Note on Quantization Non-Conditioned Periodic Systems (1924). Among most interesting problems which have presented themselves connection development Bohr theory atom, quantization unconditioned-periodic dynamical systems paramount importance, standing it does way further analytical extensions quantum theory stationary paths. following remarks, some which have already found expression elsewhere, while failing supply a final answer th question, seek contribute a few guiding ideas elucidation problem. We shall first examine problem standpoint ordinary dynamics. Notes on Dynamical Systems Non-Integrable Separation Variables on Existence “Unmechanical” Orb Atom (1924) There has been some speculation lately existence holonomic, conservative, reversible dynamical systems whose equations motion cannot integrated a separation variables Hamilton-Jacobi equation a good deal has been written about " unmechanical " orb atom. purpose th note twofold: First, two examples classical mechanics showing that, either constraining a mass­ point move freely a type coordinate space, else introducing arbitrary anisotropic fields force, dynamical systems which not integrable separation variables constructed. question physical significance such systems left present entirely untouched. Second, attempt made show that, while conclusions existence unmechanical orb atomic systems, drawn a failure obta correct values energy terms through calculus perturbations do not seem wholly justified, arguments a more general character based on theory relativity point out fact such case: unmechamcal orb do exist atom. Sommerfeld Theory Fine Structure Standpoint General Relativity (1925) present paper we intend show that, we keep mind numerical relations between magnitudes electronic mass charge, on one hand, nuclear mass charge, on other, it possible obta relativistic Keplerian conditioned-periodic orb Sommerfeld theory starting considerations general relativity. following pages we shall see we keep mind relations they exist hydrogen atom hydrogenic ions, type He+, Li++, etc,, we come conclusion such cases material field nucleus almost Euclidean (Laue conditions) almost static. a preliminary, it might not altogether out way review briefly how orbital equation obtained Sommerfeld theory. Theory Continuous X-Ray Spectrum (1925) A satisfactory theory continuous X-ray spectrum is, writer knowledge, still lacking. Classical electrodynamics leads conclusions violent clash experiment; fact former would lead expect a continuous spectrum having frequencies, whereas latter shows definitely existence a sharp maximum frequency limit. usual quantum interpretation th phenomenon, deduced Einstein-Planck law hv=eV, limit wave-length depends only on voltage applied tube varies inversely it. Th accordance experiments performed date. far writer aware, however, no quantum theory continuous X-ray spectrum has been offered, although several suggestions have been put forward. present paper we intend advance alternative quantum theory continuous X-ray spectrum, which makes use following principles based on following assumptions. Heaviside´s Proof Expansion Theorem (1926) Synopsis.-Heaviside´s proof celebrated Expansion Theorem, found scattered Electrical Papers, reconstructed. It based upon so-called "conjugate theorem,” discovered independently Rout, which establishes a relation between any two normal modes oscillation a dynamical system. Heaviside´s argument applies systems having finite number degrees freedom no repeated null roots determinantal equation system. relation between Heaviside s, Carson´s Wagner´s proofs painted out. treatment transient state electric power transmission line Heaviside operational method (1926) theory alternating current circuit steady state, since introduction vect method algebra complex quantities has reached a high degree perfection a state serve daily uses engineer. same said steady-state theory transmission line, since introduction Kennelly hyperbolic function method (1) circle diagram, nothing desired terms simplicity ease calculation. . theory transitional state, on other hand, both electrical circuit and, a much greater degree, transmission line, generally so complicated it useless, except experienced hands mathematician profession. Known, moreover, enormous importance transitory state both circuit line numerous phenomena, sometimes mysterious appearance, which it often gives rise. There therefore a great need create a method analys transitional state fills, respect latter, same role algebra complex quantities hyperbolic method respect stationary one quite simple so it Useful engineer. Essentially Heaviside operational method a method determining arbitrary constants enter solution a linear differential equation constant coefficients Commentary on Ludwig Casper work "On Heaviside formula switch-on processes (1926). Archiv für Elektrotechnik, August 6, 1925, Vol. 15, p. 95, a short note Ludwig Casper appeared, which wanted give a new method deriving Heaviside formula switch-on processes. Mr. Casper noted Heaviside gave formula without proof mentions K. W. Wagner only one who gave proof th formula. I would now like point out Heaviside own method, far auth knows, correct. It was first time detail work “On Self-Induction Wires” Parts 3 4, Philosophical Magazine. Issue October November, 1886 (Vol. 22, pp. 332/352 p. 419/442: see especially p. 335 p. 426, "Electrical Papers", vol. 2, p. 202 P. 225). formula appeared Vol. 2 “Electromagnetic Theory” was derived first time previous results (ie so-called “conjugate theorem” cited above) work “On Resistance Conductance Operators etc.” Philosophical Magazine, Vol. 2, pp. 479/502, 1887 [s. P. 501, equation (54), “Electrical Papers”, vol. 2, p. 355 very specifically p. 373]. On Conditions Validity Macromechanics (1927) recent papers, De Broglie Schrödinger have developed a remarkable theory atomic dynamics based upon conception mechanics fundamentally subject laws wave phenomena, dynamics small-scale systems (micromechanics) being related large-scale models (macro­ mechanics) much same way wave optics connected geometrical optics. As well known, dynamical trajectories macromechanics orthogonal curves surfaces constant action. micromechanics, we shall see, rays may defined curves orthogonal equiphase surfaces. It becomes some interest investigate under what conditions equiphase surfaces coincide surfaces constant action and, therefore, it possible determine dynamical trajectories methods classical mechanics. th paper we intend show macromechanics (geometrical optics) coincides micro­ mechanics (wave optics) when: (a) frequency associated wave becomes infinite or, what amounts same, Planck constant vanishes; (b) the "medium" (field) which wave propagates homogeneous. Condition (a) embodied Bohr Correspondence Principle has been studied Optics. Condition (b) leads conclusion failure macromechanics, atom must definitely traced non-homogeneous force field. On relativistic theory wave mechanics (1927) No physicist unaware enormous importance recent works De Broglie (1) Schrödinger (2) who, laying foundations a new branch modern physics, have not only placed atomic theory matter a entirely different, but have eliminated many difficulties contradictions Bohr old quantistic theory. known Einste theory relativity, so deep starts ramifications even philosophy itself must have studied problems raised it. It known disparity between relativity atomistics so far complete; how many attempts have been made find discontinuous solutions fundamental equations relativity, which correspond proton electron, have gone failure. Wave mechanics, on other hand, have, Struik Wiener (3) have noted, organic connection the. Einste relativity therefore offers possibility acting a link between it atomistic. th brief note we propose demonstrate it possible deduce more general fundamental equation Schrödinger case electron, variants, particular cases relativistic equation Fock (4) De Don der (5). a starting point, after having broadly indicated idea Wiener Struik, we will construct equation wave mechanics when, addition a Coulomb electrostatic field a magnetic field, there a gravitational field central symmetry. Comments on work Mr. G. von Gleich: On mass changeability two-body problem. (1927). a recent work, zieth G. v. Already incorrect conclusion general theory relativity leads electron orb hydrogen atom, which different those obtained when one takes in account known mass variability m = m0 (I - β2) -1 .2. consideration general relativity should, according v. Lead immediately results deviate Sommerfeld theory fine structure. J. Kudar comments on th a critique conclusions same time, “ gravitational forces electrons atomic nuclei completely neglected compared their electrostatic effect, so gravitational field atomic nucleus not noticeably influence movement electrons caused electrical forces” Unified Field Theory Electricity Gravitation (1929) MAY we permitted direct attention a certa aspect Einste three recent papers (Berliner Berichte, pp. 217, 224, 1928; Feb. 1920) on distant parallelism which came light a discussion Prof. D. J. Struik? avowed aim papers develop improved unified field theory electricity gravitation. A much more pressing need general relativity theory a harmonisation quantum -theory, particularly Dirac theory spinning electron. On bas Levi­ Civita parallelism task seems hopeless, inasmuch we have no adequate means comparing spins different points. On other hand, notion a parallelism valid whole space Einste n-uples enables us carry over Dirac theory in general relativity almost without alteration. we need do interpret Dirac Po,P1 ,p ,p 3 not differentiation respect four variables x,y,z,t defined throughout space-time, but differentiation along lines quadruple (Einste 4-Be ). is, quadruples need not integrable so furnish us a co-ordinate system throughout space, such a co-ordinate system absolutely inessential proof invariance Dirac equations under a Lorentz transformation. Statistical Interpretation Various Formulations Quantum Mechanics (1929). AT conclusion valuable paper Profess Van Vleck has concisely pointed out some outstanding difficulties present formulation quantum mechanics. first comment we would like add ma question does not seem so much choice coordinates Hamiltonian "symmetrization” difficulty Schrödinger. Cartesian coordinates enjoy a privileged position, just elementary geometry, because then square distance between two points a quadratic expression diagonal form involving only sum squares. One always find wave equation arbitrary coordinates making use variation principle Schrödinger. ultimate reason symmetrization difficulty remains however unknown. (Cf. Schrödinger, Annalen der Physik, 79, p. 747, 1926.) On Spherically Symmetrical Statical Field Einste Unified Theory Electricity Gravitation (1929). recent papers Einste has developed a theory electricity gravitation based on a fundamental asymmetrical tens ªh which serves associate each point a continuum X4 a quadruple axes ("4-Bein") consequently determines a parallelism directions valid: whole space not only infinitesimal neighborhood a point. index a ªh associated directions axes each point, while index L ordinary vect index. difference between index L index a signalized facts, first "a" indices contracted only "a" indices L indices only L indices; second, there no distinction between co- contravariant "a" indices, so any two such indices contracted: Contravariant L indices introduced familiar manner taking ah normalized minors determinant [ ªh-,... 1, coefficients g-,...,,. Riemann geometry defined g-,...,,. = ªh,. ªh,,.. On Einste Unified Field Equations Schwarzschild Solution (1929). recent papers Wiener auth have determined tensors shL Einstein’s unified theory electricity gravitation under assumptions static spherical symmetry symmetry past future. It was there shown field equations suggested Einstein’s second 1928 paper lead th case a vanishing gravitational field. purpose th paper investigate, same case, nature gravitational field obtained field equations suggested Einste first 1929 paper. On Spherically Symmetrical Statical Field Einstein’s Unified Theory: A correction (1929). a previous paper authors present note have treated case a spherically symmetrical statical field, stated conclutions: first under Einste definition electromagnetic potential electromagnetic field incompatible assumptions static spherical symmetry symmetry past future; second, one uses Hamiltonian suggested Einste second 1928 paper electromagnetic potential vanishes gravitational field vanishes. We have recently become aware method used paper incorrect because definition Au on page 353 valid only coordinates local axes taken Cartesian. th note we replace reasoning previous paper .by argument employing Cartesian local coordinate axes Cartesian Gaussian coordinates continuum. None conclusions previous paper vitiated th investigation, although some final formulas supplemented additional term. Note on Statistical Interpretation Maxwell Equations (1929). It has often been trivially noted literature Schrödinger relativistic equation a force-free field reduces m = O equation propagation light free space. accordance general principles quantum mechanics Schrödinger equation obtained energy-momentum relation - Po + Px + Pu + P,2 + m2c2 = O substituting components four-vect p operators p= (h/2,ric)a/at, Px = (h/21ri)a/ax, etc. a photon proper mass vanishes coefficients h/21ri divide out corresponding wave equation. th sense equation propagation light regarded wave equation photon just Schrödinger equation wave equation electron without spin. Unified Field Theory Schwarzschild”s Solution: A Reply (1930). I have read a great deal interest Mr. Salkover’s criticism (Phys. Rev. 35, 209 (1930)) my paper on Einste unified field theory Schwarzschild solution (Proc, Nat. Acad. 15, 784 (1929)). I agree Mr. Salkover expression electromagnetic potential which I used my paper calculation tens Vbx was not correctly copied previous paper Wiener myself there referred (Proc, Nat. Acad, 15,802 (1929): communicated May, 1929). expression latter paper course agrees calculated Mr. Salkover, value tens Auv reported my paper. Th mistake, while vitiating my results on Schwarzschild solution, no way affects our previously announced conclusions, particular non-existence unified theory electrostatic field spherical symmetry. Spherically Symmetrical Field Unified Theory (1930). th paper a systematic investigation spherically symmetrical statical field Einstein´s unified theory electromagnetic gravitational fields given. investigation carried through two cases: without time symmetry. a time symmetrical field we understand one whose Riemann interval not altered changing +dt in – dt. beg we develop a generalization formulas parallel displacement covariant affine differentiation Einstein´s geometry which enables us introduce orthogonal transformations local coordinates, so bring them formally in agreement Gaussian coordinates. We then solve Einste field equations Fµα=0 Gµα=0, first without assumption time symmetry. We determine values physical significance constants integration considering case a very weak field, obta asymptotically classical Coulomb field (e/r2) classical Newtonian potential (-m/r). Positive negative electricity appear unified theory empirical facts, just classical theory. present theory, however, Newtonian field Coulomb field separated only asymptotically. immediate vicinity a charged mass particle they inextricably bound together. Our solution gives a quadruple (“Vierbein”) 1h1 = 1 + e2/r4 , 2h2 = 1, 3h3 = 1 4h1 = - (ie/r2) (1 + e2/r4)1/2, 4h4 = 1 + m ʃαr r-2(1 + e2/r1)-3/2dr. All other components vanish. a transformation variables th shown reduce quadruple recently calculated Einste Mayer. field equations then solved assumption time symmetry. electrostatic field vanishes, but component g44 metric form has approximately Schwarzschild value, while others have Euclidean values. Because absence a law motion, no observable predictions having do path exploring particle made. It shown, however, unified field theory gives first approximation same red shift 1916 theory. it assumed orbit a planet a Riemannian geodesic, 116 theory, then advance perihelion Mercury comes out approximately 7” per century, is, about same 1905 theory. Relativity Uncertainty Principle (1932). A simple ideal experiment measure simultaneously position momentum a free electron, kinetic energy instant when th kinetic energy measured, taking in account finite velocity propagation light embodied Lorentz transformation, proposed. Th experiment follows: observer sends out a light signal which strikes a free electron returns along own track. measurements made on outgoing returning signals required quantities, uncertainties their measurement, calculated. absence detailed knowledge concerning nature interaction between signal electron, it found any electron velocity between 0 c, ΔpΔx ΔEΔt have upper a lower bound. upper bound depends only on initial electron velocity lower bound a function electron velocity outgoing frequency. bounds come close together low electronic velocities low frequencies, but their separation increases initial electron velocity increases becoming infinite limit v=c. limit c→∞ upper lower bounds coalesce frequencies electron velocities giving Heisenberg results. possibility, but not necessity, a relativistic quantum mechanics ΔpΔx=h (and ΔEΔt=h) is, we believe, established present results. Relativistic Thomas-Fermi Atom (1932) Thomas-Fermi equation determining inner atomic potential charge distribution generalized take care relativistic change mass velocity. relativistic equation then solved numerically means a first order perturbation method help differential analyzer. solution applied case mercury it shown while atomic potential changes only slightly, charge density appreciably increased immediate vicinity nucleus slightly decreased larger distances. On Compton Latitude Effect Cosmic Radiation (1933) By considering influence earth magnetic field on motion charged particles (electrons, protons, etc.) coming earth directions space, it shown experimental variation cosmic ray intensity magnetic latitude, found Compton collaborators, fully accounted for. cosmic radiation must conta charged particles energy between lim paper. experimental curve may represented a suitable mixture rays energies, but it not excluded a part radiation may consist photons neutrons. predominantly negative particles there must region rapidly varying intensity a predominant amount rays coming east, conversely positive rays. Because fact regions near magnetic equat there a predominance rays coming nearly horizontally, absorption atmosphere may increased. Finally fact Compton result definitely shows cosmic rays conta charged particles gives some support theory super-radioactive orig rays advanced one present authors. Interpretation Azimuthal Effect Cosmic Radiation (1933). connection directional experiments recently performed Mexico 29º geo­ magnetic latitude Thomas H. Johnson L. Alvarez A. H. Compton, working independently, showing cosmic radiation predominantly positively charged, following discussion, based upon theory latitude effect recently developed Lemaître writer, becomes interest. particles any energy we have shown any point on earth there a cone which directions allowed intensity cosmic radiation allowed directions each individual energy same, outside which directions forbidden consequently intensity zero. On North-South Asymmetry Cosmic Radiation (1935). development theory latitude azimuthal effects reviewed significance Johnson experiments on north-south asymmetry brought out. Störmer critical remarks commented on reference Lemaître Vallarta paper recent papers Lemaître Bouckaert. On Longitude Effect Cosmic Radiation (1935). It shown equivalent magnetic dipole describing earth magnetic field large distances placed magnetic center earth calculated Schmidt, theory latitude azimuthal effects developed Lemaître Vallarta accounts longitude effect found Clay, van Alphen Hooft and, independently, Millikan amd Neher. Further it shown distribution function magnetically deflectable particles capable reaching earth tropical belt substantially exponentially decreasing function their energy, theory quantitatively agrees experiment. On Geomagnetic Analys Cosmic Radiation (1936). results extensive study trajectories asymptotic a certa family periodic orb earth magnetic field, carried out means Bush differential analyzer, presented th paper. theory region full light, ma cone, fully discussed. Attention then restricted section ma cone plane geomagnetic meridian it shown north-south asymmetry furnishes most direct approach analys energy spectrum a wide region, independently particles sign. Further it shown general shape minimum north-south asymmetry discovered Johnson course Mexican experiments fully accounted action earth field. On Allowed Cone Cosmic Radiation (1936). Further results extensive study trajectories asymptotic a known family unstable periodic orb earth dipolar magnetic field, carried out means Bush differential analyzer, presented th paper. A detailed discussion our methods determining asymptotic trajectories means differential analyzer numerical integration a whole family a time; comparison results obtained shows absence systematic errors any consequence mechanical integrations exhib precision attained differential analyzer. families asymptotic trajectories then analyzed systematically order determine ma cones latitudes up 30º. Th leads theory azimuthal effect a study region vicinity zenith. Contributions theory effects latitude asymmetry cosmic rays. Calculation a family asymptotic orb (1936). Among trajectories charged particles animated a certa speed arriving a point in, earth sounds action magnetic field, we distinguish four species: 1º) Some come infinity, without having encountered surface earth. ones really traversed cosmic rays form cone cosmic rays energy considered. 2º) Others come another region earth. form defended region. 3º) Some come asymptotically a periodic quasi-periodic orbit. They separate trajectories coming infinity those coming earth their tangents point impact earth form generatrices limiting cone cosmic rays. We called them trajectories first kind. 4º) There finally trajectories which come infinity reach point considered only after having been tangent earth. delimit a region cone coming a kind shadow effect ground. We have called them trajectories second kind. Only first second kind trajectories need calculated precision. others only useful qualitative discussions solution special problems such theory rotation effect galaxy studied Compton Getting Longitude Effect Cosmic Radiation Position Earth´s Magnetic Centre (1937). theory allowed cone electrically charged primary cosmic rays, developed a number recent papers Lemaître Vallarta, competent account longitude effect due regard paid fact earth centre does not coincide magnetic centre. Assuming Schmidt determination latter and, Clay, exponentially decreasing energy distribution primary charged particles, it has been shown longitude effect along geomagnetic equat should roughly sinusoidal maximum geographic longitude 12º W, minimum 168° E. intensity difference between maximum minimum calculated about 8 per cent, more recent calculations position earth magnetic centre (Bruins 4, Bartels 5) have somewhat revised Schmidt earlier result, but theoretical longitude effect not much affected thereby. longitude maximum minimum depends only on coordinates earth magnetic centre, particularly on longitude ; intensity difference between maximum minimum closely connected distance between earth centre magnetic centre, a lesser extent energy distribution primary particles. Cosmic Rays Magnetic Moment Sun (1937). It generally assumed constant intensity cosmic radiation sea-level beyond a certa latitude accounted clocking effect atmosphere, which absorbs particles energy below limit latitude resulting action earth magnetic field. important consequence th assumption latitude which intensity becomes constant should displaced towards geomagnetic poles altitude above sea-level increases, since then thickness absorbing air layer decreases therefore energy absorbed. Compton mounta experiments already throw doubt on existence th displacement, th doubt strengthened Cosyns stratospheric measurements which, while admittedly inconclusive, indicate intensity remains constant north 49° even highest altitudes reached. Geographic Asymmetries Cosmic Rays Related Earth Magnetization (1937). It has been shown (see 1 references end paper] known magnetic field Earth may closely approximated a magnetic dipole strength 8.1 x 1025 Gauss­-cm magnetic center Earth. Th center situated a point 340 km geographical center. latitude 6.5 north longitude 161.5 east. extension ax eccentric dipole pierces Earth surface two points, one latitude 80° north longitude 83º west other latitude 76° south longitude 121° east. points labeled NPED SPED, respectively, Figure l. According theory Lemaître Vallarta (2, 3, 4], cosmic rays, electrical particles distributed isotropically space, bent Earths magnetic field some shot back infinity while others allowed reach Earth surface. any point a distance Earth magnetic center, particles a energy reaching point included a cone very complicated shape which forms a boundary region where directions allowed separates th region other regions where only some no directions allowed. any allowed direction, moreover, intensity just same would exist absence magnetic field hence same it infinity. Since boundaries cones may determined theory, intensity cosmic rays a energy may calculated point question, situated a distance magnetic center Earth. An Outline Theory Allowed Cone Cosmic Radiation (1938). discovery the existence electrically charged particles primary cosmic radiation has very considerable impetus study motion such particles field a magnetic dipole, a problem considerable difficulty which Störmer has devoted a great deal attention past thirty­five years, originally connection Birkeland theory aurora borealis. dynamical trajectories solutions a set non-integrable differential equations, it th feature non-integrability which imparts this, well other well-known dynamical problems, their most interesting difficult characteristics. following we shall attempt give a summary theory such motions, particularly it concerns directions which charged particles a energy may, may not, arrive a point on earth, due regard being paid fact that, first order approximation, earth magnetic field a dipole earth acts impenetrable body. allowed forbidden regions very considerable physical interest intimately related understanding experimental observations intensity cosmic radiation on earth. sequel we shall, however, confine our attention mainly mathematically interesting features problem. Present Status Theory Effect Earth Magnetic Field on Cosmic Rays (1939). present communication intended cover a brief summary results obtained through analys motion charged particles earth magnetic field includes following topics: a) ma cone, any direction which particles a energy may arrive a point on earth. Latitude azimuthal effects. Special cases latter: north-south east-west asymmetries. ratio positive negative primaries. b) shadow cone, outside which no particles a energy may arrive a point on earth. c) penumbra, i.e. region intermediate between ma cone shadow cone which certa directions allowed others forbidden. d) eccentricity earth magnetic center longitude effect. position earth magnetic center. earth magnetic quadripole. (e) earth magnetic field galactic rotation effect. (j)- sun magnetic field some consequences existence. knee latitude effect. Solar diurnal annual variations. (g) Preliminary communication on magnetic storms cosmic ray intensity. Galactic Rotation Intensity Cosmic Radiation Geomagnetic equat (1939). angle deflection experienced a primary cosmic particle moving plane geomagnetic equat calculated different energies result used find diurnal variation intensity cosmic rays arriving vertically geomagnetic equator. it assumed number primaries varies inversely cube their energy, calculation shows there should a diurnal variation vertical intensity 0.17 percent, primaries positive, a maximum 13 hr. 20 min. sidereal time. a ratio three positive one negative primary, same distribution law, amplitude diurnal variation should 0.1 percent a maximum 12 hr. 30 min. sidereal time, while primary radiation a whole neutral (one positive particle each negative) amplitude should 0.06 percent maximum 8 hr. 40 min. sidereal time, number primaries exponentially decreasing function their energy, amplitude diurnal variation should 0.24- percent maximum 18 hr. sidereal time, assuming primaries positive; primary radiation neutral amplitude should 0.19 percent maximum should occur 20 hr. 40 min. sidereal time. expected diurnal\ variations fur several! values lower limiting energy discussed. Scattering Cosmic Rays Stars a Galaxy (1939). problem dealt th note may formulated following way: imagine a galaxy N stars, each carrying a magnetic dipole moment µn (n=l, 2, …N) assume density, defined number stars per unit volume, varies according any law, while dipoles oriented random because their very weak coupling. Under th condition resultant field whole galaxy almost vanishes. Let there isotropic distribution charged cosmic particles entering galaxy outside. Our problem find intensity galaxy. lts importance arises fact distribution should prove anisotropic a means would available determining whether cosmic rays come beyond galaxy, independent galactic rotation effect already considered Compton Getting. Are There Multiple Charged Primary Particles Cosmic Radiation? (1939). William P. Jesse Piara S. Gill have recently reported a considerable latitude effect very large cosmic ray bursts (about 30 percent) existence which raises most interesting questions manner origin. If, suggested their note, burst frequency a function geomagnetic, not geographic, latitude, implication th latitude effect results action carth magnetic field on charged primary particles, essentially same manner total radiation. A Theory World-Wide Periodic Variations Intensity Cosmic Radiation (1939). existence small world-wide variations intensity cosmic radiation well known. variations two kinds: periodic nonperiodic. Notable examples of former are:. (a) diurnal variation depending on solar time, studied experimentally Hess collaborators, world-wide character which was established Thompson 2 Forbush 3. (b) diurnal variation depending on sidereal time. Fundamental problems theory cosmic radiation bound up th effect, which has been recently discussed elsewhere 4 which we shall not concerned further here. (e) world-wide seasonal variation studied Hess,5 Compton Turner, 6 Gill,7 Forbush 8 more recently Millikan Neher.9 (d) variation intensity period 27 days discussed Hess collaborators, 1 Graziadei,10 Gill,7 Kolhörster. 10 Of nonperiodic effects most important seems associated magnetic storms, world-wide nature which has been strikingly brought out Forbush.11 th paper we propose develop certa theoretical considerations bearing on three periodic effects mentioned above, particular reference problem sun permanent magnetic field.12 treatment magnetic storm effect will reserved another occasion. Determination Energy Spectrum Primary Cosmic Rays (1939) While principle many effects depending on action earth’s magnetic field used determination energy spectrum primaries, there fact two serious difficulties which stand Way a successful attack. One intensity measurements must made atmosphere which, aga because unwelcome phenomena related passage high energy primaries through matter, acts mainly 50 blur pure magnetic effect we interested know. other some best-known magnetic effects most readily accessible experiment, notably latitude longitude effects, notoriously insensitive primary energy spectrum. Th difficulty would course persist even unwanted confusions due atmosphere could eliminated making observations clear top atmosphere. Hence need devising a simple experiment satisfy following requirements. Remarks on Zwicky Paper “On Formation Clusters Nebulae Cosmological Time Scale (1940). a paper recently published PROCEEDINGS Zwicky has marshalled arguments purporting show existence large clusters nebulae inconsistent cosmological short time scale demanded theory expanding universe. ma argument appears formation clusters nebulae requires multiple close encounters; estimate interval between collisions leads him then conclusion time formation a large cluster out a random distribution would greater than 1018 years, a figure obviously so much greater than age expanding universe rule out latter. He then examines possibility clusters might already have existed early stages expansion dismisses it on ground that, according investigations Sinclair Smith, velocity distribution central part a cluster same a considerable distance center, large radial velocities rim a cluster not observed, would expected they had shared expansion universe. He finally suggests experimental verification theory expanding universe which would consist attempting find out remote clusters more condensed than nearby ones. On Birkhoff New Theory Gravitation (1944). It pointed out first place: (1) Birkhoff gravitational theory based on "flat" space-time, "red shift" accounted energy change photon it travels emitting body, whereas photon plays no especial role Einste theory; (2) solution problem two more bodies feasible new theory because simpler character. Four comments H. Weyl concerning Birkhoff theory discussed, it concluded taken much reserve. regard third those comments it pointed out "perfect fluid" used Birkhoff ultimate carrier mass electric charge characterized simplest fluid disturbance velocity light (c). It affirmed a glaring defect earlier relativistic theories disturbance velocity matter has been taken arbitrary, although gravitation electromagnetic field have been equal c. differential equations theory then set up, additional cosmological term gravitational potentials h; suggested, namely, h;¡= (K/8)(ti-x2-y2-z2)g.¡, where x, y, z, Lorentz coordinates K (small) cosmological constant. explicit formula rate advance periastron P two bodies (mass points) masses m m2 given, obtained solution two-body problem theory, possible application double stars referred to. authors propose give a detailed development theory applications papers published shortly elsewhere. Physical aspects Birkhoff theory gravitation (1944) Since Birkhoff theory gravitation presented our consideration a physical theory, it necessary, first all, remember criterion truth which every theory a physical phenomenon must conform. th regard, room distinguish between criterion truth which a mathematical theory must conform which must comply a physical theory. It said a mathematical theory true when it free internal contradictions. order able affirm properness a physical theory it necessary that, addition being free internal contradictions, is, addition meeting truth conditions a mathematical theory, it accordance experiment. It follows th last analys it experiment must decide on whether Birkhoff theory acceptable not. So far no experiment a definitive nature comes present a decision fav against th theory. plenty examples theories history science were intended apply physical phenomena were developed strict mathematical rigor, which, however, have had abandoned because they were not satisfied experiment. A satisfactory theory a physical phenomenon usually starts a simple well-defined model then studied sufficient mathematical rig until reaching conclusions confronted through experiment. Birkhoff theory does not separate th classic path Note on roots some transcendent equations (1945). relation certa physics problems, auth present note has been forced find roots some transcendent equations, which may interest other researchers. Determination Sign Energy Spectrum Primary Cosmic Radiation (1947). An experiment reported th paper measurement complete azimuthal effect. Th experiment was performed Mexico City (geomagnetic latitude 29", altitude 2242 m above sea level) constant zenith angles 20º, 40º, 60°. A characteristic feature length atmospheric path constant, hence assumption made number secondaries detected cosmic ray telescope a measure .of number primaries. analys yields energy spectrum primary radiation form K / E1•n (E=energy, K =constant). no evidence negative primary particles. results subject revision because penumbra bands th latitude only imperfectly known, because resolving power our present apparatus. possibility a bright line spectrum, such a spectrum superimposed on a continuous distribution, not ruled out. possibility negative primaries excluded lim experimental error. spectrum obtained our experimental data agrees completely determined experiment Gill, carried out Lahore, Punjab, India. result valid energy range about 350 600 millistörmers, 6 21 Bev primaries protons. On Magnetic Field Milky Way Effect on Cosmic Radiation (1947). It was discovered H. W. Babcock 1 early-type stars possess very high magnetic fields, corresponding magnetic moments order 1037 gauss-cm3, stars approximately same mass magnetic field proportional angular momentum. On assumption th relation applicable a galaxy, he calculates a dipole moment 10 59 gauss-cml Andromeda nebula (Messier 31) concludes "th should apply almost readily our galaxy Andromeda nebula. purpose th note point out that, so long condition weak magnetic coupling among stars a galaxy still obtains, stellar dipole moments still oriented random resultant field galaxy almost vanishes. Under conditions considerations developed Vallarta Feynman still hold no effect on isotropic distribution charged cosmic rays entering galaxy outside expected. Taking in account order magnitude interstellar distances galaxy, it readily shown even stellar dipole moments high 1038 gauss-cm 3 condition weak coupling still prevails. On Energy Cosmic Radiation Allowed Earth Magnetic Field (1948) Curves giving proton energies (a) below which no energy allowed earth magnetic field, (b) above which energies permitted, latitudes case (a), latitudes Oº 40º geomagnetic case (b), following directions: (1) vertical, (2) zenith angle 45°, north, south, east west azimuths. addition distance angular corrections due longitude effect geographic latitudes up 40". Cosmic Rays Magnetic Field Moon (1948) a recent communication, S. Chapman has calculated, on bas theory proportionality between magnetic moment angular momentum developed recently P.M.S. Blackett, ratio MM/ME between magnetic dipole moment moon earth 1/38400. It may readily shown th result correct magnetic field moon only affect particles primary cosmic radiation very small energy, which able reach earth only immediate vicinity geomagnetic poles. On energy spectrum primary cosmic radiation calculation rocket experiments outside atmosphere (1949) Van Allen Gangnes have measured azimuthal effect cosmic radiation outside atmosphere on Aerobee A 1O rocket launched on March 1, 1949 a height 106 kilometers a point geomagnetic equat 1,000 kilometers west coast Peru. results a similar experiment conducted White Sands, New Mexico, geomagnetic latitude 41º north, have been described Van Alien.2 geomagnetic equat zenital angles vary between 55º 35 ° they found equal intensities north south azimuth, ratio intensities west east azimuths has value 1.46. They found ratio vertical intensities geomagnetic latitudes 41º north 0 ° 2.40. geomagnetic equat intensities north south azimuths must strictly equal, according theory allowed cone cosmic radiation, because allowed cones symmetrical relation west-east plane, consequently north-south symmetry disappears • azimuthal effect outside atmosphere more particularly east-west asymmetry, KE-1 • 7 spectrum region 4 14 Bev protons stress obtained vertical latitude effect. azimuthal KE-1 • 5 region 10 24 Bev. agreement seems Hites experimental error, but it not completely excluded exponent really different different regions energy. It should stressed results deduced observations made outside atmosphere. On Mechanism Sudden Increases Cosmic Radiation Associated Solar Flares (1949). Few sudden large increases intensity cosmic radiation taking place soon after appearance a solar flare have been observed. Most solar flares not associated such increase. When two phenomena appear together, intensity cosmic radiation begins increase about hour after appearance solar flare, rises rapidly maximum value then decreases rather slowly back normal value (Fig. 1). whole process lasts about a day. It appears only high intermediate geomagnetic latitudes. About a day after appearance flare magnetic-storm felt over world, intensity cosmic radiation decreases then slowly rises back normal value. We shall not concerned th paper magnetic storm effect. Galactic Rotation Effect Orig Cosmic Radiation (1949). A fundamental question related primary cosmic radiation whether latter a local a universal phenomenon, i.e., whether it exists only immediate vicinity earth through universe. Th question has do on one hand general cosmological problems on other problem orig primary cosmic radiation. former case total energy contained form cosmic rays only a very small fraction total energy universe, while latter it constitutes a relatively large fraction, certainly not less than 1/10,000 part total energy, including mass energy. Th estimate may readily made taking in account known density primary rays our vicinity average energy each particle, assuming a power spectrum form (K/ E•)dE between 2 3. It dear two different cases source cosmic radiation must sought quite different processes. On Low Energy Spectrum Primary Cosmic Radiation Sun Magnetic Dipole Moment (1949) F various reasons which will discussed detail later, it was deemed important compare directly vertical primary cosmic ray intensity was measured Swarthmore, Pennsylvania (52° N geomagnetic latitude) Churchill, Manitoba (69° N geomagnetic latitude), both without lead absorber. measurements were obtained standard quadruple coincidence counter trains identical those used earlier. Typical results shown Figs. 1 2. records other balloon flights have not yet been analyzed, although spot checks have shown good agreement points plotted here. A total 13 flights was conducted Fort Churchill, most them morning. ratio vertical intensities no lead absorber, extrapolated top atmosphere, was found 1.46. On Energy Spectrum Heavy Nuclei Primary Cosmic Radiation (1950) Bradt Peters have obtained recently intensity heavy nuclei primary cosmic radiation reaching earth 55º-and 30º north geomagnetic latitudes. According their figures, ratio I/I intensities two latitudes 4 helium, 3.4 carbon, nitrogen, oxygen, 3.5 nuclei atomic number greater than 10. difference between ratios not significant lies experimental error. On Albedo Effect Cosmic Rays (1954). First I should like express my very deep gratitude Organization Committee th Congress very kind invitation was extended me come Tokyo. Th fulfills hope which I had many years visit Japan. I have had a great admiration physicists well known Japan. I think a I will devote th morning’s time a discussion two problems. First, question general magnetic field sun, then time permits, question albedo. On relationship between electromagnetic radiation cosmic radiation emitted Sun (1954) It well known, a result Forbush observations confirmed other physicists, that, on one hand, on certa rare occasions associated large solar eruptions, intensity cosmic radiation increases twice normal value. Th increase observed on Earth moderate high magnetic latitudes but not near equator. On other hand, always almost whenever spots appear on surface Sun a very considerable increase intensity electromagnetic radiation comes directly star. Th increase mainly perceived region wavelengths a few centimeters a few meters. We propose give here a theory two phenomena. energy spectrum primary cosmic radiation determined neutron intensities (1954) Using neutron intensity measurements J. A. Simpson assuming proportionality between number neutrons measured number primary protons, energy spectrum primary cosmic radiation proton component determined well known methods. spectrum written form n = K / JtY (n, number primary protons E, energy), analys results Y increasing value 1.18 low energy region (1.3 4.3 Bev) a 1.82 high energy region (4.3 14 Bev). Th accordance results obtained measurements secondary particles loaded heights achieved probe balloons rockets. Th indicates neutron component effectively due primary proton component. Atmospheric radioactivity radioactive precipitation (1957) Atmospheric radioactivity includes due natural causes, mainly cosmic radiation natural radioactivity environment, addition produced explosions atomic bombs. first varies height above sea level 1st latitude; second depends on particular circumstances observation site, mainly on concentration potassium 40. Mexico, radioactive level produced cosmic radiation 0.16 milliröntgens (mr) per day, approximately 60 mr per year . sea level same latitude it about half. It known intensity cosmic radiation has remained constant since long before man appeared on earth. second approximately 0.20 mr per day, 75 mr per year varies observation site. third currently average approximately 1.5 mr per year. Geomagnetic Coordinates Cosmic Radiation (1958) Waddington, McDonald Fay have found geomagnetic cutoff North America Europe does not agree values calculated standard geomagnetic theory, but rather those corresponding a change geomagnetic latitude order 4° 5°. Independent evidence has been adduced Simpson Chicago group well Rose Ottawa group 4 has led them propose a new system "effective geomagnetic coordinates" cosmic radiation. They have theoretical arguments support their proposal. On orig ring high intensity radiation surrounds Earth (1959) Four causes may responsible existence a high intensity radiation ring distances 1,000 30,000 kilometers earth surface around it, follows: 1) albedo trajectories follow secondary particles (fragments atmospheric nuclei) produced impact high-energy primary cosmic radiation particles. Th source would ohble up distances order magnitude 2,000 kilometers. 2) unstable periodic orb field Earth magnetic dipole produce regions concentration, where particles agglomerate. residence time regions depends on value 1 / M, where (M) characteristic exponent Poincaré. stable unstable periodic orb were calculated Baños, Uribe Lifschitz, values their characteristic exponents Lifschitz. regions extend between geomagnetic latitudes Ñ =: ± 30 ° produce a strong latitude effect on intensity ring. Th aspect problem has been studied annexed work Gall Lifschitz. 3) passage a primary particle through magnetic field cloud emitted sun change angular momentum respect ax Earth dipole, without modifying energy, consequently it confine it a neighboring region earth No way out. is, constant y1 Störmer theory goes from 4) Albedo particles angular momentum and,> 1 circulate a value less than 1 a value greater than l perpetually a neighboring closed region earth. 5) shadow cone opens distance ground increases contributes increase intensity. Finally contribution periodic hairp orbits, which have not been sufficiently studied. comparison experimental data obtained third Russian satellite American "explorer" cannot done moment because quantitative results have not yet come our knowledge. Theory Geomagnetic Effects Cosmic Radiation (1961) discovery latitude effect cosmic radiation simultaneously Clay [1] Compton [2], proving charged particles primary radiation, gave very considerable impetus study motion such particles earth magnetic field and, first approximation, field a magnetic dipole, a problem very considerable difficulty which Störmer [ 3] devoted a great deal attention since beginning th century, originally connection Birkeland theory aurora borealis. Today we know a good deal about composition primary radiation, which includes predominantly protons bare nuclei helium iron, positively charged particles. It only last few years attempts have been made work out motion such particles second approximation geomagnetic field, introducing both dipole quadrupole terms. On cavity formed solar plasma earth magnetic field energy field cosmic radiation (1962) Midgley Davis, Slutz have calculated, different methods, shape dimensions cavity created solar plasma Earth magnetic field. Th a problem whose exact realistic solution offers very considerable difficulties, which now have not been overcome. authors cited, one present note1 have only been able reach a solution adopting simplifying hypotheses certainly little adjusted physical reality problem. hypotheses essentially three: First. cavity only a magnetic field sources inside earth. Outside cavity only plasma without own magnetic field. Second. Plasma has axial symmetry. Third. equilibrium conditions independent time.