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750] ON THE THEORY OF RECIPROCAL SURFACES. 
Taking then further as given the 5 quantities j', w, C', B', 
equations (18) and (21) give p, a, 
equation (19) gives 2/3' + 37' 4- St', 
„ (20) „ 4/3'4- y' + 0', 
(28) „ P + W, 
so that, taking also t' as given, these last three equations determine fi', y', 6'; and 
finally 
equation (22) gives k', 
(23) „ h', 
» (24) „ q, 
„ (25) „ r, 
viz. taking as given in all 20 quantities, the remaining 26 will be determined. 
614. In the case of the general surface of the order ft, without singularities, we 
have as follow: 
n = n, 
a = n (n — 1), 
S = \n (11 — 1) (w — 2) (11 — 3), 
k — n(n — 1) (11 — 2), 
n' = n (n — l) 2 , 
a = n {n — 1), 
g' = (n — 2) (w 2 — 9), 
k = 3n (11 — 2), 
V =\n (n — 1) (n — 2) (n 3 — n 2 + n — 12), 
k' = %n(n - 2) (?i 10 - 6w 9 + 16w 8 - 54ft. 7 + 164?i 6 - 288?i 5 
+ 547ft 4 - 1058ft 3 + 1068ft 2 - 1214ft + 1464); 
t' = ift (ft — 2) (n 7 — 4ft 6 + 7ft 5 — 45ft 4 + 114?^ 3 — 111ft 2 + 548ft — 960), 
q' = ft (ft — 2) (ft — 3) (?i 2 + 2ft — 4), 
p = ft (ft — 2) (11 s — ft 2 + ft — 12), 
c' = 4ft (ft — 1) (ft — 2), 
h! = \n (w - 2) (16ft 4 - 64ft 3 4- 80ft 2 - 108ft 4-156), 
r' = 2ft (n — 2) (3ft — 4), 
a = 4ft (ft — 2), 
¡3' = 2ft (ft-2) (lift-24), 
7' = 4ft (n — 2) (ft — 3) (ft 3 — 3ft 4-16), 
the remaining quantities vanishing. 
C. XI. 
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