330
A MEMOIR ON THE THEORY Of RECIPROCAL SURFACES.
[411
2. I take account of conical and biplanar nodes, or, as I call them, cnicnodes,
and binodes; of pinch-points( J ) on the nodal curve; and of close-points and off-points
on the cuspidal curve : viz. I assume that there are
G, cnicnodes,
B, binodes,
j , pinch-points,
X, close-points,
0, off-points,
deferring for the present the explanation of these singularities. The same letters,
accented, refer to the reciprocal singularities. Or using “ trope ” as the reciprocal term
to node, these will be
O', cnictropes,
B', bitropes,
j' , pinch-planes,
X> close-planes,
0', off-planes;
but these present themselves, not in the equations above referred to, but in the
reciprocal equations.
3. The resulting alterations are that we must in the formulae write k - B, 8-G
in place of k, 8 respectively; and change the formulae for c (n — 2), [rib], [be], into
c (n — 2) = 2a + 4/8 + 7 + 0,
[ab] = cib - 2p -j,
[ac] = ac — 3 a —
respectively.
4. Making these changes, and substituting for [ab], [ac], [be] their values, the
formulae become
a (n - 2) = k — B + p + 2a,
b (n — 2) = p + 2/3 + 37 + St,
c (n — 2) = 2a + 4/3 + 7 + 0,
a (n - 2) (n - 3) = 2 (8 - G) + 3 (ac - 3<r - x) + 2 (ctb - 2p - j),
b (n — 2)(n- 3) = 4& + (ab — 2p - j ) + 3 (be - S/3 — 27 — i),
c (n — 2) (n - 3) = 6h + (ac —Sa - x)+ 2 (be — S/3 — 2<y — i),
which replace the original formulae (A) and (B).
1 This addition to the theory is in fact indicated in Salmon, see the note, p. 445 ; the i there employed,
which is of course different from the i of his text, is the j of the present Memoir.