The King William Banks buoy is situated about 12 miles east of the northern tip of the Isle of Man. It marks the eastern extremity of the bank of this name and is an important turning point, especially for the British Railway ships on the Heysham to Belfast run. This buoy, which is unlit, is a third class spherical buoy and thus presents a poor radar target. The British Railway ships are fitted with radar and always pass the buoy at night so that if it could be detected by radar regularly it would be of great assistance to navigation. In fact complaints were made about its radar reflecting properties, and accordingly a pentagonal cluster of 12 in. corner reflectors designed by the Admiralty Signal and Radar Establishment was supplied to Trinity House and mounted on the buoy on 3 March 1949 (see Notice to Mariners 619/49) at a height of 11 ft. 6 in. above the water.

The absolute measurement of fast neutron flux presents several difficult problems. Few methods have yet been described in the literature, although the experimental techniques developed by several authors for the detection of fast neutrons (Baldinger, Huber and Staub(7), Barshall and Kanner(9), Amaldi, Bocciarelli, Ferretti and Trabacchi (3), Gray (19), Barshall and Battat(8)) may easily be adapted to this type of measurement. It is, however, most important to have available methods of measuring fast neutron flux to permit the determination of cross-sections for nuclear processes induced by fast neutrons, and several such methods have been developed in the Cavendish Laboratory in recent years. They are the subjects of separate papers (Bretscher and French (13), Kinsey, Cohen and Dainty (21), Allen (l), Allen and Wilkinson (2)). The main purpose of the present paper is to describe the results of experiments carried out to compare these methods in order to test the validity of the assumptions implicit in the individual methods.

The grand partition function is derived by averaging over all partitions and numbers of molecules of a grand ensemble. The statistical parameters that arise are interpreted thermodynamically. A formula for the mean fluctuation of the numbers of molecules is deduced, and it is shown that the latter can become very great for singularities of the partial potentials. In that case, the system becomes statistically indeterminate, and separates out into several phases. Conversely, in the statistics of actual systems, the search for singularities, at which the partial potentials become independent of the composition, leads to the conditions under which a phase-change takes place. It is shown that, on certain hypotheses, the grand partition function is the generating function of the ordinary partition function, and that the latter can always be represented in product-form. The generating function of a factor in this product-form is called a reduced grand partition function. The significance of its parameters is discussed, and thoir use in the statistical thermodynamics of liquid mixtures is reviewed.

It is clear from § 1 that we shall have quite a large number of theorems to consider, as we may distribute the conditions of monotony and existence in a good many different ways. It is therefore convenient to collect together the various results for series and for transforms in tabular form, for reference and comparison; this is done at the end of this section.

The transform table is very simple. The theorems contained in it have been discussed at length in M.F.(I); they may be summarized by saying that (1) holds in all the cases considered, except perhaps in the consine case of section [B] where the truth is not known. The series results are apparently more complicated as well as more numerous, since there are extra conditions to be imposed in [A, 1], [B, 1], [C′, 1] and [C, 2]. However, these are certainly satisfied if we suppose f and g to be positive whenever they are given to be monotonic (a condition which automatically holds in the transform theorems, since in these the monotonic functions tend to zero at infinity). If we confine our attention to this case, the series results are what we should expect from the transform ones, except that the difficulty in section [B] for transforms is not reproduced for series in [B, 1] or [B′, 1]; it is reproduced in [B, 2].

In this paper we are interested in statements about Diophantine approximation which are ‘almost always’ or ‘almost never’ true. Let

be s non-negative functions of the positive integer n, and let

The theory of graduation discusses methods of obtaining a smoothed series of values of a function from a given empirical set of values. This is usually done by replacing each observation by a weighted average of it and neighbouring observations. Thus if {xi} (i = 0, ± 1, …) is a sequence of values, we can replace them by the series

where the A's are constants which are chosen in some suitable manner. If we regard the xi as the sum of a functional part fi and an error term εi we may attempt to choose the A's in such a way that the functional part is reproduced as well as possible, e.g. that

for some desired type of function. If this is so and the error terms εi are independently distributed with zero mean and finite standard deviation σ, we find that

where the ηi are a series of random variables which are no longer independent of each other but which have zero mean and a standard deviation σ1 such that

An abbreviated version of a paper read before a joint meeting of the Institute and the Royal Aeronautical Society.

The usefulness of a particular aircraft may well depend upon the availability or simultaneous development of equipment. Safe, reliable, economic and speedy operation in the civil field may rest, for instance, on facilities which enable all-weather operation to be undertaken—including navigation, traffic control, approach and landing facilities. Military operations, even more directly, will be determined by the instrument aids available.

In his Presidential Address to the Institute this year the Astronomer Royal, Sir Harold Spencer Jones, stresses the importance of an early resumption of the general survey of the Earth's magnetic field and developments in airborne magnetometers in the United States have suggested the possibility of using aircraft for this purpose. The techniques which are available for aerial survey will be discussed in this paper and an attempt will be made to evaluate the relative merits of conducting world wide magnetic surveys by sea and by air.

This paper describes a method for the reduction of simultaneous star altitude observations in pairs, whereby an unambiguous ‘fix’ is obtained without reference to the dead reckoning position and without use of the intercept procedure. The method is adequate in all latitudes. An especially swift reduction is possible for the moments at which two stars are situated at equal altitudes.

When position is determined by means of two simultaneous star altitude observations and use is made of a single chosen position it is possible for the Marcq St. Hilaire intercepts to be so short that they can be regarded as negligible. When this occurs the chosen position is, in effect, the true position as indicated by the observations and no further work is required for a fix. If, in such a case, it is desired to have the Sumner lines of position represented on the chart it will be possible to draw them through the chosen position. The selection of such a chosen position may be made with certainty, in every instance, by computing numerical values for the latitude and the longitude of the appropriate intersection of the two circles of equal altitudes corresponding to the corrected observations. If the entire computation had to be done after the observations were made, the labour of this procedure would prohibit its use in rapid navigation. It happens, however, that the greater part of the work can be precomputed and presented for the navigator's use in a tabular form. These data are extremely compact and are permanent except for the need of occasional adjustment for precessional effects and on account of proper motions of the stars. No almanac would be required for use with such a table if the navigator were not dependent on it for his G.H.A. Aries. No knowledge of the dead reckoning position is employed in this computation; star observations suffice to determine the position without ambiguity.

Let f(x, αi) be the probability density function of a distribution depending on n parameters αi(i = 1,2, …, n). Then following Jeffreys(1) we shall say that the parameters αi are orthogonal if

Probably the most important single factor governing the value which can be obtained from a given radar equipment in a ship is the presence or absence of blind or shadow sectors. This is entirely dependent, on the siting of the scanner. There is no arc of bearing on which the careful seaman does not keep a visual lookout, constant or periodic, when his ship is at sea. As radar is capable of providing him with a means of detection on all bearings, which will often be more effective than his eyes, it would be surprising if he were not prepared to go to some lengths to avoid obstructions to its ‘view’.

The necessity for an accurate instrument to indicate the height of an aircraft above the ground over which it is flying at any instant has long been recognized and a variety of types have been suggested.

In the first part of this paper a method was given for constructing a wave potential when the normal velocity is a prescribed function of the angular variable on a submerged circular cylinder. It was shown that the method breaks down for values of the parameters Ka and Kf for which a certain infinite determinant vanishes. The vanishing of this determinant implies the existence of a non-trivial velocity potential, such that the normal velocity vanishes on the cylinder and both velocity components vanish at infinity. In this part of the paper it is shown that there can be no non-trivial solution of this kind; in other words the infinite determinant does not vanish. In the absence of a general uniqueness theory for surface waves it seems worth while to establish this particular result.