686] 
ON A FUNCTIONAL EQUATION. 
301 
We have in like manner 
Ri = W t + ^Z U where W X = G( ax x + b) + D (cx x +d), 
Si = — W x — \Z X , where Zy = G (-dx x + b) + D (cx x — a). 
Substituting for x x its value, we find 
W x (cx + d)= C[(a + d) (ax + b)~ (ad — be) x] + D [(n + d) (ex + d) — (ad — be)], 
Z x (cx + d) = C[ — (ad — bc)x]+D[ — (ad — be)]: 
hence, substituting for ad — be as before, 
W x (cx + d) = {(A + l) 2 W — (a + d) A (Gx + D)}, 
a + d , 
Zy (cx + d) = 
(A +1) 2 
— (a + d) A (Gx + D)}, 
whence without difficulty 
consequently 
RySy' 
Ry'Sy 
R l , (.ea + d) = ( - a + *) X R, 
S,'(o* + <i) = ^4 S’-. 
A/ + 1 
RS' i . t t 
jga, that is, & = £ 
RJtj RR 
a cif ^ eja/ : 
» , ’h = log A +17, =2^+77, 
which are the formulae in question. 
(2) For the value of RS' — R'S, we have 
RS' - R'S = (Ac + d) (- W - XZ) — (— Ad — c) ( W + 
= A 2 + -J cZ+(X +1) {(d — c) W-dR} 
= - (A -1) j(l + A + ^ cZ + (c - d) W + dzj ; 
or substituting for A + -, Z and TF their values, this is 
A; 
= —^ ^ {(a 2 + d 2 + ad + be) c (b(7 + dD) 
+ (ad - be) [(c - d) (a G + c D) + d (bC7 + d D)]}. 
In the term in { }, the coefficient of G is 
[(a 2 + d 2 + ad + be) b + (ad -be) a] c - d (a - b) (ad - be) 
= (a + d) (c£b — bd) c — (a + d) dx (ad — be),