Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., sadlerian professor of pure mathematics in the University of Cambridge (Vol. 6)

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.
	        
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