformation can be expressed in the quaternion form 
i&\ + jy l + kz 1 = (1 + A) (ix +jy + Jcz) (1 + A) -1 , 
where A denotes the vector i\ +jp + kv. 
Quadric Surfaces (Paraboloids, Ellipsoid, Hyperboloids). 
37. It appears, by a discussion of the general equation of the second order 
{a,...\x, y, z, l) 2 = 0, that the proper quadric surfaces* represented by such an equation 
are the following five surfaces (a and b positive):— 
(1) z = Y~ + i|r> elliptic paraboloid. 
ACt AO 
(2) z = x ~yr, hyperbolic paraboloid. 
ACt AO 
n/2 ¿2 
(3) - 2 +^ 3 + g 2 = 1, ellipsoid. 
(4) °L q_ U- — — l ) hyperboloid of one sheet. 
(5) — 2 + nr —- 2 = — 1, hyperboloid of two sheets. 
CL“ 0“ C 
It is at once seen that these are distinct surfaces; and the equations also show 
very readily the general form and mode of generation of the several surfaces. 
Fig. 20. 
In the elliptic paraboloid (fig. 20), the sections by the planes of zx and zy are 
the parabolas 
z = 
x 2 
2a ’ 
* The improper quadric surfaces represented by the general equation of the second order are (1) the pair 
of planes or plane-pair, including as a special case the twice repeated plane, and (2) the cone, including as 
a special case the cylinder. There is but one form of cone; but the cylinder may be parabolic, elliptic, or 
hyperbolic.