222 
PROFESSOR CAYLEY ON RECIPROCAL SURFACES. 
points, that is, the number of points of intersection of the three surfaces is 
=iW/j'£ — + yocv + ufig — 2a/3y) + ufiyr. 
59. I represent the complete value of /3' by 
(3'= 2n(lln—24) 
-(110n-272)Z> + Mg 
— (116w— 303 )c+- 2 % 
+^f^ + 248y+1985 
—/iC—gB —xi —Xj —g>x ~~ v ® 
-h'C'-g’B'-x'i'-x'j'-M-M', 
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 5 , g, x, X, p, v, 
h 1 , g\ x\ X', gJ 
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 
fl=54:-hC-gB-x!f-(*'x!-v'0-h'C , -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— 5, 
(II) 
18=54- g- 16i/, 
(III) 
13=54 — 25—/.', 
(IV) 
OS 
II 
cn 
1 
r 
1 
1 
OO 
(VI) 
3 = 54-35-3?,', 
(VIII) 
0=54-2^-16/-/, 
(IX) 
1 = 54— 25— <7 — X'— 2jh/, 
(XIII) 
0 = 54-45-6?,', 
(XVI) 
0 = 54- h—2g — 2g}—g\ 
(XVII) 
0=54—3(7—3/, 
(XXI) 
which are all satisfied if only 
5 =24, 
(7+16/=36, 
g+ 2 ^= 12 , 
9+ 9’= 18, 
?,'= — 7. 
61. If w 7 e apply to the same surfaces the reciprocal equation for /3, or, what is the same 
