If in the expansion

0

Σ (v +8) P;” (cos 0) Jv+ (ax) £ ̧‡o (αx) = († αɑx)* £, [x√a2 — 2aa cos 0 + a2]

8=0

J,

Tv+8 © =

г (v)

[a2 - 2aα cos 0+ a2]2

-

and integrate with regard to a between 0 and ∞, we obtain

for if we write the integral on the left hand side equal to f(x), we have

hence since there is only one function f(x) which leads to a given function of t, we must have

This formula is a particular case of a more general expansion given in Nielsen's Hundbuch der Cylinderfunktionen. The simplification that occurs in this particular case appears to have been overlooked.

7. We shall now consider the problem of constructing transformations which are simply periodic.

Let L. (u) and L. (u) be two adjoint linear differential expressions and let w (s, t) be a solution of the partial differential equation*

the path of integration and the function (t) being at present arbitrary except that they must be such as to allow the integral to be differentiated a suitable number of times by the rule of Leibnitz. We then have

(}aar) J, [zNq2 − 2ɑa cos 0 + 2o) T (1)

[a2 − 2az cos 0 + a2)

and integrate with regard to a between 0 and x, we obtain

for if we write the integral on the left hand side equal to ƒ(z), we have

Now let the path of integration and the function & be chosen so that the second integral is zero for all values of s in a given interval or domain, then we have the relation

L, (ƒ) = √ w (s, t) L. ($) dt,
[

which indicates that the distributive operations L, (4) and

obey the commutative law.

In what follows we shall

W. (d) = w(s, t) $ (t) dt
W.($

f ¢

suppose the integral is taken along a real path from a to b and that the linear conditions imposed on are such that the above equation is satisfied for all values of s in the interval (a, b).

Now let (t) be a solution of the differential equation
Lt (u) +λu = 0,

and let us suppose that the linear conditions to be satisfied by (t) are such that they can only be satisfied by a solution of the above equation for certain particular isolated values of X. We shall call these the conditions "C" and shall assume that they are sufficient to determine a solution of the differential equation uniquely except for an arbitrary constant multiplier.

Further, let w (s, t) be chosen so that it satisfies the conditions C when considered as a function of s; the function f(s) will then also satisfy these conditions, provided $(t) and w(s, t) are continuous. Moreover, since

f(s) is a solution of L, (ƒ)+λƒ= 0, hence it must be a constant multiple of $(s) and so we have the relation

where the value of μ depends in some way on the corresponding value of X.

This is a homogeneous integral equation of the first kind which is satisfied by all the solutions of L, (f)+f=0 which satisfy the given conditions. The values of μ corresponding to the different singular values of X are the characteristic values of μ for the integral equation.

If the function w(s, t) be properly chosen the solutions of the adjoint integral equation