PEOEESSOE CAYLEY ON EECIPEOCAL SIJEEACES. 
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53. I assume 
j3'= 2n(n— 2)(llra— 24) 
— b ( A.n — B ) -(- Cq 
— c(Dn— E)+Fr 
-G/3-Hy-I t 
+ linear function (i, j, 0, C, B, 0, %j, C, B'), 
where it is to be remarked that, in virtue of the equations obtained No. 11, two of the 
coefficients of this form are really arbitrary : I cannot recall the considerations which led 
me to write D=116, E=303. 
54. Forming the reciprocal equation 
(3= 2ra'(ra , -2)(llra'-24) 
— b'(An' — B) + C'qr 
-c'(Dn'-R)+Wr 
-G/3'-H y'-It' 
+ linear function (i\ f, 0, C', B', i,j, 0, %, C, B), 
and substituting herein the values which belong to the surface of the order n without 
singularities, we should have identically 
0= 2ra(ra- l)\n- 2)(n 2 + l)(lln 3 - 22 n 2 + lira - 24) 
— \n(n - 1 )(ra - 2)(ra 3 -n 2 + n- 12)[Ara(ra - 1) 2 -B] 
+ n(n— 2)(n— 3)(w 2 +2ra— 4)C 
— 4 n(n — l)(ra — 2)\J)n(n— l) 2 — E] 
+ 2n(n-2)(3n-4:)F 
— 2ra(ra— 2)(llra— 24)G 
— 4ra(ra— 2)(ra— 3)(ra 3 +3ra— 16)H 
— &i(n —2 )(n 7 — 4n 6 + 7n 5 — 45w, 4 + 11 4 n 3 — 1 1 1 n 2 + 548w — 960)1 ; 
or dividing the whole by n{n— 2), this is 
0= 2(%—1) 2 (m 2 +1)(11+ ! —22% 2 + 11^—24) 
-i(ra-l)(w 3 -ra 2 +ra- 1 2)[Ara(ra-l) 2 -B] 
+ (n-3)(n 2 +2n-4:) C 
— 4(ra— l)[Dra(w— l) 2 — E] 
+2(3ra-4)F 
— 2(llra-24)G 
-4(w-3)(rc 3 +3w-16)H 
— -K^ 7 — 4ra 6 + 7w 5 — 4 5w 4 + 1 1 4w 3 — 1 1 1 ra 2 +.54 8ra — 9 60)1. 
2 h 2 
