256 
ON A MATRIX AND A FUNCTION CONNECTED THEREWITH. 
[755 
But the case m = 1 is a very remarkable one; we have here A = 1, and the 
relation between the coefficients is thus (a 4- df = 4 (ad — be), or what is the same thing 
(a — df + 46c = 0. And then determining the values for A = 1 of the vanishing fractions 
which enter into the formulae, we find 
a n x + b n = — (a + df 
{(n + 1) (ax + b) + (n — 1) (— dx + b)}, 
c n x +d n = ^(a + d) 11 - 1 {(w + 1) (cx + d) + (n- 1)( cx — a)}, 
or as these may also be written 
a n x + b n = ^ (a 4* dy 1 - 1 [x [n (a - d) + (a + rZ)] + 2nb], 
c n x + d n — ^ (a + d)^ 1 {x 
2nc + [— n (a — d) + a + cZ]}, 
which for n — 0, become as they should do a 0 x + b 0 = x, c 0 x + d 0 =l, and for n= 1 they 
become a 1 x + b 1 = ax + b, c 1 x + d 1 = cx + d. We thus do not have a ' X ^ = x, and the 
c-ipc + d x 
function is not periodic of any order. This remarkable case is noticed by Mr Moulton 
in his edition (2nd edition, 1872) of Boole’s Finite Differences. 
If to satisfy the given relation (a — dy + 4&c = 0, we write 2b = k (a — d), 2c = — ~ (a - d). 
ic 
then the function of x is 
ax + ^Jc (a — d) 
— ^hr 1 (a — d) x + d ’ 
and the formulae for the /zth function are 
a n x + b n — ~ (a + d) n ~ l {(a + d) x + n (a — d) (x + h)), 
c n x + d n =ss^ (a + dy- 1 |(a + d) -n(a- d) ^ + l^j ; 
which may be verified successively for the different values of n. 
Reverting to the general case, suppose n = go , and let u be the value of 0” (x). 
Supposing that the modulus of A is not =1, we have \ n indefinitely large or 
indefinitely small. In the former case, we obtain 
_ A (ax + b) + (— dx + b) _ (Aa — d) x + b (A + 1) 
A(c#+c0 + ( cx — a)’ c (A + 1)# + Xd — a ’ 
which, observing that the equation in A may be written 
A a — d _ b (A+ 1) 
c (A + 1) Ad — a ’