Harold Jeffreys

An invariant form for the prior probability in estimation problems.

It is shown that a certain differential form depending on the values of the parameters in a law of chance is invariant for all transformations of the parameters when the law is differentiable with regard to all parameters. For laws containing a location and a scale parameter a form with a somewhat restricted type of invariance is found even when the law is not everywhere differentiable with regard to the parameters. This form has the properties required to give a general rule for stating the prior probability in a large class of estimation problems.

The Draining of a Vertical Plate

When a vessel of liquid has been emptied and put aside, a thin film of liquid clings to the inside and gradually drains down to the bottom under the action of gravity. The layer being thin, the motion is very nearly laminar flow, and the curvature of the surface in a horizontal direction may be ignored. Thus the problem for a cylindrical vessel is reducible to that of a wet plate standing vertically.

The Thermodynamics of an Elastic Solid

1. We consider a small cube of the solid, of edge l , in the actual strained position. If the stresses were removed and the cube cooled to the absolute zero, the displacements of its particles would be (− u , − v , − w ). Then the particles may be considered to have displacements ( u , v , w ) from a standard position; and the cube has strain components of the usual forms

On isotropic tensors

1. An isotropic tensor is one the values of whose components are unaltered by any rotation of rectangular axes (with metric σ i ( dx i ) 2 ). Those up to order 4 in 2 and 3 dimensions have many applications. The results suggest a general theorem for tensors of order m in n dimensions, that any isotropic tensor can be expressed as a linear combination of products of δ and є tensors, where δ ij = 1 if i = j and 0 otherwise, and is 0 if any two of the i 1 to i n are equal, 1 if i 1 … i n is an even permutation of 1, 2, 3, …, n , and – 1 if it is an odd permutation.

On the use of asymptotic approximations of Green's type when the coefficient has zeros

Allowance is made for the error terms in the cases where χ has a simple zero at x = 0, or two simple zeros at a; = ± 1 (with χ either positive or negative for − 1 x

Dynamics of the Moon

We know the mass of the Moon very well from the amount it pulls the Earth about in the course of a month; this is measured by the resulting apparent displacements of an asteroid when it is near us. Combining this with the radius shows that the mean density is close to 3.33 g/cm3. The velocities of earthquake waves at depths of 30 km or so are too high for common surface rocks but agree with dunite, a rock composed mainly of olivine (Mg, FeII)2SiO4. This has a density of about 3.27 at ordinary pressures. The velocities increase with depth, the rate of increase being apparently a maximum at depth about 0.055R in Europe and 0.075R in Japan. It appeared at one time that there was a discontinuity in the velocities at that depth, corresponding to a transition of olivine from a rhombic to a cubic form under pressure. It now seems that the transition, though rapid, is continuous, presumably owing to impurities, but the main point is that the facts are explained by a change of state, and that the pressure at the relevant depth is reached nowhere in the Moon, on account of its smaller size. There will, however, be some compression, and we can work out how much it would be if the Moon is made of a single material. It turns out that the actual mean density of the Moon would be matched if the density at atmospheric pressure is 3.27—just agreeing with the specimen of dunite originally used for comparison. The density at the centre would be 3.41. Thus for most purposes the Moon can be treated as of uniform density. With a few small corrections the ratio 3C/2Ma2 would be 0.5956 ± 0.0010, as against 0.6 for a homogeneous body. To make it appreciably less would require a much greater thickness of lighter surface rocks than in the Earth.

The Fenno-Scandian uplift

The hypothesis that the uplift of Finland is due to viscosity, following the removal of ice at the end of the glacial period, is applied to the established difference between the Earth’s hydrostatic and actual ellipticities. It is shown to imply that there would be comparable rises at all latitudes above about 30°, and depressions nearer the equator. Known variations of height with time are inconsistent with the hypothesis that they are due to viscous flow.

Science, Logic and Philosophy*

SO long as an idealist confines himself to the description of sensations and to the construction of his ideal world, he can dispense with the theory of probability, but at the cost of having to reconstruct his world with every observation that does not happen to fit his laws exactly. An idealist that does not accept the theory of probability could expect the sun to rise in the west to-morrow, and nobody believing otherwise could make the slightest contact with him such as could alter his opinion. But he has no basis for inferring new sensations without it, because an infinite number of laws can always be made to fit any finite number of data. Without some rule for selecting the most suitable laws there is no reason to prefer any one prediction to any other, whether one is an idealist or a realist.

The use of Stokes’s formula in the adjustment of surveys

ZusammenfassungUnter den gegenwärtigen Umständen erscheint die Anwendung derStokes’schen Formel zur Ausgleichung nicht wünschenswert, sofern man nicht ein General-Abkommen über die Verteilung der Schwerewerte, das als Basis für die Berechnung angenommen wird, erzielen kann. Die Schwierigkeiten, die sich bei der Anwendung dieser Formel mit allen Gliedern ergeben, können in großem Maße verringert werden, wenn man die Glieder inP2,P3 undP4 (Zonen-Harmonische) unterdrückt.SummaryThe use ofStokes’s formula to adjust Survey data in present circumstances is definitely undesirable, unless general agreement is first obtained on the distribution of gravity to be adopted as a basis for the computation; the difficulties arising from the use of the completeStokes formula would be greatly reduced by omission of the terms inP2,P3,P4 (zonal harmonics).ResumenHay que excluir la utilización de la fórmula deStokes para la compensación de las grandes triangulaciones, a no ser que se haya adoptado previamente un conjunto coherente de medidas fundamentales de la gravedad como base de los cálculos.Las dificultades inherentes al empleo de la fórmula deStokes serían reducidas si se omitiesen la introducción en esta fórmula de las armónicas zonales enP2,P3 yP4.RésuméDans les circonstances actuelles—état du réseau gravimétrique mondial—l’emploi de la formule deStokes pour la compensation n’apparaît pas souhaitable, à moins que l’on ne puisse obtenir un accord général sur la distribution de la pesanteur à adopter comme base d’un calcul. Les difficultés que soulève l’emploi de cette formule avec tous les termes, seraient toutefois grandement réduites, si l’on en supprimait les harmoniques zonauxP2,P3 etP4.SommarioL’uso della formula diStokes per la compensazione delle grandi triangolazioni è da escludere, a meno che non si raggiunga un generale accordo preliminare sulla distribuzione della grività come fondamento per i calcoli; le difficoltà che sorgono dall’impiego della formula diStokes completa sarebbero grandemente ridotte, pur di omettere i termini inP2,P3,P4 (armoniche zonali).

A derivation of the tidal equations

As the validity of Laplace’s tidal equations is under discussion (Proudman 1942), I think it may be worth while to deal with a difficulty in their derivation pointed out once by Mr E. Gold (1926) in a discussion at the Royal Meteorological Society. It concerns the use of z, the normal distance from a standard level surface, as one of the position co-ordinates. The other level surfaces are approximately surfaces where gz is constant; but g varies by a factor of about 1/200 from pole to equator. Hence in an ocean 4 km. deep, if we take the standard level as mean sea-level, z on the level surfaces near the bottom will vary by about 20 m., which is much more than the height of the equilibrium tide. We need some explanation of why differences of pressure for constant z can be ignored, especially since the usual form of the equations suggests that the condition for equilibrium is that the pressure is constant over surfaces of constant z. Similar considerations apply to atmospheric motions. It is clear from hydrostatics that equilibrium can exist if the pressure and density are constant over the level surfaces, and this would be true even if we took into account the small departures of the level surfaces from symmetry about the axis. The difficulty arises entirely from the fact that when the ellipticity is taken into account the three directions of increasing θ (co-latitude), Φ (longitude) and z (height from a given surface) are not strictly orthogonal; when we are above or below the standard surface the direction of the normal to the standard surface from the point is not at right angles to the direction of the meridian in the level surface, and if we use z as a co-ordinate this difference should be allowed for.