[ 83 ] 
V. Corrections and Additions to the Memoir on the Theory of Reciprocal Surfaces 
(Philosophical Transactions, vol. clix. 1869). By Professor Cayley, F.R.S. 
Received July 22, — Read November 16, 1871. 
1. I am indebted to Dr. Zeuthen for the remark that although the “ off-points” and 
“ off-planes,” as explained in the memoir, are real singularities, they are not the singu- 
larities to which the $, & of the formulae refer. The most convenient way of correcting 
this is to retain all the formulae with Q, & as they stand, but to write co, oj for the number 
of “ off-points” and “ off-planes” respectively ; viz. we thus have 
co, off-points, 
6, unexplained singular points, 
and 
oJ , off-planes, 
S', unexplained singular planes, 
the formulae as they stand, taking account of the unexplained singularities 6 and 0, but 
not taking any account at all of the off-points and off-planes co, oJ . The extended for- 
mulae in which these are taken into account are : — 
a(n — 2)=% — B ^ -\~cjco, 
h(n— 2) = £+2/3 + 3y-f-3 1, 
c(n — 2) = 2c -j- 4/3 y S -{- co, 
a{n - 2)(n - 3) = 2(& - C - 3* ) + 3(ac - 3a - x - 3«) + 2 (ah - 2 § -j), 
h(n — 2)(n — 3)= 4 k -\-(ab — 2q—j) +3(^6'— 3/3 — 2y — i), 
c(n—2)(n—3)= 64 -\-(ac— Sa—^—oco) +2(fo— 3/3 — 2y— i), 
which replace Salmon’s original formulae (A) and (B). 
2. In the formulae 
q=h 2 — b — 2k— 3y— 6t, 
r^c 1 — c — 21i — 3/3, 
it is assumed that the nodal curve has no actual multiple points other than the t triple 
points, and no stationary points other than the y points which lie on the cuspidal curve ; 
and similarly that the cuspidal curve has no actual multiple points, and no stationary 
points other than the /3 points which lie on the nodal curve; and this being so, q is the 
class of the nodal curve and r that of the cuspidal curve. But we may take the formulae 
as universally true ; viz. q may be considered as standing for b 2 — h — 2Jc — 3y— 6£, and r 
m 2 
