523]
ON THE TRANSFORMATION OF UNICURSAL SURFACES.
389
For this purpose, imagine through the nodal curve a surface of the order k, which
therefore meets the surface besides in a curve of the order nk — 2b; this curve I call the
&-thic residue of the nodal curve, or simply the “ residue.” The projection (Abbildung) of
the complete intersection hi is a curve of the order kn' passing kr times through each
of the points a r : this is made up of the projection of the nodal curve once, and of
the projection of the residue. But as shown by Dr Clebsch the projection of the
nodal curve is of the order (n — 4) n' + 3, and it passes (n — 4) r + 1 times through each
of the points a,.; hence the projection of the residue is of the order (k — n+ 4) ri — 3,
and it passes (k — n + 4) r — 1 times through each of the points ai r . I assume that
the projection of the residue is the general curve which satisfies the foregoing con
ditions, viz. that the residue, and its projection as defined by the foregoing conditions,
depend each of them on the same number of constants. The necessity for this is I
confess by no means obvious: but take as an illustration Steiner’s quartic surface as
transformed by the equations x : y : z : w = x' 2 : y' 2 : z' 2 : (pc' + y' + z') 2 : the nodal curve
consists of three lines meeting in a point, the quadric residue is the remaining inter
section of the surface by a quadric cone passing through the three lines; and the
projection thereof is a line; the quadric cone, and therefore the conic, each depend
upon 2 constants; and the line which is the projection of the conic depends upon
the same number (2) of constants: at all events I make the assumption provisionally.
Now in the projection of the residue, we have twice the number of constants
= [(& — n + 4) n' — 3] (k — n + 4) n! — X [(& — n + 4) r — 1] (k — n + 4) ra r ,
viz. this is
= (k — n + 4) 2 (n 2 — ’Er 2 a r ) + (Ic — n + 4) (— 3n' + %ra r ),
or, what is the same thing, it is
= {k — n + 4>) 2 n +(k — n + 4) (w — 6 — 2©),
viz. reducing, and replacing © by its value =—^ (n—2)(n— 3)+b, the number in
question is
= k 2 n + k (— n 2 + 4m — 2b) + 2 (pi — 4) b.
Now k being = or >n- 3, the curve of intersection of a given surface n by a
surface k depends on
%(k + 1) (k + 2)(k + 3) - $(k-n +1) (k-n + 2)(k-n + 3)- 1
constants ; and making the surface k to pass through the curve b we have to subtract
herefrom (k -1-1) b — \g — 2t; that is, for the residue, twice the number of constants is
= }(k + 1) (k + 2) (k + 3) - £ (k - n +1) (k - n + 2) (k - n + 3) - 2 - 2 (k + 1) b + q + 41,
viz. this is
= k 2 n + k (— n 2 + 46 — 2b) +^(n — 1) (n — 2) (n — 3) — 2b + q + 41.
