PROFESSOR CAYLEY ON RECIPROCAL SURFACES. 
227 
Addition, August 3, 1869. 
As in the theory of Curves, so in that of Surfaces, there are certain functions of the 
order, class, &c. and singularities which have the same values in the original and the reci- 
procal figures respectively ; for convenience I represent any such identity by means of the 
symbol 2, viz. $(n, a, b, . . .)=2 denotes that the function <p(n, a , b, . . .) is equal to the 
same function <p(n', a', b\ . . .) of the accented letters. By what precedes we have a=%; 
and it is moreover clear that any function of the unaccented letters which is =0, or which 
is equal to a symmetrical function of any of the accented and unaccented letters, or to 
a function of a, is =2 : for instance, from the equations of No. 5 we have 3 a! —x' = 3 n—c, 
and thence 3 n — c—x=?>a! — x — that is, 3 n — c — £=2 ; and from one of the equations 
of No. 11 we have n— '2C— <r— 2j— 3x=zn-\-n'— a, =2; we have thus the 
system of eight equations, 
a 
3 n — c—x 
a(n— 2)— x-\- B— §— 2a 
b{n- 2)- §-2/3-3y-3 1 
c(n— 2)— 2a — 4/3— y— 6 
n + *- <r-2C-4B-2/-3 x =2, 
2q-2g+ (3+3i + j =2, 
3 r+ c — 5a — /3 — 45 +2 \i-\- %=%. 
Or if from these we eliminate x , g>, <r, then the system of five equations, 
a =2, 
w(c— 8) — 4/3 — y — fl + 4C + 8B + 6 X + 4/ =2, 
(a-J)(»-2)-llw+3c+40+9B+2/3+3y+3#+6x+4;=2, 
3r-20w+6c-j3 + 2i+10C+20B+16 5C +lQ;-4fl =2, 
2i-2&(»-2j+6^ + 6y+6«+3i-h? =2. 
By means of a theorem of Dr. Clebsch’s I was led to the following expression for the 
“ deficiency ” of a surface of the order n having the singularities considered in the fore- 
going Memoir : 
Deficiency =i(n- - l)(n- — 2)(n — 3) — (n - 3)(6 + c) +£(?+*■)■ +.2#+ 1/3 +fy+?'— £0 ; 
This should be equal to the deficiency of the reciprocal surface, viz. we must have 
2(w-l)(w— 2)(w— 3)— 12(n-3)(5+c)+6^+6r+24#+42]3+30y-fl2e-i |fl = 2; 
but from a combination of the last mentioned five equations we have 
— 2 n 3 + 6 n 2 -f in -+- [12n — 36)5 -f- (12 n — 48)c — Qq — 6r — 2 it 
— 41/3 — 30y— 13^—7/— 8 x +20 — 40— 10B=2 ; 
2 i 2 
