586 
OjST the theory of reciprocal surfaces. 
[416 
Also, reciprocal to these 
(15) a = V(V — 1) — 26' — 3c'. 
(16) « = 3ri (V - 2) - 66' - 8c'. 
(17) 8 = ^(V - 2) (n' 2 - 9) -(V 2 -n'-6)(26' + 3c') + 26' (6' -1) + 66V + fc' (o' - 1). 
(18) a’ (n # - 2) = V - 5' + p + 2V + 3V. 
(19) 6'(V-2) = p'+2/3' + 3 7 '+3i'. 
(20) c' (ri - 2) = 2 V + 4/3' + 7 + 0' + «'. 
(21) a' (V - 2) (V - 3) = 2 (S' - (7' - 3«') + 3 (aV - 3V - %' - 3V) + 2 (a'6' - 2/ -/). 
(22) 6'(V-2)(V-3) = 4k' + (a'6' — 2/ —/) + 3 (6V - 3/3' - 2/ - i). 
(23) c' (V - 2) (V - 3) = 6V + (aV - 3V - *' - 3«') + 2 (6V - 3/3' - 2/ - ¿'). 
(24) = 6' 2 - 6' - 2k' - 2/' - 3 7 ' - 6('. 
(25) r' = c 72 - c 7 - 2h' - 3/3', 
together with one other independent relation, in all 26 relations between the 46 quantities. 
628. 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) (6 + c) + 6q + 6r + 24i + 42/3 + 30 7 - f 0 = 2 ; 
(27) 26?i-12c-4(7-105 + /3-7j-8 % + 10-4« =2, 
in each of which two equations 2 is used to denote the same function of the accented 
letters that the left-hand side is of the unaccented letters. 
(28) /3' + |0' = 2n(n-2)(lhi-24 ; ) 
+ (- 66 n + 184) 6 
+ (— 93 n + 252) c 
+ 22 (2/3 + 3 7 + St) 
+ 27 (4/3 + 7 + 0) 
+ /3 + 2 0 
- 24(7 - 285 - 21 j - 38 X - 73« 
+ 4(7' +105'+ 7/ + 8 X '- 4V. 
Or, reciprocally, 
(29) /3 + |0 = 2V (V — 2) (11V — 24) 
+ (— 66V + 184) 6' 
+ (- 93V+ 252) o' 
+ 22 (2/3' + 3 7 ' + St') 
+ 27 (4/3 7 + 7 7 + 0' ) 
+ £'+¿0' 
- 24(7' - 285' - 27/ - 38*' - 73«' 
+ 4(7 + 105 + 1j + 8% - 4«.