300 
[686 
ON A FUNCTIONAL EQUATION. 
and the value is thus 
1 1 
-* 
(RS'-R'S)X 
_ R' ' - R’ (\R' + S')(\R+S)' 
1 ”f* A/ X “j” A ^ 
Hence, from the assumed value of (f>x, we obtain 
— d) 2 + 4be] (A 
G(dC-cD) (\R' + S') (XR + S) 
* _ * - £ - „) - v«“ - <*>■ +w ' !l - 1/ ' 
We have 
or since 
this is 
= (X ad-bc^ (d ° ~ CjD) ^ + ( d - a ) æ ~ 6 1> 
RX + S = (A 2 - 1) (cx + d), see post, (2), 
R'X + S'=(X -l)(a + d)(Gx + D), 
cx 2 + (d — a) x — b 
' 1 rp rp 
1 ’— tv tAs~t . 
cx -f- d 
ÿ. - * - £ - «0 - +46c -i^ J - gc > 
(a + d) A 
But from the value of A, 
G (ad — bc)(X 2 —l)(Gx + D) 
ad — be 
(x — x,). 
A 2 — 1 (a + d) \J{(a — d) 2 + 46c} ’ 
and the equation thus is 
, . , N (A AD — BGI , N Ax + B 
$x - (fix, -(x- X,) j £ - > ~( x ~ Cx + D ’ 
as it should be. 
(1) For the foregoing values of t},, we require R„ S lt R,', S,', the values 
which R, S, R', S' assume on writing therein x, for x. We have 
R, = X (cx, + d) + (cx, — a), 
S, — (cx, + d) — A (cx, — a): 
substituting for x, its value, we find 
R, (cx + d) = (a + d) A (cx + d) — (ad — be) (A + 1), 
or writing herein 
this is 
ad 6c _ («+<*)*»■ 
(X+l) 1 ’ 
R 1 (cx + d) = ( - a +yy , R i 
S,(cæ + d) = pOfs. 
A + I 
and similarly