411] A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 331 
5. For convenience I annex the remaining equations; viz. these are 
a' = n (n — 1) — 26 — 3c, 
k = 3n (n — 2) — 66 — 8c, 
8' = \n (n - 2) (w 2 - 9) - (w 2 - n - 6) (26 + 3c) + 26 (6 -1) + 66c + fc (c -1); 
the equations 
q — b~ — 6 — 26 — 87 — 6t, 
r —c 2 — c — 2/i — 3/8, 
(2, r in place of Salmon’s R, S respectively); the equation 
a = a'; 
and the corresponding equations, interchanging the accented and unaccented letters, in 
all 23 equations between the 42 quantities 
11, 
a, c 
) , K 
n', 
a', i 
, K 
6, k, t, q, p, j ; c,h,r,a-,0,x‘, 
b', k', t\ q\ p, j'; c', h', r, 0\ x 5 
/3, j, i ; 
7 > ; 
B, G, 
B', c. 
Article Nos. 6 to 12. Developments. 
6. We have 
(a — 6 — c) (n — 2) =(tc — B — 6) — 6/3 — I7 — 3i, 
(a-2b- 3c) (n - 2) (n - 3) = 2 (6 - 0) 
- 8Jfe - 186 - 6 (6c - 3/8 - 2 7 -»); 
and substituting these values of 8, k in the formula 
n' = a (a — 1) — 28 — 3«, 
and for a its value, = n (n — 1) — 26 — 3c, we find 
n' = 11 (n — l) 2 — n (76 + 12c) + 46 2 + 86 + 9c 2 + 15c 
- 8k - 18h + 18/3 +127 + 12i - 91 
— 2C — SB — 3d, 
where the foregoing equations for a — b — c and a - 26 — 3c show clearly the origin of 
the new terms —2C— SB — SO; these express that there is in the value of n' a reduction 
= 2 for each cnicnode, = 3 for each binode, and = 3 for each off-point. 
7. We have (n — 2) (w — 3) = n 1 — n + (— 4m + 6) = a + 26 + 3c + (— 4<n + 6); and making 
this substitution in the equations which contain (n — 2) (n — 3), these become 
a (— 4n + 6) = 2 (8 — G) — a? — 4tp — 9a — 2j — 3^, 
6 (— 4n + 6) = 4k — 26 2 — 9/3 — 67 — Si + 2p — j, 
c (— 4>7i + 6) = 66 — 3c 2 — 6/8 — 47 — 2% — 3cr — x> 
42—2