411] 
A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
351 
and (observing that the Table of Singularities in my Memoir on Cubic Surfaces was 
obtained without the aid of the formula now in question) I endeavour by means of the 
results therein contained to find the values of the unknown coefficients h, g, x, X, g, v, 
h, g> æ, fi, v. 
60. For a cubic surface n = 3, and for a cubic surface without singular lines (in 
fact for all the cases except the cubic scrolls XXII and XXIII), the formula is 
¡3' = 54 - hC - gB - X'j' - g' X ' - v’ff- KG' - g'B' ; 
and applying this to the several cases of cubic surfaces as grouped together in the 
Table, and referred to by the affixed roman numbers, the resulting equations are 
54 = 54, 
(I) 
30 = 54 - h, 
(II) 
18 = 54 - g- 16*/, 
(III) 
13 = 54 — 2h — X', 
(IV) 
6 = 54 — h — g — ¡i — 8v, 
(VI) 
3 = 54 - 3h - 3V, 
(VIII) 
0 = 54 — 2g — 16 F — g', 
(IX) 
1 = 54 — 2h — g — X' — 2g, 
(XIII) 
0 = 54 — 4/* — 6 à/, 
(XVI) 
0 = 54 — h — 2g — 2g — g, 
(XVII) 
0 = 54 — 3g — 3g', 
(XXI) 
which are all satisfied if only 
h =24, 
g + 16*/ = 36, 
g + 2g =12, 
9 + 9' = 18. 
V = — 7. 
61. If we apply to the same surfaces the reciprocal equation for ¡3, or, what is 
the same thing, apply the original equation to the reciprocal surfaces, as given by 
interchanging the upper and lower halves of the Table of Singularities, we have 
another series of equations, viz. this is 
0 = 54432 - 
54432, 
(I) 
0 = 27851 - 
27846- K, 
K 
5, 
(II) 
0 = 18180- 
18318- g'-lQv, 
g’ + 16v = 
- 138, 
(III) 
0 = 11765 - 
11756 - 2h'-X, 
2 K + X 
9, 
(IV) 
0= 6917- 
6584 — K — g' — g — Sv, 
K + g' + g -f- 8v = 
45, 
(VI) 
0= 3534 - 
3522 - 3K - 3X, 
3h' + 3A 
12, 
(VIII) 
0= 3024- 
3144 — 2g' — 16^ — g, 
2g 16i> 4" g 
- 120, 
(IX) 
0= 1433 - 
1386 — 2K — g' — X — 2g, 
2 K + g' + A -f- 2 g — 
47, 
(XIII) 
0= 518- 
504 — 4>K — 6\, 
4>K + 6A 
14, 
(XVI) 
0 = 383 - 
322 — K — 2g' — g — 2g, 
K + 2g' + g + 2 g = 
61, 
(XVII) 
0 = 54 
-3^7 -3g, 
3g + 3g' — 
54, 
(XXI)