[ 201 j 
YI . A Memoir on the Theory of Reciprocal Surfaces. By Professor Cayley, F.B.S. 
Received November 12, 1868, — Read January 14, 1869. 
The present Memoir contains some extensions of Dr. Salmon’s theory of Keciprocal 
Surfaces. I wish to put the formulae on record, in order to be able to refer to them in 
a “ Memoir on Cubic Surfaces,” but without at present attempting to completely deve- 
lope the theory. 
Extension of Salmon’s Fundamental Equations. Article Nos. 1 to 5. 
1. The notation made use of is that of Salmon’s ‘ Geometry,’ pp. 450-459, with the 
additions presently referred to ; the significations of all the symbols are explained by 
way of recapitulation at the end of the Memoir. I remark that my chief addition to 
Salmon’s theory consists in a modification of his fundamental formulae (A) and (B) ; 
these in their original form are 
a(n— 2)= z-\- 
50-2)= *+2j3 + 3y+3$, 
cO—2)=2ff-f4 ( 6+ y, 
a(n- 2)0— 3)=25 +3[«c]+2[«5], 
h(n — 2)0 — 3) = 4£+ [aQ+3[5c], 
c(n— 2)0— 3)=6A+ [m?] + 2[5c], 
where 
[« 5 ]=« 5 — 2 ^, 
[ac\=ac— 3<r, 
[5c]=5c— 3/3— 2y— «. 
2. I take account of conical and biplanar nodes, or, as I call them, cnicnodes, and 
binodes; of pinch-points* on the nodal curve; and of close-points and off-points on the 
cuspidal curve : viz. I assume that there are 
C, cnicnodes, 
B, binodes, 
j , pinch-points, 
X, close points, 
& , off-points, 
* 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. 
MHCCCLXIX. 2 P 
