750] 
ON THE THEORY OF RECIPROCAL SURFACES. 
229 
Also, reciprocal to these, 
(15) a' = n' (n — 1) — 26' — 3c'. 
(16) k = 3n’ (ft - 2) - 66' - 8c'. 
(17) 8 = (n' - 2) (to' 3 - 9) - (?/ 2 - n! - 6) (25' + 3c') + 26' (6' -1) + 66V + fc' (o' - 1). 
(18) a (n' — 2) = k — B' + p' + 2a + 3w . 
(19) 6'(to'- 2)= />' + 2/3'+3 7 '+3/. 
(20) c (n' — 2) = 2a-' + 4/8' + 7 ' + 0' + (o'. 
(21) a (to' - 2) (n - 3) = 2 (8' - O' - 3 ©') + 3 (a'c' - 3</ - - 3«') + 2 (a'6' - 2/>' -/). 
(22) b' (n - 2) (to' - 3) = 4&' + (a'6'- 2p' -/ ) + 3(6'c'-3/3'-2 7 '-/). 
(23) o' (»' - 2) (to' - 3) = 6/i' + (aV - 3<r' 3«') + 2(6'c'-3/3'-2 7 '-/). 
(24) g' = 5' 2 - 6' - 2&' - 2/ - 3/ - 6/. 
(25) r' = c' 2 — c' — 2 // — 3/3', 
together with one other independent relation: in all 26 relations between the 46 
quantities. 
608. The new relation may be presented under several different forms, equivalent 
to each other in virtue of the foregoing 25 relations; these are 
(26) 2 {n — 1) (n — 2) (n — 3) — 12 (n - 3)(b + c) + 6q + Gr + 241 + 42/3 + 30 7 — § 6 = 2, 
(27) 26to- 12c-4C'-10£ + /3—7j —8 % +i<9 —4ft, = :S; 
in each of which two equations X is used to denote the same function of the accented 
letters that the left-hand side is of the unaccented letters. 
(28) £' + £0'= 2to(to-2)(11to-24) 
+ (- 6 6n + 184) b 
+ (— 93?i + 252) c 
+ 22(2/3+3 7 + 3£) 
+ 27 (4/3 + 7 + O') 
+ /3 + 10 
- 240 - 285 - 21 j - 38 x - 73« 
+ 40' +105' + 7/ + 8*' - 4 
Or, reciprocally, 
(29) /3 + $0 = 2to' (to' - 2) (lb/ - 24) 
+ (- 66n' + 184) b' 
+ (- 93?/ + 252) c' 
+ 22 (2/3' + 3 7 ' + 3 if) 
+ 27 (4/8' + 7 '+0') 
+ /3' + 4^ 
- 240' - 285' - 27j' - 38 % ' - 73«' 
+ 40 + 105 + 7j + 8^ - 4&).