306 
ON A FUNCTIONAL EQUATION. 
[686 
for the nth repetition of ^x, = ——,, is a very interesting one. It is to be 
remembered that, when n is even the numerator and denominator each divide by 
1, but when n is odd they each divide by A 2 —1; after such division, then further 
dividing by a power of A, they each consist of terms of the form A a + —, that is, 
they are each of them a rational function of A + ^. Substituting and multiplying 
by the proper power of ad — be, the numerator and denominator become each of 
them a rational and integral function of a, b, c, d of the order w + 1 when n is even, 
but of the order n when n is odd; in the former case, however, the numerator and 
denominator each divide by a + d, so that ultimately, whether n be even or odd, the 
order is — n as it should be. 
For example, when n = 2, the value is 
(A 3 — 1) a + (A J — A) b (A 2 + A -j- 1) a + Ab 
(A 3 — 1) c + (A 2 — A) d (A' + A + 1) c + Ad 
or, as this may be written, 
where, observing that 
A + ^ + 2 = ^ , - a + b = — (a + d) x, — c + d = — (a + d), 
A ad — be 
the numerator and denominator each divide by a + d, and the final value is 
(a + d) {ax -f b) — {ad — be)x _ (a 2 + be) x + b {a + d) 
{a + d) {ex + d) — {ad — be) ’ c {a + d) x + bc +d 2 ’ 
which is the proper value of ^ 2 #. But, when n — 3, the value is 
(A 4 — 1) a + (A 3 — A) b _ (A 2 + 1) a + Ab 
(A 4 - 1) c + (A 3 - À) d ’ “ [A 2 + 1) c + Ad ’ 
and this is 
(a 2 + d? + 2bc) {ax + b) + {ad — be) {— dx + b) 
(a 2 + d 2 + 2 be) {ex +d)+ {ad — be) { ex — a)’ 
or finally 
{a 3 + 2 abc + bed) x + b {a 3 + ad + bc + d 3 ) 
c {a 2 + ad + bc + d 2 ) x + {abc + 2bed + d 3 ) ’ 
which is the proper value of ^\ 3 x.