356 A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. [411 
and from one of the equations of No. 11 we have n — 2(7 — 45 + k — a — 2j—S X —n+n' — a, = 2 ; 
we have thus the system of eight equations, 
a 
= 2, 
3n — c — k 
= 2, 
a (n — 2) — k 4- 5 — p — 2a 
— Y 
b (n— 2) — p — 2/3 — Sy — St 
— V 
c (n — 2) — 2a — 4/3 — 7 — 0 
= 2, 
?? 4- « — er — 2(7 —45 — 2J — 
OJ 
II 
M 
2q — 2p 4- /3 4- 3» 4- j 
= 2, 
3r 4- c — 5a — /3 — 40 4- 2» 4- 
X = 2; 
or if from these we eliminate k, p, a, then the system of five equations, 
a = 2, 
n (c — 8) — 4/3 — 7 — 0 4- 4(7 4- 85 + 6^ 4- 4j = 2, 
(a - b) (n - 2) - \\n + 3c + 4(7 + 95 + 2/3 4- 3 7 + 3t + 6 x + 4j = 2, 
3r - 20w + 6c - /3 + 2» + 10(7 4- 205 + 16 x + 10j -46» =2, 
2q — 2b (n — 2) 4- 5/3 + 67 4- 66 4- Si +j = 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 
foregoing Memoir: 
Deficiency = £ (n — 1) (n — 2) (n — 3) — (n — 3) (b 4- c) + £ (q 4- r) 4- 2t + |y3 4- + i — £0. 
This should be equal to the deficiency of the reciprocal surface, viz. we must have 
2 (n — 1) (n -2)(n- 3)- 12 (n - 3) (64- c) + Qq 4- 6r + 24i 4- 42/3 + 30 7 + 12» -|0 = 2 ; 
but from a combination of the last-mentioned five equations we have 
— 2n 3 4- Qn 2 4- 4n 4- (12n — 36) b 4- (12n — 48) c — 6q — 6r — 241 
- 41/3 - 30 7 - 13» - 7j - 8 X 4- 20 - 4(7 - 105 = 2 ; 
and adding to the last preceding equation we have 
26w - 12c 4- /3 - i - 7j - S x 4- \0 - 4(7 -105 = 2. 
Substituting for 2 its value in terms of the accented letters, we obtain for /3' the value 
/3' = /3 4- 26w - 12c 4-»' 4- 7/ 4- 8 X ' - £0' 4- 4(7' 4- 105' 
- 26?/ 4- 12c' - i - 7j - 8 X + \6 - 4(7 - 105. 
c = — Si» 4* k 4" 311, 
We have 
and thence 
12c' - 26?/ = - 36a + 12/c 4-10?/ ;