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In a communication published in the ‘Proceedings,’ Mr. Marconi has given the results observed when a straight horizontal conductor is substituted for the usual vertical conductor employed as a transmitter or receiver at a wireless telegraph station. The object of the following note is to consider the theory of such an arrangement, or at any rate one aspect of it. The receiver, as being the more important, will be considered first. Let AB (fig. 1) represent the horizontal receiver, consisting of a straight conductor having the end A connected to a spark-gap CC 1 or other wave-detector. The electric oscillations in AB can be represented by a distribution of Hertzian oscillators along AB, and, if L denotes the current strength at any point of AB, it must satisfy the conditions L= 0 at B, the free end, and d L/ ds = 0 at A, since the electric force perpendicular to AB at A must vanish. If the distance of AB from the earth is not too small, the effect of the oscillations belonging to the image in the earth of AB on those in AB may be neglected, the radiation from the free end B will be approximately symmetrical with respect to AB, and the oscillations in AB are then approximately the same as if BA formed part .of a semi-infinite straight conductor in which a system of oscillations is being maintained, B being the free end and A. the first node from the free end; the wave-length of these oscillations is very approximately five times the length of AB, and therefore the receiver is of maximum efficiency when its length is one-fifth of the length of the transmitted wave, a result observed by Marconi. When the distance of AB from the earth is so small that the effect of the oscillations in the image of AB in the earth on the oscillations in AB is not negligible, the radiation from the free end B will not be symmetrical with respect to AB, but may be taken as being approximately symmetrical with respect to some line through B making an angle with BA; the wave-length of the oscillations in AB is therefore equal to the wave-length of the oscillations in a bent conductor joining AB; that is greater than five times the length of AB, and, therefore, in this case the receiving conductor has its maximum efficiency when its Length is somewhat less than one-fifth of the length of the transmitted wave, result also observed by Marconi. To examine the effect of the orientation of the receiver, consider a straight conductor BAB' twice the length of AB (fig. 2) and its image B 1 A 1 B 1 ' in the horizontal plane, A and A 1 being their middle points respectively.

Professor Wood has found that the spectra of an incandescent lamp formed by a grating can have bright and dark lines in them. These occur at the same place in each order of spectrum, and move through the spectrum when the inclination of the grating is altered. The phenomenon has been investigated by Bayleigh,f who discusses the case of diffraction by a grooved reflecting surface. On account of the analytical difficulties he uses approximations. In the present paper a theoretical grating is discussed, the phenomena of which can be investigated accurately, an advantage that is, however, coupled with the drawback that we cannot feel certain how far the phenomena agree with those of the ordinary metallic reflection grating. A theorem that gives the sum of certain infinite series in a finite form, and transforms others into new series, is found in 5.

A standard mutual inductance, after the design recently described by the writer, has now been constructed at the National Physical Laboratory. As the details of its construction will be published later, it is sufficient here to mention that its value calculated from the dimensions is 10.0178 millihenries. It forms an extremely accurate standard, against which both mutual and self inductances can be readily tested. In addition to this, it affords a means of obtaining values of resistance coils in absolute measure, and thus evaluating the ohm. This can be done in an indirect way by finding the capacity of a condenser in terms of resistance and time by Maxwell’s Commutator Method, and in terms of resistance and mutual inductance by Heydweiller’s modification of Carey Foster’s method. The comparison of resistance with mutual inductance can, however, be made far more simply and directly by the use of two-phase alternating currents in the method which I proceed to describe. I shall first take the ideal simple case, and afterwards notice some of the difficulties that may arise in practice. (2) Theory of the Medhod. In fig. 1 let M be the mutual inductance (a small fraction of it being adjustable) and R the resistance; and let A cos pt and B sin pt be currents in quadrature, e. g ., from a two-phase alternator or a phasesplitting device. Let G be a vibration galvanometer tuned to frequency n where p = 2 πn .

Diffusion weighted imaging (DWI) is a non-invasive method for investigating the brain white matter structure. It can be used to evaluate fiber bundles in the brain. However, clinical acquisitions are often low resolution. This paper proposes a method for improving the resolution using sparse representation. In this method a non-diffusion weighted image (bO) is utilized to learn the patches and then diffusion weighted images are reconstructed based on the trained dictionary. Our method is compared with bilinear, nearest neighbor and bicubic interpolation methods. The proposed method shows improvement in terms of Peak Signal-to-Noise Ratio (PSNR) and Structural SIMilarity (SSIM).

Having recently had to photograph some small solids on an enlarged scale, I was led to enquire into the conditions as to the angular aperture and focal length of the lens which would give the best definition in the picture obtained. If the actual distance apart of the closest points in the object which must be separated in the picture is greater than several wave-lengths of the light employed, the problem may be solved by geometrical optics.

In this paper, we propose Visibility Index Functions (VIFs), that enable evaluating and predicting the legibility of sentences quantitatively and conveniently. In light of nonlinearity attached to suprathreshold phenomena, we analyzed the data of the series of legibility evaluation experiments of Sato and Hara(1999), and constructed two formulations : VIFVL(VL, A) and VIFΔL(ΔL, Lb, A), as functions of Visibility Level VL, luminance difference ΔL[cd/m2] between a visual target and its background, background luminance Lb[cd/m2] and character size A[min]. We showed that both the VIFs can well describe both the experimental results of Sato and Hara and the legibility data derived by Inoue and Akizuki (1997) with no parameter change except low luminance conditions for background.

Fuzzy systems that can automatically derive fuzzy if then rules and membership functions from numeric data have been developed previously. In this paper, we propose two new fuzzy learning methods for automatically deriving membership functions and fuzzy if-then rules from a set of given training examples. The proposed methods first select relevant attributes and build appropriate initial membership functions. They then simplify the intervals and the membership functions of each attribute before the decision table is formed. These attributes and membership functions are then used in a decision table to derive the final fuzzy if-then rules and membership functions. Experimental results on Iris data show that our methods can achieve a high accuracy. The proposed methods are thus useful in constructing membership functions and in managing uncertainty and vagueness. They can also reduce the time and effort needed to develop a fuzzy knowledge base.

This paper presents a quasi-static analysis of a metama- terial consisting of a three-dimensional array of small tunable stacked split ring resonators (SSRRs). The resonance frequency of the pro- posed resonator structure can be controlled by adjusting the auxiliary lumped elements which are inserted between the split of each ring. In addition, the size of the ring resonator can be reduced to an order of 0.01λ that is one tenth that of the split ring resonator (SRR) by choosing the proper lumped elements. The analysis is based on the quasi-static Lorentz theory, and the generalized matrix representation of the macroscopic constitutive relations of the composite medium is calculated.

1. Introduction.— 1·0. The object of this paper is to develop methods where by the differential equations of physics may be applied more freely than hitherto in the approximate form of difference equations to problems concerning irregular bodies. Though very different in method, it is in purpose a continuation of a former paper by the author, on a “Freehand Graphic Way of Determining Stream Lines and Equipotentials” (‘Phil. Mag.,’February, 1908; also ‘Proc. Physical Soc.,’ London, vol. xxi.). And all that was there said, as to the need for new methods, may be taken to apply here also. In brief, analytical methods are the foundation of the whole subject, and in practice they are the most accurate when they will work, but in the integration of partial equations, with reference to irregular-shaped boundaries, their field of application is very limited.

Hitherto the use made of the so-called “entrance and exit pupil” of an optical system has been chiefly in regard to aperture and the questions which are related thereto. I have been led to investigate this conception more completely, and I have now found that it may be used with advantage as an additional factor, in connection with the Gauss planes, for explaining the action of optical instruments as regards the perspective of the images formed. The Gauss planes enable us to refer the action of a complicated optical system to an equivalent single lens placed in two positions—the entrance equivalent plane and the exit equivalent plane. In fig. 1, the upper half of the diagram shows the course of a ray of light from X passing through the three lenses to X', each lens bending the light to a certain extent. The lower half of the diagram shows a lens of the correct equivalent focus placed at E to receive the light, and at E' to discharge it. This will act in a similar manner to the complete system, and many of the properties of the optical system may be correctly studied by confining our attention to an equivalent single lens, placed in the plane E to receive the light, and shifted to the plane E' to discharge it. By this means, assuming that the optical system is corrected in such a manner that the oblique rays and those far from the axis act in the same manner as the direct axial rays, the position and size of images can be determined with accuracy; but the perspective of the image cannot be correctly explained by aid of the Gauss planes alone. The “pupils” used as a modifying factor in connection with the Gauss planes will explain this apparent discrepancy.

The medium is supposed to be such that its properties are everywhere a function of but one co-ordinate x , being of one uniform quality where x is less than a certain value x 1 , and of another uniform quality (in general, different from the first) where x exceeds a greater value x m -1; and the principal problem is the investigation of the reflection which in general ensues when plane waves in the first medium are incident upon the strati-fications. For the present we suppose the quality to be uniform through strata of finite thickness, the first transition occurring when x = x 1 , the second at x = x 2 , and the last at x = x m -1. The expressions for the waves in the various media in order may be taken to be ϕ 1 = A 1 e i [ ct + by - a 1 ( x - x 1 )] + B 1 e i [ ct + by + a 1 ( x - x 1 )] , ϕ 2 = A 2 e i [ ct + by - a 2 ( x - x 1 )] + B 2 e i [ ct + by + a 2 ( x - x 1 )] , ϕ 3 = A 3 e i [ ct + by - a 3 ( x - x 2 )] + B 3 e i [ ct + by + a 3 ( x - x 2 )] , and so on, the A's and B's denoting arbitrary constants. The first terms represent the waves travelling in the positive direction, the second those travelling in the negative direction; and our principal aim is the determina­tion of the ratio B 1 /A 1 imposed by the conditions of the problem, including the requirement that in the final medium there shall be no negative wave.

In this paper, 3D face recognition under isometric deformation (induced by facial expressions) is considered. The main objective is to employ the shape descriptors that are invariant to (isometric) deformations to provide an efficient face recognition algorithm. Two methods of the correspondence are utilized for automatic landmark assignment to the query face. One is based on the conventional iterative closest point (ICP) method and another is based upon the geometrical/topological features of the human face. The shape descriptor is chosen to be the well-known geodesic distance (GD) measure. The recognition task is performed on SHREC08 database for both correspondence methods and the effect of feature (GD) vector size as well as landmark positions on the recognition accuracy were argued.Copyright © 2014 by ASME

That fatigue of the retina by a colour will change the hue of the spectrum colours is well known. In this paper a method of measuring the amount of fatigue is described, showing that by fatiguing the retina colour-blindness is produced which is the same as that found when one of the three sensations is deficient in sensitiveness. In the first part of this communication the fatigue is produced in one eye, and the change in hue is registered by the unfatigued eye, and from these observations what we may call the factor of the fatigue is calculated, which is the ratio in sensitiveness of the unfatigued sensation to that of the fatigued. In the second part of the communication a method is described of measuring the luminosity of what, for short, we will call the fatigue spectrum, which is, of course, the luminosity of the spectrum as seen by the fatigued eye. In previous papers the value of the three sensations in luminosity to the normal eye has been given, and the red, green, and blue sensations have been indicated in equations as R. S., G. S., and B. S. In this paper the luminosities of the three sensations of the spectrum have to be shown as curves of equal stimulation, and the percentage values of the different colours have been derived from these curves. The curves of equal stimulation of the spectrum are the ordinary sensation curves, two of which (the blue and the green) have to be multiplied by factors to make them of equal area. Equal ordinates in these curves will then give the amount of white light existing in the different colours. The sensations of equal stimulus are indicated for the sake of brevity as R’S., G’S., and B’S.

THE present paper is a supplement to the mernoir of the same title issued last year*. It is not proposed to give here any account of the work which has been done in the attempt to elicit information on the more difficult subject of the natutre of selection in man, but only to publish the correlations and regressions obtained by using an alternative measure of environment, and by varying the periods in which the effects of a selective death-rate can be detected.

This paper may be regarded as a continuation of that communicated in January, 1912, wherein I described a new optical apparatus by means of which it is possible to obtain load-extension diagrams free from the inertia effects of the parts of the testing machine in which the materials are strained and broken. The apparatus is automatic in its action and gives a true record of the physical properties of the materials. The curve corresponding to the straining and breaking of a specimen is drawn by means of a spot of light moving over a photographic plate placed in the camera which forms part of the instrument. There is no practical limit to the speed at which a diagram can be taken other than that imposed by the "rapidity" of the plate.

A new method of investigating the structure of a crystal has been afforded by the work of Laue* and his collaborators on the diffraction of X-rays by crystals. The phenomena which they were the first to investigate, and which have since been observed by many others, lend themselves readily to the explanation proposed by Laue, who supposed that electromagnetic waves of very short wave-lengths were diffracted by a set of small obstacles arranged on a regular point system in space. In analysing the interference pattern obtained with a zincblende crystal, Laue, in his original memoir, came to the conclusion that the primary radiation possessed a spectrum consisting of narrow bands, in fact, that it was composed of a series of six or seven approximately homogeneous wave trains. In a recent paper I tried to show that the need for assuming this complexity was avoided by the adoption of a point system for the cubic crystal of zincblende which differed from the system considered by Laue. I supposed the diffracting centres to be arranged in a simple cubic space lattice, the element of the pattern being a cube with a point at each corner, and one at the centre of each cube face. A simpler conception of the radiation then became possible. It might be looked on as continuous over a wide range of wave-lengths, or as a series of independent pulses, and there was no longer any need to assume the existence of lines or narrow bands in its spectrum.

The authors have each separately dealt with the question of complete and incomplete colour-blindness caused by the absence of, or decrease in, the response to stimulation of the red or green perceiving apparatus which is functional in the case of vision. We have shown that a large number of cases of defective perception of colour are simply explained on this hypothesis. Abnormality of colour vision may also be due to a shift in, or an alteration in form of, one of the sensation curves. In the present paper we discuss the effect of one type of shift on the colour perception and give the results of a series of measurements which show that such a shift, without any alteration of form, does sometimes occur. In a paper which appeared in the ‘Proceedings of the Royal Society,’ one of us indicated how a shift, that is a displacement of the whole curve so that the maximum is displaced to a different wave-length, of one of the sensation curves could be detected by a simple spectrum test, which is as follows. When yellow light of wave-length 5760 Å. U. is mixed with blue light of wave-length 4610 in suitable proportions the mixed light looks to the normal eye exactly the same as the white light from the crater of the electric arc both in hue and brightness. If a person who has a shift in one of the sensation curves is shown this match it will not appear correct to him nor can it be made correct by any alteration in the proportions of the yellow and blue lights. There will, however, in every case be found a position of the slit through which the yellow light is obtained with which a satisfactory match can be obtained. In other words the wave-length of the light which is complementary to the blue will be different to that of normal vision.

Prof. Fleming exhibited an apparatus for the production of vibrations on strings loaded or unloaded. This method is an improvement on that of Melde, in which transverse vibrations are created on a string attached to the prong of a tuning fork. Melde's method has the disadvantage that it is not possible to put great tension on the string without stopping the fork, or to use very heavy or loaded strings without using forks of great size. Moreover, the frequency cannot be varied except by changing the fork. In the Author's method the vibrations are produced on a string by attaching one end to the shaft of a small continuous-current motor of about 1/8 H.P. The motor has on one end of its shaft a mechanism for counting revolutions, and on the other end a disc which has a crank pin inserted in its outer face, and to this pin is attached a light crankshaft connected at its outer extremity with a rocking lever. The string is fastened to a hook on the crankshaft near to the crank pin. The other end of the string is attached to a fixed point which can be moved by means of a screw, in some cases a spring balance being interposed to measure the tension. When the motor is started the string has a circular motion given to its end which is equivalent to two simple harmonic motions at right angles to each other. If the tension is rightly adjusted the string then vibrates in sections, and the number of sections can be adjusted by altering the tension. The distance from node to node can then easily be measured and the frequency determined from the speed of the motor. In this way the velocity of the wave is measured, and can be compared with the velocity determined by taking the square root of the quotient of the tension by the linear density of the string. A number of tables were exhibited showing the velocity of the wave motion on strings of various kinds confirming the known laws of string vibrations. This method is particularly useful in studying the properties of loaded strings. Strings made of flexible cotton cord can be loaded with glass or wooden beads and set in vibration. In this manner it can be shown experimentally that when the wave length on the string extends over a distance of more than 8 or 10 loads, the string vibrates as if the loading matter were uniformly distributed, but the string cannot propagate vibrations when the half wave length approaches equality to the distance between two loads. It is also possible to show by this apparatus very prettily the reflection of a wave at a load placed at any point on the string, and also that this reflection is reduced by tapering off the loading. We can thus imitate with this loaded vibrating string all the phenomena of inductive loading in telephone cables on the Pupin system. The theory of the vibrations of a loaded string has been studied by many mathematicians - for example, by Lagrange, Lord Rayleigh, Prof. P. G. Tait, and a very interesting Paper was published on the subject by Mr. Charles Godfrey in the "Philosophical Magazine" for April, 1898. This method of causing a string to vibrate by a motor affords a simple method of demonstrating all the laws of vibrating strings to a class

A method of drawing the cusped wave-fronts produced by refraction and reflection at a spherical surface and a graphic treatment of the elementary theory of the rainbow are described in the Paper. The method is based upon the properties of the centre of junction. In the case of reflection at a spherical surface in which aplanatic points are not available the position of the junction-centre is obtained by elementary geometry. By the use of junction-centres the distribution of the successive foci obtained on refraction and reflection at a spherical surface is readily plotted. If in the case of a rainbow it is assumed that the emergent portion of the effective pencil, as well as the incident portion, is one of parallel rays, then the distribution of foci with regard to the drop is a symmetrical one. The caustics on refraction and reflection are also readily drawn as loci of points and envelopes of rays. Hence the wave-surfaces can be obtained. These in some regions are cusped and the caustics are the loci of the cusps. In the case of pencils that are effective in producing rainbows, the lengths of the incident portion of the chief ray intercepted by the spherical drop of radius, r, and a concentric sphere of radius, r. μ, are in the ratio 1 to n + 1, where n is the number of internal reflections. This leads to a geometrical construction for finding the points of incidence of the chief rays of the effective pencils. These rays are at minimum deviation, and the same chief ray remains at minimum deviation although the associated pencil may be converging or diverging.

There are many methods by which a resistance can be measured absolutely in the electromagnetic system of units, and all of these necessarily involve absolute measure­ments of a length and of a time. The length may be the axial length or radius of a coil, or the radius of a disc, or it may involve all of these, and the time may be the time of vibration of a magnet, or of a rotation of a coil, or of the period of an alternating current. In any case, the precision secured in the measurement of a resistance depends primarily on the accuracy obtained in these measurements of length and of time. The first absolute measurements of a resistance were made by Kirchhoff in 1849, but it is to W. Weber that we owe the first distinct proposal (in 1851) of a definite system of electrical measurements according to which resistance can be measured in terms of an absolute velocity.

In the year 1867 A. Wöhler, locomotive superintendent of a railway company in Berlin, exhibited at the Paris Exhibition the results of some experiments on the endurance of metals, and was thereupon engaged by the Prussian Government to carry out the more exhaustive enquiry into this subject with which his name is always associated. The results of his labours were published in 1871, and were highly appreciated, but few additional experiments were made until the subject was again taken up successively by Sir Benjamin Baker, Reynolds and Smith, Rogers, Stanton and bairstow, Eden, Rose and Cunningham, and Prof. Hopkinson. All these experiments are confined either to fatigue bending or to push and pull tests, using only steel or iron, whereas the present ones include a large number of torsion fatigue tests on various metals. Until comparatively recently there was no satisfactory standard of comparison for fatigue tests, the determination of the asymptote or limiting fatigue stress for an infinite number of revolutions from a few irregular test results leading to very uncertain conclusions, so much so that by some it was considered very doubtful whether there were any real fatigue limits, while others adopted as standards of comparison the fatigue stresses which would cause fractures at the millionth repetition. The first problem which had to be investigated was therefore to ascertain the relationship between the intensities of fatigue stresses and the numbers of repetitions of these stresses which would cause fracture; and, should this relationship be found to indicate the existence of a limiting stress for an infinite number of revolutions, or more briefly of a fatigue limit, then the next step would have to be its exact determination.

Having occasion to examine a large number of persons who have abnormal colour vision, the author has found that the so-called anomalous trichromates are clearly divisible into two groups, which groups can be sharply differentiated by quite simple experimental means. By anomalous trichromate is meant a person who, while not a dichromate, i. e ., a person who can match all colours of the spectrum by a mixture of two suitably chosen monochromatic lights, has not got normal colour vision. Such cases were first investigated by Lord Rayleigh, and have since been investigated by many observers. It has already been noticed by several investigators that there are two classes of anomalous trichromates. These are distinguished by the fact that when a monochromatic orange (D-light) is matched by a mixture of green and red light one class, the protanopes, take more red in the mixture than the normal, while the other class, the deuteranopes, take more green than the normal. The following paper is intended to prove that both of these classes require to be subdivided, or rather that anomalous trichromates ought preferably to be divided into two other classes, and then each of these classes subdivided into protanopes and deuteranopes. As has been pointed out by Sir W. Abney, if white light is transmitted through a thickness of about 0·8 cm. of a saturated solution of chromate of potash the colour of the resulting light can be matched by a single spectral colour, and, further, in the case of persons having abnormal colour vision the wave-length of the spectral colour differs from the one which matches the chromate light for the normal. As a result of applying this test to a large number of persons who have not got normal colour vision, the author noticed that, while for some observers the monochromatic colour which matches the chromate light is of quite definite wave-length, for others a band of the spectrum matches the chromate light; further, that one edge of this band always coincides with the position which matches the chromate light to the normal eye. Thus if a slit is traversed through the spectrum and the light which passes is received on a screen alongside the light which has traversed the chromate solution and the position noted when the two lights match, the slit being moved alternately first from the red end of the spectrum and then from the green end, in the case of persons having normal colour vision the settings will lie between 578 and 580 μμ when the source of the white light is the electric arc. With one class of abnormals, which will be called class A, the setting coming in one direction will agree with the normal setting, but the setting coming in the other direction will differ markedly from the normal setting. This class is divided into two subdivisions, one, A r , in which the abnormal setting, and hence the whole of the band of the spectrum which matches the chromate light, lies on the green side of the normal match, and the other, A g , where the band lies entirely on the red side of the normal. It must be remembered that the match with the chromate appears to be equally good throughout the band included between the extreme settings, and, further, that one edge of the band always coincides with the normal setting.

It does not seem to be sufficiently realised that most of the theory usually classed under “Centre of Gravity” is purely geometrical in cliaracter ; and, on the other hand, such impossible terms as “centre of gravity of an area” still offend the eye. This paper suggests a presentation of the subject free from such defects ; but its main object is to establish a sounder transition from the treatment of discrete point-systems, which is the essential groundwork of the theory, to the much further reaching treatment of continuous point-systems. The fact that this transition remains in an unsatisfactory state is an instance of the reluctance to face the necessity for explicit recognition of certain fundamental ideas of the Infinitesimal Calculus, in elementary Mathematics. It is not long since the proper conception of tangent was admitted as suitable for an elementary course of Geometry. Still more recently the more abstruse conception of arc-length has been faced. From what follows it may appear that the conception of centroid awaits its turn.

The aim of the investigations here entered upon is statistical analysis of the yield of agricultural crops in terms of the stages of the plant’s development. When the yield is a fruit, then precedent stages determining the yield must be the flower and, before this, the development of flowering branches. In such a case, for full analysis of the final yield, careful records of these stages must first be taken all through the development season of the crop. The senior author has elaborated, during experiments on cotton in Egypt, the procedure for carrying out such analysis and the points to be attended to. Armed with this method of analysis the effects of environmental conditions on the development of the crop can be satisfactorily distinguished. The chief factors of environment capable of control in field work are—the space allowed to each plant, the date of sowing, the supply of water, of manures, and the choice of the year in which the crop is grown.

In the general introduction to these analyses we defined our purpose as being statistical investigation of the yield, in terms of the different stages of the plant’s development, by means of careful records of all these stages throughout the develop­ment season. In the spacing experiment we examined the effect of providing the cotton plant with various volumes of soil, and found that the method of field cultivation practised in this respect by the Egyptian native cultivator gives the highest yield obtainable from a given area, though the space allowed to each individual plant is very much less than it could profitably utilise if growing isolated.

The object of this paper is to study the groups formed by the residue classes of a certain type of Kronecker modular system and some closely related generalizations of well-known theorems in number theory. The type of modular system to be studied is of the form 9) = (mn, mn-1, , Mnl, m). Here m, defined by (ml, m2, ***, Mk), is an ideal in the algebraic domain Q of degree k. Each term mi, i = 1, 2, *.., n, belongs to the domain of integrity of Qi = ( Q, xl, x2, * , xi), and is defined by the fundamental system ((1t/, t'2j), ***, i,t')). The various 1(i), j = 1, 2, *.., jI, are rational integral functions of xi with coefficients that are in turn rational integral functions of xl, x2, ... , xj_i, with coefficients that are algebraic integers in Q . In every case the coefficient of the highest power of xi in each of the 't') shall be equal to 1. We shall see later that the developments of this paper also apply to modular systems where the last restriction here cited is omitted, being replaced by another admitting more systems, these new systems in every case being equivalent to a system in the standard form as here defined. Any expression that fulfills all of the conditions placed upon each {() with the possible exception of the last one, we shall call a polynomial, and no other expression shall be so designated. This definition includes all of the algebraic integers of Q. Throughout this paper we shall deal exclusively with polynomials as here defined. The first part of this paper will contain the introduction with the necessary definitions and a discussion concerning the factoring of the system 9). The second section will then be devoted to setting up necessary and sufficient conditions that a set of residue classes belonging to 9) form a group when taken modulo 91. In the third section we shall study the structure of such a group with respect to groups belonging to certain modular factors of 9), besides

Before presenting this paper I wish, first, to correct a clerical error in the topic as it appears on the program. It should read “A Comparison of Digestible Coefficients for Cattle, Swine, and Poultry as Suggesting a More Accurate Basis for Computing Poultry Rations.” In other words, I wish to offer this merely as a suggestion, with the hope that it may eventually lead to the use of a more uniform method of computing nutritive ratios by the different Poultry Departments. Due to the difficulty of obtaining coefficients of digestion of feeds by poultry, which has resulted in the very meager data now available, we have come to freely use the coefficients for other animals, and in these calculations. If I am correctly informed, the station computes the nutritive ratio in all instances from the total composition of the grain, disregarding digestibility entirely. Most of the stations, however, in the …

The absence of coma from an optical system free from spherical aberration can be established by means of the sine condition, from the refraction of rays which originate at the centre of the object. In actual systems spherical aberration is always present to some extent, and the conditions which have to be secured in practical cases do not involve mathematical freedom from spherical aberration and coma, but rather their confinement within predetermined limits. Previous investigations on this subject have been published by Conrady and by Chalmers; the present Paper shows that the conclusions of both require some qualification. Conrady's conclusion, that the sine ratio for any zone gives the exact magnification, holds only for rays in a secondary plane, and one of the further conditions (a) that the zone is free from spherical aberration, or (b) that the stop is at the principal focus for rays travelling in the negative direction, must also be satisfied. Under these conditions the magnification for rays in a primary plane is (1 + tan θ' d/dθ') times that for the secondary rays, where 2θ' is the angle subtended by the zone at the centre of the image. The conditions assumed by Chalmers involve the satisfaction of a relation between various quantities, including the curvature and astigmatism of the system; his results may in consequence not apply strictly where very large apertures are involved. Analysis of the general problem shows that the primary and secondary comatic displacements can be accurately derived from the properties of thin oblique pencils of rays. If a central incident ray inclined at θ to the axis is met by an oblique ray in the same axial plane at a distance r from its point of origin, and is refracted at an angle θ' with the axis and meets the image plane at a distance r' from the image of its intersection with the oblique ray, the primary comatic displacement of the oblique ray plus the aberrationless first order displacement is proportional to r' sec θ'/sr sec θ, where μs/μ' is the magnification in a primary plane for a small normal object at the point of intersection of the two rays. The corresponding secondary displacement is ρ'/σρ, where the Greek letters have the same meaning for secondary rays as the corresponding Roman letters for primary rays. These two quantities determine the coma exactly whatever the spherical aberration may be. It follows that coma is dependent upon the principal surfaces of a lens. When the object is at infinity the coma is determined by the locus of the point of intersection of a ray, incident parallel to the axis, with its refracted portion. The results established in the Paper form one of a series of reciprocal relations which exist between the aberrations of an object and those of the effective stop.

The subject-matter of this paper may be briefly summarized in the statement that a method of calculating critical speeds of disappearance of colour flicker from colour sensation and hue discriminat...

Equations for tracing rays in an axial plane through an optical system have the normal refraction terms separated from those representing aberrations. By expressing the latter as a fraction with the first order aberration as the numerator and a correcting factor, which may take various forms, as the denominator, rays may be traced exactly through the system using a short table of cosines in terms of sines in place of the extensive tables, giving sines in terms of angles generally employed. A considerable saving of time is effected in the calculations, and the estimation, without calculation, of the aberrations of other rays is facilitated.

The Paper discusses the reason why satisfactory telescope objectives are obtained by neglecting thicknesses and solving for freedom from first order aberrations. It is shown that the introduction of thicknesses into such an objective without any alteration in the curvatures of the surfaces yields a lens corrected for aberration for a zone which is a constant fraction of the full aperture obtainable. For objectives of the usual type this zone is very approximately the one that would be selected for correction to obtain the most favourable balance between first and second order aberrations. It follows that objectives calculated from first order formulae in which thicknesses are neglected do not require trigonometrical verification or correction unless the conditions are very abnormal.

The Paper describes approximate methods of treatment of the first order aberrations of a centred optical system. Two methods are used, one primarily suited to the case where the separation of the surface is small, and the other more suited for use where the separations vitally affect the design. The aberrational defects are expressed as lateral aberrations - i.e., as defects measured in the focal plane of the system. The procedure adopted is to express the aberrational defect of a single surface in terms of the constants of the surface, and the perpendicular distance of the ray considered from the centre of curvature of the surface. The value of this perpendicular can be expressed in terms of the co-ordinates of the ray in any chosen medium, and thus the aberration due to each surface can be expressed in terms of the co-ordinates of the chosen ray, in such a way that the aberrations of the individual surfaces can be summed.

Published in Petroleum Transactions, AIME, Volume 57, 1917, pages 1010–1033. The Oklahoma legislature recently passed a bill providing for "the creation of an oil and gas department under the jurisdiction of the Corporation Commission, authorizing the Corporation Commission to appoint a chief oil and gas conservation agent, and conferring exclusive jurisdiction on the Corporation Commission in reference to the conservation of oil and gas, and the inspection of gasoline and oil, the product of crude petroleum, and repealing all acts or parts of acts in conflict there- with and declaring an emergency." Some of the functions of the new bureau include the receiving and filing of all well logs, and the direction and supervision of plugging all abandoned oil and gas wells, under rules prescribed by the Corporation Commission. The two leading petroleum-producing States of the country, Oklahoma and California, have now established oil and gas bureaus with a staff of inspectors to see that the important rules relating to the conservation of oil and gas are thoroughly enforced. The economic significance and the value of well logs in the conservation work has been recognized, and the necessary legal action taken to insure and procure them. William B. Heroy, of the Government Land Classification Board, in a personal communication to the writer, says that the drilling of wells is also regulated by public authority in the Roswell artesian basin in southeastern New Mexico, where a permit must be obtained for the drilling of any well and a certified log of the well recorded with a county official. The method is said to give fairly satisfactory results. In a recent article 1 a National instead of a State Bureau was suggested for collecting and filing well logs. In discussing the paper, L. L. Hutchison characterized such a plan as impractical because the average driller or contractor will often juggle or guess at the log and will not keep the record sufficiently close or accurate, making it necessary to maintain a special man at each well for the sole purpose of obtaining such data correctly, a procedure that is expensive. T.P. 057–50

This paper describes some corona tests which were made in Peru on a 70-mile transmission line located at an average altitude of 13,300 feet. The voltages range from 40,000 to about 72,000. The method employed in the tests is described and the calculations are given. The corona losses were found by subtracting the core and copper losses of the transformers and the line copper losses due to the line charging currents from the total loss measured. Core-loss tests were made on the transformers in order to check these against the shop tests. The highest voltage experimented with was 72,300 when a corona loss of 153 kw. was found, or a little over two kw. per mile of line. The results are compared to ``Peek's law'' and a wide divergence found, amounting to over 800 per cent at about 66,000 volts and a loss of 58 kw. Close correspondence is found between these tests and those made by Faccioli on the Shoshone-Leadville line, and the equation of the loss-voltage curve obtained is the same as was established by Faccioli, and shows this relation to be logarithmic and not quadratic for the voltage range investigated. It is shown that the results are in good agreement with the formulas deduced by Professor Ryan from extensive laboratory tests. Finally test data are given for a test made during rainy weather and this shows no appreciable difference from tests made during fair weather.

The present communication aims at presenting, in a necessarily curtailed and somewhat preliminary form, the results of an enquiry into the comparative efficiency of the wood as a water-conducting tissue in about sixty species of plants. These consisted for the most part of trees and shrubs, but a few herbaceous forms were studied as well. The investigation was under­ taken in the first instance in order to find out whether the efficiency (regarded from the standpoint of its water-conductivity) of the wood could be usefully expressed for a given species in a quantitative form, and if so, what kind of deviation from the average or mean was to be expected, and to what extent separate species might differ among themselves in this respect; secondly, whether the mean conductivity can be correlated with any obvious character such as deciduous or evergreen habit; thirdly, to ascertain if possible whether definite changes of external conditions can evoke corre­sponding change in the water-conductivity. It is a remarkable circumstance that, although the absorption of water by the roots and its elimination during transpiration from the leaves and other green surfaces are processes which have served as the starting-points of a vast number of investigations, the behaviour of the wood as the intervening conducting channel has been almost entirely neglected. It is true that the path actually followed by the water has long been known, and that attempts to discover an adequate physical explanation of the ascent had been repeatedly made for many years before the fine researches of Dixon and Joly showed clearly where the solution of this problem was to be sought. But the limitation which the structure of the wood may impose on the volume of water transmitted appears to have attracted no attention at all. And yet the life and habits of a plant are so closely linked up with the problem of water supply that every factor that can influence the complex adjustment of supply of and demand for water deserves serious consideration. It cannot be devoid of interest for those concerned in arboriculture and forestry, for not only may it determine whether a particular species can flourish under given natural conditions, but it also concerns the inter-relations between those tissues in the wood which materially determine the nature and commercial value of timber.

INTRODUCTORY NOTE. The author whose name appears at the head of this paper was a graduate student at Harvard when the United States entered the war. He went to an officers' training camp at the earliest opportunity, received a commission, and has been in France since the fall of 1917. Although his mathematical work was thereby interrupted before he was ready to regard even a first undertaking as definitely accomplished, some of the results that he had obtained are so striking, and, in their field, important, that I have asked and obtained his permission to publish them without further delay. Even of the work that he had done, only a part is contained in the manuscript that I now have, the rest having been lost in the confusion of his departure. The part that is preserved, forming the substance of the present paper, deals with a particular differential system, the place of which in a more general theory is indicated below. Mr. Hopkins had treated other particular systems in a similar way, and, when the work was suspended, was trying to raise the discussion a step higher in the scale of generality; he met with considerable difficulty in this attempt, and it is to be hoped that he will have an opportunity of coming back to the problem. As the manuscript that I have is only a first rough draft, written before the declaration of war, with no thought of immediate publication, it would have required a considerable amount of editing, in any event. In view of this, I have taken the liberty of rewriting it altogether, in quite a different form,t which seems to me more elegant for the problem in hand, though less adapted to the sort of generalization that Mr. Hopkins had ultimately in view. The exposition, then, is the work of the undersigned, who must in particular assume responsibility for any imperfections that may be found in it; the substance, however, is essen-tially the work of Mr. Hopkins, and is rightfully published under his name. D-UNHAM JACKSON. August 1, 1918.

The following method of calculating the stresses in the spars of an aeroplane wing is essentially a simplification in form of the method given in the paper “Some Contributions to the Theory of Engineering Structures, with special reference to the Problem of the Aeroplane,” by Messrs. H. Booth and H. Bolas, issued by the Air Department of the Admiralty in April, 1915.

The auxanometers usually employed for recording longitudinal growth allow a magnification of only 10 or 20 times. The growth of a plant is, however, so slow that it requires several hours to detect the normal rate of growth and its variation under any changed condition. It is, moreover, impossible to maintain the external conditions absolutely constant throughout the experiment; even if this were possible, there would be some autonomous variation of the rate of growth during such lengthy periods. Hence, there must always be some uncertainty in the results obtained by a method which requires long time for observation. The elements of uncertainty can only be eliminated by reducing the period of the experiment to a few minutes, but that would necessitate devising a method of very high magnification and automatic record of the magnified rate of growth. I attempted to solve this problem by the employment of the optic lever, where an axis carrying a mirror underwent rotation proportional to the growth-elongation. The reflected spot of light magnified the movement of growth from 1000 to 10,000 times. The vertical movement of the spot of light was converted into a horizontal movement by means of a mirror suitably inclined. The excursion of the spot of light was followed by means of a pen on a drum revolving at a known rate; or the record was obtained automatically by photography. Hence a curve was obtained whose ordinate gave growth-movement, and abcissa time.

WITH reference to the paragraph on the method of maceration by tryptic digestion in NATURE of March 6, p. 9, it may be of interest to your readers to learn that further work on the process has shown that equally good results are obtained by the use of Messrs. Allen and Hanbury's Liquor Trypsini Co. This costs only 3s. 9d. a bottle, and the requisite strength is obtained by adding 1 c.c. to a litre of water. The procedure is in other respects identical with that previously described. The trouble of dissolving the powder is thus avoided, and the cost is reduced from is. per litre to rather less than ½d., so that the method becomes practicable for use on a large scale.

Synopsis.---This paper outlines a symmetric form of exposition of the principles of thermodynamics made possible through recognizing on a parity from the outset the two kinds of conservation naturally called after Black and Carnot, each of which is physically valid under proper conditions, and which lead to the energy and entropy scales of measurement of heat; thus exhibiting both of these, together with the thermodynamic scale of temperature, as calorimetric concepts. In the two parts of the paper are given respectively an experimental and a deductive treatment.In part I. the experimental meaning of conservation is considered in terms of certain conditions of consistency; first in connection with single-temperature transfers of heat, where those conditions underlie the uniqueness of meaning of an arbitrary scale of heat for various temperatures; second in connection with passages of heat between bodies at different temperatures, which reveal the dual nature of a quantity of heat, here illustrated by some analogies.In part II. are investigated the deductive consequences of Carnot's hypothesis of the universal ratio of efficiency. With only the single-temperature conservation of heat assumed to be known experimentally (probably the mildest basis on which this principle is intelligible), it is shown that there follows practically the entire theory of reversible thermodynamics, in particular the existence of thermal energy and entropy and the equivalence of thermal energy and work.

A readable summary of existing formal concomitant theory can be given, perhaps, by referring to the scope and general trend of my former papers in this field.t The first of these gave the results of an attempt to correlate the modular and the algebraic invariant principles by emphasizing the process of transvection between a given form and the universal covariants of the modular group of binary transformations G(p2-) (psL1). The chief difficulty encountered was the apparent one that this process is not alone definitive for the construction of complete systems. It is also not the simplest algorism for the generation of modular concomitants. A second paper was an improvement from the latter point of view in that polar and transvectant operations characteristic of the modular theory were developed, but lack of definitiveness remained and yet remains, as each complete system which I have developed contains forms which were constructed by purely empirical methods. In subsequent papers I emphasized what seems to me at present to be the most favorable method upon which to ground a general theory by which it results that a modular covariant is a member of a finite scale of derived modular concomitants constructed by processes theoretically analogous to the important algebraical method of successive convolution.

If we consider a pencil of rays issuing from a point on the axis of a symmetrical optical system ( i. e ., a system of refracting spherical surfaces, the centres of which lie on a straight line called the axis of the system), it is well known that, if the pencil be a thin one, of which the mean ray is along the axis, the first approximation to the emergent pencil is another punctual pencil, of which the rays pass through an image point, also situated on the axis. The general method of treatment of such image points, which are usually referred to as “geometrical” images, is due to Gauss, and is developed in any text book of Geometrical Optics. When, however, the pencil considered is one of finite aperture, the outlying rays do not, after emergence, pass through the Gaussian image point, nor do they have the inclination assigned to them by the Gaussian calculation. The emergent rays lying in any one axial plane touch an envelope or caustic, which has one cusp at the Gaussian image, with the axis as proper tangent. The intercepts of any given emergent ray upon the axis and the image plane, measured from the Gaussian image, are known as the longitudinal and transverse spherical aberrations of that ray.

The paper indicates how a whitened cube can be made to give accurate results when used in the determination of the average candle-powers of lamps having different polar distributions. As a preliminary, the relative effects of different portions of the interior of the cube in producing illumination on the window are determined, and the results used to calculate the illuminations produced by different types of light distribution. The ratios of these illuminations to that furnished by a uniform point source of unit intensity give apparent average candle-powers in the cube. It is then shown by experiment that candle-powers obtained by the usual methods of integrating photometry agree with these calculated values. Thus it is possible to determine by how much these apparent candle-powers differ from true candlepowers deduced by point-to-point methods, and hence to obtain true candle-powers from experimentally determined apparent values. In the special case when a comparison is made between lamps having identical polar distributions, it is shown that the usual method gives accurate results.

In his recent book A History of the Conceptions of Limits and Fluxions in Great Britain from Neuton to Woodhouse (Chicago and London: The Open Court Publishing Company, 1919), as well as in a series of articles in the American Mathematical Monthly for 1915 on The History of Zeno's Arguments on Motion, Mr Cajori discusses certain aspects of the conception of a limit, and treats in considerable detail the controversy between Jurin and Robins that arose out of the publication of Berkeley's Analyst. I gave an account of the controversy in a paper that appears in Volume XVII. of our Proceedings, and as Mr Cajori's estimate of the respective merits of the contributions by Jurin and Robins differs greatly from mine, and as the conception of a limit is fundamental in modern mathematics I venture to draw the attention of the Society to the matter.

In this paper the authors discuss the possibilities, for work of high precision, of the critical angle method of measuring the refractive index of a prism. For general work the method is inferior in convenience and accuracy to the method of minimum deviation, but by a suitable arrangement of the apparatus it can be used to determine refractive indices near 1.53 without the use of a spectrometer, and serves as a useful check on more general methods which depend on the accuracy of a spectrometer circle. In the second part of the paper the errors of setting in critical angle refractometry are discussed, and a simple method of determining the necessary correction is described.

General Reluirements are stated for mat, bench walls, checker work, regenerator korbels, main flues, division walls, oven chambers and jamb blocks. The following Tests are described:—inspection, reheating tests on three qualities, bending test. Requirements and specifications are discussed. The latter part ofthe paper deals with Apparent density and Permanent expansion. It is shown that the relation between them is a linear one and a chart is given. The average of thirty-nine determinations gives an apparent density, for Western brick, 0.016 higher than for Pennsylvania brick, due to the fact that the Western ganister is slower to transform than the Eastern.