The geometrical or vectorial representation of a sample as a vector with n mutually perpendicular components corresponding to the n observations in the sample was introduced into statistics by Fisher (1), and led to the solution of many theoretical problems of statistical distributions. Subsequently Fisher (2) gave an alternative algebraic method applicable to finding distributions in connection with regression and the analysis of variance. The actual use of symbolic vector notation in deducing some of the properties of the sample vector—besides tending to stress the complementary character of these two different methods of proof, and the common principle underlying the analysis of any sample into its components—indicates also some points in connection with the assumption of the normal law in which it seems almost essential to retain the geometrical side of this vector representation. Moreover, it will be seen that the vector theory used here in reviewing briefly the analysis of a sample of one dependent variate can readily be extended to cover the case of correlated variates.

1·1. Let f (z) be an integral function of order ρ, and let

It is well known that, if ρ < 1,

This result was first conjectured by Littlewood, who proved it with cos 2πρ in place of cosπρ; it was afterwards proved independently by Wiman and Valiron about the same time. Many other authors have written on the subject, considering among other things the set of valves of r for which m (r) is comparatively large.

1. Elsewhere I have expressed the opinion that a linearly measurable plane set of points, of upper density ½, has zero projection (i.e. projection of measure zero) on almost all directions. Though unable to prove this I have there managed to obtain a partial result. I would not attempt to guess to what extent this is true of a general irregular linearly measurable plane set; these sets are not amenable to the methods used in the case of those of upper density ½, the latter having a remarkable and simple structure. It is a quite trivial matter to construct an irregular set, of upper density 1, with positive projection on some one direction; and from it we can deduce another such set with positive projections on a denumerable every-where-dense set of directions. But, working on a suggestion due to Mr Besicovitch, I have obtained an irregular linearly measurable plane set, of upper density 1, with positive projection on a set P of directions, where the members of P in any sector have cardinal t, though P has measure zero. It may be that the set of points has positive projection on directions other than those of P, but there are no obvious indications of any such directions.

The test of significance of the difference between two means was extended by means of Fisher's methods of analysis of variance to test jointly the differences between several means, the test being essentially the same as the test of significance of two variances. But though it is true in many cases that we are chiefly concerned with two independent estimates of variance, and the method of analysis of variance is available, the problem of jointly testing the variances obtained from several independent samples does not fit obviously into this scheme.

This article deals with the currents set up in thin-walled cylinders by two-dimensional field variation in the plane of the section, and is applied particularly to the problem of the distribution of alternating currents in strip conductors, a matter of some technical importance.

It is trivial that the fundamental equations of electromagnetism are invariant under orthogonal transformations of space. This invariance can be brought into evidence by using the calculus adapted to the orthogonal group, viz. the vector-calculus. The equations can be written either in their integral form

or in their differential form

When any metal film is deposited on the surface of a massive copper oxide slab and illuminated with visible light, a current flows in such a direction that the electrons pass from the Cu2O to the metal in greater numbers than from the metal to the Cu2O. This is the reverse direction to the easier flow of electrons under an applied voltage when the contact is allowed to act as a rectifier. This effect is sometimes called the “Vorderwandeffekt” to distinguish it from the similar group of phenomena which occur at the natural interface between massive copper and Cu2O when the surface of the copper has been oxidized in situ (“Hinterwandeffekt”).

The surfaces whose prime-sections are hyperelliptic curves of genus p have been classified by G. Castelnuovo. If p > 1, they are the surfaces which contain a (rational) pencil of conics, which traces the on the prime-sections. Thus, if we exclude ruled surfaces, they are rational surfaces. The supernormal surfaces are of order 4p + 4 and lie in space [3p + 5]. The minimum directrix curve to the pencil of conics—that is, the curve of minimum order which meets each conic in one point—may be of any order k, where 0 ≤ k ≤ p + 1. The prime-sections of these surfaces are conveniently represented on the normal rational ruled surfaces, either by quadric sections, or by quadric sections residual to a generator, according as k is even or odd.

It has long been realised that there are difficulties in understanding the continuous β-ray spectra emitted by certain radioactive bodies, if the conservation of energy, as applied to nuclear processes, is to be retained. In the case of radium E, for example, the energy of the disintegration β particles varies from quite low values to about a million volts, the mean being nearly 400,000 volts.

In recent years Cantor manifolds have assumed great importance in topology. This is due chiefly to the work of Urysohn and to the later work of Alexandroff. Urysohn, in the paper in which he introduced the theory of dimensions, showed that a 2-dimensional closed surface is a 2-dimensional Cantor manifold. Alexandroff has not only extended this result to n dimensions but has also established the existence on the cutting set Z of the surface F of generalized cycles homologous to zero on the various parts of F − Z. These results show that the deepest structural characteristics of the closed surface come to light when one dissects it.

Many of the most frequently used applications of the theory of statistics, such for example as the methods of analysis of variance and covariance, the general test of multiple regression and the test of a regression coefficient, depend essentially on the joint distribution of several quadratic forms in a univariate normal system. The object of this paper is to prove the main-relevant results about this distribution. As an application of these results, the theory involved in the method of analysis of covariance will be investigated.

Boron and carbon activated by diplon and proton bombardment have been used as positron emitting sources in experiments on the annihilation radiation.

By coincidence counting with two Geiger-Müller tubes it has been established that the radiation is emitted in pairs.

By the coincidence method of Becker and Bothe it has been shown that the annihilation radiation consists only of “soft” quanta.

By ordinary absorption measurements it has been shown that the annihilation radiation is homogeneous with a hardness corresponding to 0·5 million e-volt.

1. Many insulating crystals, such as rock salt, after exposure to high energy radiation, e.g. X-rays, β-rays, ultra-violet light or γ rays, acquire, as a result of this treatment, two new properties: (i) a new absorption band, usually situated in the visible spectrum, well separated from the continuous absorption in the far ultra-violet, and, when intense, giving visible coloration to the crystal in ordinary light; (ii) the power of showing what has been termed the inner photo-electric effect, i.e. when subjected to an electric field and then illuminated with light in the region of the new absorption band, the “activated” crystal gives an instantaneous response in the form of a small electronic conduction; in the dark or for light outside this band it is still non-conducting (Fig. 1).

Two perpendicular intersecting straight lines will be said to be normal to each other. Then the Petersen-Morley theorem is: Given arbitrarily three lines a, b, c, let a′ be the unique normal to b and c, b′ that to c and a, and c′ that to a and b; then the lines x, y, z, normal respectively to the pairs of lines a and a′, b and b′, c and c′, have a common normal.

The hydrogen discharge has for some time been established as a convenient source of continuous spectrum in the ultra-violet. The continuum extends from nearly 4000 Å. to the limit of a quartz spectrograph and provides a very good background for absorption measurements. A number of tubes have been described for producing this spectrum, but the following advantages are claimed for the form here described.

1. Die Fassung des lokalen Dimensionsbegriffes führt zu verschiedenen neuen, z. Teil schwierigen dimensionstheoretischen Strukturproblemen. In einer ganz natürlichen Weise ergeben sich dabei mehrere Gesichtspunkte je nach der Dimension des Raumes, in welchem die Gebilde eingebettet sind. Einer dieser Gesichtspunkte betrifft die n-dimens. Cantorschen Mannigfaltigkeiten (c.m.) in einem n-dimens. Raum; wir beginnen hier deren Untersuchung im einfachsten wesentlichen Fall—zweidimens. ebener Mannigfaltigkeiten. In dieser ersten Note werden wir beweisen, dass eine zweidimensionale c.m., welche von einer einzigen geschlossenen Kurve begrenzt wird, 1-dimensionale Mannigfaltigkeitsstellen dann und nur dann enthält, wenn die Begrenzungskurve keine einfache (Jordan-) Kurve ist. Damit wird eine Bedingung für das Auftreten 1-dimensionaler Mannigfaltigkeitsstellen gewonnen. Diese Stellen sind nämlich mit den nicht erreichbaren Randstellen des offenen Teiles der Mannigfaltigkeit identisch. Es ist bekannt, dass die hier behandelten speziellen c.m. noch schärfere Eigenschaften besitzen und deshalb sich mit viel schwächeren Begriffsbildungen erfassen lassen, nämlich unter dem Gesichtspunkt des Zusammenhangs im Kleinen. Demgegenüber sind die allgemeinen ebenen Mannigfaltigkeiten nicht erforscht und dort gelangt der lokale Dimensionsbegriff zu seiner Geltung. Die allgemeineren c.m. können ausserordentlich kompliziert sein, ja bereits im Falle, wo jeder Punkt der Mannigfaltigkeit ein 2-dimens. konzentrischer m.-Punkt ist. Die Gesichtspunkte, unter welchen sich solche c.m. behandeln lassen, sind vor allem der Zusammenhang einer ebenen c.m., die Anzahlihrer offenen Komponenten und insbesondere die Mehrfachheitihrer Punkte, je nach der (endl. oder unendlichen) Anzahl der nichtäquivalenten Folgen von c.m., welche einem 2-dimens.

The object of the present paper is to report upon a case of neutron-produced disintegration of a type which has not previously been observed. The nucleus disintegrated was that of carbon and the peculiarity of the disintegration lies in the fact that three heavy particles resulted from the transformation. Hitherto no example of such disintegration resulting in more than two heavy particles has been obtained, and with carbon, in particular, several experiments agree in showing that this more usual type of disintegration is very rare indeed. In similar circumstances, and with neutrons of less than 12 × 106 electron volts energy, disintegration phenomena occur in oxygen or in nitrogen at least ten times as frequently as in carbon. Thus Harkins, Gans and Newson obtained only two examples of disintegration in 3200 pairs of photographs taken with a source of radiothorium and beryllium and an expansion chamber filled with ethylene, and concluded that in each case an atom of oxygen or some other impurity must have been involved. Likewise, Feather, in 2210 pairs of photographs with the neutrons of polonium-beryllium and an expansion chamber filled with a mixture of acetylene and helium, found only one example of a disintegration which could reasonably be ascribed to the nuclear reaction,

In a paper by Mr Lyons and myself an elementary proof of the following theorem was given:

Given three lines a, b, c in space, if a′ is the line of shortest distance of b and c, b′ that of c and a, and c′ that of a and b; and if x is the line of shortest distance of a and a′, y that of b and b′, and z that of c and c′, then x, y, z have a common line of shortest distance.

An account is given of attempts to detect radiation in thunder clouds by means of photographic plates attached to balloons. Eleven balloons were sent up and the plates attached to ten were recovered and examined.

The result was negative in every case.

A small ionisation chamber with quartz condenser intended for detecting radiation in thunder clouds is described.

These observations were made at the Solar Physics Observatory, Cambridge, and I have to thank the Director for giving me every facility.

I am also greatly indebted to Professor C. T. R. Wilson for his continued advice and interest.

By an isolated singularity of an algebraic surface in [r] (i.e. space of r dimensions) I shall mean one which not merely is not upon any branch of a multiple curve of the surface, but has also the property that when the surface is projected into [3] from a general space [r − 4] the singular point remains in isolation, i.e. no branch of the double curve created by the projection will of necessity pass through it.