1. The problems of stress distributions in an infinite plate containing a circular hole were solved in a general manner by Bickley(1) for isotropic materials. An alternative method, using complex stress functions, was later given by Green(5) and extended so as to apply to aeolotropic materials. In the present paper Green's method is employed to determine the stress distributions that arise when the stresses are produced by isolated forces acting at points on, or near to, the edge of the circular hole, and though some of the solutions are cumbersome they are all obtained in finite form.

1. Let a, b be positive constants; and let y1, y2, …, yn be real exponents, not all equal, having arithmetic mean y defined by

(here, and in what follows, the summation ∑ extends over the values i = 1, 2, …, n). Then it is clear that

since the right-hand sides are the geometric means of the positive numbers whose arithmetic means stand on the left-hand sides. I know of no results, however, which relate the ratios and and I have had occasion recently to require such results. This note gives an inequality between these ratios, subject to certain restrictions on a and b.

1. It is trivial that

for all integers n ≥ 0 implies sn = 0 (n = 0, 1, 2, …). This conclusion is no longer true if it is only known that (1·1) is true for an infinity of n. But we shall show that the truth of (1·1) for, roughly speaking, one-half of all positive integers, together with an order condition on the magnitude of sn, ensures sn = 0 for all n. This follows from

Theorem 1. Let a1 < a2 < … be positive integers, n(R) the number of a's not exceeding R, sn a sequence of complex numbers. If

(ii) sn = ο(nK)as n tends to infinity

then sn = P(n), where P(x) is a polynomial of degree less than K.

3. The essential requirement that the nova must satisfy in the above theory is that the total mass in the form of diffuse gaseous material must be of the order of 1/10 the solar mass, which requirement seems to be consistent with the observations of novae. Thus, although the theory applies explicitly to novae that break into two pieces of stellar mass having a gaseous filament drawn out between them (the pieces may have a mass ratio as great as 5/1), it seems clear that the discussion could be adjusted to include the case of novae following other models.

The process described above can be applied to the formation of non-solar planets. The mass of available planetary material depends upon such factors as the mass of the nova and its separation from the companion before outburst. It is to be expected that variations in these factors can lead to planetary masses that vary over a fairly wide range. It follows that since an appreciable fraction of the stars of mass comparable or greater than the solar mass are known to be members of binary systems, the number of planetary systems must be at least of the same order as the number of novae that have occurred in this class of star.

Stokes and Wilson(1,2) have shown that the integral breadths of the Debye-Scherrer lines produced by small or imperfect crystals of the cubic system are given by

or, what is the same thing, that the apparent particle sizes are given by

In these equations λ is the X-ray wave-length, θ is the Bragg angle, Vt is the volume common to the crystal and its ‘ghost’ shifted a distance t in the hkl direction(1), and Jt is the mean value of the product FF of the structure amplitudes of two cells separated by a distance t in the hkl direction (2). The purpose of this note is to outline an argument which establishes the same result for crystals of any symmetry.

There are several mistakes and misprints in the above paper.

(1) In introducing Voigt's notation for the strain components the well-known factor 2 in the normal components has been omitted, and (3·11) should read

The following formulae in which this notation is used are correct as they stand.