447] ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. 231 
Y' = 0, Z' — 0, W' = 0 are quadric surfaces passing through three fixed points (say the 
principal points) and through a fixed line (say the principal line) in the second figure. 
Taking x' = 0, y' = 0 for the planes passing through the principal line and through 
two of the principal points respectively; z' = 0 for the plane passing through the 
three principal points, w' = 0 for an arbitrary plane passing through the first mentioned 
two principal points, the implicit factors of x, y', w may be so determined that for 
the third principal point x' — y' — — w'. That is, we shall have 
for principal line x' = 0, y' = 0, 
for principal points {x' = 0, z' = 0, w' = 0), 
» (y' =0, / = 0, w' = 0), 
„ {x' = y' = -w', z = 0), 
and this being so, the equation of a quadric surface through the principal points and 
line will be 
{ax' + ¡3y') z' + yx' {y' 4- w') + By' {x' + w'), 
and the equations of transformation may be taken to be 
x : y : z : w — x'z' : y'z' : x' {y' + w') : y' {x' + w). 
94. Writing these in the extended form 
x : y : z : w : x — y : z — w = x'z' : y'z' : x' {y' + w') : y' {x + w') : z' {x — y') : w' {x — y') 
and forming also the equation 
xy : {xw — yz) — z' : x' — y', 
we at once derive the reciprocal system of equations 
x' : y : / : w' = x {xw — yz) : y {xw — yz) : (x — y) xy : {z — w) xy, 
so that this is a cubic transformation. And the cubic surface in the first space 
(corresponding to an arbitrary plane ax' + by' + cz + dw' = 0 of the second space) is 
{ax + by) {xw —yz) + c {x — y)xy + d{z — w) xy—Q ; viz., this is a cubic surface having 
the fixed double line {x = 0, y = 0), the fixed simple lines {x = 0, z — 0), {y — 0, w = 0), 
and {x — y — 0, z — w = 0); it has also the variable simple line {dz + cx = 0, dw + cy = 0). 
The principal figure of the first space thus consists of the three simple lines {x = 0, 
z = 0), {y = 0, w = 0), {x — y = 0, z — w — 0), and of the line {x = 0, y = 0), a double line 
counting four times in the intersection of two of the cubic surfaces. 
95. The cubic surface as having the double line (x = 0, y = 0) is a cubic scroll, 
and this line is the nodal directrix thereof; the line {dz -\-cx = 0, dw + cy = 0) is the 
simple directrix ; the lines {x — 0, ^ = 0), {y= 0, w — 0), {x — y — 0, z — w= 0) are at once 
seen to be lines meeting each of these directrix lines; and they are generating lines 
of the scroll. To explain the generation of the scroll, observe that the section by 
any plane is a cubic curve having a given double point (viz., the intersection of the 
plane with the nodal directrix); and three other given points (viz., the intersections of