302 
ON A FUNCTIONAL EQUATION. 
[686 
and similarly the coefficient of D is 
[(a 2 + d 2 + ad + be) d + (ad — be) c] c — d (c — d) (ad — be) 
= (a + d) (ad - cb) c - (a + d) d (ad — be). 
Hence the whole term in { } is 
= (a + d) {[(db - bd) c — d (ad — be) x]G + [(ad — cb) c — d (ad — be)] D], 
which is readily 
also 
so that we have 
reduced to 
(a + d) (ad — be) (— dC + cD) ; 
ad — be = (a + d) [c# 2 + (d — a) x — 
b] 
RS' - R'S = ad-bc d ~ (d0 - cD) № +(<*-<*)«-&]> 
which is the required value of RS' - R'S; and there is no difficulty in obtaining 
the other two formulae, 
R\ + S — (X 2 — 1) (cx + d), 
R'\ + S' = (\ - 1) (a + d) (Gx + D); 
the verification is thus completed. 
To show how the formula was directly obtained, we have 
Ax+B_A AD-BG 1 
Gx + D~G G Cx+D 
A 
the equation then is 
= -g + fix suppose ; 
A 
6x — <f)x 1 = (x — xfi) + (x — aq) fix. 
0 
Hence, if x 1} x 2 , x 3 , ... denote the successive functions S-x, Wx, Wx, &c., we have 
A 
(j)Xi (fix 2 — -jj (X\ X 2 ) "b (Xi X 2 ) fiX\, 
A 
(fix 2 <fix3 = -Q (xo x 3 ) -f- (x 2 x 3 ) fix2, 
whence adding, and neglecting cfix x and x X} we have 
A 
<fix= Jj X + [(x — X-fi fix + (&! — X 2 ) fix Y + (X 2 — X 3 ) fix2 +...], 
where the term in [ ], regarding therein x ly x 2 , x 3 , ... as given functions of x, is 
itself a given function of x\ and it only remains to sum the series. 
Starting from 
and writing 
X 1 — ^sx = 
ax + b 
cx + d’ 
a 2 + d 2 + 2 be 
ad —be