348 A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. [411 
54. Forming the reciprocal equation 
/3 = 2n' (n — 2) (11m' — 24) 
-V (An'-B) + C'q 
— c' (Dn — E) + F'r 
— G/3' — Hy — It' 
+ linear function (%, j', 6', x> O', B', i, j, 6, %, G, B), 
and substituting herein the values which belong to the surface of the order n without 
singularities, we should have identically 
0 = 2 n(n — l) 2 (n — 2) (n 2 + 1) (llw 8 — 22m 2 + 11m — 24) 
— \n (m — 1) (m — 2) (m 3 — m 2 + m — 12) [An (m — l) 2 — B] 
+ m (m — 2) (m — 3) (m 2 + 2m — 4) G 
— 4m (m — 1) (m — 2) [Dn (m — l) 2 — E] 
+ 2m (m — 2) (3m — 4) F 
— 2m (m — 2) (11m — 24) G 
— 4m (m — 2) (m — 3) (m 3 + 3m — 16) H 
— j?n (m — 2) (m 7 — 4m 6 4- 7m 8 — 45?i 4 + 114/? 3 — 111?*. 2 + 548m — 960) I; 
or dividing the whole by n (n — 2), this is 
0 = 2 (m - l) 2 (m 2 + 1) (11m 3 - 22m 2 + 11m - 24) 
— k (m — 1) ( n3 — m 2 + m — 12) [An (m — l) 2 — B] 
+ (m — 3) (m 2 + 2m — 4) G 
— 4 (m — 1) [Dm (m — 1 ) 2 — E] 
+ 2 (3m - 4) F 
- 2 (11m- 24) G 
— 4 (?2 — 3) (m 3 + 3m — 16) H 
— j;(n 7 — 4m 6 + 7m 5 — 45?i 4 + 114m 3 — 111m 2 + 548?i- — 960) I. 
55. And then, expanding in powers of n and equating to zero the coefficients 
of the several powers n 7 ,...n°, we obtain 
22 
-88 
+ 154 
-224 
+ 250 
- 184 
+ 118 
— 
48 
U 
+ 2 A 
+ ^A 
- 20A 
+ ^A 
- 6A 
+ ^B 
- B 
+ B 
— 1 3 R 
+ 
6D 
+ C 
- G 
- 10C 
+ 
12C 
- 4D 
+ 12 D 
- 12 D 
+ 4 D 
+ 4D 
4 E 
+ 6D 
- 
8 F 
- 22C 
+ 
48 G 
- 4F 
+ 12 H 
— 12ZT 
+ 
i—* 
o 
o 
tq 
— 
192 H 
' 
+11 
-K 
+ 
-19/ 
/ 
+ 160/ 
II 
II 
II 
ii 
11 
ii 
ii 
II 
0 
0 
0 
0 
0 
0 
0 
0