ON HAMILTON’S NUMBERS. 
301 
we obtain 
(1 — x) Sn+l F„ + 1 (a?) = (1 — x) Sn F„(ir) -f- x n ~ 1 (i — £c)*»+i +1 — x n ~ l {l — x)' s '* + J , 
which gives, when we write successively n — 1, n — 2, n — 3, ... 0 in the place of n, 
(1 — x) Sn F„(a?) = (1 — x) s>i -i- F /i _ 1 (a;) + x n ~' 2 (l — x) s,i + 1 — x n ~ 2 (l — x)*»-i + 1 ; 
(1 — x )^- 1 F ;< _. x (x) = (l — £c )*»-2 F n _ 2 {x) + x n ~ 3 (l — a?)^-i + 1 — x ,l ~ 3 (1 — x) Sn ~ 2+1 ; 
(1 — x) Sl Ffa) = (1 — x) So F 0 (x) + a; -1 (l — x) ,?l + 1 — — x) So+1 . 
Hence, by addition of these n equations, we find 
(1 — x) Sn F n (x) = (1 — a ;)* 0 F 0 (*) + x n “ 2 ( 1 — a ;)^ 41 — a: -1 (l — x )" 0+1 
+ af~ 3 (l — x) Sn ~ l +2 + x n ~\l — xW - 2+2 cc 1 (1 — a;)* 1 "' -2 , 
where it has been assumed that it is possible to assign to s 0 (previously undefined) 
such a value as will make the last of the above n equations, viz., 
(I — x) Sl Fj(£c) = (1 — x) So F 0 (x) + x~ l {\ — £c) Jl + 1 — a: -1 (l — x) Sa + 1 , 
identically true. That this can be done is obvious ; for, if in that equation we write 
for F 1 (x), F 0 (a:), and their values, viz., 
F^cc) = (L — x)~ 2 — 1 , F 0 (a:) = (1 — x)~ l , and = a 0 = 1 , 
then, on making s 0 = 0 , the equation becomes 
(1 — x)~ l — (1 — x) = (1 — a ;) -1 + a; -1 (l — x) (1 — x — 1 ). 
Thus the general value of F„(a:) is given by the equation 
(1 — x) Sn F„(£c) = (L — £c ) -1 + X"“ 2 (l — x) Sn + l — X~ 1 {1 — x) 
+ x n ~ 3 (l — x) s,i - l + 2 -|- £C m_4 (l — x )' f « _s+3 + a? _1 (l — a;)^ +2 , 
which is equivalent to 
(1 — x) Sn F„(£c) — (1 — x)~~ l + X~ l {\ — x) ~ x n ~ l (l — x) s > t+l 
— x n ~' 2 (l — x) s,i + 2 + x““ 3 (l — x) Sn 1 + 2 x n ~\1 — x) % “ 2 + 2 -f a:“ 1 (l —* a:)' ?l4 ' 2 , 
where, a 0 , a l} a 2 , a 3 , . . . being the Hypothenusal Numbers 1 , 2, 6 , 36, . . . we have 
