Actual Deviation Research Articles

Systematic sampling

Systematic sampling

The Development of Locomotive Power at Speed

The power developed per unit of cylinder volume of locomotives in the speed range from 250 to 400 r.p.m. has in recent years been doubled. Forty years ago it was often the case that the diminution of the mean effective pressure with increase of speed, rather than the evaporative capacity of the boiler, limited the power of the engine. The most advanced designs maintain the mean pressure at a high percentage of the calculated value to the highest speeds, and the effect of valve events and clearance volume on the calculated mean pressure thus becomes of practical importance, especially as designers are endeavouring to use considerably higher steam pressures in single-expansion cylinders. The first part of this paper sets out the effect of these factors over a wide range of steam pressures in the form of basic data graphs of mean pressure, relative efficiency, and steam consumption. It is shown that without the decrease of clearance volume and increase of expansion ratio which compound expansion affords, the improved thermal efficiency which higher steam pressure offers cannot be fully realized above about 250 lb. per sq. in. boiler pressure, though greater power can be obtained. The second part of the paper is devoted to an examination of the actual deviation of mean pressure with speed for a wide range of locomotives, including some of the most recent designs, and the reasons for the radical improvement are indicated. In 1905 the late Professor Dalby suggested a simple proportional relationship between mean pressure and speed. Such a relationship does not hold for modern locomotives, and at the end of the paper a simple exponential law is proposed with a single coefficient characteristic of the locomotive. This law may be used to estimate the power of a future design at any speed, or as a criterion for assessing the performance of an existing locomotive in respect of power developed, which is generally of more importance to railway companies than thermal efficiency.

The diffraction theory of optical aberrations*)Part I: General discussion of the geometrical aberrations

In this paper the geometrical aberrations of a rotationally symmetrical optical system are dealt with in a new way, suggested by the wave-optical treatment of the image errors to be published subsequently. A spherical surface is introduced with its centre at the Gaussian image (situated at a radial distance σ from the axis) and passing through the centre of the exit-pupil. The actual wave-front deviates by an amount V (σ, γ, ) from this sphere, where γ andare plane polar co-ordinates upon the latter. The aberration function” V is expanded in terms γn cosm, whereas usually the characteristic function analogous to V is expanded in terms containing γn cosm . Thus a single aberration is defined now by an aberration function blnmσ2l+mγn cos m. As a consequence of this new definition the discussion of the aberration figures is simplified considerably. Also a simple classification of image errors of all orders is found, according to which the value of m determines the general type to which a given single aberration belongs (spherical aberration for m = 0, coma for m = 1, astigmatism for m = 2).

The diffraction theory of optical aberrations: Part I: General discussion of the geometrical aberrations

In this paper the geometrical aberrations of a rotationally symmetrical optical system are dealt with in a new way, suggested by the wave-optical treatment of the image errors to be published subsequently. A spherical surface is introduced with its centre at the Gaussian image (situated at a radial distance σ from the axis) and passing through the centre of the exit-pupil. The actual wave-front deviates by an amount V (σ, γ, ) from this sphere, where γ andare plane polar co-ordinates upon the latter. The aberration function” V is expanded in terms γ n cos m , whereas usually the characteristic function analogous to V is expanded in terms containing γ n cos m . Thus a single aberration is defined now by an aberration function b lnm σ 2l+m γ n cos m . As a consequence of this new definition the discussion of the aberration figures is simplified considerably. Also a simple classification of image errors of all orders is found, according to which the value of m determines the general type to which a given single aberration belongs (spherical aberration for m = 0, coma for m = 1, astigmatism for m = 2).

A Peculiarity in Rainfall Variability

A SOMEWHAT perplexing and highly interesting peculiarity in the regimen of rainfall variation is this: all over the world, there is a much greater degree of uniformity in the relative variability of annual rainfall expressed as a percentage of the normal than in the absolute variability expressed as the actual deviation from the normal in inches or millimetres. As regards Great Britain, the late Mr. Carle Salter pointed out in his “Rainfall of the British Isles” that though the percentage variability is rather greater in the dry eastern than in the wet western districts, it is of the same order of magnitude everywhere, which of course implies that the actual differences in the quantity of rain from year to year are much larger in the wet parts. In fact, the deviations in the actual amount of rain above and below normal increase so systematically with the rainfall itself that they swamp the percentage values and become quite useless in comparative statistics.

XVII.—Tests for Randomness in a Series of Numerical Observations

It is often a matter of considerable importance to decide whether the fluctuations exhibited by a series of observations are random in character, or whether some factor operating according to a definite law must be assumed to exist. One general method of testing a statistical hypothesis is to work out results which would be expected theoretically, and then to compare these with the observations. A statistic such as the mean or standard deviation to be observed on a particular hypothesis may be calculated and compared with the actual mean or standard deviation as evaluated from the observed figures. Agreement does not necessarily imply that the hypothesis is a true one, or even that it is consistent with the observations, because it might be found that when another statistic was calculated, the agreement was no longer satisfactory. Obviously the greater the number of tests applied and found satisfactory, the greater the confidence which can be placed in the conclusions.

International and Domestic Commodities and the Theory of Prices

The assumption of independent national price systems, in theory and in treatment of current problems, 409. — I. No and no prices; only local prices, more or less sensitive to external conditions, 410; merchandise and services, wholesale and retail prices, not differentiated in the terms international and domestic commodities, 413. — II. Parallelism in movement of wholesale price indices, 413; Internationalization of the commodity price structure, 414. — III. Illusion of price structure for wholesale merchandise: large or sustained movement not essential to price interdependence, 422; defective market data and apparent price deviations, 432; limitations upon actual deviation; a priori indications of high correlation, 433. — IV. Reformulation: the local category consists mainly of services and retail prices, 436; merchandise at wholesale is international, 437; regional prices structurally related, 442. — V. Practical applications: farm relief, 452; export combinations, 453; tariff and monetary problems, 455. — VI. Regional prices of goods: as affected by foreign exchange, 456; by movement of specie or changes in purchasing power, 457; a possible reconciliation with orthodox theory, 458; price stabilization through credit policies of central banks, 459.

The valve filament at constant voltage

When the filament of a thermionic valve emits and a thermionic current flows from the filament to the anode and through an external circuit back to the filament, the filament current is necessarily different at all points along the filament. An ammeter in each filament lead and a voltmeter across them will all read differently when the valve emits, the actual deviations depending on the electronic emission and the circuit used. The only exceptions are the case of a filament connected directly to a source of P.D. with no intermediate rheostat, when the voltage will be maintained constant by the supply, but the currents will vary in any case. The other exception is that of the valve with the separately heated cathode, the potential of which is uniform. The variations of the reading of the filament voltmeter and ammeters have been investigated experimentally, and a circuit has been devised in which there is no change of reading of the voltmeter or either ammeter when the anode current is switched on and off. The current read is the filament battery current, and with this circuit tests may be repeated on valves with the certainty of the filament conditions being constant for different tests. An investigation was also made of the validity of the practice of readjusting the filament voltage after switching on the anode potential, and it was found that this practice is not correct as regards maintaining the filament under constant conditions, except for those valves intended to be run without a rheostat. All tests were made with valves used as diodes, as in this case the emission is usually greater than the anode current when the valve is used as a triode. For standard tests on valves used as triodes the precautions necessary to maintain constant filament conditions are of less importance. For standard valve tests the circuit devised should be adhered to, and the practical means of ensuring constant conditions are indicated. The method of reducing the asymmetry of the temperature gradient of large valves used as rectifiers is of some interest here, and the conditions have been calculated. The experimental work necessary shows how fine precision measurements may be made with commercial instruments, as numerous checks are possible, most of which show up the accuracy of the majority of the readings, which in most cases involved a change of less than one scale division. Finally, a valve was tested with its filament supplied with A.C., and various circuits tried. It was found immaterial to which point of the filament circuit the anode return was connected, on account of the symmetry of the temperature gradient, at any rate at 50 similar and above.

The cause and elimination of night errors in radio direction-finding

The paper describes experiments which have been carried out with a view to obtaining more conclusive evidence as to the causes of the apparent variations in bearings observed under certain conditions on wireless direction-finders. In the course of the experiments the Adcock “four-aerial” direction-finder has been developed, and with its aid it has been shown that the actual deviation in azimuth of wireless waves is practically negligible. These experiments thus constitute a proof that the variable errors observed on closed-coil direction-finders at night are caused by downcoming waves arriving from the upper atmosphere and polarized with the electric force in a horizontal plane. The investigation also indicates the possibility of the Adcock system being developed into a practical direction-finder which is free from night errors, and those errors associated with observations on aircraft transmissions made at a, ground direction-finding station.