238 
ON THE RATIONAL TRANSFORMATION BETWEEN TWO SPACES. [447 
of the first space is a scroll (a theorem the analytical verification of which seems by 
no means easy). But without assuming the identity of the scroll II with this Jacobian, 
or taking the order of the scroll to be known, I proceed to show that the scroll II 
is the scroll generated by the lines each of which meets the curve 2 three times; 
it will thereby appear that the order is = 8, and that the curve is a triple line on 
the scroll. 
Consider a point P' on 2', and the corresponding line L of the first space: take 
©' a plane in the second space; corresponding to it the cubic surface © in the first 
space. By imposing a single relation on the coefficients {a, b, c, d) in the equation 
ax' + by' + cz' + dw' = 0 of the plane ©', we make it pass through the point P'; 
therefore by imposing this same single relation on the coefficients (a, b, c, d) of the 
cubic surface ©, we make it pass through the line L ; © is a cubic surface through 2; 
and it is easy to see that the effect will be as above only if the line L cuts the 
curve 2 three times; this being so, the general cubic surface © meets L in three 
points (viz., the three intersections of L with 2), and if © be made to pass through 
a fourth point on the line L, it will pass through the line L; it thus appears that 
the line L meets 2 three times, and consequently that the scroll II is generated by 
the lines which meet 2 three times. 
110. The theory of a scroll so generated is considered in my “Memoir on Skew Surfaces, 
otherwise Scrolls”^). Writing to = 6, h = 7 and therefore M [= — \m{m — 1) + h\ = — 8, 
the order of the scroll is [to] 3 + (to — 2) M = 40 — 32) = 8 ; but calculating the values of 
NG (m 3 ) = 1 [to] 4 + 6to + M (3 [to] 2 - 12to + 33) + M\ 3, 
NR (to 3 ) = [to] 6 + f [to] 5 — ^ [to] 3 — 3to + M(J$ [to] 4 — ^ [to] 3 — f to 2 + 8to — 20)+M 2 (^ [to] 2 —2to) ; 
these are found to be respectively = 0; viz., there are no nodal generators, and no 
nodal residue; the sextic curve 2 is a triple curve on the surface, and there is not 
any other multiple line. 
111. It may be remarked that any plane ©' meets the sextic curve 2' in six 
points; hence the corresponding cubic surface © contains six lines, generatrices of II, 
and, therefore, each meeting the curve 2 three times; say six lines L. Through one 
of these lines L, draw to the cubic surface © a triple tangent plane meeting it in the 
line L and in two other lines, say M, N; this plane must meet 2 in three new points 
which must lie on the lines M, N; viz., one of these lines must pass through two 
of the points, and the other line through the third point. 
Addition—September, 1870. 
[Some corrections have been made in accordance with the concluding paragraph of a 
paper “Note on the Rational Transformation and on Special Systems of Points,” 450.] 
The formulse of No. 84 are included in the following more general formulae; viz., 
if the principal system consist of points, each a simple point, or 2 points each a 
1 Phil. Trans, vol. cun. 1863, pp. 453—483, [339]. See the Table S (m 3 ) &c., p. 457; in the value of 
NR (m :J ) instead of term + 3m read - 3m. [This correction should have been made in the present Reprint.]