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686] ON A FUNCTIONAL EQUATION. 303 
then the nth function is given by the formula 
(V*+i — 1) (ax + b) + (A n — A) (— dx + b) 
x • = *+ = (X»«-l)( n . + d) + (x»-X)(a.-B) 
_ (A w+1 — 1) a + (A n — A) b 
“ (A K+1 - 1) c + (A n — A) d 
_V l P + Q 
~\ n R + S’ 
if P = Aa + b, Q = — a — Ab, and as before P = Ac + d, S = — c — Ad. 
I stop to remark that A being real, then if A > 1 we have \ n very large for n 
P 
very large, and x n = ^ which is independent of n; the value in question is 
_ A (ax + b) + (— dx + b) 
Xn A (cx + d) + ( cx — a)’ 
which, observing that the equation in A may be written 
A a —d _ 6 (A + 1) 
c(A + l) Ad — a ’ 
is, in fact, independent of x, and is = or ^ ^ + ^; we have x n ^ — x n , or 
" '* - 1 - ' A a — a 
c (A -t- 1) 
calling each of these two equal values x, we have 
x = 
ax + b 
cx + d’ 
which is the same equation as is obtainable by the elimination of A from the equations 
A a — d b( A + l) 
x = 
c (A + 1) A d — a ' 
Q 
The same result is obtained by taking A < 1 and consequently x n = 
We find 
A n-1 P + Q A n P + Q 
u n—i «"m 
X n — 
A n ~ l R + S A n R + S ’ 
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