411] A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 353 
where x, x', g are constants which remain to be determined. The cubic surfaces fail 
to determine them, for the reason that in all of them we have i = 0, i! = 0; and 
165 + 8^ + 6 — 16-6' + 8 X + 0': this last is a very remarkable relation, for the existence 
of which I do not perceive any a priori reason. 
Substituting herein for q, r their values from No. 11, this may be written in the 
form 
/3' = 2ft (ft - 2)(lift - 24) + 6 (- 66ft + 184) + c (- if*» + 240) 
+ 141/3 + ^7 + 66« 
x' i! + 7j' - 6 X ' - \&- 50' - 18B' 
-(# + 87)i -2lj -^ x + ^0-24>O 
+ j\g (-16B-8 x -0 + 16B' + 8 X + ff). 
64. We have of course by interchanging the unaccented and accented letters, the 
reciprocal equation giving the value of ¡3. 
Article Nos. 65 to 68. Recapitulation. 
65. In recapitulation, I say that we have between the 42 quantities 
ft, a, S, k ; 
/ / W / 
n, a } o, k ; 
b,k,t,q,p,j; c, h, r, 
b', k', t', q', p, j'; c', h\ /, 
J, O', x\ /3', ï; B', O', 
in all 25 equations, viz. these are 
a = a', 
a' = ft (ft — 1) — 26 — 3c, 
k = 3ft (ft — 2) — 66 — 8c, 
S' = \n(n— 2) (ft 2 — 9) — (ft 2 — ft — 6) (26 + 3c) + 26 (6 — 1) + 66c + fc (c — 1), 
a (n — 2) — k — B + p + 2<r, 
6 (ft - 2) = p + 2/3 + 3y + St, 
c (ft — 2) = 2cr + 4/3 + 7 + 6, 
a (ft — 2) (ft — 3) = 2 (3 — (7) + 3 (ac — 8a - x ) + 2 (a6 — 2p — j), 
6 (ft — 2) (ft — 3) = 4& + («6 — 2p - j) + 3 (6c — 3/3 — 2y — ¿), 
c (ft — 2) (ft — 3) = 6h + (ac — Sa — x ) + 2 (6c — S/3 — 2y — i), 
q = b 2 - b — 2k — Sy — 6t, 
r — c~ — c — 2h — 3/3, 
a — n (ft/ — 1) — 26' — 3c', 
k = 3ft' (ft' - 2) - 66' - 8c', 
S = \ri (ft' - 2) (ft' 2 - 9) - (ft' 2 - ft' - 6) (26' + 3c') + 26' (6' - 1) + 66V + fc' (o' - 1), 
45 
c. VI.