T

C2

=

m2 — bm — a = 0.

.b;

or, putting C for and a +ẞ for b, and reducing,

aß.

(5),

and continuing the process and expressing the result in the usual notation of continued fractions,

the number of simple fractions being x. Of the value of this continued fraction the right-hand member of (7) is therefore the finite expression. And the method employed shews how the calculus of finite differences may be applied to the finite evaluation of various other functions involving definite repetitions of given functional operations.

and applying to this the same method as before, we find

and in order to satisfy the condition u。=t,

When a and B are imaginary, the exponential forms must be replaced by trigonometrical ones. We may, however, so integrate the equation (8) as to arrive directly at the trigonometrical solution.

For let that equation be placed in the form

8,- tan (C-x tan^2)

But tvs, and ut2+', where

=

for the general expression of yet.

This expression is evidently reducible to the form

A+ Bt
C+ Et'

the coefficients A, B, C, E being functions of x.

t given in (9), it

Reverting to the exponential form of appears from (10) that it is real if the function

is positive. But this is the same as 4. The trigonometrical solution therefore applies when the expression represented by 2 is positive, the exponential one when it is negative.

In the case of v=0 the difference-equation (12) becomes tx tx + 1 + μ (tx+1 — t2) = 0,

a result which may also be deduced from the trigonometrical solution by the method proper to indeterminate functions.

Periodical Functions.

3. It is thus seen, and it is indeed evident a priori, that in the above cases the form of t is similar to that of t, but with altered constants. The only functions which are known to possess this property are

On this account they are of great importance in connexion with the general problem of the determination of the possible forms of periodical functions, particular examples of which will now be given.

Ex. 7. Under what conditions is a+bt a periodical function of the xth order?

and this, for the particular value of x in question, must reduce to t. Hence

cquations which require that b should be any ath root of unity except 1 when a is not equal to 0, and any ath root of unity when a is equal to 0.

Hence if we confine ourselves to real forms the only periodic forms of a+bt are t and a-t, the former being of every order, the latter of every even order.

Ex. 8. Required the conditions under which

periodical function of the ath order.

In the following investigation we exclude the supposition of e = 0, which merely leads to the case last considered.