curve and the photo-electric current curve must be significant. It indicates that if the anomalous variation in curve (c) is due to absorption, the anomalous absorption affects only the shorter wave radiations, and then it is difficult to see why the appearance of the discharge should alter as the pressure is reduced. On the other hand, it agrees well with the hypothesis that the maximum intensity of the radiations moves towards the infra-red, for then the photo-electric effect, being due to radiations of longer average wave-length than those causing the ionization effects, would suffer little change. On the whole, the evidence indicates that the difference between the curves for nitrogen and hydrogen is not due to the gases obeying different absorption laws, but is due rather to the difference between the modes of variation with pressure of the radiations emitted by the gases.

Summary.

Preliminary experiments are described showing:1. The variation with pressure of the ionizing and photoelectric radiations from hydrogen and nitrogen when the gases are excited by a discharge.

2. The variation of the intensity of these radiations when the pressure is kept constant and the discharge current is varied.

The curves obtained depend upon both absorption and emission variations, and an attempt is made to estimate the relative importance of the two effects, and so to separate them. Tentative explanations are offered of the various phenomena, and it will be a matter for further investigation to decide between them.

Reasons are put forward suggesting that the radiations are molecular in origin.

A spontaneous ionization phenomenon in oxygen is described and an explanation is offered.

In conclusion, the writer again wishes to express his thanks to Professor Taylor Jones for his encouragement and advice, which have proved invaluable. The work was performed in the Research Laboratories of the Natural Philosophy Department of the University of Glasgow.

July 1928.

is known from theoretical considerations to be of the form

(1)

(2)

where (x) has the period. The two main problems which arise are (a) the determination of n, for a given 0, such that the equation shall have a solution of assigned periodicity, and (b) the determination of μ when and are given. The first problem has been solved by the present writer, and tables of the functions of periods π and 2π will in due course be available. The second problem will now be considered.

When is a pure imaginary, the solution is said to be stable; otherwise it is unstable. When is zero, a stable solution exists for every positive value of ŋ, but as increases, the probability that a solution with 7 chosen at random shall be stable diminishes rapidly. This is shown very clearly in fig. 1. The curves ao, b1, a1, b2,... represent the determinations of corresponding to the elliptic cylinder functions ce(x), se ̧(x), ce̟(x), seq(x),.... The regions between a, and b1, between a and b, and so on, for positive, and the symmetrical regions for negative, are the regions of stability, and since the bounding curves tend rapidly to coincidence, the above statement is justified. The diagram furnishes a rough guide as to whether the solution corresponding to any value-pair (0,ŋ) is stable or not. Naturally, in physical problems the stable solutions are the more interesting.

Several methods of determining the index μ have been

* Communicated by the Author.

This form of the Mathieu equation differs slightly from the form generally in vogue, but it has been adopted as the standard in the computation of the elliptic cylinder functions which is now in progress. In the majority of physical applications is positive.

Proc. Roy. Soc. Edin. xlvi. pp. 20-29, 316-322 (1925–26); xlvii. pp. 294-301 (1927).

devised; the latest is that due to Whittaker*.

The essential

point of his method is to express n and μ in the forms

n=F(o), μ=G(o),

where F(a) and G(o) are respectively even and odd Fourier series in whose coefficients depend only on 0. The first equation is solved numerically for σ; the second then gives

μ immediately. This method is of practical value, and has. in fact, been applied to G. W. Hill's classical problem in the lunar theory. The numerical work is, however, laborious, and is complicated by the fact that in the stable case o is either purely imaginary or a complex number of the form

+Ti. For that reason the following method is to be preferred in practice whenever the solution is known to be stable

*Proc. Edin. Math. Soc. xxxii. pp. 75-80 (1914).

2. The General Solution.

The Mathieu equation is satisfied by a solution of the form

provided that the coefficients e, satisfy the recurrencerelations

{n−(2r+p)3}e,= 0 (er-1+er+1)

(4)

for all integral values of r, and provided that these relations are consistent. There is evidently no loss of generality in assuming that 0<R(p) ≤1, where R(p) denotes the real part of p. When the solution is stable, p is real.

Equation (1) is unchanged if x is replaced by π-x. Consequently, when the solution (3) exists, there also exists the solution

00

y = Σe, cos (2r+p)(π−x)

81

= cos pπe, cos (2r+p). + sin pπ Î e, sin (2r+p).',

which is distinct from (3) except when p is an integer or

zero.

With these exceptions, therefore, there is always associated with (3) the solution.

y = Σe, sin (2r+p)x..

81

(5)

Conversely (3) is associated with (5) except when p is an integer or zero. These solutions are evidently distinct and form a fundamental pair. Thus, setting aside the special cases which arise when p is an integer or zero, equation (1) has a general solution of the form

= A Σ e, cos (2r+p) x + B Σ e, sin (2r+p)x,

y =

which is not essentially different from (2). The number p is determined by the condition that the recurrence-relations be consistent.

If p=1, the recurrence formula for n is the same as that for r-n-1, and it is an easy matter to prove that

*

Express e/e, as a continued fraction in two distinct ways as in the following section; this shows that e-te, and the rest follows immediately.

e_r-1=+er. If e_r_1=er, then (5) becomes identically zero, and (3) may be written

y = c, cos (2r+1)x,

0

where c=2e,. For r 1, c, follows the same recurrencerelation as e,, but for r=0 the relation becomes

(n−0—1)c。 = Oc1.

The solution is thus a multiple of an elliptic cylinder function of the type ce2m+1 (x, 0). If e_r-1=-er, (3) becomes

identically zero, and (5) may be written

where c=2e,.

∞

y = c,' sin (2r+1)x,

0

The recurrence-relation for c,' is the same

as that for c, when

≥1, but when r=0 it becomes

(n+0−1)c,' = Oc1'.

Thus the solution is a multiple of an elliptic cylinder function of the type sem+1(~, 0). The case p=1 virtually includes the case where p is any odd integer.

If p=0, the recurrence-relation for rn is the same as that for r=n, and it is easily proved that e_,= ±er. If e_r=er=cr, (5) becomes identically zero and (3) may be written

For 1, c, follows the same recurrence-relation as e,, but for r=0 the relation becomes

The solution is thus a multiple of an elliptic cylinder function of type ceam(x, 0). Similarly, it e,=—e_r=c,', (3) disappears and (5) may be written

2m

For r1 the recurrence-relation is as before, but with co=0. The solution is thus a multiple of an elliptic cylinder function of type se2m(x, 0). The case p=0 virtually includes the case where p is an even integer.

It is clear that in the general case Σe, 0, for if the contrary were true, then both (3) and its first derivative would vanish for a=0, which is impossible since the origin is an ordinary point of equation (1). Thus, if necessary, the solutions (3) and (5) may be made definite by the assumption Σε=1.