220 
PROFESSOR CAYLEY ON RECIPROCAL SURFACES. 
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 
A A 
+2A 
— |A 
+ i 2 £ A 
— 20A 
+¥A 
— 6A 
-B 
+B 
+ 6B 
+c 
-C 
-100 
+12C 
-4D 
+12D 
— 12D 
+4D 
+4E 
— 4E 
+ 6F 
— 8F 
— 22G 
+48G 
— 4H 
+12H 
— 12H 
+100H 
-192H 
+|I 
|-3i 
+¥i 
-191 
+¥i 
-H A i 
+1601 
II 
II 
11 
II 
II 
II 
ll 
II 
0 
0 
0 
0 
0 
0 
0 
0 
viz. the equations are read vertically downwards. The first, second, and third equations, 
and the sum of the fourth and fifth, all give the same relation, 132 — 3A — 1=0; there 
are consequently, inclusive of this, five independent relations. By combining the equa- 
tions so as to simplify the numbers, I find these to be 
3A+I -132=0, 
4A-B-2C -80=0, 
7A-B + 2E + 2F+ 2G- 476=0, 
26A-B+8D + 8H —1532 = 0, 
6A-E-2F +12G-48H+40I— 132 = 0. 
56. I found, as presently mentioned, A=110, B = 272, C=44; values which satisfy 
(as they should do) the second equation ; and then assuming D = 116 and E=3Q3, we 
have E=- 2 -, G=— H=-248, I=— 198 ; and the formula is 
/3'=2m(m— 2)(11m— 24) 
-(110*1-272)5 + 44? 
— (116w— 303)c+^r 
+- e -| i /3+248y+1985 
+ linear function (i, j, 6, %, C, B, i\ /, 5', yj, C', B'), 
the process not enabling the determination of the coefficients of the linear function. 
57. The values of A, B, C were found from the general theorem that if three surfaces 
of the orders v, g respectively intersect in a curve of the order m and class r which is 
a- tuple on uj, /3-tuple on v, and y- tuple on g, then the number of the points of inter- 
