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Since the Iast papers the research has been continued at intervals through the nine years. Lome numerical errors on pp. 325 and 327 in the last paper have been detested by the author, but nothing which in the least decree alters the main results of that paper, viz., that the number of revolutions per second which a disc having a white sector w , and a black sector 360° — w , must make in order that the flicker may just vanish, under an illumination I, is given by an equation of the form n = a+b log w (360- w ), a being a constant, and b of the form c+d log I, where c and d are constants. If the disc is considered to be under unit illumination when lighted by a standard stearins candle, burning 8·273 grm. of stearins per hour, at a distance of 4 metres; that is, if I is taken to be 1 under these conditions, then, from the combination of the series of observations described in papers I and II, with man others made since, it follows that the numerical value of n is determined with a very fair decree of accuracy to within at most threes revolutions per second, and generally within one, from the equations n = 10·7+12·57 log I, for discs half white, half black, and n = -27·83+(8·57+20·79 log I) log w (360 - w ), when tbs angle of the white sector is w °.

The argument of this paper map be regarded as centred round the ascertainment of two linear differential equations, those denoted by (P′ - p ) η -1 Ψ = 0, (Q′ — q ) η -1 Ψ = 0; these are, in fact, differential equations satisfied by a quotient of theta functions of several arguments when all of these arguments except one are taken constant, and the theory of the paper is in close connection with the classical method for approaching the definition of the theta function from the algebraical side. It would seem to be important to justify this assertion in precise terms, so as to make clear the general bearings of the results arrived at by the authors. The elementary algebraic method which they follow has in my opinion great interest, especially as it enables them to deal in part with the theory even when the curve has unknown multiple points; but one is not satisfied until the functions involved are defined by their behaviour and not by their algebraical form. I hope, therefore, that the writers will allow me to make the following remarks; and that I may be pardoned for explicit references to my own volume, ‘Abel’s Theorem,’ 1897. This was witten before the appearance of Weierstrass’ lectures (‘Werke,’ vol. 4, 1902; 600 quarto pages), and is very imperfect; but it is briefer in one respect than Weierstrass’ theory, by the use of the fundamental integral functions (Kronecker, Dedekind and Weber, Hensel). What there is of novelty in the following remarks relates mainly to these. The general ideas are expounded also in Clebsch and Gordan’s ‘Abelsche Functionen’ (1866)—to which, as to Weierstrass’ volumes, explicit references are given below. I consider in the first place a curve of which the multiple points are known and allowed for; but exemplify in examples how the theory can be applied when this is not so, the existence of a new multiple point being regarded as a particular case.

Brief mention is made of early work in the design of experiments by Galen, Avicenna, and Laplace, after which a seemingly unknown 1815 paper of Gergonne's on polynomial regression is introduced and discussed, and a translation of the paper presented.

The advantages of using a newly available square root graph paper for nuclear spectra are illustrated by means of a sample spectrum plotted on linear, logarithmic, and square root scales.

Both paper and author (D. R. McNeil, (1968)) will be referred to below as DRM. The said paper deals with the following situation: an intersection is controlled by a traffic light with a fixed cycle time, T; the possibility of other delays, e.g., due to turning vehicles, is ignored; arrivals at the light form a compound Poisson process; if vehicles arrive to find the light green and the queue empty they are not delayed, while in the contrary case they depart when they reach the head of the queue, providing the light is green, each vehicle taking a constant time to move off. The length of the effective red period is R. For further details and discussion, DRM may be consulted.

In a paper which appeared a few years ago the authors investigated the exponential representation of functions analytic in the upper half-plane with positive imaginary part there [1]. We refer to that paper in the sequel as A-D. One of the principal results of A-D, there called Theorem A, can be extended to a considerably more general result, the proof of which is perhaps simpler than that given in A-D. We give the extended version of Theorem A here. We will use the notations and results of A-D without further explanations. Before we present the extension and its proof we would like to add some information that by oversight was omitted from the list of fundamental properties of the functions in the class P given in Section 1 of A-D. In such a comprehensive review one should mention that the classical theorem on representation of a positive harmonic function in a circle by a Poisson-Stieltjes integral is due to G. Herglotz [2]. The following results of L. H. Loomis [3] were not given: XVII. For all ~ for which /t[~] = O, the limits lira Im[~b(~ + iq)] ~l~ O and lira hOg(A) exist and are f ini te simultaneously and are equal. Their ~ 0 common value is the symmetric derivative of #(A) at 2 = ~ multiplied by ~r. XVIII. I f for two values of 0 in the interval 0 < 0 < ~

I regret to announce that there is a serious error in my paper in these Transactions, volume 84, pp. 430-443. The error was discovered by Louis Kokoris who found that on line 8 of page 434 the expression given as 4[g(bz)](az) should have been 4[g(az)](bz). As a consequence the computation of P(z, g, az, b) yields nothing, the proof of formula (30) is not valid, and the important Lemma 9 is not proved. Thus the paper does not give a proof of its major result stated as Theorem 1. Nevertheless, the theorems of the paper are all correct and we shall provide a revision of the proof here. This revised proof has been checked by Louis Kokoris to whom the author wishes to express his great thanks. We observe first that the equation

Measuring principle of Bekk's smoothness is analysed and it is clarified that there exists in measuring Bekk's smoothness excessive leakage of air which depends on permeability of paper and the roughness of paper of the opposite side of the tested surface. And so the excessive leakage is measured experimentally to be found out that it can not neglected for determining surface roughness of paper by Bekk's method.A direct reading instrument of smoothness of paper are designed applying a similar principle of air micrometer. Theoretical and experimental analyses show that a flow of air through spaces between a test surface and a flatly finished surface obeys Hagen-Poiseuille's law. From this respect the relation between a value of Bekk's smoothness and pressure drop of the new instrument can be given theoretically.Experimental researches show that the observations of smoothness of paper, which is obtained by the new instrument, coincide with those of Bekk's instrument when they are corrected by estimating the excessive leakage.

Three cases and sources of inaccuracy in finding the value of E in the former paper (Proc. Roy. Soc. vol. xv. p. 429) having been pointed out by Mr. Glaisher, and some other minor errors not noticed by him having since been detected by the author, and these having vitiated the results, but only in a slight degree the inferences drawn from them (for in the former paper the last and leading conclusion as to the value of E, though limited , was certainly correct ), the author has been led, from a deep sense of obligation to the Royal Society, to revise, correct, and extend what he had previously done. And it will be seen, from comparing Mr. Glaisher’s remarks and results with what follows in this paper, that the supplementary matter herein given, including the extension of E &c. to 110 places of decimals, can scarcely be without interest to mathematicians, and especially as regards the summation of the harmonic series in the formula for finding the value of E. Not having seen M. Oettinger’s article in Crelle’s ‘Journal,’ “On Computing the value of E,” the author is unable to state what artifices he used in summing the harmonic series. Mr. Glaisher gives a very simple and obvious one from M. Oettinger, which the author could not but see and employ for calculating the values of the reciprocals of the even numbers.

Hysteresis in the equibrium moisture content and differences in the paper strength caused hereby, were measured and at the same time the error variances were calculated.The results were summarized as follows ;(1) The air circulated over the sample for 6 hours at the rate of not less than 1.8m/sec.. it is sufficient to conditioning except highly moisture-resistant paper such as hard-sized paper and paperboard.(2) The greatest difference between the equiribrium moisture contents was observed in groundwood as much as 2 %.(3) Consequently, the tensile strength differs not less than 10% in newsprint.(4) The differences in strength as well as in equiribrium moisture content were reduced according to the degree of pulping process.(5) Almost all papers were signified the variance in strength at 5 % level except the folding endurance.(6) The coefficient of variation was grown up in proportion to the difference in strength caused by hysteresis.(7) Increased the measuring area of a test piece decided by tester type (such as folding endurance, tearing strength, bursting strength and tensile strength.) decreased the coefficient of variation.

The proof of a theorem in the above-mentioned paper (Smith, W. L. 1955 Proc. Roy. Soc . A, 232, 6) assumes more than it should and a new proof is given. For certain (unusual) kinds of Semi-Markor process the definition given in the aforementioned paper is inadequate to define the process for all values of the time parameter, and this weakness is removed. Furthermore, a ‘regularity’ condition for an S. M.- process to have step-function realizations (almost certainly) was given incorrectly; this error is now corrected, and certain other ‘regularity’ conditions are discussed.

For Many Years teachers have been composing examinations in algebra and in geometry and pupils have been writing the answers to the questions. For many years also the teachers have been permitting the pupils to look over the corrected papers and have been explaining to the class as a whole the errors made by the individuals, while the pupils have been trying to find out why the mark recorded in the upper left hand corner in red or in blue is not more than it happens to be. The teachers have religiously entered the marks in their rating books, in order to consult them most conscientiously in determining the numerical evaluation of the pupils' achievement at the end of the set marking periods in accordance with the regulations of the schools. Individual teachers have at times, no doubt, introduced innovations into their school routine and have tried to learn as much from the examinations as possible; but, in general, from the point of view both of the teacher and of the pupil the examinations have been useless gestures, i.e., as far as the use of the written papers is concerned as an integral part of the classroom work.

The method of plotting observations on probability paper becomes clear when the curve on probability paper is regarded as an integral of the histogram. The observations are divided into intervals, which may often conveniently be the natural interval of measurement, and the total frequency of observations up to and including the nth interval is plotted against the upper end of the nth interval. It is shown that in drawing a straight line to fit the observations, less weight should be attributed to deviations from the line towards its ends than in the middle regions.

The Article in the March number of the Mathematics Teacher contained a summary of the correct and the incorrect answers toquestions on tests in algebra and of typical errors found in the papers written by the pupils. This paper contains similar tabulations for geometry.

his test shows how the rigidity is worked out by applying those figures to the formulas which are calculated with Clark's Rigidity-tester, when several sorts of Washi are left in the atmosphere of the different temperatures and humidities and then balanced.The rigidity of Washi is less influenced by the temperature within the limits around the ordinary temperature ; however, it is influenced by the change of the atmospheric relative humidity.In general, the bigger becomes the humidity, the smaller becomes the rigidity.But, this change can hardly be recognized in case of machine-made foreign-style paper, while in cas of Nashio-paper, for instance, mixed with clay, humidity of the mixtures is considered to be greatly influential on.The phase of paper's textile-arrangements generally enables us to imagine that the vertical course differs from the horizontal course in rigidity ; so it is, in reality ; the horizontal rigidity is smaller than the vertical one.

IN TUNING the tones of a scale the ear wrestles with some fascinating prob lems concerning vibration ratios. This paper describes a convenient method for displaying to the eye the mathematical relationships involved.1 The proportions embodied in musical intervals and the essential features of three tuning systems?Pythagorean, Just (or Pure), and Equal Temperament?can be demonstrated by a succession of paper folding operations. Musicians may be de lighted?and mathematicians amused?to discover that even such fabulous numbers as the "Pythagorean comma" of ratio 524288/531441 or "the twelth root of 2" 0V2)? generally simplified to 1.059463+ ?will exhibit their meaning with as much accuracy as anyone deserves?that is, with as much as he can hear?after a rela tively few folds. The procedure outlined below approaches the elegance of geom etry by relying upon proportional division rather than on mathematical computation. The folding procedure is a variant of the old monochord demonstration in which frequency is inversely proportional to string length; a strip of paper?calculating machine tape proves convenient?is sub stituted for the vibrating string. If a

The results obtained in Section B may be summarized as follows: 1. (a) The brightness contrast in the object chosen as a “typical landscape” (Fig. 5) was found to be 39, the highest light having a brightness of 2500 millilamberts and the lowest 64 millilamberts. 2. (b) A panchromatic material was used in making the required negatives in order to minimize the distortion of the visual tone relations due to color effects. By developing a series of sensitometric strips the time of development required to give any desired gamma was determined. The curves obtained show that the exposure latitude of this material is 35, whjch is approximately sufficient for the rendition of the entire object on the straight line portion of the characteristic curve. 3. (c) A group of nine negatives was made, each having identical exposures, the development time being varied to give gamma values ranging from .38 to 1.32. The densitometric characteristics (D max., D min., and γ) of these negatives were determined by density measurements made directly thereon and by density measurements made on sensitometric strips developed with each negative. 4. (d) Using the negatives thus obtained groups of prints were made from the appropriate negatives on thirty-seven different developing-out papers selected to cover the entire range of contrast available in materials of this type. From the prints made on each developing-out paper judgement was made by seventy-five observers on the negative producing the optimal photographic quality. The data thus obtained have been examined critically, and from it certain conclusions have been drawn relating to the characteristics of the observers, the repeatability of judgments of print quality, and the average photographic quality considered as optimal by observers differing radically in their training for the judgment of print quality. li ](e) A summary of the statistical data thus obtained is given, from which is obtained the gamma of the optimal negative for each of the developing-out papers used. The probable error to be remarkably small and indicates that these values may be relied upon as truly indicative of the negative quality which will give the best possible print of the particular test-object used in this work on each of the positive materials. 5. (f) From density measurements made directly on the positives, values of maximum and minimum density for the optimal positive are derived, the difference between these giving the density scale for the optimal positives. These values relative to the optimal positives, together with the densitometric measurement relative to the optimal negatives, constitute the data from which, in connection with the sensitometric data to be obtained in the following section, values of contrast for the positive materials are computed.

The Barkas analysis is applied to a series of nine papers of known visual gloss. With some reservations, the analysis is found to hold approximately and the distribution of mirror facets so derived is found to accord well with the surface structure of the papers as found by microscopic examination. It is concluded that, whilst the analysis is of negligible value for practical gloss determinations, it appears to present a useful method of investigating the surface structure and "finish" of paper and similar delicate materials. The gloss of the papers examined appears to be assessed solely by the intensity of the light reflected at the specular angle and not by the contrast noted by examination at different angles nor by the ratio of specularly-reflected to diffusely-reflected light. Some suggestions for future research are given.

Introduction. In a recent paper [2] R. H. Bruck has introduced the concept of right neoring and discussed some properties of these systems. In particular, he has considered analogues of certain properties of the ring of integers. This paper is essentially a commentary on Bruck's paper and we generalize some of his results as follows. The construction of the universal right neoring in [2] is applied to the free monogenic $3-loop in any subvariety $3 of the variety of loops and a complete analogue of Theorem 4.1 of [2] is obtained for any one of these subvarieties. Then, using a result similar to those obtained in [5], it is shown that this construction yields uncountably many right neorings with an identity which generates the additive loop of the right neoring. Conversely, every right neoring with an identity which generates its additive loop can be obtained from a free monogenic $3-loop by the above construction. Each of these right neorings has some properties resembling those of the ring of integers. One possible answer is given to the question raised by Bruck concerning the existence of universal right neorings with free additive loop of arbitrary rank. A brief proof is given, using the results of [4; 5], of the cancellation properties of the monogenic universal right neoring. Finally, we discuss briefly the relationship between right neorings and the logarithmetics of Etherington.

It has been proved by Pepis that any formula of the first-order predicate calculus is equivalent (in respect of being satisfiable) to another with a prefix of the formcontaining a single existential quantifier. In this paper, we shall improve this theorem in the like manner as the Ackermann and the Gödel reduction theorems have been improved in the preceding papers of the same main title. More explicitly, we shall prove theTheorem 1. To any given first-order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.An analogous theorem, but producing a prefix of the formhas been proved in the meantime by Surányi; some modifications in the proof, suggested by Kalmár, led to the above form.

In a recent paper Papadopoulos [1]1 claims to have obtained solutions to the problem of sound propagation from an impulsive point source in a semi-infinite elastic solid which differ in certain respects from solutions previously obtained by Pekeris [2], [3], and by Pekeris and Lifson [4]. The results of Papadopoulos are based on a singular integral source representation developed by him [5]. By tests based on simple special cases we show that, in a number of cases where Papadopoulos claims a result different from that of Pekeris, the results of Papadopoulos are unacceptable whereas the results of Pekeris are acceptable. Reasons for these errors in Papadopoulos's work are suggested. A number of other errors in the paper of Papadopoulos are also noted. 1 Numbers in brackets refer to references at rear.

The three experimental processes and results are described on this paper by using the paper impact strength tester.The first is an appendix report of regular impact strength test which has been described in 1954.At the second report, the four creasing style sample of paper were measured by impact strength tester.There were good correlation between drop number of bags and impact strength about creasing parts of bags.The third report is on the mechanical properties of paper during the repeated impact loading.The very interesting results were obtained about the repeated impact strength and various moisture content of paper

In a previous paper the hypothesis propounded by Kapteyn that the stars are moving in two streams was examined by considering the proper motions of stars which were moving more than 20″ a century. By a graphical method the directions of these two streams were found with results in fair agreement with those found by Kapteyn and Eddington by other methods and with different material. It was evident from that paper that it would be possible in a large percentage of cases to say with tolerable certainty to which stream individual stars of large proper motion belonged.

In the technique of Paper-electrophoresis, the size of the serum sample is an important factor, because a linear relation between protein concentration and densitometer reading depends upon it considerably.While the original technique of applying the serum to an electrophoresis paper by a micropipet requires a skill acquired only by repeated practice, the use of an applicator strip which was used in this experiment, makes it easy to apply a constant amount of the serum sample. Furthermore, the above mentioned technique presents a well-separated pattern of serum protein, the separation of α1- and α2-globulin fractions being especially distinct.The standard errors of the fractions are smaller compared with those by the micropipet method.

Experiments were conducted over three years to determine the effect on yield of storing seed potatoes in paper sacks, with or without the use of the sprout-suppressant tetrachloronitrobenzene (TCNB). Treatment with TCNB increased the proportion of small tubers and decreased the proportion of large ones, even when the crop was grown at wide spacing. There was insufficient evidence to confirm any advantage which paper sacks may have over jute sacks as containers for TCNB-treated potatoes. In comparison with storage in jute sacks under the same conditions, the storage of untreated potatoes in plain paper sacks was found to have no detrimental effect on either total yield or the yield of large tubers.

Wavy-edged paper sheets will result in “fanning in” at the back edges as they pass through the squeeze impression of the offset press, which makes a dominant source of misregister and wrinkling and is closely related to the handling of paper. It has been tried in this investigation to express the degree of waviness in a quantitative manner by defining the wavy edge number as the product of the average height of waves and their number existing in a unit length of paper edge. The dependence of the wavy edge number upon the difference of relative humidity between paper and atmosphere determined on paper samples of various degree of hygro-expansivity has indicated that the safety limit of deviation of atmospheric humidity over paper humidity, within which only practically tolerable degree of waviness is found, can be greatly affected by the quality of paper.

The breaking strength of paper yarn can be calculated by Formula (64) of Report No. 30, Formula (75) of Report No. 31 and Formula (78) of Report No. 32. Therefore, the following methods can be adopted in the case of the chemical treatment.1. The method to increase P0 : (Breaking Strength of Raw Paper) by : a. treating raw fibers b. treating raw papers c. treating tapes2. The method to increace μ (Friction-Coefficient) by : d. treating fibers e. treating tapes f. plastering fibers together g. plastering tapes together h. plastering yarns together all of which can be re-arranged as follows : 1) The method to treat raw fibers2) The method to treat raw papers3) The method to give the cut materical the treatment liquid instead of water, to twist and to heat the twisted yarn in order to attain the treatment effect4) The method to give the cut material the treatment liquid, instead of water, to strengthen in the tape stage, and to give water and to twist5) The method to treat yarns.By the comparative study on the above 5 methods, 3) method was found to have the best treatment effect.

Two new Weibull probability plotting papers are presented. Their use is explained, and they are compared with other Weibull papers...

The paper describes a theoretical and experimental investigation into unsteady flows in simple branch systems. In order to simplify the analysis one-dimensional unsteady flow theory is used to analyse the wave action in the pipes. The method of characteristics is used for this purpose, and both homentropic and non-homentropic flows are investigated. The boundary conditions at the branch are represented by quasi-steady flow theories. Two such theories are examined, the first called the constant pressure theory assumed no pressure change across the branch at the junction. A full graphical treatment is given elsewhere (4). In the paper the assumptions and the formulation of the equations for an electronic computer are briefly discussed. The second method assumes a pressure drop across the branch at the junction. An approximate theoretical method of estimating the pressure drop with certain types of flow is given. Using experimentally determined steady flow loss coefficients at the junction the expressions used in the second method are formulated for direct computation using a digital computer. The results are given of an extensive experimental investigation to test the validity and limits of the theories described above. Steady flow experiments to evaluate the loss coefficients at the junction over the range of pressure and flow directions covered by the unsteady tests are described. The unsteady flow tests on a special multi-cylinder pulse generator are discussed in detail. The results show that under certain conditions the calculated pressure diagrams using constant pressure theory do not agree with the experimental diagrams, both the pulse shape and amplitude being incorrect. Using the pressure drop theory together with experimentally determined steady flow loss coefficients there is considerable improvement in the theoretical diagrams. Agreement is obtained over a wide range of flows.

VERBAL arithmetic problems to be solved without the use of pencil and paper have practically disappeared from the present-day curriculum. However, the need for practice in performing calculations without pencil and paper is much in evidence. Many pupils, as well as adults, are completely dependent on pencil and paper in solving arithmetic problems arising in their daily lives. The purpose of the investigation reported in this article was to ascertain the ability of a group of sixth-grade children to solve certain verbal arithmetic problems without using pencil and paper.

Let A be an m × n matrix of complex numbers. Let Aτ and A* denote the transpose and conjugate transpose, respectively, of A. We say A is diagonal if A contains only zeros in all positions (i, j) with i ≠ j. In a recently published paper [4], M.H. Pearl established the following fact: There exist real orthogonal matrices O1 and O2 (O1 m-square, O2 n-square) such that O1AO2 is diagonal, if and only if both AA* and A*A are real. It is the purpose of this paper to show that a theorem substantially stronger than this result of Pearl's is included in the real case of a theorem of N.A. Wiegmann [2]. (For other papers related to Wiegmann's, see [l; 3].)

Mitsumata trees may be cut down any day in autumn when they have shed their leaves. The bark of the trees which are cut down in autumn or March of the following year is called fuyu-kawa or the winter bark, while that of the trees are cut down since April is called haru-kawa, or the spring bark. According to tradition, as a matter of course, the winter bark makes better paper, but the spring bark gives higher retention. The raw mitsumata is steamed, and the bark is stripped and dried. This dried bark is called kuro-kawa, or the black bark. In order to make highh-grade paper, kuro-kawa must be turned into shiro-kawa, or the white bark. According to tradition, paper after boiling the black bark with alkaline solution gives higher retention than that of the white bark. For the purpose of investigating these tradition, analyses were made at the various stages of the process of mitsumata paper-making. Mitsumata-Edgeworthia Papirifera Sieb..-, grown in Kochi-ken. Shizuoka race, was employed in the present study. The results of experiments are shown in Table, 1, 2, 3, 4 and may be summarized as follows : (1) The spring bark gives higher retention than the winter bark at the stage of the stripping off the black bark, while the winter bark gives highe retention than the spring bark at the process of paper-making.(2) It is not a fact that the black bark treated by the soda process gives high er retention than that of the white bark at the process of paper-paking.(3) The quality of mitsumata-paper ;1) Strength and stiffness of boilling the white bark is more than that of the black bark, so the winter bark is more than the spring bark.2) Opacity is the most in the boiling of the black bark of the spring bark.3) Brightness or lustre is the most in the boiling of the winter bark.

1) As regards viscoelastic properties of paper of z-direction (perpendicular to paper surface), it was showed that its properties were able to same treated as other two directions.In compressed case, reciprocal curse for tensile curse was obtained. Apparent Young's modulus of Z-direction was known 109 dyne/cm2 as order, less than x, y direction and from this fact simple consideration about the calendering was made.2) About the smoothness of paper, assuming the strain due to the press N^hich was added to paper surface is proportional to smoothness, straight line relation of compressed stressismoothness was explained, and adaptable region of this straight line relation was derived from the s aloe of apparent Young's modulus and second Young's modulus, fits experimental data considerably.

In spite of significant increases in yield of potatoes from the use of mulch paper, its use on a commercial scale is not likely because of the cost of the paper. The paper used in this experiment cost at retail $7 a roll of 300 linear yards or at the rate of over $100 an acre. Even though it could be bought in large quantity at half this cost, the difference between this and the value of the increased crop at 40 cents a bushel would still be in favor of cultivation. The average cost per acre of cultivating potatoes on 100 western New York farms in 1931 was $6.75. Under present methods of applying mulch paper, the saving in cost of cultivation would be far more than outweighed by the cost of applying the paper.

It is shown that the rings under the title admit an order-theoretical characterization as in the commutative case studied by Abian. Introduction. Let R be an associative ring equipped with the binary relation (^) defined by xay if and only if xy = x2 in R. In this paper, it is shown that ( ^ ) is an order relation on R if and only if, R has no nilpotent elements i9*0). Conditions on the binary relation (g) in order that R split into a direct product of division rings are then studied in the light of Abian's result (l, Theorem l). Using similar argumentation and using certain subdirect representation of rings with no nilpotent elements, one obtains a complete similarity with the commutative case (yet, no extra complication in the computa- tions). Conventions. R is an associative ring which is, unless otherwise stated, with no nilpotent elements (other than 0). As a result of (2), R can be embedded into a direct product of skewdomains, R—* YLiei £i (that is to say, rings R, having no one-sided divisors of zero). The former embedding is fixed throughout the paper. It is therefore legiti- mate to identify any element x in R with the family consisting of all its projections (xj.e/. Finally, all definitions in (l) are extended (verbatim) to the present case (of a noncommutative ring R) and are freely used throughout.

Normal probability paper is so designed that the cumulative distribution function of a normally distributed chance variable appears as a straight line. It is a common practice to plot the observations of a sample on this paper to obtain a graphical check for normality or to obtain a graphical estimate of the mean and variance of the population. Textbooks, however, are not very specific about methods for plotting, for, although the ordered observations are plotted along the abscissa, some uncertainties about the corresponding ordinates are left unresolved. The purpose of this paper is to indicate, with a special example, that any graphical technique should depend to a large extent on the purpose for which the graph is drawn. In particular, it presents tables covering sample sizes up to 10, for selecting the ordinates on normal probability paper so as to obtain “optimum” graphical estimates of the mean ζ and the standard deviation σ of a normal distribution. The somewhat more complicated problem of...

The measurement methods of paper by the porosity testers of six kinds, Gurley densometer, Emiel Greiner porositymeter, Schopper air permeability meter, Emanueli porositymeter, Oken type denso-aspero meter and smoother, were examined, and further the relationships between these testers were theoretically and experimentally obtained.(1) In the relation between the Gurley densometer (G) and the Emiel Greiner porositymeter (E), the theoretical formulas agreed well with the experimental values. But, in the range of low air resistance, the experimental values by the Emiel Greiner porositymeter were smaller than the theoretical ones, whereas in the range of high air resistance, that relation was reverse.The relationship between two testers was shown by the following theoretical formulas.Ve=83046.6/0.803tG+513.34 … (1) VE= (979+0.675Ve) /1033 Ve … (2)were Ve and VE in cc/5mintG in sec/100cc (2) In the relation between the Schopper air permeability meter (S) and the Gurley densometer or the Emiel Greiner porositymeter, the theoretical formulas agreed very well with the experimental values.Especially, the theoretical formula of relation between the Schopper and the Gurley was in better agreement with the experimental values than that relation between the Emiel Greiner and the Gurley.The relationship between the Schopper and the Gurley was shown by the following theoretical formula.tG=24096.38/Vs … (3)where tG in sec/100ccVs in cc/5in(3) In the relation between the Emanueli porositymeter (M) and the Gurley densometer, the theoretical formula also agreed well with the experimental values, and this formula was as followsAM=43.478×tG/d … (4) ortG=523.023×P1/P2… (5)where AM in a unit of Emanuelid is thickness in cmtG in sec/100ccP1 or P2 is a differernt pressure (4) The Oken type denso-asperometer (O) and the Smoother (SM) have close relation with usual tester as the Gurley densometer etc.These testers were suitable for a manufacturing process test of high air resistance paper because these testers have superior accuracy and especially made the rapid test possible.The relationship between the Gurley and the Smoother was shown by follows experimental formulas;in the case which standard head was used, log HSM=-1.146log tG+5.122…(6) in the case which cramp type head was used, log Hsm=-0.638 log tG+4.247… (7)The relationship between the Gurley densometer and the Oken type denso-aspero meter was shown by the following experimental formula, log T0=0.965 log tG+0.0818… (8)

Examination of recent papers on dimensions and physical quantities suggests a need for more careful definition of these quantities. This matter was dealt with in an earlier paper, but is now expanded and treated in more detail. It is maintained that if definitions were more precise there would not be the same confusion about the dimensions of certain quantities. There may still be differences of opinion on the most appropriate definition of a particular quantity, but if the definition is adequate then there is no doubt about the dimensions of the quantity as deduced from the definition. In this connection there is discussed the use of quantities defined as ratios, such as specific gravity (or relative density), and their more fundamental equivalents, such as density. Further, a logical set of definitions leads to the conclusion that the introduction of certain universal constants is unnecessary - for example, the Newtonian constant N in the equation for force, F=Nmf. Finally, there is discussed briefly the definition of the intensive quantity, temperature, in terms of the extensive quantity, energy.

The performance of the Bausch and Lomb and Sheen glossmeters is examined. Comparison of gradings of nine printing papers according to visual and instrumental means shows that, despite high correlation between them, there are nevertheless important discrepancies not attributable to sampling, instrumental or personal errors. The effect of angle of incidence and viewing on grading is investigated; best results are noted at 45°, and the discrepancies become more pronounced at greater angles. Gradings of the same papers according to the Jones, Askania, Goerz, Ostwald and Klughardt methods are examined; all are found unsatisfactory. It is concluded that, for the limited range of papers tested, the most satisfactory grading is given by measurements of the intensity of light received when angles of incidence and viewing are 45°.

The flux I 0 reflected at specular angles 45°–45° from the surface of glossy paper and the assumed value of its diffuse component I 2 defined experimentally are measured with a goniophotometer. The ratio of I 0 to I 2 is taken as a measure of gloss, and denoted by G N F according to an earlier paper of E. Fukushima. 1) For a series of machine glazed papers differing considerably in colour, the values of C N F are compared with the visual gradings of 36 Japanese observers, and a linear correlation of the logarithm of G N F with the visual gloss is found with a correlation coefficient +0.986 except for two papers. The method of determining the value of I 2 is also introduced and the voting distributions of the observers for the gloss of the papers are discussed.

Discussion of the problem of an elastic half-space with spherical cavity is continued in respect of Rayleigh waves on the plane boundary. Displacements in the initial and first group of higher order Rayleigh waves are derived by using the time-harmonic solution developed in part I of this series with attention confined to the case of time-harmonic normal stress at the cavity. These are employed to find also the response to an exponential shock at the cavity and graphs are presented showing the surface motion due to the initial Rayleigh waves. Finally, in an appendix to the paper, some comments are given on a recent paper by R. D. Gregory on the problem of the half-space with cavity.

For a range of papers of luminance factor varying from 20% upwards and of gloss varying from almost complete matt to gloss high enough to give fair mirror images the equation G = S(a + 10/L) is proposed, where G is a measure of visual gloss, S is the relative intensity of reflected light under 45° - 45° conditions of illuminating and viewing, L is the luminance factor expressed as a percentage and a is a constant which, under the conditions used by the authors, is approximately 0.20. The visual gradings of eight papers out of ten are not markedly affected by the conditions of observation, but for the remaining two there is a highly significant shift in grading according to whether these conditions permit formation of mirror images or not. Gloss is considered to be a gestalt, corresponding to no single physical property of a surface, being judged by appraisal of a physical situation taken as a whole. The validity of the method used in establishing the psychological gloss scale is checked by experiments in weight grading.

Summary. Examiners are often advised that, if one examination paper is to have twice the weight of another, it is only by the former's having twice the S.D. of the latter that the required effect will be achieved. But precisely what is the effect which is called ‘having twice the weight’? And will doubling the S.D., in fact, achieve it? The answers call for a definition of Weight. Some established definitions are examined, and it is shown that, on these, doubling the S.D. will only rarely have the effect promised. A simple definition is proposed, which appears to be in accord with examiners' intentions, and which is such that a paper's weight is directly proportional to its S.D.

Summary Damp paper towels, damp cloths and dry bare hands were compared in effectiveness of stimulating milk let-down and in cleaning the udder prior to machine milking. No significant differences were obtained among these methods in stimulating milk let-down, as measured by graphic milk flow records obtained during the milking process. Sediment ratings of the milk obtained under three different udder treatments averaged 0.19mg. for paper towels, 0.15mg. for cloths, and 0.43mg. for dry bare hands. Differences between the dry hand method and either cloth or paper were highly significant, whereas differences between the paper towels and cloths were not statistically significant. Differences in standard plate counts and coliform counts of milk obtained under the three different udder treatments were not statistically significant.

A method is described for making a projection drawing from any point of view in which the drawing is set out directly from the dimensions of the object without the use of intermediate plans and elevations. Seven key figures are given with the necessary numerical factors which give directly the proportions for forty-two points of view. By the use of square ruled paper and these figures, aided by the slide rule, figures may be drawn in which the dimensions are found by means of the rulings on the paper only, so that scales of length are not needed. An example of the use of the method is an illustration of two pentagonal dodecahedrons, one that of Euclid and the other that of a crystal of iron pyrites. The last part of the paper deals with the solution of the spherical triangles on which the method depends.

er demonstrating case, I finally invented the form of folding paper display case shown in Figure 1. This case is easily made by using a No. 160 white card paper. The design is mapped out, cut and creased, as inFig. 2.—Plan of demonstrating case. A 160-pound white bristol cardboard is used. The design is mapped out according to the dimensions indicated. The cardboard is cut on all solid lines except slits marked A, B, C and D, also 1, 1, 1 and 2, 2. 2, which are to be done later. All dotted lines are creased with some blunt instrument, the cardboard being placed over soft wood. As many apertures as are desired are punched out, 1*4 inches for each being allowed. All creased lines are bent forward except a, b, c, d, e and /, which are to be bent backward, and are folded as in illustration. The flaps A, B and C, D should overlap; slits are cut in opposite direction, as indicated, so that they will properly lock. With a sharp instrument are cut small slits 1 and 2, which will go through three thicknesses of paper that make up the back of the case, and by which the whole is fastened with ordinary paper fasteners.

Proceeding from a unifying framework, this paper reviews recent developments in lexicographic orders, preferences, utilities, probabilities and decision rules. Several new results are included.