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ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. [447 
the plane with the three generating lines respectively); this cubic also passes through 
the intersection of the plane with the simple directrix. Conversely, if the plane be 
assumed at pleasure, and if, taking for the simple directrix any line which meets the 
given generating lines, we draw a cubic as above, then the scroll is the scroll 
generated by a line which meets each of the directrix lines, and also the cubic. 
If the plane be taken to pass through any generating line, then the cubic section 
breaks up into this line, and a conic; the conic does not meet the simple directrix, 
but it meets the nodal directrix; and any such conic will serve as a directrix; viz., 
the scroll is generated by the lines which meet the two directrix lines and the conic. 
96. Any two scrolls as above meet in the three fixed generating lines, and in 
the nodal directrix counting four times; they consequently meet besides in a curve of 
the second order, which is a conic (one of the conics just referred to). In order to 
further explain the theory, suppose for a moment that the two scrolls had only a 
common nodal directrix; they would besides meet in a quintic curve; this curve would 
meet the nodal directrix in four points, viz., the points at which the two scrolls have 
a common tangent plane. Now if at any point of the nodal directrix the two scrolls 
have a common generating line, then the plane through this line and the nodal line 
is one of the two tangent planes of each scroll; that is, the scrolls have this plane 
for a common tangent plane. Hence, in the case of the common three generating 
lines, the points where these meet the nodal line are three of the four points just 
referred to; there remains therefore one point, which is the point where the conic 
meets the nodal line; through this point there are for each of the scrolls two 
generating lines; one of these for the first scroll, and one for the second scroll, lie in 
a plane with the nodal line; the other two determine the plane of the conic; and 
the tangent to the conic at its intersection with the nodal line is the intersection of 
the plane of the conic with the plane of the first-mentioned two generating lines. 
97. Analytically we have the two equations 
c (x — y)xy + (ax + by ) (xw — yz) + d (z — w) xy= 0, 
c (x — y)xy + (a'x + b'y) (xw — yz) + d' (z — w) xy = 0 ; 
or, combining these equations so as to eliminate successively the terms in x (xw — yz) 
and y (xw — yz), and for this purpose writing 
(be' — b'c, ca' — c'a, ah' — a'b, ad' — a'd, bd' — b'd, cd' — c'd) = (a, b, c, f, g, h), 
and therefore 
af + bg + ch = 0, 
we have 
b (x — y) x — c (xw — yz) — f (z — w) x = 0, 
— a (x — y) y + c (xw — yz) — g(z — w)y = 0, 
and multiplying the first of these by c + g and the second by c — f, and adding, the 
whole divides by x — y, and the final result is 
(c + g) (bx - fz) - (c - f) (ay + gw) = 0 ; 
viz., this is the equation of the plane of the conic.