An equation of state for a system with a single type of transformation

Based on theory of a previous paper, the writer has developed an equation of state for a system with a single type of transformation. This equation is of the form

$\text{h=A+Bv+Cp+Dpv?T(E+Fv+Gp+Hpv)}$

where h = ε + pv is the total heat, p the pressure, v the specific volume, T the temperature, and p, v, T are considered independent variables. A, B, C, etc., are constants for the system. The latent eat at constant (p, T) is given by

$\text{λp,T=(v}{}_2\text{?v}{}_1\text{)(}\text{?h}\text{?v}\text{)P,T= (v}{}_2\text{?v}{}_1\text{)[(B?TF)+p(D?TH)]}$

. These equations are checked with data on saturated and superheated ammonia, and the agreement is good to within a few tenths of a per cent. Also, checks with data on saturated and superheated steam show agreement within several per cent.     

The porous structure of paper in relation to drying and impregnation

It is shown that water adsorbed by paper cannot be completely removed by pumping while immersed in an impregnating compound which wets the paper.A theory which explains this effect is presented, according to which the equilibrium moisture content of paper pumped under these conditions does not correspond to the external pressure but to one $\text{2λ}\text{R}$ greater than this where λ is the surface tension of the impregnant and R the effective pore radius.The application of these findings and the theory to the preparation of experimental samples of impregnated paper of controlled moisture content; to the determination of moisture in impregnated paper; to the drying and impregnation of paper and to the investigation of the pore structure of solids is discussed.     

Report on the work of the Bartol Research Foundation, 1933–1934

Using the velocity analyzer of Zartman with improved technique the combined velocity spectrum of Bi atoms and Bi₂ molecules was obtained at 827°, 851°, 875°, 899°, 922°, 947° C. From the spectral distribution curves the relative abundance of Bi atoms and Bi₂ molecules in the beams at the above temperatures could be determined to 1 per cent. The vapor pressure curve of Bi was obtained experimentally by the method of effusion and the values so obtained were combined with the degree of dissociation of the vapor as computed from the beams to give the heat of dissociation. The heat of dissociation was computed from the data, assuming the pressure to be given by the temperature of the crucible T_c. In calculating the heat of dissociation, the equilibrium temperature was taken as that of the slit chamber T_s which was 24° above T_c. The results of these calculations plotted with log₁₀K_p as ordinates against $\text{1}\text{T}{}_{\mathrm{s}}$ give a straight line whose slope yields the value of the heat of dissociation as 77,100±1200 calories. The curves for the distribution of velocities observed and computed on the assumption of a given ratio of Bi atoms to Bi₂ molecules in the beam were compared in an attempt to test the law of distribution of velocities. On the high velocity side agreement in two curves was obtained within the limits of experimental accuracy. On the low velocity side important deviations were noted of such a sort that the observed curves below a velocity $\text{α}\text{2}$, (α is the most probable velocity) gave more molecules than the theory demanded. Other deviations were observed on some of the runs taken with a fourth slit in which a deficiency of molecules was observed between velocities of .75α and $\text{α}\text{2}$. This deviation was probably due to a warping of the fourth slit carriage due to heat. The nature of the variation at velocities less than $\text{α}\text{2}$ indicated the presence of molecules of greater mass than Bi₂ in the beam and at the lower temperatures a distinct peak corresponding to Bi₈ molecules was observed which were present to less than 2 per cent. The vapor pressure curve for Bi was determined by least square reduction of the observed points to be given by $\text{log}{}_{10}\text{P = ? 52.23 ×}\text{195.26}\text{T}\text{+ 8.56}$ between 1100° and 1220° abs. It lies very close to the extrapolated curve given in the International Critical Tables.     

Air flow separation at high speed

The resistance coefficient of a body moving in a fluid depends on Reynolds Number R, Mach Number M and the parameter $\text{gL}\text{U}{}^2$, which is customarily neglected in view of small weight of the air. Here L denotes a characteristic length; U denotes the body's speed of translation. The author points that dimensional deduction of this parameter does not limit it to the acceleration of gravity, and that the resistance coefficient is affected by the general acceleration to which the air is subjected. Evaluation of the acceleration of the air flowing about spheres puts this parameter in the form $\text{L}\text{R}$, where the characteristic length L is interpreted as the mean free molecular path. Large and small spheres were found to have widely different values of the pressure coefficient $\text{Δp}\text{q}$ for the same Reynolds Number or Mach Number. Here Δp denotes the difference in pressure between front stagnation point and the rear portion of the sphere, and q denotes the dynamic pressure. The plot of $\text{Δp}\text{q}$ against the parameter $\text{L}\text{R}$ removes this confusion. The low values of $\text{Δp}\text{q}$ are found to be associated with $\text{L}\text{R}$ below a certain critical value, and high values of $\text{Δp}\text{q}$ with $\text{L}\text{R}$ above the critical value, which apparently indicates the condition under which the flow separation takes place. Attention is called to the effect of air pressure on the separation as shown by the parameter $\text{L}\text{R}$, and its possible bearing on the drag in high altitude flying.     

Stabilization of carrier-frequency servomechanisms. II. Design formulae for proportional-derivative networks

In an alternating current servomechanism, the error is proportional to the modulation envelope of a modulated-carrier error signal. It is shown in part I that for stability and fidelity of the servo, it is highly desirable that the effect of the controller includes a proportional-derivative action on the modulation envelope. This action may be obtained with various forms of RC networks, including the parallel “T,” bridge “T,” and Wien Bridge forms.This part contains detailed design procedures and tables of values for the various types of proportional-derivative networks. Several forms of parallel “T” networks arise from the fact that there are five independent time constants in the network, while in order to realize the desired transfer characteristic it is necessary to impose only four conditions. It is indicated how the remaining degree of freedom may be used to obtain the most suitable input and output impedances for the source and load impedances with which the parallel “T” is to be used. The derivations for the parallel “T” formulae are given in an Appendix.Tolerance requirements on the components of parallel “T” and bridge “T” networks are derived. If ±1 per cent components are used at 60 cycles, the resonant frequency will lie between 56.4 and 63.6 cycles, and the notch width (rejection band width) will be within ±0.99 cps. of the correct value. In order to guarantee that the phase shift at 60 cycles is within ±10°, the percentage deviation of each part must be less than ($\text{9.0}\text{T}{}_{\mathrm{d}}\text{ω}{}_0$), where ω₀ is the carrier angular frequency, T_d the derivative time constant.     

A system for rapid production of photographic records

In the land-based radar direction of aircraft it was found necessary to transfer to a large plotting screen a 10X magnification of the PPI cathode-ray tube face. An instrument was developed to do this photographically, with the projected image lagging only 15 seconds behind the latest information. A single 360-degree sweep on the PPI tube is reduced to a $\text{1}\text{2}$-inch diameter image on Eastman Type 5302 Fine Grain Release Positive Film. The resultant image is rapid-processed by the application of hot solutions in a special processing cup which restricts the liquids to a small circular area on the film.The spent solutions are quickly removed by a vacuum line and the near dry image is indexed into the projection system where an air pressure gate completes the drying, cools the film, and holds it flat during projection.The projected image, at a magnification of over 300X, has sufficient quality to resolve all the pertinent detail of the tube face.The method is felt to have some general application.     

Perturbation solutions of the Carson–Cambi equation

The periodic differential equation (1+ε cos t)y&#x030B; + py = 0, hereby termed the Carson–Cambi equation, is the simplest second-order differential equation having a periodic coefficient associated with the second derivative. Provided |ε|<1, which is the case we examine, then the differential equation is a Hill's equation and thus possesses regions of stability and instability in the p–ε plane. Ordinary perturbation theory is employed to obtain the stable (periodic) solutions to ε³. Two-timing theory is employed to obtain solutions for values of k near the critical points k = ±$\text{1}\text{2}$, ±$\text{3}\text{2}$, ±$\text{5}\text{2}$. Three-timing is employed to extend the solution near k = ±$\text{1}\text{2}$. The solutions of the Carson–Cambi equation are compared with the solutions of the corresponding Mathieu equation.     

Angular region: a measure of underdamped behavior

The natural modes of an underdamped dynamical system are given by the characteristic numbers of the quadratic operator pencil

$\text{P(s)=s}{}^2\text{I+sB+A,}$

where the operator A depends on the dissipative and reactive elements of the system, while B depends solely on the reactive elements. The operator P(s) for every applied stimulus vector signal x must satisfy:

$\text{(Bx,x)}{}^2\text{<4(Ax,x).}$

A measure of underdamped behaviour is suggested by predetermining an angular region |φ| containing all natural modes of the system,

$\text{|}\text{tan}\text{φ|?}\text{[4(Ax,x)?(Bx,x)}{}^2\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(Bx,x)}\text{.}$

When a comparison between positive operators A and B is available, say B²=KA, then

$\text{|}\text{tan}\text{φ|?}\text{√(4?K}{}^2\text{)}\text{K}\text{.}$

The paper is motivated by Duffin-Krein-Gohberg's earlier mathematical contributions.     

The scattering of X-rays

Present Status of the Problem.—The scattering of X-rays is one of the outstanding problems of electromagnetic radiation which has not been solved satisfactorily. All theories (based on classical electrodynamics) presented thus far do not explain either the diminution in the scattering coefficient, or the observed asymmetry in the scattering, or both. Among such theories we may mention:J. J. Thompson's Theory.—Assuming that the scattering is done by a point electron, and making use of certain additional hypotheses, Thomson showed that the scattering coefficient of any substance is given by

$\text{σ=}\text{8πNpe}{}^4\text{3m}{}^2\text{c}{}^4$

and that the intensity of the scattered radiation is given by

$\text{I}{}_{\mathrm{\theta }}\text{=I}\text{e}{}^4\text{(I+cos}{}^2\text{θ)}\text{2r}{}^2\text{m}{}^2\text{c}{}^4$

where N is the number of atoms per c.c., p the number of electrons per atom, e the electronic charge, m the electronic mass, c the velocity of light, I_θ the intensity of the scattered radiation at an angle θ between the incident beam and the radius vector joining the centre of the electron and the point P distant r from the electron, and I is the intensity of the incident beam. This theory explains neither the asymmetry nor the decrease in the coefficient of scattering.Schott's Theory.—Among other things, the assumption is here made that the atom consists of coaxal rings of electron. The electrons in each ring are spaced at equal intervals and revolve with a uniform angular velocity, which, however, may be different for different rings. This theory fails to explain the observed diminution in the scattering coefficient.Debye's Theory.—In its essentials, Debye's theory has the same merits and demerits as that of Schott. Debye assumes that all the electrons in an atom are arranged in a single ring, and that they are spaced at equal intervals. This theory (and also Schott's theory) explains the asymmetry and the “excess scattering,” but is altogether unable to explain the diminution in the scattering coefficient.Modification of the Classical Theory.—The present paper presents a discussion of the possibility of modifying the classical theory (that of J. J. Thomson) so as to account for the decrease in the scattering coefficient as well as the dissymmetry. By assuming that the electron is made up of a number of parts—for simplicity, of two parts—it has been found possible to account for the diminution in the scattering coefficient without, at the same time, explaining the observed asymmetry. To accomplish both objects is what was aimed at in the combination of the present work with that of Debye. In this research the goal has not been perfection between predicted and observed results, but rather to discuss some possible modifications of the classical theory and their consequences.     

The representation of radiation reaction in wave mechanics

It is well known that the wave mechanical ψ equation leads to the conclusion that the centroid of the wave mechanical electron should move according to the classical electrodynamic equation of motion in which, however, the terms representing what is commonly called radiation reaction are absent. If v is the velocity of the electron, the classical rate of change of momentum is $\text{md}\text{dt}\text{{}\text{v}\text{(}\text{I}\text{?}\text{v}{}^2\text{c}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{}}$. The equation of motion including radiation reaction terms may be regarded as obtainable by replacing this quantity by one obtained by operating upon it with the operator P^?1

$\text{P={I?}\text{α}{}_1\text{kd}\text{dt}\text{+}\text{α}{}_2\text{d}\text{dt}\text{(}\text{kd}\text{dt}\text{)?·}}{}^{\mathrm{?}}$

where α₁, α₂, etc., are constants and $\text{k = (}\text{I}\text{?}\text{v}{}^2\text{c}{}^2\text{)}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$. The main purpose of the paper is to show that if there be any relativistically invariant ψ equation which leads to the classical equation of motion without radiation reaction terms, then by replacing the vector and scalar potentials U and ? in that equation by P(U) and P(?), a relativistically invariant equation of motion will be obtained including the radiation reaction terms, provided that the $\text{d}\text{dt}$ in P be now regarded as $\text{?}\text{?t}$ + u · grad, where u is the velocity of the wave mechanical density distribution at a point. The purpose is to use the power to produce the equation of motion as a criterion for suggesting the proper modification of the ψ equation to apply in those cases where, on the classical theory, the electron would suffer great acceleration, as in ionization by rapidly moving corpuscles.     

The reflection of hydrogen atoms from lithium fluoride

The paper describes the phenomena associated with the reflection of a sharply defined beam of hydrogen atoms from a crystal of LiF. Of primary interest is the fact that the atoms show interference effects in agreement with the wave mechanics theory and plane grating diffraction patterns are photographed. Evidence of the thermal agitation of the surface ions is obtained from the diffuse reflection with surrounds the specular beam.The Schrödinger wave equation for the motion of a free particle of mass m is

$\text{▽}{}^2\text{ψ ?}\text{4πmi}\text{h}\text{?ψ}\text{?t}\text{= 0}\text{(I)}$

. The solution of this equation corresponding to the kinetic energy $\text{mv}{}^2\text{2}$ is

where

$\text{v }\text{mv}{}^2\text{2}\text{and}\text{σ}\text{mv}\text{h}$

. The motion of such a particle should have the characteristics of a plane wave of frequency ν and wave-length $\text{λ =}\text{1}\text{σ}$. The experiments of various investigators¹ have shown the validity of the wave theory of the motion of the free electron and have given values of the wave-length in agreement with the theory.The free motion of atoms, ions and molecules should likewise have wave characteristics. In the case of the hydrogen atom, as the simplest example, the complete wave equation may be written in the form

$\text{I}\text{m}\text{▽}{}^2\text{x,y,zψ +}\text{I}\text{μ}\text{▽}{}^2\text{η,μζψ ?}\text{8π}{}^2\text{μ?ψ}\text{mh}{}^2\text{η}{}^2\text{+ μ}{}^2\text{+ ζ}{}^2$

$\text{?}\text{4πi}\text{h}\text{?ψ}\text{?t}\text{= 0,}\text{(3)}$

where x, y, z, are the coördinates of the center of mass of the atom and ξ, η, ζ the coördinates of the electron with respect to the center of mass. If m_? and m₊ are the masses of electron and proton, m and μ have the significance

$\text{m = m}{}_{\mathrm{?}}\text{+ m}{}_{\mathrm{+}}\text{and}\text{I}\text{μ}\text{=}\text{I}\text{m}{}_{\mathrm{?}}\text{+}\text{I}\text{m}{}_{\mathrm{+}}$

. Equation (3) is solved by

$\text{ψ = U}{}_1\text{(x,y,z) U}{}_2\text{(η, ν ζ) ?}{}^{\mathrm{<mtext>2\pi iEt</mtext><mtext>h</mtext>}}$

, where E may have a continuous set of values and represents the total energy. U₁ and U₂ must satisfy the equations

$\text{▽}{}_1{}^2\text{U}{}_1\text{+}\text{8π}{}^2\text{mβU}{}_1\text{h}{}^2\text{= 0,}\text{(4)}$

and

$\text{▽}{}_2{}^2\text{U}{}_2\text{+}\text{8π}{}^2\text{μ}\text{h}{}^2\text{(α ?}\text{μ?}\text{m}\text{η}{}^2\text{+ ν}{}^2\text{+ ζ}{}^2\text{)}\text{U}{}_2\text{= 0}\text{(5)}$

, where

$\text{α + β + E}$

.     

Nuclear Resonance Flourescence Studies of Levels in 206Pb, 207Pb, 208Pb and 209Bi below about 5MeV

Capabilities of the resonance flourescence technique are discussed. Specific reference is given to the study of ground state dipole and electric quadrople transitions below about 5 MeV in ²⁰⁶Pb, ²⁰⁷Pb, ²⁰⁸Pb and ²⁰⁹Bi. It is suggested that many of the states observed in ²⁰⁷Pb and ²⁰⁹Bi arise from the weak-coupling of a p_{$\text{1}\text{2}$} neutron hole or an h_{$\text{9}\text{2}$} proton to collective levels of ²⁰⁸Pb.     

Mathematical models of information system use

This report presents the results from a study of mathematical models relating to the usage of information systems. For each of four models, the papers developed during the study provide three types of analyses: reviews of the literature relevant to the model, analytical studies, and tests of the models with data drawn from specific operational situations. (1) The Cobb-Douglas model: x₀ = ax₁^bx₂^(1?b).This classic production model, normally interpreted as applying to the relationship between production, labor, and capital, is applied to a number of information related contexts. These include specifically the performance of libraries, both public and academic, and the use of information resources by the nation's industry. The results confirm not only the utility of the Cobb-Douglas model in evaluation of the use of information resources, but demonstrate the extent to which those resources currently are being used at significantly less than optimum levels. (2) Mixture of Poissons:

$\text{χ}{}_0\text{=}\text{∑}\text{i=0}\text{n}\text{i}\text{∑}\text{j=0}\text{p}\text{n}{}_{\mathrm{j}}\text{e}{}^{\mathrm{mj}}\text{(m}{}_{\mathrm{j}}\text{)′/i!}$

where x₀ is the usage and (n_j,m_j),j = 0 to p, are the p + 1 components of the distribution. This model of heterogeneity is applied to the usage of library materials and of thesaurus terms. In each case, both the applicability and the analytical value of the model are demonstrated. (3) Inverse effects of distance: x = a e^?md if c(d) = rdx = ad^?m if c(d) = r log(d).These two models reflect different inverse effects of distance, the choice depending upon the cost of transportation. If the cost,c(d), is linear, the usage is inverse exponential; if logarithmic, the usage is inverse power. The literature that discusses the relationship between usage of facilities and the distance from them is reviewed. The models are tested with data from the usage of the Los Angeles Public Library, both Central Library and branches, based on a survey of 3662 users. (4) Weighted entropy:

$\text{S(x}{}_1\text{,x}{}_2\text{,...,x}{}_{\mathrm{n}}\text{)= -}\text{∑}\text{i=1}\text{n}\text{r(x}{}_{\mathrm{i}}\text{P(x}{}_{\mathrm{i}}\text{)}\text{log}\text{(p(x}{}_{\mathrm{i}}\text{)).}$

This generalization of the “entropy measure of information” is designed to accommodate the effects of “relevancy”, as measured by r(x), upon the performance of information retrieval systems. The relevant literature is reviewed and the application to retrieval systems is considered.     

The theory of the electrical breakdown of gases at atmospheric pressures

After a brief summary and discussion of Townsend's theory of the electrical breakdown of gases in the light of certain recent criticisms, it is shown that these are compatible with his theory except for one serious discrepancy. This discrepancy lies in the fact that at the assumed fields at which breakdown occurs in air, at atmospheric pressure, and in inert gases at low pressures, the $\text{β}\text{P}$ of Townsend's theory cannot have the significance given it by Townsend, as the positive ions are incapable at these fields of acquiring the ionizing energy. Various solutions proposed are discussed and found inadequate. It seems necessary, in order to keep this otherwise successful theory, to doubt the validity of the assumption generally made for plane parallel electrodes, that the potential drop between the plates is uniform before the spark passes. If fields about ten times as great as those calculated from the uniform drop existed, the theory could be applied. It is shown that such fields are possible under the conditions of the spark potential experiments, due to space charges resulting from the difference of ionic and electronic velocities. The existence of such fields requires a finite spark lag interval of about 10^?4 seconds, as yet not definitely observed. Experiments should be undertaken in laboratories equipped for the purpose to look for both the spark lag and the non-uniform field.     

An experimental investigation of forced vibrations

The solution of the differential equation y″ + 2Ry′ + n²y = E cos pt is written in a new form which clearly exhibits many important facts thus far overlooked by theoretical and experimental investigators. Writing s = n ? p, and Δn = n ? √n² ? R², it is found: (a) When s ≠ Δn, there are “beats,” and the first “beat” maximum is greater than any later maximum while the first “beat” minimum is less than any later “beat” minimum. The “beat” frequency is $\text{(s ?}\text{Δ}\text{n)}\text{2π}$. (b) When n² ? p² = R², there are no “beats,” and the resultant amplitude grows monotonically from zero to the amplitude of the forced vibration, (c) At resonance, when n = p, we still have maxima which occur with a frequency $\text{Δ}\text{n}\text{2π}$ in a damped system. (d) The absence of “beats” is neither a sufficient nor a necessary condition for resonance in a damped system.In the experimental investigation the upper extremity of a simple pendulum was moved in simple harmonic motion and photographic records obtained of the motion of the pendulum bob. Different degrees of damping were used, ranging from very small to critical.The experimental results are in excellent agreement with theory.