416] 
ON THE THEORY OF RECIPROCAL SURFACES. 
587 
The foregoing equation (26) in fact expresses that the surface and its reciprocal have 
the same deficiency; viz. the expression for the deficiency is 
(30) Deficiency = £ (n - l)(n - 2)(n - 3) - (n - 3)(b + c) + %{q + r) + 2t + f/3 4- §y + i - %9 t 
= K» / -l)(» # -2)(n , -3)- &c. 
629. The equation (28) (due to Prof. Cayley) is the correct form of an expression 
for /3', first obtained by him (with some errors in the numerical coefficients) from 
independent considerations, but which is best obtained by means of the equation (26); 
and (27) is a relation presenting itself in the investigation. In fact, considering a as 
standing for its value n (n — 1) — 2b — 3c, we have from the first 25 equations 
6 
a 
= 2 
+ 2 
3 n — c — k 
= 2 
- 2 
a(n — 2) — k + B — p — 2 a — 3<w 
= 2 
- 4 
b (n - 2) — p — 2/3 — 3y — 31 
= 2 
- 6 
c (n — 2) — 2a — 4/3 — 7 — 9 — q) 
= 2 
+ 2 
n + k — a — 2G — 4 B — 2j — 3p£ — 
Sco = 2 
- 3 
2q-2p + /3+j 
= 2 
- 2 
Sr + c — 5a — /3 — 4# + % — (o 
= 2 
and multiplying these equations by the numbers set opposite to them respectively, and 
adding, we find 
— 2 n 3 + 12 n 3 + 4 n + b (12 n — 36) + c (12 m — 48) 
- 6q-6r-4G - 105-41/3 - 30y - 24i - 7j - 8 X + 29 - 4© = 2, 
and adding thereto (26) we have the equation (27); and from this (28), or by a like 
process, (29), is obtained without much difficulty. As to the 8 ^-equations or symmetries, 
observe that the first, third, fourth, and fifth are in fact included among the original 
equations (for an expression which vanishes is in fact =2); we have from them 
moreover 3n — c= 3a' — k, and thence 3n — c — k= 3a' — k — k, which is = 2, or we have 
thus the second equation; but the sixth, seventh, and eighth equations have yet to 
be obtained. 
630. The equations (15), (16), (17) give 
n' = a (a — 1) — 28 — 3/c, 
d = 3a (a - 2) - 68 - 8k, 
V = \a(a - 2)(a 2 -9)- (a 2 - a- 6)(28 + 3/c) + 28 (8 - 1) + 68* + f* (* - 1); 
from (7), (8), (9) we have 
(a— b — c) (n — 2) = k — B — 6/3 — 4y — St — 6 + 2&>, 
(a-2b- 3c) (n - 2) (n - 3) = 2 (8 - C) - 8k - 18h - Qbc +18/3 + 12y + 6i - 6<u, 
74—2