atljata, SJeto ^atk BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE 1891 The date shows when this volume was taken. To renew this book copy the call No. and give to Uie Ubrarian. QC ^7^ eS^f"®"^"'^^'^^'*^'"''''^^'"^ HOME USE RULES ^'^SiimMiK??'^ All Books subject to Recall All borrowers must regis- ter in the library to borrow books for home use. All books must be re- turned at end of college year for inspection and repairs. Limited books must be re- turned within the four week limit and not renewed. Students must return all books before leaving town. Officers should arrange for the return of books wanted during their absence from town. Volumes of periodicals ' and of pamphlets are held in the library as much as I>ossible. For special pur- poses they are given out for a limited time. Borrowers should not use their library privileges for . the benefit of other persons. Books of special value and gift books, when the giver wishes it, are not allowed to circulate. Readers are asked to re- port all cases of books • marked or mutilated. 1924 012 325 381 Do not deface books by marks and writing. ^^1 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924012325381 ELECTRODYNAMIC WAVE -THEORY OF PHYSICAL FORCES VOLUME I BULLETINS 1 to 6 INCLUSIVE ANNOUNCING THE DISCOVERY OF THE PHYSICAL CAUSE OF MAGNETISM, OF ELECTRODYNAMIC ACTION. AND OF UNIVERSAL GRAVITATION. WITH PROOF THAT THESE FUNDAMENTAL FOBCBS OF NATURE ABE D"DE TO THE INTEBPENETBATION OF WAVES PBOPAGATED WITH THE VELOCITY OF LIGHT THROUGH THE FREE ABTHEB, BUT MORE SLOWLT THBOUGH THE SOLID MASSES, WHENCE ABISES ALSO THE BEFRACTION, DIS- PEBSION AND PERHAPS ABSORPTION OF PART OF THE WAVE ENBBGT, AND THUS THE HITHERTO UNEX- PLAINED FLUCTUATIONS OF THE MOON's MEAN MOTION ESTABLISHED BY NEWCOMB IN 1909. AND JUSTLY PRONOUNCED THE MOST ENIGMATICAL PHENOMENON PRESENTED BY THE CELESTIAL MOTIONS. BY T. J. J. SEE, A. M., Lt. M., Sc. M. (Missou.); A. M., Ph. D. (Berol.); Professor of Mathematics, U. S. Navy, Formerly in Charge of the 26-Inch Equatorial Telescope of the U. S. Naval Obser- vatory, Washington, D.C, More Recently in Charge of the U. S. Naval Observatory, Mare Island, California; Fellow of the Royal Astronomical Society; Mitglied der Astronomischen Gesellschaft; Member of the London Mathematical Society; American Mathematical Society; Deutsche Mathematiker Vereinigung; SocieteMathematiquede France; Circolo Mathematico di Palermo; Calcutta Mathematical Society; American Philosophical Society held at Philadelphia; Washington Academy of Sciences; Philosophical Society of Washington; Academy of Sciences of St. Louis; American Physical Society; Societe Francaise de Physique; Fellow of the Amer- ican and British Associations for the Advancement of Science; Member of the British Astronom- ical Association; Societe Astronomique de France; Astronomical Society of the Pacific; California Academy of Sciences; Seismological Society of America; National Geograph- ical Society; Honorary Member of Accademia di Scienze Lettere ed Arti de' Zelanti di Aci-Reale, Sicily; Sociedad Astronomica de Mexico; etc.. Author of Researches ON THE Evolution of the Stellar Systems, Vol. I. 1896, Vol. II, 1910, and of other Investigations in Theoretical and Practical Astronomy, and of Five Memoirs on the New Theory of Earthquakes, Mountain Forma- tion and Kindred Phenomena Connected with the Physics of the Earth, Published in the Proceedings of the American Philosophical Society held at Philadelphia, 1906-13; and OF THE Discovery of the Physical Cause of the Terrestrial Land and Ocean Hemispheres, 1916. O ®eos del ■yew/xeVpet — Plaio. 1917 THOS. P. NICHOLS & SON CO., Lynn, Mass., U.S.A. WM. WESLEY & SON, London A. HERMANN ET FILS, Paris COPYRIGHT, 1917, BY T. J. J. SEE DEDICATED TO THE MEMORY OF MY REVERED MASTER Dr. GEORGE WILLIAM HILL Most Illustrious of Astronomical Geometers WHO DID not hesitate TO DEPART FROM BEATEN PATHS, AND IN ORDER TO PRESERVE HIS INDEPENDENCE EVEN PUBLISHED AT HIS OWN EXPENSE THE IMMORTAL RESEARCHES IN THE LUNAR THEORY, 1877. CONTENTS Dedication iii Introduction by the Author vii BULLETIN No. 1 Announcement of Two Important Astronomical Discoveries . . 1 I The Physical Cause of the Hitherto Unexplained Fluctuations in the Moon's Mean Motion, Dec. 10, 1916. II The Physical Cause of Universal Gravitation, 1914-17. BULLETIN No. 2 Discovery of the Physical Cause of Magnetism .... 9 BULLETIN No. 3 Discoveries in Cosmical Magnetism 33 BULLETIN No. 4 Theory of the Transmission of Physical Forces, Based on the Propagation of Electrodynamic Waves in Right Lines or in Curves Conforming to Fermat's Law of Minimum Path and THE Principle of Least Action 57 BULLETIN No. 5 Theory of the Fluctuations of the Moon's Mean Motion Deduced from Phenomena: Results Explained by the Refraction, Dis- persion AND Perhaps Absorption of Part of the Sun's Elec- trodynamic Wave Energy in Propagation Through the Earth. 97 BULLETIN No. 6 Definite Confirmation of the Discovery op the Physical Cause of Universal Gravitation 127 INTRODUCTION The Science of Astronomy is essentially one of Geometry on a great scale. It consists in the tracing out of regular orbits from exact observations, the deduc- tion of the masses under whose central forces the orbits are described, and the calculation of the perturbations to which these principal motions may be sub- jected. Cosmogony goes further yet, and attempts to discover the foundations of the permanent Geometry of the Heavens. It explains the gradual origin of the system of the world through excessively slow Processes of Capture known to be going on in Nature, and made possible by the primordial diffusion of Cosmical Dust under the action of repulsive forces for the formation of nebulae in the vacant regions of starless space. Both Astronomy and Cosmogony thus require an insight into the physical cause which underlies the observed central forces. This primordial agency, com- bined with the Process of Capture, in the course of millions of ages, has established the recognized order of the heavens. The astronomer therefore labors to throw light on the Cause of Universal Gravitation, so as to find out whether the received Law of Newton is rigorously exact, and wiU hold true for all time, or only a close approximation which may give rise to no large errors of calculation inside of sev- eral centuries. Sir Isaac Newton himself was greatly occupied with these questions, while composing the Principia, 1687, and not very much additional light has been shed on them since, because the Cause of Universal Gravitation has remained as completely hidden from our view as in the days of the formulation of the Law of Gravitation. Notwithstanding this long Stationary Period, the time has now come for a great and significant advance, if we can correctly make out the nature of the physical forces governing the Universe by connecting them with Magnetism, which is simpler and more easily understood. Forty years have now elapsed since the writer's attention was attracted to the Physical Cause of Universal Gravitation as a problem awaiting investigation. The central force which governs the motion of the Moon came under special con- sideration in 1881, and in 1882 the Transit of Venus and the Great Comet which passed so near the Sun, and for months remained visible in the morning sky tiU about sunrise, gave these early studies a more definite direction. Yet the solu- tion of the problem still appeared remote, and for a third of a century, indeed, it was not forthcoming, so that heretofore nothing has been published bearing on the subject. (vii) VIU INTEODUCTION But in April, 1914, a mode of attacking this baffling problem, through the related phenomenon of Magnetism, finally presented itseK; and there seemed to be a reasonable hope that if the nature of Magnetism could be clearly made out, the physical cause of Universal Gravitation also could be established. During the past three years much attention has been given to the subject, with the result that the problem of Magnetism now seems to be solved in so clear and simple a manner that little remains to be desired in the way of improvement. The discovery of a Physical Mechanism, conformable to the well established laws of Dynamics, by which stresses in the medium can be exerted across space, so as to give physical forces acting in right lines, when the transmission is through the free Mther, has been a most urgent desideratum of Science for many centuries. Up to very recent times, however, such a mechanism had not been disclosed. For in an Address on the "iEther of Space," at Bedford College, England, in February, 1914, Sir Oliver Lodge, declared that there is a mysterious force, which he did not understand in the least, the force of gravity. It puts the force of cohesion absolutely to shame. This force of gravity binds the Cosmos together. It must be transmitted by the Mther, but as there is no mechanical connection between Mther and Matter, the nature of the attractive process has not been made out, though Sir Oliver believed it to be electrical. Now whatever shades of opinion may develop among natural philosophers as to the details of the processes disclosed in these Bulletins, it cannot, I think, be denied that Electrodynamic Waves will explain Magnetic, Electrodynamic and Gravitational Attraction, in accordance with the definitely established Laws of Nature. The historical difficulty of finding a physical mechanism between bodies under which they will mutually attract, owing to the interpenetration of waves changing the separate stresses and thus generating forces pulling in straight lines, seems therefore to be overcome. The Electrodynamic Wave-Theory of Universal Gravitation is a necessary consequence of the discovery of the Physical Cause of Magnetism. For some time, it has seemed to me that there could be no mode of attacking the problem of Gravitation except through the more familiar attraction of Magnetism; and the line of reasoning here adopted has therefore been based on the laws of wave action shown to be operative in Electrodynamics. The explanation of Magnetic and of Gravitational Attraction now put forth has the indispensable merit of simplicity and of conformity to Faraday's great experimental discovery that all bodies are magnetic. The magnetism of all matter means, in other words, that it was shown by Faraday's careful Laboratory INTRODUCTION IX Experiments that all bodies emit waves which may be more or less coerced into paral- lelism by the action of suitable electric currents. This comprehensive Electrodynamic Wave-Theory obviously is sufficient to account for the phenomena of Universal Gravitation, and it has the additional advantage of illustrating clearly the unity and correlation of aU the forces of Nature, with which Faraday had become so deeply impressed. Whether the Electrodynamic Wave-Theory also fulfills the necessary condition of being the only possible explanation of gravitation is a question which may be fairly left to the judgment of the reader. The philosophic methods of Archimedes and of Newton naturally explain Gravitational Attraction by the same mechanism which accounts for Magnetic Attraction; and as this Electrodynamic Wave-Theory seems definitely established by the rigorous geometrical and physical conditions fulfilled by the forces pro- ducing the Lunar Fluctuations natural philosophers doubtless will hold that it is also the only possible Theory of Gravitation. It seems certain that no other theory will explain the Attraction operating between bodies, which we ascribe to forces pulling in right lines. Such tension in the medium becomes a maximum directly between the masses, because along these connecting right lines the oppositely directed waves always mutually interpenetrate with double the velocity of Light, thus putting the medium under extraordinary tension, like that of a stretched mass of India rubber. Heretofore, no such natural mechanism as is now outlined has been known to the investigator. And from the author's studies of the rigorous geometrical and physical conditions fulfilled by the forces producing the Lunar Fluctuations he believes that no explanation other than that now offered is mathematically possible; so that the Lunar Fluctuations appear to constitute an observational experimentum crucis of the physical cause underlying Universal Gravitation. The discovery of the physical cause of the hitherto Unexplained Fluctuations of the Moon's Mean Motion has therefore confirmed the Electrodynamic Wave- Theory of Gravitation, and given this theory both a mathematical and physical basis, which will, I think, deserve the earnest consideration of natural philosophers. Accordingly there results the Electrodynamic Wave-Theory of Magnetism, of Electrodynamic Action, and of Universal Gravitation, with proof that these Fundamental Forces of Nature are due to waves propagated with the velocity of Light through the free ^ther, but more slowly through solid masses — whence arises also Refraction, Dispersion, and perhaps Absorption of part of the wave- energy, and thus the hitherto Unexplained Fluctuations of the Moon's Mean Motion. X INTRODUCTION The Lunar Fluctuations discovered by Newcomb were justly regarded as the most enigmatical phenomenon presented by the Celestial Motions. It is remarkable that they now become an experimentum cruets for establish- ing the undulatory nature of Gravitation, owing to the perturbative effects actually observed to be exerted upon the Moon when near the shadow of the Earth. The Greek astronomer Hipparchus initiated the investigation of the Lunar Inequalities through the study of the Evection, about 140 B.C., but left it to be more fuUy worked out by Ptolemy some 300 years later. After steady progress extending over 2000 years and culminating in the recent exhaustive researches of Professor E. W. Brown, the Lunar Theory was almost perfected, — when at last the cause of the Lunar Fluctuations unexpectedly comes to light, and every sensible Inequality in the Motion of the Moon thus finally disappears! These results are of no small interest, and I gladly would have offered them to one of the numerous Learned Societies which exist for the promotion of Science. Several of the foremost mathematicians of Europe and America have generously urged this course, and laid me under lasting obligations for their kindness. But there are too many well known instances of discoveries being misunderstood to justify the belief that the reception accorded to such new work would be suffi- ciently favorable to ensure its prompt publication by conservative committees having no part in its development. It is a somewhat melancholy reflection, but one quite justified by History, that it is chiefly individual discoverers who have the vision to perceive and the moral courage to support new Truth in advance of its triumph. For example, it is well known that the Royal Society delayed unduly and thereby evaded the publication of Newton's Principia, 1686; and to prevent the loss of the work Halley had to print at his private expense the most immortal production of the human intellect. Some things even worse than the smothering of priceless discoveries occasion- ally occur; for in his History of Astronomy, Laplace gives an account of the per- secution of Anaxagoras, at Athens, and justly observes: "To estabUsh itself on earth, Truth has often had to combat accredited errors, which more than once has proved fatal to those who have discovered it." In the well known case of Galileo the discoveries which he made and promul- gated did not quite prove fatal, indeed, yet this first philosopher of Europe, in a blind old age, after a laborious life devoted to the founding of Modern Science, was thrown into prison, because he persisted in publishing the discoveries which established the Copemican System of the world. INTRODUCTION XI Laplace comments on the difficulties of Kepler as follows: "With so many claims to admiration this great man lived in misery, while judicial Astrology, everywhere honored, was magnificently recompensed. Fortunately, the enjoy- ment which a man of genius receives from the truths which he discovers, and the prospect of a just and grateful posterity, console him for the ingratitude of his contemporaries . ' ' Thus the labor of the discoverer always is unduly increased, yet the investigator who succeeds, — simply because he is a labor-loving and truth-loving man, such as HiPPARCHUs was, according to the description given by Ptolemy in the Almagest, Lib. Ill, Ch. I, — is not without the noblest of rewards. And such a true discoverer naturally is unwilling to condescend to secure publication of his priceless treasures. As is weU known, in past struggles for new ideas I have often stood entirely alone; I can again do so, cheerful and hopeful of the ultimate triumph of reason and justice. It took the Capture Theory of Cosmogony eight years to triumph in some unprogressive countries, yet that period is short in the long History of Astronomy. If the];e be those who hesitate to welcome the present results on the nature and mode of propagation of Physical Forces, let us hope that they will not lay themselves open to the grave charge that there are none so blind as those who do not wish to see. Galileo has related how certain learned professors in Padua refused to look through the newly-invented Telescope lest they see Jupiter's sat- ellites, and thus be compelled to admit the truth of his discovery. Unjust criticism is not only useless, but philosophically beneath contempt. He who is deserving of the name of natural philosopher shows a genuine interest in Truth, and labors to make it more accessible to those worthy to take part in such studies. That moral integrity is a prerequisite to the successful cultivation of Science is impressively emphasized by Plato {Republic, Book VII, §535 of Jowett's Translation), and specifically stated by Apollonius of Perga in his second letter to Eudemus. It is conceded that very little progress in regard to the Cause of Universal Gravitation has been made during the past 250 years. As the laws of Electro- dynamic Action were quite unknown in the time of Newton, that great philos- opher naturally was unable to solve the problem; yet this is no reason why we should idly fold our hands today and ignore aU the progress made since the time of Ampere. We should rather assume the responsibility of leadership in the new advance which is demanded by the state of Science in our time. XU INTRODUCTION Probably the present work could have been somewhat improved by still fur- ther delay in publication; but I deemed this course unjust to contemporary in- vestigators, and will cheerfully take the blame for any incompleteness in the present development. It is only by the united efforts of many minds that a new subject may be rounded out in all its aspects. In the course of this thorough test the discoverer himself must make many sacrifices. In the first instance, he has the choice, on the one hand, between following the beaten path, which is the course of least resistance, — safe and conservative, yet involves no step forward; and of opening a new path, on the other, which may lead through thorns and underbrush, and caU for the energy and force of character appropriate to the pioneer. Enormous are the difficulties of the ex- plorer, but great also may be the reward of his daring courage! In the contemplation of the world the philosopher finds that wonders are many, and thus he never ceases to marvel at the mysteries of Nature disclosed to his searching vision. Yet sublime as are the wonders opened to us in the past, it seems that they are as nothing compared to the unseen but stupendous power operative in the process of Universal Gravitation. This power results from the interpenetration of waves traveling from aU bodies with the Velocity of Light and by decreasing the stress in the ^Ether between them thus generating the tre- mendous physical forces required for holding the planets in their orbits ! Lest we forget the stupendous power of Gravitation, in terms of the Strength of Steel, we may recall two instances of the mighty forces now traced to the inter- penetration of Electrodynamic Waves traveling with the Velocity of Light: 1. In the JEther of Space, 1909, pp. 112-126, Sir Oliver Lodge shows from the recognized laws of Dynamics that the pull of the Earth on the Moon is equiva- lent to the breaking strength of a steel column 400 miles in diameter, or of a forest of five million million weightless pillars each a square foot in cross section — the tenacity of the steel being thirty tons to the square inch. 2. In the same way Sir Oliver Lodge shows (p. 130) that the pull of the Sun on the Earth is equivalent to the tenacity of a forest of weightless steel pillars each eleven inches in diameter, acting on every square foot of a hemispherical section of the globe — the steel again having the breaking strength of thirty tons to the square inch. To inquire into the unseen mechanism for producing these wonderful forces would have appealed to the genius of Archimedes or of Newton! No physical problem could be more worthy of the meditation of the natural philosopher! INTEODUCTION XIU With uncovered heads we stand in silent awe at the amazing mysteries un- folded to our contemplation! The curtain of the ages never before was thus parted to open such a vision to the human mind! The revelation of the previously unseen mechanism of the physical world must be pronounced, in Sophoclean phrase, beyond mortals wonderful! It may not be inappropriate to point out a certain Analogy between the Infinite Series of Higher Analysis, so well known to the geometer, and indispensable to the numerical evaluation of the curves traced by the heavenly bodies in their motions and mutual perturbations; and the Eledrodynamic Waves which by mutual interpenetration generate the forces for holding the planets in their orbits and thus cause the phenomenon of Universal Gravitation. The Series of Analysis represent numerical oscillations by which finite and definite values of transcendental functions may be approximated; the Electro- djoiamic Waves are the physical oscillations for generating the corresponding continuous forces under which the heavenly bodies describe the higher curves recognized in the Geometry of the Heavens. If the nature of Series has been mysterious to the geometer, so also are the Waves to the natural philosopher; yet with this Analogy before us the problem of the philosopher is somewhat illum- inated by the close correspondence and unbroken continuity shown to pervade the order of Nature. The author is much indebted to his former Secretary, Mr. D. R. Adams, for efl&cient service in preparing the first outline of the Electrodynamic Wave- Theory, 1914; and to Mr. Joseph Pedeoni, his present Secretary, for a like service in completing this Volume, 1916-17. But of aU those who have contributed to the entire development I owe most to Mrs. See, for steadfast support and patient interest in a labor presenting many difl&culties. It is not yet quite finished, it is true, but with the appearance of the present Volume the ultimate triumph of these discoveries seems fully assured; and when the significance of the advance made comes to be appreciated, it wiU be only right that those who aided in overcoming almost insurmountable obstacles, in a period of universal distress and world darkness, should be gratefully remem- bered as bearers of the Light, T. J. J. SEE. Starlight on Lontre, Montgomery City, Missouri, September 19, 1917. ELECTEODYIAMIC WAYE-THEOBY OF PHYSICAL FOECES BULLETINS I, II, III, IV, V, VI RESULTS OF RESEARCHES ON THE ELECTRODYNAMIC WAVE-THEORY OF PHYSICAL FORCES BULLETIN NO. 1 ANNOUNCEMENT OF TWO IMPORTANT ASTRONOMICAL DISCOVERIES: I. The Physical Cause of the Hitherto Unexplained Fluctuations IN THE Moon's Mean Motion, Dec. 10, 1916. II, The Physical Cause of Universal Gravitation, 1914-17. By T. J. J. SEE After most thorough investigation, — including the independent confirma- tion of the results from both the mathematical and physical points of view, as well as by comparison of the new Physical Theory with the observed Fluctuations, which are the outstanding differences between pure Gravitational Theory and actual observation of the Moon's Mean Motion, as set forth by Newcomb, 1909 — the author of this Bulletin announces that on December 10, 1916, he was able to discover the Physical Cause of the hitherto Unexplained Fluctuations in the Moon's Mean Motion. This discovery will prove to be remarkable afs reveahng the existence of a New Type of Long Inequahty in the Theories of the Celestial Motions. These Inequahties depend on the Electrodynamic Nature of Gravitation, and may occasionally become sensible to observation, but heretofore their existence has scarcely been suspected and never considered probable. The late Professor Simon Newcomb regarded the Unexplained Fluctuations in the Moon's Mean Motion as the most enigmatical phenomenon presented by the celestial motions, and thus the discovery of the cause of these perplexing in- equalities will no doubt occasion surprise among investigators. There naturally will be curiosity to know in what way the discovery was made. Unfortunately, owing to the serious problem of publication for new work, it is not practicable at present to give details of the processes of investigation. I ANNOUNCEMENT OF TWO IMPORTANT ASTRONOMICAL DISCOVERIES The results here set forth, however, will leave no doubt of the rigor of the indi- cated laws of the Moon's Mean Motion. In explanation of this statement it remains to add that the discovery of the Cause of the Moon's Fluctuations was made in the extension of researches on the Cause of Universal Gravitation, with which the author has never ceased to be occupied since 1881, and first had his attention directed to in 1878. It is well known that in April, 1892, he had the honor of discussing the Cause of Gravita- tion in some detail with Professor Sir George Darwin on the occasion of a visit to this eminent mathematician at Cambridge. In spite of all the attention previously given to the subject no conclusion could be reached at that time; and although the problem was kept steadily in view it was twenty-two years later, April, 1914, before any definite promise of a solution opened up. Even then the greater part of the spare moments of the past three years had to be given to perfecting the development of the new Elec- trodynamic Wave-Theory. Early in 1915 a first outline of the work on the Cause of Gravitation was sent to one of the oldest and most celebrated of the Scientific Societies of Europe. Although not published, it seems to have been given much consideration, and in spite of the evidence that it was not fully understood, was found worth filing away in the Archives for future reference. New conceptions seldom are welcomed at first, and besides the conditions in Europe during the past three years have been little suited to philosophic contemplation. The researches on the Cause of Gravitation necessarily have extended over all the related investigations of the past two hundred fifty years, and thus cov- ered a very wide field. Accordingly they cannot be adequately described in a Bulletin, and it is deemed best to defer details of the new Electrodynamic Wave- Theory, in the hope that before many years the work may be brought out as a whole. The intervening time may be utilized for additional researches, and for the preparation and publication of the results. But as the Cause of Gravitation seems clearly and definitely made out, and confirmed by the discovery of the cause of the hitherto V^iexplained Fluctuations in the Moon's Mean Motion, it appears advisable to announce also this discovery in the present Bulletin. The first outline of the Electrodynamic Theory, ascribing the stresses for transmitting forces across space to wave action, in an unpubhshed Manuscript, 570 pages, December 10, 1914, bears the title: "Electrodynamic Theory of Mag- netism and of Universal Gravitation: Discovery of the Cause of Gravitation, with proof that this Fundamental Force of Nature is proi^agated with the velocity of Light." ANNOUNCEMENT OP TWO IMPORTANT ASTRONOMICAL DISCOVERIES 3 Weber's Fundamental Electrodynamic Law of 1846 is shown to be the Law of Nature. Thus the formula of Weber where m and m' are the masses, r the distance, and c the velocity of transmission for the wave action, reduces to Newton's Law of 1686 „ _ mm' U/T d T when the orbit is circular, so that j; = > im ~ ^ ' ^^^ there is no relative motion between the bodies of the system. All the periodic phenomena of Terrestrial Magnetism are explained, including Earth Currents, Auroras, "Magnetic Storms," and the dependence of "Mag- netic Storms" on the sun spot cycle, and a connection is shown to exist between the Magnetism of the Earth and the Electrodynamic Forces which control the motions of the Planets in their orbits. Within the past few years Professor Einstein, who had previously failed to connect Gravitation with Electrodynamics, has published a Relativity Theory for explaining the true law of Gravitation, including the motion of Mercury's Perihelion, etc.; but so far as I know he makes no attempt to assign the Cause of Gravitation. Indeed, under Einstein's vague and chimerical theory, Gravity is not a "force," but "a property of space!" This Relativity Theory of Einstein has appealed to some English mathematicians, but they evidently are misled by de- ceptive analysis, and reasoning on false premises, as so frequently happens when appropriate physical conditions are overlooked. The present author has now shown that at least half a dozen different ex- planations of the motion of Mercury's perihelion are possible. Any one of them a priori is admissible, yet not one of them is established by the necessary and sufficient conditions required to make a conclusive argument. Moreover each of these independent explanations may be varied and combined with the others, so as to multiply indefinitely the possible solutions. The motion oi Mercury's perihelion is therefore in no way decisive for or against any theory of Gravitation _ The Fluctuations of the Moon's Mean Motion, on the other hand, have been generally regarded as essentially incapable of theoretical explanation. In 1909 Professor Newcomb wrote: "I regard these fluctuations as the most enigmatical phenomenon presented by the celestial motions, being so difficult to account for 4 ANNOUNCEMENT OF TWO IMPORTANT ASTRONOMICAL DISCOATERIES by the action of any known causes, that we cannot but suspect them to arise from some action in nature hitherto unknown" (Monthly Notices, January, 1909). This prediction was verified through the discovery of a cause not hitherto suspected, comiected with the Electrodynamic Wave-Theory of Gravitation, by which alone the present author was able to solve the mystery of the Moon's Fluctuations. The accompanying table and plate illustrates the completeness of the new physical theory. It seems quite evident that no outstanding fluctuation now remains in the Moon's motion in excess of 1".0; for the average residual shown for the past eighty years is about 0".70, and on that basis we know from the theory of prob- ability that a periodic inequality of 1".0 cannot remain undetected in the Moon's Mean Motion. This quantity is so small as to be near the limit of vision with a 6-inch tele- scope such as a transit circle. Thus it appears that on the basis of true physical theory, resulting from the extension of the work on Gravitation, I have been able to improve the accuracy of the Moon's predicted mean longitude at least a dozen fold. There are in all three hitherto undiscovered terms of sensible magnitude in the Moon's Mean Motion as follows: 1. The SJiort Period Fluctuation, recurring in 18.0293 years, with a coefficient of 1".0: ALi = 1".0 sin 119°.9675 {t - 1800.0) -|-239°.42S ... (1) 2. The Large Fluctuation of 61.7006 years' period, with coefficient of 3".0, and having an exceedingly close resemblance to the Great Inequality in the mean motion of Jupiter and Saturn, of which the physical cause was discovered by Laplace in 1785: ^U = 3".0 sin !5°.83597 {t - 1800.0) -hl26°.35j ... (2) 3. The Great Fluctuation, estimated by Newcomb to have a period of 275 years, but now found theoretically to have a period of 277.590 years. This involves also the auxiliary period of 17.9971 years. This Great Fluctuation likewise has some analogy to the Great Inequality of Jupiter and Saturn. The coefficient is 13".0 and the formula: AL3 = 13".0 sin 1 1°. 29691 it - 1800.0) -f- 100°.6j ... (3) ■ o " O ~ir 9 An. m. p O o o Zd ^a. 1900 * o ? & "A o 5 •^ti..7>^ i^>: Q h ''9^7% ^/.>r^' o o © V PiscovERY OP THfc Physical CAUSfc OP Trt& f LUCTUATI0;^5 OF* TH£ M00>Hb JAtAH MOlloH 3Y T.J.J.5££ D££.10,191^. Explanation: The very small circles represent the Observed Fluctuations, the larger circles the new Theoretical Fluctuations. + 14.0 -t 12.0 + 10.0 + 6.0 ^ -I- 6.0 H -^4.0< o -t- 2.0^P - 6.0 2 "^"^-.IE.0 I -)40^ jia - 2.0 -4.0 -le.o ANNOUNCEMENT OF TWO IMPORTANT ASTRONOMICAL DISCOVERIES 5 Any Theory of Gravitation which explains the cause of the Moon's Fluctua- tions must be regarded as meeting the conditions of an experimentum crucis. As the Electrodynamic Wave-Theory triumphs under this severe test it must be held to rest on a true Law of Nature. On the other hand the light afforded by the Electrodynamic Wave-Theory for making this difficult discovery in the Lunar Theory, which successfully chal- lenged the greatest mathematicians for more than a century, shows the exceeding usefulness of the unpublished researches on the Cause of Gravitation. Having first drawn attention, in 1869, to the departure of the Moon's actual motion from the theoretical places indicated by pure Gravitational Theory, New- COMB worked out the chief Fluctuations in his Researches on the Motion of the Moon, 1878, and finished his final memoir on the subject just before his death in 1909. Hill became deeply interested in the Lunar Theory in the early seventies, and in 1877 published his celebrated Researches in the Lunar Theory. To the date of his death in 1914, Dr. Hill never ceased to be interested in the Motion of the Moon. Like Newcomb he was occupied with the subject over forty years, yet he was unable to throw any light on these mysterious Fluctuations. It was thus left to Professor E. W. Brown to make the most exhaustive researches on the Lunar Theory yet attempted. He has now worked on the subject for more than a quarter of a century, but has not found any satisfactory explanation of the Fluctuations established by Newcomb. Accordingly it would appear that the discovery of the Physical Cause of these mysterious Fluctuations may be regarded as no ordinary achievement. If there be those who hesitate to welcome the new results, they will still have the oppor- tunity of improving on them in their own way, without embarrassment with the details of the author's processes of discovery. Universal Gravitation is now shown to be due to Electrodynamic Waves traveling with the velocity of light, and reacting on matter through stresses in the medium due to such waves, which are shown to be decidedly different from those conceived by Maxwell in his Electromagnetic Theory of Light. In cor- recting the researches of Maxwell and removing a traditional defect in the Wave- Theory of Light, the author has been able to harmonize them essentially with some curious views of Newton, on the cause of Gravitation, dating from 1721. Ampere's explanation of Magnetic Forces, as due to elementary currents of electricity circulating around the atoms is also harmonized with the Electrody- namic Wave-Theory; and altogether the unity and harmony introduced into our Theories of the Physical Universe is remarkable. 6 ANNOUNCEMENT OF TWO IMPORTANT ASTRONOMICAL DISCOVERIES Not only does the assigned Cause, of Gravitation explain all known celestial phenomena, and triumph under the experimentuvi crucis of the Lunar Fluctuations, the explanation of which appears to be unique, but it seems thereby effectually to exclude every other conceivable explanation. For mere coincidence could not give this perfect physical theory of the intricate phenomena of the Lunar Fluctua- tions. It is only for these weighty reasons that the author has finally decided to announce the discovery of the cause of the Moon's Fluctuations and of Universal Gravitation. These two discoveries necessarily are associated, one and inseparable. Details of the more elaborate researches on the Cause of Gravitation must await publication, but the accompanying proof of the value of the discovery in the Lunar Theory is ample, and it should be of interest to investigators in more than one of the Physical Sciences. T. J. J. SEE Starlight on Loutre, Montgomery City, Missouri, February 19, 1917. ANNOUNCEMENT OF TWO IMPORTANT ASTRONOMICAL DISCOVERIES Physical Cause of the Fluctuations of the Moon's Mean Motion T. J. J. See, Dec. 10, 1916 t 4, 1".0 sin Ai Ai 3".0sinA2 = ^u vl.-l 13".0 sin Ai = AL3 Ob8. Fl. Cal. Fl. i = 3 1=1 Final Resid. O-C Wt. New- comb o " ^^ „ ^1 n ^, 1829.5 108.53 + 0.95 298.51 -2.34 138.86 + 8.55 + 7.0 + 7.16 -0.2 20.0 1833.5 188.42 -0.15 321.85 -1.85 144.05 + 7.63 + 6.1 + 5.63 +0.5 10.0 1838.5 308.27 -0.95 351.03 -0.47 150..53 + 6.40 + 5.8 + 4.98 +0.8 30.0 1843.0 18.14 +0.31 17.30 +0.89 156.37 + 5.21 + 6.3 + 6.41 -0.1 20.0 1846.5 88.04 + 1.00 37.72 + 1.83 160.91 + 4.25 + 5.3 + 7.08 -1.8 10.0 1848.5 127.84 + 0.79 49.39 + 2.28 163..50 + 3.69 + 5.6 + 6.76 -1.1 8.0 1849.5 147.81 + 0..53 .55.23 + 2.46 164.80 + 3.41 + 3.5 + 6.40 -2.9 15.0 1850.5 167.78 +0.21 61.07 + 2.62 166.10 + 3.12 + 4.2 + 5.94 -1.7 18.0 1851.5 187.75 -0.13 66.91 +2.76 167.40 + 2.84 + 3.7 + 5.47 -1.7 12.0 1852.5 207.72 -0.46 72.75 + 2.86 168.70 + 2.55 + 3.5 + 4.95 -1.4 8.0 1853.5 227.69 -0.74 78.59 + 2.94 170.00 + 2.26 + 3.0 + 4.46 -1.4 7.0 1854.5 247.66 -0.92 84.43 + 2.98 171.30 + 1.97 + 3.4 + 4.03 -0.6 14.0 1855.5 267.63 -1.00 90.27 +3.00 172.60 + 1.77 + 3.8 + 3.77 ±0.0 6.0 1856.5 287.60 -0.95 96.11 +2.98 173.90 + 1.38 + 3.3 + 3.41 -0.1 6.0 1857.5 307.57 -0.79 101.95 +2.93 175.20 + 1.09 + 3.2 + 3.23 ±0.0 9.0 1858.5 327.54 -0.54 107.75 + 2.86 176.47 + 0.80 + 4.3 + 3.12 + 1.2 8.0 1859.5 347.51 -0.22 113.59 + 2.75 177.77 + 0.51 + 4.4 + 3.04 + 1.4 6.0 1860.5 7.48 +0.13 119.43 + 2.61 179.07 + 0.21 + 4.1 + 2.95 + 1.1 12.0 1861.5 27.45 +0.46 125.27 +2.45 180.37 - 0.08 + 3.2 + 2.83 +0.4 5.0 1862.5 47.42 +0.74 131.11 +2.26 181.67 - 0.28 + 3.5 + 2.72 + 0.8 7.0 1863.5 67.39 +0.92 136.95 +2.05 182.97 - 0.67 + 2.3 + 2.30 ±0.0 6.0 1864.5 87.36 + 1.00 142.79 + 1.81 184.27 - 0.97 + 1.9 + 1.84 +0.1 10.0 1865.5 107.33 +0.95 148.63 + 1.56 185.57 - 1.26 + 1.3 + 1.25 ±0.0 5.0 1866.5 127.30 +0.79 154.47 + 1.29 186.87 - 1.56 + 1.2 + 0.52 +0.7 6.0 1867.5 147.27 +0.54 160.31 + 1.01 188.17 - 1.85 — 0.8 - 0.30 -0.5 5.0 1868.5 167.24 +0.22 166.11 +0.72 189.44 - 2.13 — 1.2 - 1.19 ±0.0 10.0 1869.5 187.21 -0.12 171.95 +0.42 190.74 - 2.42 - 0.9 - 2.12 + 1.2 9.0 1870.5 207.18 -0.46 177.79 +0.11 192.04 - 2.71 - 2.1 - 3.06 + 1.0 6.0 1871.5 227.15 -0.73 183.63 -0.19 193.34 - 3.00 - 4.4 - 3.92 -0.5 10.0 1872.5 247.06 -0.92 189.47 -0.49 194.64 - 3.29 — 5.1 - 4.70 -0.4 16.0 .1873.5 267.03 -1.00 195.31 -0.79 195.94 - 3.57 - 5.4 - 5.36 ±0.0 12.0 1874.5 287.00 -0.95 201.15 -1.08 197.24 - 3.85 - 6.1 - 5.88 -0.2 8.0 1875.5 306.97 -0.80 206.99 -1.36 198.54 - 4.13 — 6.5 - 6.29 -0.2 8.0 1876.5 326.95 -0.54 212.83 -1.63 199.54 - 4.41 - 6.5 - 6.58 +0.1 30.0 1878.5 6.88 +0.12 224.47 -2.10 202.41 - 4.96 - 6.6 - 6.94 +0.3 18.0 1879.5 26.83 +0.45 230.31 -2.21 203.71 - 5.23 - 5.7 - 6.99 + 1.3 14.0 1880.5 46.82 +0.72 236.15 -2.49 205.01 - 5.49 - 6.9 - 7.26 +0.3 20.0 1881.5 66.79 +0.92 241.99 -2.65 206.31 - 5.76 - 7.3 - 7.49 +0.2 12.0 1882.5 86.76 + 1.00 247.83 -2.78 207.61 - 6.03 - 7.4 - 7.81 +0.4 8.0 1883.5 106.73 +0.96 253.67 -2.88 208.91 - 6.29 - 8.4 - 8.21 -0.2 7.0 1884.5 126.70 +0.80 259.51 -2.95 210.21 - 6.54 - 8.6 - 8.69 +0.1 30.0 1885.5 146.67 +0.54 265.35 -2.99 211.51 - 6.79 - 9.2 - 9.24 ±0.0 50.0 1886.5 166.64 +0.23 271.19 -3.00 212.81 - 7.04 — 9.8 - 9.81 ±0.0 18.0 1887.5 186.61 -0.11 277.03 -2.98 214.11 - 7.29 — 9.8 -10.38 +0.6 20.0 8 ANNOUNCEMENT OF TWO IMPORTANT ASTRONOMICAL DISCOVERIES Physical Cause of the Fluctuations of the Moon's Mean Motion (Continued) I ' Ai l".0 8inAi = ALi A2 3".0 sin Ai = AL2 A, 13".0 sin Az = ALs Obs. Fl. Cal. Fl. i=S = SAL,. i = l Final Resid. O-C Wt. Nbw- COMB „ O // O ^, ,, ,, „ 1888.5 206.58 ^-0.45 282.83 -2.92 215.38 - 7.53 -11.0 -10.90 -0.1 8.0 1889.5 226.55 5-0.72 288.67 -2.84 216.68 - 7.76 -11.2 -11.32 +0.1 7.0 1890.5 246.52 "-0.92 294.51 -2.73 217.98 - 8.00 -11.4 -11.65 +0.2 10.0 1891.5 266.49 -1.00 300.35 -2.59 219.28 - 8.23 -11.3 -11.82 +0.2 16.0 1892.5 286.46 1-0.96 306.19 -2.43 220.58 - 8.46 -11.0 -11.85 +0.8 17.0 1893.5 306.43 1-0.80 312.03 -2.22 221.88 - 8.68 -11.3 -11.70 +0.4 15.0 1894.5 326.40 -0.55 317.87 -2.01 223.18 - 8.90 -11.8 -11.46 -0.3 30.0 1895.5 346.37 ?-0.23 323.71 -1.78 224.48 - 9.11 -11.2 -11.12 -0.1 60.0 1896.5 6.34 >0.11 329.35 -1.52 223.78 - 9.32 -10.4 -10.73 +0.3 60.0 1897.5 26.31 Vo.44 335.39 -1.25 227.08 - 9.52 -11.5 -10.33 -1.2 20.0 1898.5 46.28 ;+0.72 341.19 -0.97 228.35 - 9.71 -11.2 - 9.96 -1.2 28.0 1899.5 66.25 ^+0.91 347.03 -0.67 229.65 - 9.91 -11.3 - 9.67 -1.6 12.0 1900.5 86.15 "+1.00 352.87 -0.37 230.95 -10.10 -10.1 - 9.47 -0.6 15.0 1901.5 106.12 +0.90 358.71 -0.07 232.25 -10.28 -10.4 - 9.45 -0.9 16.0 1902.5 126.09 +0.81 4.55 +0.24 233.55 - 10.46 -10.2 - 9.41 -0.8 18.0 1903.5 146.06 1+0.56 10.39 +0.54 234.85 -10.63 -10.0 - 9.53 -0.5 12.0 1904.5 166.03 '+0.24 16.23 +0.84 236.15 - 10.80 - 9.7 - 9.72 ±0.0 20.0 1905.5 186.00 +0.10 22.07 + 1.13 237.45 - 10.96 - 9.5 - 9.73 +0.2 20.0 1906.5 205.97 -0.44 27.91 + 1.40 238.75 -11.12 - 9.8 -10.16 +0.3 16.0 1907.5 225.94 -0.72 33.75 + 1.66 240.05 -11.26 - 9.3 -10.32 + 1.0 15.0 1908.3 241.94 -0.88 38.38 + 1.86 241.06 -11.38 - 9.3 -10.40 + 1.1 9.0 hsued May 8, 1917 PUBLISHERS : THOS. P. NICHOLS & SOX CO., Lynn, Mass., U. S. A. © WM. WESLEY & SON, London A. HERMANN ET FILS, Paris RESULTS OF RESEARCHES ON THE ELECTRODYNAMIC WAVE-THEORY OF PHYSICAL FORCES BULLETIN NO. 2 DISCOVERY OF THE PHYSICAL CAUSE OF MAGNETISM Bt T. J. J. SEE I Geneeal Introductory Remarks Having announced in the first of this series of Bulletins the discovery of the physical cause of the hitherto Unexplained Fluctuations in the Moon's Mean Motion, and of Universal Gravitation, and indicated the main features of the new theoretical explanation of the Lunar Fluctuations, which are shown to constitute an experimentum crucis for establishing the truth of the Electrodynamic Wave- Theory of Universal Gravitation, the author is able to follow these significant results with the announcement of the discovery of the Cause of Magnetism, — including the Magnetism of the Earth and the periodic changes therein depending on the Sun and Moon, — in the action of Electrodynamic Waves of a New Type. These waves proceed from the equators of the atoms, are flat in those planes and in all planes normal to the lines of force, and propagated with the Velocity of Light. Notwithstanding the enormous development of the modern physical sciences which deal with Electricity and Magnetism, beginning with the publication of Gilbert's Treatise De Magnete, 1600, it is a recognized fact that up to the present time the nature of Magnetism has successfully challenged the ingenuity of the natural philosopher. About half a century ago, however. Maxwell put into mathematical language some penetrating observations of Faraday (Experimental Researches in Electricity, §1297, etc., Nov., 1837) in regard to the properties of the fines of force: and was able to attribute the properties of the magnetic field to the effects of rotations around the lines of force. Indeed, at an earlier date Lord Kelvin had treated of Faraday's discovery of 1845, that there is sensible rotation of a beam of polar- (9) 10 DISCOVERY or THE PHYSICAL CAUSE OF MAGNETISM ized light when it is passed through heavy glass, carbon disulphide, etc., along the path of a magnetic hne of force (Proc. Roy. Soc, Vol. VIII, June, 1856, Phil. Mag., March, 1857, Baltimore Lectures, 1904, Appendix F.) Lord Kelvin inferred that "the magnetic influence on light discovered by Faraday depends on the direction of motion of moving particles," and that "Faraday's optical discovery affords a demonstration of the reahty of Ampere's explanation of the ultimate nature of magnetism." After discussing Rankine's hypothesis of molecular vortices, which he had himself developed at length, Lord Kelvin finally concludes: "I think we have good evidence for the opinion that some phenomenon of rotation is going on in the magnetic field, that this rotation is performed by a great number of very small portions of matter, each rotating on its own axis, this axis being parallel to the direction of the magnetic force, and that the rotations of these different vortices are made to depend on one another by means of some kind of mechanism connecting them." Maxwell followed up Kelvin's researches, and, having become associated with Faraday in 1861, formulated a mathematical theory of the physical lines of force, 1861-2, and applied the theory of molecular vortices to Magnetic phenomena and to Statical Electricity. Notwithstanding the originality, elasticity and vigor of his powers at thirty, Maxwell had great difficulty in understanding the mechanism of molecular vortices, because he had no suspicion that they represented flat waves, having common direction and velocity and thus would be devoid of friction and dissipation of the Energy of motion into heat. This difficulty indeed has continued to the present day. For in his Mathe- matical Theory of Electricity and Magnetism, third edition, 1915, p. 486, Dr. Jeans dwells at length on "our utter ignorance" of the mechanism by which action is transmitted through the aether. Accordingly, in his celebrated Memoir "On the Physical Lines of Force," 1861, (cf. Scientific Papers, Vol. I, pp. 468, 486) Maxwell, by default of the hypothesis of Waves, had to imagine a very artificial constitution of the aether: "I have found great difficulty in conceiving of the existence of vortices in a medium, side by side, revolving in the same direction about parallel axes. The contiguous portions of consecutive vortices must be moving in opposite direc- tions; and it is difficult to understand how the motion of one part of the medium can coexist with, and even produce, an opposite motion of a part in contact with it." "The only conception which has at all aided me in conceiving of this kind of motion is that of the vortices being separated by a layer of particles, revolving DISCOVERY OF THE PHYSICAL CAUSE OP MAGNETISM 11 each on its own axis in the opposite direction to that of the vortices, so that the contiguous surfaces of the particles of the vortices have the same motion." "In mechanism, when two wheels are intended to revolve in the same direc- tion, a wheel is placed between them so as to be in gear with both, and this is called an 'idle wheel'. The hypothesis about the vortices which I have to suggest is that a layer of particles, acting as idle wheels, is interposed between each vortex and the next, so that each vortex has a tendency to make the neighboring vortices revolve in the same direction with itself." It was from a new point of view, attained in 1914, that the present writer was able to refer Magnetic phenomena to waves imagined as flat in planes normal to the lines of force, and thus producing the observed rotation of the plane of polarized light discovered by Faraday in 1845. As this view of magnetism overcomes entirely the difficulties just described by Maxwell, and has since been fully confirmed, it is now deemed advisable to communicate some of the results to other investigators, in the hope that they may find the new lines of inquiry worthy of attention in their researches. The present treatment is restricted to brief notice by Bulletin, yet this may suffice to elucidate the chief phenomena. Accordingly, anyone may join in tliese studies who has the power of independent thinking, if his mind is not too much biased by misleading traditions, which unfortunately still are very numerous. II Analysis of Maxwell's Final Conclusions As we have seen above it was in 1861 that Maxwell confirmed Kelvin's conclusions from strict dynamical reasoning, 1856, that the transmission of Mag- netic Force is associated with a rotatory motion of the small parts of the medium, (cf. Maxwell's Article Attraction, Encyc. Brit., 9th edition, 1875, reprinted in the Scientific Papers, Vol. II, p. 488, and Treatise on Electricity and Magnetism, sec- tions 791, 813, 816, 821, 822, 830 and 831.) From the first Maxwell saw clearly that there is rotatory motion going on in the medium and that the axis of rotation is in the direction of the magnetic force. In his Treatise, §821, he concludes as follows: "The only resemblance which we can trace between a medium through which circularly polarized light is propagated and a medium through which lines of magnetic force pass, is that in both there is a motion of rotation about an axis. But here the resemblance stops ..." "There is nothing, therefore, in the magnetic phenomenon which corresponds to the wave length and the wave-propagation in the optical phenomenon. A 12 DISCOVERY OF THE PHYSICAL CAUSE OP MAGNETISM medium in which a constant magnetic force is acting is not in consequence of that force, filled with waves traveling in one direction, as when light is propagated through it." Accordingly, whilst Maxwell's great mathematical and physical intuition enabled him to forecast and to outline the Electromagnetic Theory of Light, which was afterwards so brilliantly confirmed by Hertz, he was never able to make out the undulatory nature of Magnetism. It is evident that Maxwell sought for waves along the lines of force instead of at right angles to these lines; and, not finding any trace of such waves along the lines of force, was led to the conclusion that they do not exist in Magnetic phenom- ena, as shown above in the quotation from §821 of the Treatise. Apparently it never occurred to him to consider magnetic waves in planes normal to the lines of force, although he knew the observed rotations were in these planes, and rec- ognized that the magnetic force consists in pressure normal to and tension along these lines. In §§641-46 of Maxwell's Treatise on Electricity and Magnetism, it is shown how electric and magnetic stresses may be analyzed, and the phenomena referred to the state of the medium. In the Article Attraction, 1875, Maxwell describes the theory as follows: "It is there shown that, if we assume that the medium is in a state of stress, con- sisting of tension along the lines of force, and pressure in all directions at right angles to the lines of force, the tension and the pressure being equal in numerical value and proportional to the square of the intensity of the field at the given point, the observed electrostatic and electromagnetic forces will be completely accounted for." "The next step is to account for this state of stress in the medium. In the case of electromagnetic force we avail ourselves of Thomson's deduction from Faraday'^ discovery stated above. We assume that the small parts of the medium are rotating about axes parallel to the lines of force. The centrifugal force due to this rotation produces the excess of pressure perpendicular to the lines of force. The explanation of electrostatic stress is less satisfactory, but there can be no doubt that a path is now open by which we may trace to the action of a medium all forces which, like the electric and magnetic forces, vary inversely as the square of the distance, and are attractive between bodies of different names, and re- pulsive between bodies of the same names." Maxwell then turns to the consideration of the attraction of gravitation but as in all ca.s(vs he imagines the pressure to be along the lines of the gravitative force, combined with a tension in aU directions at right angles to these lines, he DISCOVERY OF THE PHYSICAL CAUSE OF MAGNETISM 13 is unable to make progress. This view of the stresses is incomplete, and thus it is not remarkable that those who have followed him have also failed. (Cf. Mm- CHiN, Treatise on Statics, Vol. II, 1886, pp. 448-458). Maxwell adds in conclusion that stresses of the kind he imagined "would no doubt account for the observed effects of gravitation. We have not, however, been able hitherto to imagine any physical cause for such a state of stress." It may be worth noting that Professor Challis of Cambridge long gave con- siderable attention to the problems of Attraction, and had ascribed the observed forces to wave action. In reviewing Challis' "Essay on the Mathematical Prin- ciples of Physics," 1873, Maxwell was not convinced by the author's argument, yet adds that gravity is the most universal and mysterious of all actions. "What- ever theory of the constitution of bodies holds out a prospect of the ultimate explanation of the process by which gravitation is effected, men of science will be found ready to devote the whole remainder of their hves to the development of that theory." If so much importance avowedly was attached to the solution of the problem of Attraction by the greatest mathematician and natural philosopher of the past age, the present effort to give an accurate account of the successive steps in our progress will not be considered superfluous. Ill The Cause of the Attraction of Two Magnets 1. With this explanation of Maxwell's researches we shall consider the cause of the mutual action* of two magnets, a chief magnet A, and a secondary magnet or needle B. Under the hypothesis of Electrodynamic Waves, we see that if the chief magnet acts alone, the waves proceeding from it give rise to in- crease of pressure along the plane of xy, which is here taken at right angles to the axis of the magnet. 2. At any distance r = Vx^ + y^ from the axis of the magnet, the stress on a small rectangle dxdy may be denoted by m. And the stress over a larger area S = fj'dxdy of any figure in the x2/-plane, will be given the double integral jfmdxdy (1) *We are here concerned with the Physical Cause of the Stresses in Magnetism. The full mathematical theory of these stresses, without explanation of the mechanism by which they are produced, is given by Maxwell (Treatise, §§641-645) Minchin (Treatise on Statics, Vol. II, 1886, §396) Jeans (Mathematical Theory of Electricity and Magnetism, 1915, §§157-168, 193- 203) and other authorities. The present discussion is restricted to the proof of the mech- anism involved; for as Dr. Jeans says (p. 486) we have heretofore been "in utter ignorance of the ultimate laws which govern action in the ether." 14 DISCOVERY OF THE PHYSICAL CAUSE OF MAGNETISM This resultant stress always has a definite value, depending on the variation of the element of stress on the surface elements dx dy and the limits of the double integral. 3. In general the mean intensity of the stress is given by the equation m P = ^ _ ffmdxdy ^ ' " S ffdxdy And this may be applied to a surface of any extent or figure. Tension is equivalent to negative pressure. And the two types of stress on equal elements of the surface of the medium are numerically of equal magnitude. In the equations (1) and (2) the stress on the xy-plane is really a tension, but its value is identical with the pressure in the direction of the xy-plane. 4. Let us now consider the cause of Attraction in the First Case, where the Magnets present unlike poles. We may take the medium already stressed, as Maxwell showed it is, by the chief Magnet A, having increase of pressure normal to the lines of force and corresponding tension along these lines. At the centre of the Magnet the increase of pressure is in the direction of the xy-p\ane, and the tension therefore along the s-co-ordinate. 5. Now imagine a Needle or Secondary Magnet introduced into the field of the chief magnet A, with its center lying exactly in the plane of xy, as shown in the figure, which requires brief explanation. The curve of wave amplitude is greatly magnified, so as to render the law of the variation with the distance visible to the eye. The curve is a rectangular k k hyperbola referred to its asymptotes, y = - , or a = - . It is well known that in wave motion the amplitude varies inversely as the distance, but the energy of the wave action varies as the square of the amplitude, k^ or E = y^'i which corresponds to the forces following the law of the inverse squares, established for the attraction of Magnets by Gauss, Intensitas Vis Magneticce, 1833, §21, and for the attraction of the Planets under Universal Gravitation by Newton, Principia, 1687. Returning now to the stress in the medium between the magnets A and B, we perceive that the rotatory elements of the new waves arising from B will at every point of the plane tend to counteract those from the chief Magnet A. The pressure parallel to the plane of xy is thus decreased, as if by a new tension super- posed in that plane, with the value t = — m on any element dxdy. DISCOVEEY OF THE PHYSICAL CAUSE OF MAGNETISM 15 6. The resulting modified mean stress or mean new pressure in the direction of the xy-Tplane becomes P' = P-p=//&dxdy -ffvsdxdy (3) which necessarily is less than P. Accordingly, as the Medium is less stressed when the Needle acts than when it does not, there results between the two ponder- able bodies decrease of pressure in the direction of the plane xy. 7. The Pressure normal to the lines of force, as Maxwell shows, gives rise to a tendency in the Medium to expand, just as the parts of compressed India rubber do. Hence Faeaday's observation that the lines of force tend to repel one another and to separate. Thus the tendency to expansion of the compressed medium inevitably leads to the result observed by Faraday. As the pressure is decreased by the action of the Needle, it is equivalent to a new tension in the direction of the xy-Tphjxe. Hence we inevitably have attraction between the chief magnet and the needle. Like a stretched India rubber layer contracting in the xy-plane, the Medium contracts in that direction, thus drawing the magnets A and B together. 8. It is readily understood that although the introduction of the needle relaxes the pressure at every point of xy and decreases correspondingly the tension along the z-co-ordinate, the new stress, thereby superposed, is numerically greatest at the least distances from both bodies, and thus directly between the two magnets. Accordingly in the contraction of the Medium, like the relaxing of a stretched sheet of India rubber, the magnets with unlike poles are drawn together. This ex- planation is simple, and in view of Maxwell's analysis leaves nothing to be desired either mathematically or physically. IV The Cause of the Repulsion of Two Magnets 9. To reach the case of repulsion we now reverse the ends of the secondary magnet, so that the poles presented are like. In this case the mean pressure parallel to the plane of xy will become everywhere a maximum: P' = P + p = f/Mxdy + ff^dxdy (4) So far from developing tension between the two ponderable bodies, this case offers increase of stress, increase of mean pressure parallel to the xy-plane, and in- crease of mean tension parallel to the z-axis. 16 DISCOVERY OF THE PHYSICAL CAUSE OF MAGNETISM 10. The Medium is thus given an augmented stress and the two ponderable bodies are forced apart, as actually observed. The stress appropriate to each body separately is augmented by the pressure and tension superposed by the other. Accordingly, either body, by the wave action of the other, can exert a given stress at a greater distance than without it, so that the double Magnetic system reacts as if by mutual repulsion of the masses. 11. In this way we explain very simply the celebrated phenomenon of the attraction and repulsion of two Magnets. The explanation involves no hypotheses except the single one of the stressing of the medium by plane waves for producing the rotations of the elements around the lines of force, in accordance with Faraday's experiment of 1845, on the rotation of a beam of polarized light by magnetism. 12. It follows from the Electrodynamic Wave-Theory of Magnetism, that such waves proceed from every atom in each magnet. At any point of space the resulting rotations of the medium generally are not in the same plane, but in planes mutually inclined at the angle e. The intensities of the wave energies are I = —^ > J'= —75- , and therefore in a limited region the effect upon the medium from pairs of masses of atoms dnii = 0-2 dxi dy-i dz^ , dmy = o-i dxi dyi dzi , would lead to the stress element due to the resultant forces (5) dP — 5 — cos € 5 — dXi dVi dz^ dx\ dvi dzi j.^2 j.^2 And for the whole of the two magnets* exerting stresses in the medium throughout all space the sextuple integration would give for the resulting attraction +00 -H» +00 -1-00 -(-00 +w (6) '^'^Jjjjjj [p-^-T^'ios. e'^^-^-^dxidyidzidxidyidzAdx'dy'dz'dxdydz —00 —00 —00 —00 — 00 —CO In this equation we have put ri^ = {x—x^^ + {y—yi)^ + {z—z^"^ for the squared distance of the point {x, y, z) from the atoms of the primary mass, and equally *In many works on Magnetism, such as that of Dr. Jeans, §§421-2, p. 379, it is shown that the force between two magnets varies as the inverse fourth power of the distance. If the waves acting from each magnet be imagined concentrated at the mean distance of the other — which is essentially the view taken in an integral formula, where each set of waves is referred to its originating centre, — then in this expression r' = r, and we get the law of the inverse fourth power characteristic of magnetic mass action. DISCOVERY OF THE PHYSICAL CAUSE OF MAGNETISM 17 Ti^ = {x' — x^^ + {y' — Vi)"^ + {z' — z^"^, for the squared distance from the atoms of the secondary magnet. The co-ordinates {xi, yi, Zi), (Xi, y^, z^ relate to the spaces occupied by the atoms within the surfaces of the two bodies, while {x, y, z), (x', y', z') represent all space referred to the centres of gravity of these bodies as wave centres. The sextuple integral (6) determines the action due to two magnetic systems, with their waves interpenetrating throughout all space, and by the stressing of the medium, thus causing the attraction observed. If the matter of two Magnets be imagined expanded in to infinite cosmical clouds or nebulae filling unlimited space, and interpenetrating each other in any manner, the mutual potential energy of their matter under Universal Gravita- tion would be + 00 4-00 +05 +00 +00 +00 _ c c c c c c '"^' ^^ '^y' ^^' ^^ ^^ ^^ ^ J J J J J J ^/(x-x'Y + {y-y'Y + (3^^^ ^'^^ ftn _rrt no re, en rw ^ ^ • \u %f ^ • \ t -CO _00 —00 —00 Let I and I' be the intensities of the magnetization due to the two magnets, and X, [X, v, X', \i! , v' the direction cosines of their axes; then the mutual poten- tial energy of the two magnets, imagined as nebulae of unlimited extent and inter- penetrating each other in any manner, will be + 03 +CO +0S +00 +00 +05 ^ = ///i7/ ^i'^^h^'i>i''^^''^'^'^i^4y^'^''^'^'^ (8) _00 —00 —00 —00 —00 —00 When the two magnets are distinct and confined within finite surfaces, the above equations (6) and (8) may be reduced to the form of a triple integral appro- priate for the mutual action of two finite magnetic systems each external to the other, but with their waves interpenetrating throughout space, and causing attrac- tion by the stresses developed in the medium. Let us denote by R and R' the resultant forces at any point of space from all the atoms in each of the two magnets. Then this equation becomes the triple integral : + 00 +00 +05 ^ = T- S S S^^' ^°^ ^ dxdydz , (9) _00 _00 —05 where is the angle between the direction of the resultant forces. By its infinite limits this integral includes all space, and thus expresses the action resulting from the stressing of the aethereal medium by the independent systems of electro- dynamic waves emitted by the atoms of the two magnets and propagated through- out infinite space with the velocity of light. 18 DISCOVERY OP THE PHYSICAL CAUSE OP MAGNETISM In concluding this analysis of the attraction and repulsion of magnets, it only remains to point out, that the poles are positive or negative, according as the molecular rotations are positive or negative. The action thus depends on the side from which we view the vortices or the plane waves emitted by the atoms of the magnets. And as electro-magnets acquire their properties under the action of currents which line up their atoms in parallel planes, held by Ampere, 1822, the electrodynamic wave-theory impHes that mutual interactions of waves alone are concerned in magnetic attraction. Accordingly the old conception of two fluids finally disappears from the theory of magnetism. Whether this wiU lead also to the disappearance of the terms positive and negative as applied to electricity remains to be decided; but it is not easy to see how our views on this hitherto obscure subject can escape material modification. V New Theory op Oersted's Experiment op 1819 In view of the foregoing theory of Magnetic Attraction by stresses in the Medium, due to waves, we are now able to give a simple dynamical explanation of Oersted's celebrated experiment of 1819, as follows. 1. Imagine a wire bearing a current from South to North, as shown in the accompanying Figure 2, and let a magnetic needle be suspended below it. 2. Owing to the rotations in the earth's field it will be seen that looking north right handed rotations make up the elements of the electrodynamic waves from the wire, while viewed from the south pole left handed rotations proceed from the needle below the wire, as shown in the figure. This gives a general view of the nature of the mutual wave action. 3. The result is that at every point of space the stress in the medium, due to the current, is decreased by the waves due to the magnet; and when the pressure and tension are thus decreased, the medium contracts like a stretched mass of India rubber. The contraction is greatest between the needle and the wire, and thus the needle is drawn to the wire. This will apply to a copper wire, or wire of any conducting material, and hence the explanation is general, applying directly to Arago's experiment of 1820. 4. It has been held from the days of Oersted that the current only directs the needle, but neither attracts nor repels it (cf. Maxwell's address on Action at a Distance, Scientific Papers, Vol. II, p. 317). Maxwell apparently was unable to detect the error in this reasoning, and, with his endorsement of the view of Oer- DISCOVERY OP THE PHYSICAL CAUSE OP MAGNETISM 19 STED, it seems to have continued to the present time.* But the above reasoning shows the error in the traditional theory of this celebrated experiment, which was the very beginning of Electrodynamics. VI Reconciliation of Ampere's Theory op Electric Currents Cir- culating ABOUT THE AtOMS WITH THE ElECTRODYNAMIC WaVE-ThEORY OF Magnetism From experiments made in 1822, Ampere found that when the wire con- ducting a positive current in a solenoid is wound right handed about the steel bar within, that end of it becomes the south pole. Instead of attributing magnetic phenomena to two fluids Ampere assumed that each individual molecule of a magnetic substance is transversed by a closed electric current, and further that these molecular currents are free to move about their centers, and thus forced to line up in parallel planes by the action of the electric current. They are sur- rounded by an infinite system of plane waves traveling outward through space in all directions, and thus called a polarized or magnetic field. Ampere's Theory of Magnetism was proposed in 1822 {Journal de Physique, Tome XCIII, or Recueil d' Observations, Paris, 1822, pp. 164-174), and ascribes the power of magnetizing to the turnable character of the molecular currents about the molecules. He cites various considerations in support of this view, and in paragraph 7 says they "confirm the opinion, founded moreover on a compari- son of all the facts, that the properties of magnets are really due to the continued movement of two electric fluids around their particles." (Theorie Mathematique des Phenomeones Electro-dynamiques, A. M. Ampere, 2d edition, Paris, 1883, p. 151). In the accompanying diagram. Fig. 3, we give the mathematical conditions which prove that the Theory of Ampere is identical with the Wave-Theory: there is thus shown to exist an exact agreement of the two theories, as implied in equa- tion (10), which underlies the whole theory of wave motion. 1. Imagine a series of plane waves propagated along a horizontal line, as shown in Fig. 3. All the waves y = A sin | y {Vt - x) + a, [ (10) *In recognizing that by the action of its poles the needle tends to bend itself around a charged wire Maxwell impHes the attraction of the needle to the wire, but this view of the action of poles is so unsatisfactory that we proceed in a very different way. 20 DISCOVERY OP THE PHYSICAL CAUSE OF MAGNETISM are taken to have the same length (X) though of different phase (a,) as if coming from successive atoms of a magnet at different distances, but otherwise acting in concert. The successive waves are numbered a cos I a\ dV Z = Htane = "^ ; dr where V is the Magnetic potential, H the horizontal force, 8 the magnetic declina- tion, 6 the dip, I the latitude of the place of observation, on a spherical earth of radius a, and X the longitude, and r the distance from the earth's centre. Gauss takes d\x for the element of the magnetism at (n, Ua, Xo), within the Earth, and finds the action on a point in space (r, u, X) distant p from d\i: dti _ c dfi w y-ri'f-. P •>' y/r^ — 2rTo (cos u cos Mo + sin m sin Mo cos (X — Xo) + r^ Accordingly with these spherical co-ordinates the function V may be ex- panded into a converging series of solid spherical harmonics, involving sines and cosines of u and X. "The foundation of our investigation," says Gauss, "is the assumption that the terrestrial magnetic force is the combined action of the magnetized particles of the body of the earth," (p. 6). He conceives the magnetism to be due to the separation of the fluids, but adds that Ampere's conception of galvanic currents circulating about the atoms would serve equally well to explain the phenomena, which are thus somewhat independent of the particular hypothesis involved. By calculating the total magnetic moment of the globe Gauss finds that the Earth has 8,464 trillion times more magnetism than the one pound standard bar magnet used in his experiments. There would thus be required in the Earth's in- terior 8,464 trillion of such magnets, aU having parallel magnetic axes, to produce the magnetic action of the Earth in external space. With uniform distribution throughout the, whole body of the globe this would amount to very nearly eight bar magnets, more exactly 7.831, for each cubic meter of the Earth's matter. Gauss expresses surprise at this result, and adds that the more the bar magnets lack in parallelism, the stronger would have to be the average magnetization of their parts, in order to produce the actual total magnetic moment for the Earth shown by observation. DISCOVERIES IN COSMICAL MAGNETISM 37 If the mean density of the Earth be taken at 5.5, the average cubic meter of the matter will weigh 5,500 kilograms. And as Gauss' eight one pound bar mag- nets would thus weigh 4 kilograms, we may compute that the fractional part of the Earth acting as magnetized bars with parallel axes is 5,500/4 = 1/1375. The magnetism of the Earth is therefore exceedingly powerful, owing to the large size of our planet, yet the intensity is feeble compared to that of some artificial mag- nets. It thus appears that a large fraction of the Earth's mass has the property of saturated bar magnets; and hence we perceive that the Earth's magnetic field ought to exert a strong influence over very considerable regions of space. The magnetic field is thus everywhere well defined. On general principles we may be quite sure that the Earth is not filled with an artificial distribution of magnetized steel bar magnets having an average density of eight pounds per cubic meter throughout the globe. We are rather to view this average portion of the Earth's mass (1/1375) as having magnetic qualities in virtue of the atoms lined up in parallel planes, just as in Ampere's theory of Magnetism. In that case we may regard the magnetism of the Earth as due to the waves proceeding from these atoms in parallel planes. If the waves penetrate through the globe, all the observed results will follow, and we shall have a complete theory of the Earth's magnetism, based on the Electrodynamic Wave-Theory. This modern Wave-Theory of Terrestrial Magnetism is illustrated by the accompanying figure, in which the directions of the rotations of the wave elements are shown at every point of the globe. To get a view of the waves going out of the Earth at any point, it suffices to extend the rotations here shown, till they can be connected into waves of the type discussed in Bulletin No. 2. Accordingly, the Electrodynamic Wave-Theory gives a satisfactory ex- planation of Gauss' theory of Terrestrial Magnetism. And nothing now remains but to examine the conditions underlying that theory, and then to apply the theory to the elucidation of particular phenomena. Heretofore such phenomena as the Aurora and Magnetic Storms have almost completely bewildered the in- vestigator. Owing to the lack of connection with any common cause, it has been impossible for the investigator to open up an effective line of attack on the puzzling problems presented. Thus it is not remarkable that ^Cosmical Mag- netism has been a subject of the utmost obscurity, and the greatest natural phil- osophers have labored in vain to penetrate the mystery of the underlying physical forces. Even to the luminous mind of Gauss the Aurora Borealis always pre- sented a puzzling appearance — eine rdthselhofte ErscJwinung. (p. 50.) 38 DISCOVERIES IN COSMICAL MAGNETISM III The Theory of a Uniformly Magnetized Sphere Shows that the Magnetism of the Earth Depends upon the Combined Action of THE Magnetic Matter Throughout the Globe The magnetism of the Earth is roughly that of a uniformly magnetized sphere, as we see by simple calculation. 1. li I, m, n be the direction cosines of the direction of magnetization at any point, we have for components of the magnetization due to the element Idxdydz w ' I'fx© +"!(') +4© i*"** And by integration we obtain the potential of the whole body at the external point p <^) " = ///' \ 'W + -I© + »£© \ ^^*^ 2. By introducing the components of magnetization at (x, y, z) (6) A = II , B = Im , C = In , we obtain an expression in terms of the magnetic components A, B, C: 3. Let x' , y' , z' denote the co-ordinates of the external point p, so that {x - x'Y +{y- y'Y + (2 - z'y'Y Then, the expression (7) may be written (9) ^ Q = _ (^ _|, + B A, + C^)jffldxdyd. ■ since changes in x, y, z do not affect the terms A, B, C, and the operators e-tj j-/ > r-7 here placed outside of the signs of integration, DISCOVERIES IN COSMICAL MAGNETISM 39 4. Let V denote the potential of a uniform distribution of electricity of volume density unity throughout the region occupied by the magnet. Then by the usual expression for the potential we have for any external point p V = fff I dxdydz (10) 5. And therefore (9) becomes r, ,dV j,dV „dV ,,,, y = - A^-, - B—-, - C-T-, (11) dx dy dz Now let I', m', n' be the direction cosines of the outward drawn normal to the magnet at any element dS of its surface, and we may write the potential at the external point: Q = // (Al' + Bm' + Cn') -/S = f f l^^^ (12) For by (6) we have Al' + Bm' + Cn' = 1(11' + mm' + nn') = I cosd , (13) where 6 is the angle between the direction of magnetization and the outward nor- mal to the element dS of surface. 6. If the uniformly magnetized body be a sphere, of magnetic intensity Z, equation (10) reduces to the volume integral for a sphere of radius a, acting on a point distant r, F = J^a^Q (14) 7. When the magnetization is taken to be in the direction of the axis of x we have to calculate the intensity by taking the differential coefficient of the potential in respect to x: QT ^y ^ ■!T/'2;^ . ,r COS 6 ,^^^ since cos 6 dx \rj r X r 8. Accordingly it appears that the potential at any external point is the same as that of a magnetic particle of moment ^-n-a^I at the centre of the sphere. 40 DISCOVERIES IN COSMICAL MAGNETISM But as applied to the Earth this result has no well defined physical meaning. For since the Electrodynamic Wave-Theory shows that magnetism is due to the action of waves, it is natural to hold that these waves can proceed only from the magnetic matter of the Earth, and therefore not from that at the centre only, but from magnetic elements scattered throughout the mass of the globe, and every- where acting in concert, owing to the parallelism of their atomic planes. 9. Thus any hypothesis that the magnetism of the globe can rest on the properties of the matter located in a particular region is inadmissible. For al- though Gauss showed that mathematically the external magnetic potential might result from an unlimited number of internal distributions of magnetic density, yet the physical conditions now imposed by the Electrodynamic Wave-Theory exclude every distribution not consistent with the laws of the internal density of the Earth. 10. For if Magnetism be due to wave action, it must depend on the laws of the internal density of the globe for the required parallel atomic waves; and as the planes of the atoms in cosmical masses must follow the laws of chance, the magnetic field about such bodies can only be determined by the laws of internal density for all the matter. Thus we see that no other view can be held than that of Gauss, who justly assumed that the terrestrial magnetic force represents the combined action of the magnetized particles of the body of the Earth. IV The Relation between "Magnetic Storms" and the Sunspot Cycle Explained by Electrodynamic Disturbances Incident to Mass-Movements in the Sun 1. Hoping to find an intra-mercurial planet in transit over the Sun's disc, Hbinrich Schwabe of Dessau in 1826 began to make systematic daily counts of sunspots, and after seventeen years of patient watching (1843) was able to recognize in these fluctuating phenomena a regular periodicity of about ten years' duration. This is the sun spot cycle of which subsequent investigators have fixed the period more accurately at about 11.1 years. 2. In 1851-52 four independent investigators — Dr. John Lamont of Munich, General Edward Sabine of London, Dr. Alfred Gautier of Geneva, and Dr. Rudolf Wolf of Berne, — each recognized and aided in establishing a cycle in the fluctuations of Terrestrial Magnetism, also having a period of about ten years. It was found to have the same length as Schwabe's Sunspot cycle, and agreed DISCOVERIES IN COSMICAL MAGNETISM 41 with it phase for phase over more than a century. Subsequent investigators have fixed this magnetic period also at 11.1 years. 3. The phases are shown to be in such perfect agreement with those of the sunspot cycle that it has long been regarded as certain that the two phenomena — Sunspots and "Magnetic Storms" — are physically connected and mutually dependent on a common cause. The problem presented to the natural phil- osopher has been to find the physical cause underlying both phenomena. 4. The accompanying graphical illustration of the sunspot cycle and asso- ciated "Magnetic Storms," from Young's General Astronomy, represents the results of the researches of Dr. Rudolph Wolf of Zurich, who long studied the relations between these phenomena. 5. The agreement between the frequency of sunspots and the average daily range of the fluctuations in the horizontal component of the Earth's magnetism is seen to be complete. This means that when sunspots are most frequent, the range of the daily fluctuations of the Magnetic needle augments, and vice versa. And this dependence is found to be estabhshed by records extending over many sunspot cycles, since the year 1770, with some records runninp: as far back as 1680. 6. Accordingly, it cannot be denied that a physical connection exists be- tween sunspots and "Magnetic Storms." The question then arises: What is the nature of this connection? Many investigators have sought the physical cause for such a relationship, but as yet no one has succeeded in finding one which is rational and satisfactory. The explanation now put forward was developed in 1914, but has not here- tofore been published, although communicated to one of the oldest and most learned of the European Scientific Societies in 1915. 7. It is now shown that Gravity is an electrodynamic action, depending on waves traveling with the velocity of Light. The minor terms of Weber's law provide for the induction and the change of the induction, with variation of dis- tance. From the form of this law, we see that relative motion will involve in- duction* between different parts of the system. And this will apply not only to the orbital motion of the Earth, but also to internal mass-movements in the Sun. * This effect of relative motion is largely analogous to that resulting from Doppler's Principle, but in addition to the change in the wave-length, there are waves developed which may be parallel and have a common direction, as in Magnetism and in the Induced Current. 42 DISCOVERIES IN COSMICAL MAGNETISM 8. If a solar prominence of large size is ejected, so that a considerable mass of matter is in relative motion, as respects the Earth, it is obvious that the Sun's magnetic field will be disturbed; and as the Sun's magnetic field exerts an influ- ence on the Earth's magnetic field, through the infinite system of interpenetrating waves, there will thus arise a trembling of our magnetic needles while the waves due to the solar disturbance are passing the Earth. 9. Not only will the solar disturbance directly derange the Sun's magnetic field and thus affect the Earth's field, but also send irregular gravitational elec- trodynamic waves to the Earth. The passing of these irregular electrodynamic waves through the Earth — combining as they do at every point with the waves due to the atoms which produce the Earth's magnetic field - — wiU indirectly dis- turb the Earth's magnetic field, because of the mere irregularity of the wave commotions going on. 10. In this way we explain the greater amplitude of the daily range of the magnetic needle during the maximum of a sunspot period. For the more violent the solar outbreaks, the less damping of the waves from the matter above the reversing layer, and the more violent and intense the wave commotions passing the Earth; and thus the greater range in the amplitude of the daily fluctuations of the magnetic needle. Hence the exact correspondence between the curves for the "magnetic storms" and the sunspot cycle, as shown by Wolf's investigations. 11. It might be objected that solar prominence outbreaks are not always followed by magnetic storms on the Earth, and that the theory thus proves too much. Our theory does not, indeed, require this, but only specifies such response in the Earth's magnetic field when the electrodynamic waves passing the Earth are so irregular and develop such irregular induction in the air as to produce un- symmetrical magnetic stress in the Earth's field, so that the needle swings to and fro, to wider amplitudes than usual. It is clear that not all solar outbursts would produce such terrestrial effects. In many cases the matter in motion would have little or no relative motion, as respects the Earth — being largely normal to the Earth's radius vector. In other cases the outbursts would produce only minor derangement of the lines of force — or axes of wave propagation — ■ in the Sun's magnetic field, and thus exert little effect on the magnetic field of the Earth. 12. Accordingly, the fluctuation in the average daily range of the Magnetic Needle should have only a very general connection with the curve of the sunspot cycle — not a direct connection in all details, as some have supposed should occur. The Wolf curves are therefore in satisfactory agreement with the Electrodynamic DISCOVERIES IN COSMICAL MAGNETISM 43 Wave-Theory. It is very interesting to notice that this Theory would explain Sunspots themselves (with corresponding extension of the Corona) as eruptive outbreaks, due to the Combined Inductive Action of the Planets and Comets, under Weber's Law, the cycle sometimes exceeding the period of Jupiter, and sometimes falling short of it, owing to the modifying influence of the other Planets. This cause of sunspots was suggested, in the unpubUshed Memoir of 1914, as an extension of the researches of Hertz, 1880, " On Induction in Rotating Spheres." The analogy is very striking, and the proof has seemed more and more conclusive, but the subject of the physical cause of sunspots is beyond the scope of the present Bulletin. V W. Grylls Adams' Researches of 1881 on "Magnetic Storms" in Various Parts of the Earth Show that These Commotions are Simultaneous as if Depending on Mass-Movements in the Sun 1. Having now shown why the amplitude of the average daily range of the horizontal component of the magnetic needle is greater* at sunspot maximum and smaller at sunspot minimum than during the years between maximum and minimum, it only remains to prove by terrestrial observations that such wave commotions actually come from the Sun. If they do, these abnormal tremblings of the needle should occur almost simultaneously throughout the globe. 2. The accompanying plate is from the Philosophical Transactions of the Royal Society for 1881, which contains an important paper by the late Professor W. Grylls Adams, brother of the celebrated J. C. Adams, theoretical discoverer of Neptune. It will be noticed that Adams has here set to common Greenwich Mean Time the records of magnetic disturbances taken at many stations in all parts of the world. The suddenness of the changes in several instances is re- markable. The tremors affect all the components of the Earth's Magnetism, within a few minutes of the same instant, throughout the globe. This record shows that there is a magnetic wave commotion going on in every part of the globe, as if it originated from disturbances in the Sun, and had been propagated to the Earth with the velocity of light. 3. It is from just such "Magnetic Storms" as this that Dr. Rudolph Wolf derived his curve of correspondence between "Magnetic Storms" and the sunspot cycle, as given in the preceding section. The nature of the phenomenon is thus made clear, and the physical connection firmly established. * Because of the Induction Effects, described above. 44 DISCOVERIES IN COSMICAL MAGNETISM 4. Probably this reasoning will be convincing to most persons, but if there be those who doubt our conclusions they may be referred to the account of the spectroscopic and magnetic observations taken by Professor C. A. Young's eclipse party, Aug. 3-5, 1872, at Sherman, Colorado. Certain eruptions in the Sun gave bright lines in the spectrum and were simultaneous with tremblings of the magnetic needle independently noted by the magnetic observer at Sher- man, and afterwards found to have been noted also at Greenwich, England, five thousand miles away. 5. As Young and the magnetic observers worked quite independently of one another (cf. The Sun, 1902, pp. 166-168) this coincidence in the visual appear- ance of spectral bright lines, when incandescent matter is raised above the reversing layer, and the coming on of a notable "Magnetic Storm" is not accidental. It really shows the true physical connection between "magnetic storms" and the eruptions in the solar atmosphere. 6. It should be noted that just as the Frauenhofer lines become bright when the emitting matter is raised above the reversing layer, so also will the mag- netic waves from the Sun's atoms be of increased intensity when the emitting matter is above the reversing layer. This increased freedom from damping, and the relative motion of the disturbed matter causes a readjustment in the infinite systems of waves, with a trembling of the needle on the Earth, which may be recorded on the magnetographs. 7. As an observational proof that magnetic waves regularly come from the Sun, yet may have their induction effects damped and reduced in intensity by clouds and similar terrestrial objects, we refer to the well-known observations by Professor Nipher {Transactions of the Academy of Sciences, of St. Louis, 1913-16). These observations are so definite that they admit of not the slightest doubt. 8. If the clouds may exert an appreciable influence in reducing atmospheric induction and thus damping these magnetic waves, it follows that the Earth's shadow at night will be still more effective; yet the greater wave commotions in the Sun are powerful enough to give a conspicuous terrestrial effect in the Aurora, "Magnetic Storms," and "Earth Currents," as explained below. VI The Physical Cause op the Aurora Borealis The Aurora Borealis is a very striking phenomenon, but remarkably obscure as to the physical cause on which it depends. In order to get at the probable cause of the Aurora, we point out a series of interesting facts. Fig. 1. Graphical Representation of Gauss' Theory that Terrestrial Magnetism dei)ends on the combined action of the magnetized particles of the body of the Earth. Fig. 2. The Magnetic Field about the Earth, showing contour lines of equal Dip. ■■■■■■■■«■■■■■■■■■■■■■■■! ■■■■■■■■■■■■■■■■■»«■■■■■! ■■■■■■■■JBiiBaBaaaHriBaBBBB BBBBBBBIianBBaBBBBBBaBBBB iiBBBBBIIBHBBBBBBIVHBBBBBB BBBBBBBIIBBKBEg^aBIIBBBIIBBB ■■iHHniaHmainHHaKMBi nUBBBBHBBBIIBBBIinBBBBKBBI lubbbbbbbbbbbbbbbbbbbkbi -Jbsbbbbbbbbbbbvbbbbbbbki bbbbmbiibbbbbbbbubbbbbbbbi IBBBllBilBBBBBBaBMBBBBBaBBI 1BBUB!IBBBBBBKWflBBBBBBB~ ■BBBBBBBBBBBBBBBBBBBBB-. iBBBBBBBiasaBBBBBBEraS'BBBI BBBBBBBBBBBBBBSIBBBBBBBBBI BBBBBBBBBBBBBBBBBBBBBBBBI BBBBBBBBBBBBBHBBBBBBBBBBI BBBBBBBBBBBBBUBMBBBBBBBBI ■BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBi! BBBRnnaa3lfni9,!rniiyfB9!nBBBBBBBI BBBBBBBBBIIBfiBBIIililBliBBBBBBBI BBBBBBBBBBBBBBBQ^SDaXCdfiBBBI BBBBBBBBBSIBBBBBBBBBBBBBBBil BBBBBBBBBBBBBBBBBBBBBBBBBBI IBBBBBBBBBBBBBBBBBBBBBSSIBBBI BBBBBBfiBBBBBBBBBBBBBriBiSIB!! BIBBBBRBBBBBBBBBBBBBBH!^^. ^ BSIBBBUBBBBBBaBBBBBBaflBBBBO BBSSBBBBBBBBBBBBRaBEQX^lUIBBi BBBBI^^BBBBBBBEQaBBBBBBBBBBill BBBBBI BBBBBI HBBBBI IBBXEEBI BBBBB! BBBBUI aBBQIBI issaaai aflaaBBBaaaBBaBaaBBBBBBNiaaBi klBBBBBBBBBBBBBBBBBBBBBUBBBI KBBBBBBBBBBBBBBBBBBBBBBMBBI ■flBBBII ■bibbbI ■BBBBBI raBBB■BBBBBBB'flB!(IBBBBBBBBIiBi1BBBBBBBBBBBBBBBBBB■BBB!BBBBBBBI EZZSnBBBBBBBBBNiBBBBBBBrfiHnBBBBBBBBBRSBBBBBBBBnBBJSBBBBBBI BBBBBHBBBBBBnBBKBBBZil^SiaBrMBnBBBBBBBBBrjniilBBaSi^BBUBBBBBBBBBI BBBBBHBBBBBBlJBBBIiliBBBBBBCIBBBBBBBBBBnBBUBBBBBBBVBBBnBBBBBL ■BBBBBBBIBBBBilBBBBMBBBBBnBBBKSIBBBBBBnBBBSBBBBBBBBBBlNBBBBBBI BBKlBBBBBBBMBBBBnaBBBBMBBBBnnBBBBBfiBBBISBBBBBnBBBBHBBBBBBl MBBBBBP nBBBBBi ^BBBBBKI ■BaBBBBBBlBiaBBBBBB|BaSiBBBBB|flBBBnBBBBBBBBBBn«BBBBBBBBBBBKil ngiEa«iaBBBBBE0a!lBBBBBHiSy31 RIIBBBBBBBBIiBBBBBBBflBBBBBBBBBBBBBIBaBBBBBBBBBBaiBBBBBBI IB^iriill iBEia'ji IBBBBgiE !■■■■■■ Fig. 3. Wolf's Sun-Spot Numbers, with associated "Magnetic Storms." Fig. 4. A typical view of the Aurora Borealis, showing extensive curtain, with ribbon folds at the lower border, and stars \'i8ible through the illumination. An Aurora of this general aspect was observed by the author while measuring double stars at the Washburn Observatorj-, Madison, Wisconsin, Sept. 29, 1895. TorofUM H p 7,1,-k^-WBU H P BoLauuiL. H F CoLabau H.P — TcronXx). V P Melbourne H.F Son FernajTjjU).H.F- Vherinxx. H.F He^' H F Aew V F St Petersburg V.P SU"V/>UjrsL VF Bojljiwiay V F mfU Fig. (i. Comparison of the observed Rays in a Tj'pical Solar Corona with the Lines of Force in Gavhw' Theory. In his studies on the Motion of Mars, 1609, Kepleh concluded that the Hun riid Planets are [Magnetic (Opera Omnia, III, p. 37), and the evidence has since become decisive. DISCOVERIES IN COSMICAL MAGNETISM 45 1. The coincidence of the rays of the Aurora with the direction of the mag- netic needle. This is one of the most striking proofs of the relationship between the Aurora and "Magnetic Storms," which are also observed to recur in similar periodic cycles. So long ago as 1714 Dr. Edmund Halley regarded the Aurora as a magnetic phenomenon (Phil. Trans., 1714-16. No. 341). Naturally modern data were not then available to lend such general support to the theory as might establish its truth. 2. When a great "Magnetic Storm" occurs it usually is felt throughout a large part of the globe, and the Aurora frequently is visible in different countries, and over wide areas of continental extent, sometimes in both hemispheres. It is found that the "Magnetic Storms" are associated with "Earth Currents," or electrical disturbances so strong that occasionally the telegraph lines may be operated by them without the use of the regular batteries. 3. The "Earth Currents" associated with "Magnetic Storms" and Aurora Borealis may be most easily explained by supposing that there is electric induction in the globe due to passing electrodynamic waves incident to mass-movements in the Sun. This induction would generate galvanic currents in closed circuits — such as telegraph and cable lines — and also produce intensified waves from the atoms of magnetic matter such as iron, steel, and similar substances. 4. Now Faraday showed that all substances are magnetic, and thus the result of the passing of waves incident to mass-movements in the Sun would in- volve the whole globe in a ceaseless magnetic tremor. And owing to the unequal conductivity of the globe these moving waves would thus lead to an apparent development of free electricity. In the atmosphere the electric disturbances incident to the passing waves would take on the form of light; for the incessant wave flow, with the resulting electric inductions on large masses of air and clouds, would lead to slight discharges of free electricity through the highly rarified air, as in the Geissler tube experiments. 5. It is impossible to doubt that when the air is traversed by such waves, of surging electrical mass oscillations, the disturbances or stresses produced would generate light in some form. The Geissler tube experiments show not only the general type of these oscillations, but also their enormous variety, with changes of color, which are also characteristic of the Aurora. This is one of the surest proofs that the Aurora is an electrical oscillation with visible discharge due to the passage of electrodynamic waves through the higher regions of the Atmosphere. 46 DISCOVERIES IN COSMICAL MAGNETISM 6. An additional proof not only that the Aurora is an electrical phenomenon but that it depends on commotions in the Sun may be gathered from the periodici- ties in the two phenomena. (a) The Aurora has the 11 -year cycle characteristic of sunspots and "Mag- netic Storms." (b) The semi-annual period in the Aurora, with greatest Auroral Activity in March and September, and least activity in June and December, seems to in- dicate that this fluctuation depends on the spots in the equatorial region of the Sun which are most directly exposed to the Earth in March and September. (c) There is also a periodicity in the Aurora of 25.93 days, which corresponds so closely to the period of the Sun's mean axial rotation, that we are compelled to believe it depends on the return of spotted areas of the Sun to our meridian, and thus disturbs the electrical equilibrium of the globe in this average period. The researches of Maunder {Monthly Notices, 1904) confirm this result from another point of view. Carrington found from the observation of sunspots at the solar equator a period of 24.9 days, but that in latitude 30°, the period was lengthened to about 26.4 days. Spectroscopic measurements of the Sun's rotation period by Duner, Crew, and other observers agree closely with Carrington's periods from sun- spots. As the area of maximum solar activity is somewhat nearer the equator than 30° heliocentric latitude, we may take the Auroral period of 25.93 days as the time of the mean solar rotation. 7. Now suppose wave commotions to occur from bodily movements of matter in the Sun. The magnetic field of the Sun will thereby be disturbed, and a reaction generated in the Earth's magnetic field. The resulting solar wave commotion wiU travel towards the Earth with the velocity of light; and the magnetic waves incessantly receding from the Earth wiU also be set into irregular motion and the normal calm of the magnetic field of the Earth wiU be disturbed. 8. As the magnetic potential of the globe is greatest towards the magnetic poles, in accordance with Humboldt's law of the increase in higher latitudes (cf. Cosmos, Vol. I, p. 179, Bohn's Translation) it will follow that when the wave commotion incident to the outbreaks in the Sun are violent enough to elicit luminos- ity in our rarified atmosphere — as in electric discharge through the rare gases of a Geissler tube — the fight wiU appear on our northern horizon and move rapidly southward along the fines of force of the Earth's magnetic field. DISCOVERIES IN C08MICAL MAGNETISM 47 9. For the wave commotion will naturally have a rise in intensity to a max- imum, then a decline; and the luminosity will first appear in the region of greatest magnetic potential furthest north; and as the maximum approaches, the luminosity will travel southward to the fields of smaller magnetic potential. And as the disturbance declines, the streamers of the Aurora may drop back to the northern horizon. 10. This is exactly what is frequently observed in the celebrated phenomenon of the Aurora Borealis. We thus conclude that an Aurora is due to passing mag- netic waves incident to internal commotions in the Sun. Humboldt gives the following description of a typical Aurora, based largely on Argelander's obser- vations at Konigsberg: "Low down in the distant horizon, about the part of the heavens which is intersected by the magnetic meridian, the sky which was previously clear is at once overcast. A dense wall or bank of cloud seems to rise gradually higher and higher until it attains an elevation of eight or ten degrees. The colour of the dark segment passes into brown or violet ; and stars are visible through the cloudy stratum, as when a dense smoke darkens the sky. A broad brightly luminous arch, first white, then yeUow, encircles the dark segment; but as the brilliant arch appears subsequently to the smoky gray segment, we cannot agree with Argelander in ascribing the latter to the effect of mere contrast with the bright luminous margin." . . . "The luminous arch remains sometimes for hours together flashing and kindling in ever-varying undulations, before rays and streamers emanate from it, and shoot up to the zenith. The more intense the discharges of the northern light, the more bright is the play of colours, through all the varying gradations from violet and bluish white to green and crimson. Even in ordinary electricity excited by friction the sparks are only coloured in cases where the explosion is very violent after great tension. The magnetic columns of flame rise either singly from the luminous arch, blended with black rays similar to thick smoke, or simul- taneously in many opposite points of the horizon, uniting together to form a flickering sea of flame, whose brilliant beauty admits of no adequate description, as the luminous waves are every moment assuming new and varying forms. The intensity of this light is at times so great, that Lowenorn (on the 29th of June, 1786) recognized the coruscation of the polar light in bright sunshine." {Cosmos, Vol. I pp. 189-190 — Bohn's Translation). 11. From this description, which corresponds with the results of my own observations, it seems clear that the well-known curtain appearance is due to the 48 DISCOVERIES IN COSMICAL MAGNETISM escape of slight charges of free electricity under the varying electrical potential produced by the passing of electrodynamic waves incident to bodily disturb- ances in the Sun. The crown sometimes developed in the Aurora is a particular form of this luminosity so projected as to make a central canopy high in the heavens. 12. Accordingly, it appears that all the leading phenomena of the Aurora Borealis may be easily explained by the present Electrodynamic Wave-Theory. The result is a harmonious view of many puzzling phenomena of Terrestrial Magnetism. It accords with Weber's fundamental law of Electrodynamic Action, and confirms Halley's suggestion of 1714 that the Aurora is a magnetic phenomenon. 13. It has long been recognized that in its geographical distribution the Aurora hovers about the cold poles of the Earth — doubtless because, as Exner showed, the fall of the atmospheric electrical potential is some fifteen times greater in the zones of the cold poles than at the terrestrial equator. An electric phenomenon de- pending on this fall of the potential, like the change of the Induction, under the varying influence of the Sun's Electrodynamic Waves, is thus more intense in winter and in high latitudes than in summer and in the equatorial regions of the globe. 14. This view of the Aurora is borne out by the very delicate photographic researches of Dr. Slipher recently recorded in Lowell Observatory Bulletin No. 79, showing by actual observation that from June to November, 1916, there was almost persistent Aurora in the sky near the horizon. It was recorded satisfactorily even in Moonlight, but appeared plainest near the horizon, because this region of the air is of greatest depth. The strength of the luminosity is increased by this depth, because on the photographic plate the integral effect of the radiation accumulates, when the electrical potential is changing in any layer, under the progress of the Sun's diurnal influence. 15. The Aurora is therefore much more widely distributed geographically, and a much more persistent phenomenon than was previously believed. Obviously its full extent and persistence can be ascertained only by the sensitive plate of the modern camera. The more spectacular classic phenomenon becomes visible to the eye chiefly under the cold climates of the polar regions, and fluctuates in cycles corresponding to the electrodynamic wave agitations in the Sun, as above set forth. DISCOVERIES IN COSMICAL MAGNETISM 49 VII Kreil of Prague, 1841, and John Allen Broun, 1845, Independently Detect a Lunar Influence upon the Magnetism of the Earth, which Is Semi-Diurnal and Otherwise Follows the Laws of the Tides: the Researches of Lloyd and Airy Confirm the Existence of a Lunar Magnetic Tide, but Lloyd Misinterprets It In 1841 Kreil of Prague laid before the Bohemian Society of Sciences evi- dence of slight semi-diurnal changes in the magnetism of the Earth depending on the influence of the Moon. He thus anticipated Schwabe's discovery of sunspot periodicity by two years, and preceded Lamont's discovery of the corresponding cycle in Terrestrial Magnetism by ten years; so that the new lunar magnetic inequality was unexpected, and appeared the more remarkable because the period was only half a lunar day. Shortly after Kreil's discovery the same fact of a semi-diurnal inequality in the Earth's magnetism depending on the Moon was independently noted by the English investigator John Allen Broun, 1845, and fully confirmed by the researches of Sir Edward Sabine. When attention was directed to the effect of the changing distance of the Moon from the Earth, both Sabine and Broun found the variation is greater for perigee than for apogee. According to Broun's careful analysis the mean ratio of the apogee effect to the perigee effect is as 1 to 1.24 nearly. Looking into the cause of this difference Broun notices that the distance of the half orbit near apogee is to that about perigee very nearly as 1.07 to 1. He then remarks that the cube of 1.07 is 1.23 nearly; hence it follows that the mean ranges of the curves for the two halves of the lunar orbit are in the ratio of the inverse cube of the distances from the Earth, as in the Theory of the Tides of the Sea. Lloyd's Misleading Discussion of the Direct Magnetic Influence of the Sun and Moon upon the Diurnal Variations of the Earth's Magnetism In the Philosophical Magazine for March, 1858, Dr. Humphrey Lloyd, Professor of Natural Philosophy in the University, and subsequently Provost of Trinity College, Dublin, has a learned discussion, "On the Direct Magnetic In- fluence of a Distant Luminary upon the Diurnal Variations of the Magnetic Force at the Earth's Surface," which was afterwards reprinted as the concluding chapter of his "Treatise on Magnetism," London, 1874 (Longmans, Green & Co.). Dr. Lloyd's Treatise is a very useful work and doubtless has been of great service to Science, but we shall here direct attention to certain false conclusions in the 50 DISCOVERIES IN COSMICAL MAGNETISM article of 1858, which apparently misled Maxwell (Treatise on Electricity and Magnetism, edition, 1892, §474, Vol. II. p. 136) and his successors in dealing with problems of Terrestrial Magnetism. Lloyd develops his analysis at some length, and shows from the equations that the magnetic effect of the distant body "consists of two parts, one of which is constant throughout the day, while the other varies with the hour angle of the luminary." Although each of the parts varies inversely as the cube of the dis- tance, of the magnetic body, he concludes that the variable part depending on the hour angle of the luminary wiU give rise to a diurnal inequality. Since the lunar variation has two maxima and two minima of nearly equal magnitude in the twenty-four lunar hours, Lloyd concludes that "the 'phenomena of the diurnal variation are not caused by the direct magnetic action of the Sun and Moon." (p. 238.) A just criticism of Lloyd's analysis may be made, in that he retained the hour angle 6, in his expressions, (p. 235) instead of the angle 20 which occurs in the expressions for the tide-generating potential, (Cf. Sir George Darwin's Article Tides, Ency. Brit, 9th edition, §7, p. 357); but as Lloyd doubtless did not suspect that there could be such a thing as a magnetic tide, probably this over- sight was unavoidable in the state of the subject existing more than half a century ago. Lloyd closes his discussion by calling attention to Stoney's investigation (Philosophical Magazine, Oct., 1861) showing that the maximum effect of the Moon, supposed equally magnetic with the Earth bulk for bulk, upon a declina- tion magnet would be less than one-tenth of a second. The following tables embody the results of Lloyd's discussion of the mag- netic observations at Dublin. Owing to the above error of analysis, Lloyd could not interpret these results, holding that the observed diurnal variations, with two maxima and two minima of nearly equal magnitude in twenty-four lunar hours, are not caused by the direct magnetic action of the Moon. He was thus misled, and in turn misled Maxwell (Treatise, §474) and his successors. Lunar Inequality of the Easterly Force at Dublin (Lloyd) Lunar Hours Summer Lunations Winter Lunations Year -12 -0.19 -0.09 -0.14 -10 -0.21 -0.08 -0.15 - 8 -0.06 -1-0.02 -0.02 DISCOVERIES IN COSMICAL MAGNETISM 51 Lunar Inequality op the Easterly Force at Dublin (Lloyd) — Continued Year Lunar Hours — 6 — 4 — 2 + 2 + 4 + 6 + 8 Summer Winter Lunations Lunations +0.09 +0.09 +0.09 +0.12 +0.13 +0.13 +0.08 -0.01 +0.03 -0.06 -0.09 -0.08 -0.04 -0.07 -0.05 +0.05 -0.02 +0.01 +0.17 +0.08 +0.12 +0.06 +0.07 +0.07 + 10 0.00 -0.05 -0.03 Lunar Inequality of the Northerly Force at Dublin (Lloyd) Year -0.03 +0.01 +0.07 +0.04 -0.01 +0.01 +0.01 +0.08 +0.03 -0.02 -0.11 -0.08 Lunar Inequality of the Vertical Force at Dublin (Lloyd) Lunar Hours -12 -10 - 8 - 6 - 4 - 2 + 2 + 4 + 6 + 8 + 10 Sunamer Winter Lunations Lunations -0.06 +0.01 -0.01 +0.02 +0.08 +0.07 +0.04 +0.05 -0.02 0.00 -0.03 +0.04 -0.01 +0.02 0.00 +0.05 +0.08 -0.03 -0.01 -0.03 -0.10 -0.12 -0.08 -0.07 Lunar Hours f Lunar Hours f -12 +0.05 +0.02 -10 +0.00 + 2 +0.02 - 8 0.00 + 4 -0.01 - 6 -0.02 + 6 -0.06 - 4 -0.07 + 8 -0.01 - 2 +0.10 + 10 +0.01 52 DISCOVEKIES IN COSMICAL MAGNETISM How remarkable an argument Lloyd has here developed to establish the existence of a magnetic Tide in the Earth may be readily seen by anyone familiar with Sir George Darwin's discussion of the Path of a Pendulum as deflected from the vertical by the tidal forces of the Moon, (Tides and Kindred Phenomena of the Solar System, edition, 1898, p. 112). We reproduce here, for comparison, Darwin's diagram of the path of the pendulum, when the Moon is north latitude 15°, and the observer in North latitude 30°. 12 S Fig. 7. Darwin's Diagram of the semi-diurnal movements of a pendulum The lunar hours are indicated by the numbers marked along the successive posi- tions of the pendulum. The comparison however is somewhat incomplete, because Darwin's diagram gives the total forces which deflect the pendulum, while the analysis of Lloyd gives only the components into which the Lunar Inequality of the Earth's Mag- netism usually are divided. Sir George Airy Alone Recognized the Lunar Magnetic Influence as a True Magnetic Tide It appears that among all his eminent contemporaries Sir George Airy alone correctly interpreted this inequality, which he says is "a true lunar tide of magnetism, occurring twice in the lunar day, and showing magnetic attraction backward and forward in the line from the Red Sea to Hudson's Bay." (Treatise on Magnetism, 1870, p. 206.) DISCOVERIES IN COSMICAL MAGNETISM 53 Airy adds that the lunar forces are considerably less than those which follow the law of solar hours; the mean diurnal solar inequaUty being about 1/600 of the horizontal force, while the lunar is about 1/12000. According to this, the solar influence is approximately twenty times more powerful than that depending on the Moon. This will cause no surprise in view of the magnetic waves now shown to proceed from certain fields about the sunspots, and to come to us in the form of undamped vibrations when the prominences project above the reversing layer. We have seen that this ebb and flow in the magnetism of the globe has been actually observed, but its semi-diurnal character has been misinterpreted by Lloyd and several other eminent investigators. Now Newton's law of Gravitation will explain the ebb and flow of the sea, but not the ebb and flow of the Earth's magnetism discovered by Keeil and John Allen Broun. It is necessary to explain both phenomena, for in his First Rule of Philosophy Newton says that we are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. And thus, according to Newton's Rule, we are obliged to adopt Weber's Electro- dynamic Law, to explain the Magnetic Tide, by wave action in reference to poles and propagated in time, thus giving what Gauss called a construirbare Vorstellung of the established phenomena of Nature. VIII The Series of Periodic Fluctuations in the Magnetism op the Earth Shown to Depend on the Sun and Moon, and Explicable BY the Influence op their Cosmical Magnetism To convince ourselves that there are numerous fluctuations in the Earth's Magnetism depending on the Sun and Moon, we need only read attentively the well-known discussion of Terrestrial Magnetism by the late Professor Balfour Stewart, in the Article Meteorology, Encyclopedia Britannica, 9th edition. 1. It is there shown that there is a solar diurnal variation of declination changing slowly through the year, but with the form of the curve depending on the latitude of the place of observation. There is also brought out the semi- annual inequality or difference from the whole year's mean of the two half yearly means. 2. Among the recognized cosmical periods established by investigators are: (a) A yearly period, (6) A period of 11 years — the sunspot cycle, 54 DISCOVERIES IN COSMICAL MAGNETISM (c) The 18.6 year period — which is the cycle of the Moon's nodes, (d) A period of a Lunar Synodic Month, 29.53 days, (e) A period of 25.93 days, the Sun's mean rotation period, (/) A semi-diurnal magnetic tide depending on the Sun; and also a smaller but very definite semi-diurnal magnetic tide depending on the Moon, discovered by Kreil at Prague in 1841, and independently detected by John Allen Broun, 1845, as set forth above. 3. All these periodic changes are small compared to the residual constant of the Earth's magnetism, which changes but slowly or not at all. Thus as Gauss first showed in 1838, the body of the magnetism of the globe depends on the par- ticles within the Earth. The small periodic variations, on the other hand, arise mainly from the external influence exerted by the Sun and Moon, through their systems of magnetic waves. 4. It is commonly stated that the daily variation of the Earth's magnetism is not such as could be explained by the magnetism of the Sun itself, (cf. Jeans, Mathematical Theory of Electricity and Magnetism, p. 402.) As shown in the fore- going section, this original view of Lloyd is based on a false premise. It seems certain that with this correction of Lloyd's mode of analysis, the daily variation in the Earth's magnetism may be fully explained. Thus the difficulties encoun- tered by Maxwell, Chree, Stewart, Schuster, Jeans, and others entirely disappear. 5. This result confirms John Allen Broun's discovery that the diurnal variation depending on the Moon follows very accurately the law of the inverse cube of the Moon's distance. Broun remarks that "the ratio of the Moon's mean distance from the Earth in the half orbit about apogee is to that in the half orbit about perigee nearly as 1.07 to 1; as the cube of 1.07 is 1.23 nearly, we see that the mean range of the curves for the two distances are in the ratio of the inverse cubes of the Moon's distance from the Earth, as in the theory of the tides." (Stewart's Article Meteorology, Ency. Brit., 9th edition, p. 179.) As Broun had observed the Lunar magnetic effects to be as 1 to 1.24, and Sabine had found similar results, he naturally regarded the verification of this tidal law in the lunar semi-diurnal variation as very important. With the above correction of Lloyd's error of analysis, this result of Broun shows conclusively that all the diurnal effects observed can be explained by the magnetism of the Sim and Moon. 6. It is not strange therefore that in his celebrated Article on Terrestrial Magnetism, §139, Balfour Stewart recognized that as the Moon's magnetic DISCOVERIES IN COSMICAL MAGNETISM 55 influence follows as nearly as possible Broun's law of the inverse cube of the distance from the Earth, it is impossible to refrain from associating this magnetic influence either directly or indirectly with something having the type of tidal action. Stewart points out that Airy found a similar semi-diurnal inequaUty depending on the Sun in the Greenwich records, and A. Adams found correspond- ing "Earth Currents" to be induced in the crust of the globe at the corresponding hours. Conclusion After this review of many mysteries in Cosmical Magnetism, it only remains to add that as the Electrodynamic Wave-Theory fully accounts for the periodic fluctuations in the magnetism of the Earth, it must be held to rest on true laws of Nature. Otherwise it would not be possible to explain so many observed periodic changes in the magnetism of the globe. Heretofore these variations have seemed so utterly mysterious as to completely bewilder the natural philosopher. It is now clearly established that Magnetism is due to electrodynamic waves, which travel in free space with the velocity of light, but are propagated less rapidly through solid masses. Any view that Gravitation is an isolated force, entirely disassociated from the other forces of Nature, is wholly untenable. Weber's law explains how electrodynamic action is propagated in time by means of the waves we have shown to underly Magnetism in general. As Magnetism is a form of attraction to poles while Gravitation is central, the two actions are related, and it is obvious that the waves from the atoms of the heavenly bodies should give rise to both Cosmical Magnetism and Universal Gravitation. T. J. J. SEE Starlight on Loutre, Montgomery City, Missouri, June 6, 1917. Issued July 9, 1917 PUBLISHERS : THOS. p. NICHOLS & SON CO., Lynn, Mass., U. S. A. © WM. WESLEY & SON, London A. HERMANN ET FILS, Paris RESULTS OF RESEARCHES ON THE ELECTRODYNAMIC WAVE-THEORY OF PHYSICAL FORCES BULLETIN NO. 4 THEORY OF THE TRANSMISSION OP PHYSICAL FORCES, BASED ON THE PROPAGIATION OF ELECTRODYNAMIC WAVES IN RIGHT LINES OR IN CURVES CONFORMING TO FERMAT'S LAW OF MINIMUM PATH AND THE PRINCIPLE OF LEAST ACTION. By T. J. J. SEE I Introductory Remarks In his celebrated History of the Inductive Sciences, Volume I, Whewell shows that the chief conditions 'prerequisite for the development of true Sciences are clearly defined ideas, which are appropriate to the phenomena of Nature, steadily applied to well determined facts of observation. It was the neglect of these conditions that caused the failure of the physical sciences among the Greeks. He also shows that during the Middle Ages there were enormous Stationary Periods, sometimes amounting to more than a thousand years, during which no real advance was made, owing to the indistinctness of ideas then prevalent among the Arabian commentators on Greek Science, and among European Schoolmen long accustomed to the mere pohsh of scholastic erudition and the unprofit- able subtleties of dialectic rather than real research in physical science such as would have appealed to Archimedes, Newton or Gauss. In view of the deep impression made by Whewell upon his own age, through his luminous exposition of the methods of Archimedes and Newton, one would not have anticipated a reappearance of such indistinctness of ideas, because this would be more appropriate to the epoch of the Arabians than to our own time. Least of all should we have expected the reappearance of these indistinct ideas in Astronomy, the history of which, for thousands of years, is so well known and (57) 58 THEORY OF THE TRANSMISSION OP PHYSICAL FORCES SO instructive to the investigator. Yet, wonderful to relate, this unexpected event has happened! We refer to the mystical speculations of Einstein in his General Theory of Relativity. He proposes a modification of the law of Gravitation, but does it in such patch-work fashion, that it leaves Gravitation without appropriate con- nection with the recognized laws of Electrodynamics, and thus introduces into Nature the most complete discontinuity. As Electrodynamic Action rests on universal experience in Electricity and Magnetism, it cannot be that we have one kind of attraction in such cases, and a totally different kind of attraction for the heavenly bodies. This would be like the practice of inventing special aethers for special needs, which Maxwell justly denounces as unphilosophic {Treatise on Electricity and Magnetism, 3d edition, Vol. II, §761, p. 431). As regards Einstein's Theory some of his English followers seem to have overlooked what Bottlinger points out in his Munich Prize Inaugural Disser- tation, p. 48, (Troemer's Universitats Buchhandlung, Freiburg, 1912) namely, that recently the view was prevalent among physicists that Gravitation is an Electro- dynamic phenomenon, but no positive results have attended the efforts to con- ceive Gravitation as an Electrodynamic Action, and Professor Einstein found Gravity so fundamentally different from Electrodynamics that more recently he had quite turned away from an Electrodynamic explanation. Thus having failed to connect this chief force of Nature with Electrodynamics, as he long sought to do, Einstein developed a made-to-order General Theory of Relativity, couched in such seductive analysis that it received some favor in Ger- many, was thence transplanted to Holland, and finally welcomed in England — notwithstanding the fact that the underlying ideas clearly are inconsistent with the sound physical basis considered essential by both Newton and Whewell, but now overlooked by certain mathematicians in the Royal Astronomical So- ciety. For example, it is stated by Professor De Sitter in his authoritative review of Einstein's Theory {Monthly Notices, Oct., 1916, p. 702) that Gravitation is "not a force" but a "property of space." It would appear that the present author's well known aversion to such mys- ticism as this must be shared also by the more experienced investigators in Eng- land; for on January 28, 1917, one of the most eminent of the Cambridge math- ematicians wrote as follows: "I wish the perihelion of Mercury could be resolved similarly (to the new work on the Lunar Fluctuations). Otherwise we have an unlimited number of THEORY OF THE TRANSMISSION OP PHYSICAL FORCES 59 ingenious kinds of relativity on our hands; which will be remarkable for self- contradiction of the principle that everything is relative." It is just such confusion as this that I am laboring to get rid of. In the Observatory for December, 1916, the present writer has made some criticisms which should be emphasized. It is pointed out that Professor De Sitter of Leiden dwells on the analysis of Einstein, but quite ignores the physical problem of explaining the stupendous forces at work. "1. In the Aether of Space, pp. 112-126, Sir Oliver Lodge shows from the recognized laws of mechanics that the pull of the Earth on the Moon is equiva- lent to the breaking strength of a steel column 400 miles in diameter, or a forest of five million million weightless pillars each a square foot in cross section — the tenacity of the steel being thirty tons to the square inch." "2. In the same way Sir Oliver Lodge shows (p. 130) that the pull of the Sun on the Earth is equivalent to the tenacity of a forest of weightless steel pillars each eleven inches in diameter, acting on every square foot of a hemispherical section of the globe — the steel again having the breaking strength of thirty tons to the square inch." "Do these impressive results indicate that gravitation is 'not a force' but only a 'property of space'?" "Newton was weU aware of the difficulty of explaining gravitation, and although he was unable to assign the cause of this force, he very properly says in the General Scholium to the Principia (1713) that it must proceed from a cause which penetrates to the very centres of the Sun and planets without suffering the least diminution of its force. If it were a 'property of space', how could it be definitely and accurately directed to the centres of the Sun and planets? It obviously is so clearly an influence exerted by matter that one is astonished at this talk of a 'property of space'.* Such discussion shows the extent to which purely mathematical reasoning may be misapplied by those who ignore appro- priate physical considerations." "In this connection the reader might be referred to Whewell's History of the Inductive Sciences, Vol. I, Chapters I and II, where he shows the cause of the failure of the physical science of the Greeks: namely, that although they were keen observers of nature and reasoned very acutely, their ideas were not appro- priate to the facts. No mischief is more wide spread in modern science than *If gravity were a "property of space," there would be no reason why the attraction should increase with the mass, and thus Newton's law of direct proportionality to mass would be violated, so as to change entirely the recognized foundations of Physical Science. 60 THEORY OF THE TRANSMISSION OF PHYSICAL FORCES reasoning on false premises, though Aristotle warned us against this practice 2,200 years ago, and Newton emphasized the same caution again and again." "It is the belief of many experienced investigators that the whole Doctrine of Relativity rests on a false basis, and will some day be cited as an illustration of foundations laid in quicksand. Dozens of books have appeared on the sub- ject. Thus our problem to-day is not merely to discover Truth, but to discover a way out of an ensnaring mesh of errors." 'Tn view of these facts even the most aetherial pure mathematician ought to realize the impossibility of successfully treating gravitation from the analytical point of view, by merely transforming the equations, and ignoring the physical conditions which alone enable us to understand these stupendous forces of Nature. From a study of these problems running over many years, I venture to think that the chief difficulty consists not in Analysis or its transformation — since we know the law of the attractive force — but in discovering some physical agency by which we can consistently explain the enormous forces actually operative in gravitation." The accompanying diagram of the Newtonian theory of a central force of attraction constantly directed towards the Sun's centre and thus curving the Fig. 1. The Newtonian Theory, of the Central Force directed to the Sun's centre, incessantly acting on a planet and curving the path at every point of the orbit. planet's motion at every point of the orbit will make the central forces discussed in the above extract from the Observatory a little more intelligible than they other- wise would be. The stupendous power of these forces and the exactness with which they are directed to the Sun's centre must be clearly borne in mind by those who wish to form ideas appropriate to the facts of Nature. Moreover, if the Force of Gravita- tion is to be done away with, so also must the Force of Magnetism; yet we do not hear THEORY OF THE TRANSMISSION OP PHYSICAL FORCES 61 0/ such a theory of Magnetic Relativity, and it is doubtful if Einstein, De Sitter and their associates will propose one. In view of the surprising fact that so learned a mathematician as Dr. J. H. Jeans (in the Observatory for January, 1917, pp. 57-58) attempts to reply to the above criticisms, and speaks of Fermat's condition, it seems advisable to make known part of my investigations into that problem and others relating to the path of transmission of gravitational force, whether in right lines or in curves which conform to Fermat's Law of minimum path, in accordance with the Principle of Least Action. We shall therefore outline without further delay certain geomet- rical conditions, with the appropriate physical foundations, which underly the Electrodynamic Wave-Theory of Universal Gravitation, as applied to the Fluc- tuations of the Moon's Mean Motion. II Electrodynamic Waves Propagated in Free Aether with the Velocity OF Light the Physical Cause of Those Actions in Nature Which Involve the Principle of Least Time. In Bulletin No. 2 we have given a direct and simple proof that Magnetism is due to the action of plane waves originating in the atoms and propagated through- out aU space. Two magnets mutually attract when the rotations of the elements in the waves from one magnet undo the rotations of the elements in the waves from the other magnet: for this decreases the stress in the medium, and it then con- tracts just as a stretched mass of India rubber does when released. It was one of the most remarkable of Faraday's experimental discoveries that all bodies are magnetic. Thus it is estabUshed that all bodies emit waves of the type described in Bulletin No. 2. And if the waves from a second body are so directed as to decrease the stress in the medium between that and the first body, then the two masses will mutually attract. Upon this general theory rests the discovery of the cause of Magnetism, of Electrodynamic Action, and of Universal Gravitation. In the sixth Bulletin we shall examine with more detail, and enter with all due rigor into the cause of Universal Gravitation. It will be shown that the Medi- um suffers decrease of stress and is therefore under tension between two heavenly bodies, while beyond them there is increase of stress and therefore increase of pressure, both influences operating to hold the bodies together. The increase of tension between the masses and the increase of pressure beyond them fully explains Universal Gravita- tion, through mutually interpenetrating waves emitted from the atoms. Thus the geometrical and physical laws of wave action require rigorous investigation. The theory of physical forces developed in these Bulletins is so fundamentally 62 THEORY OF THE TRANSMISSION OF PHYSICAL FORCES different from that heretofore current, if indeed any uniform conception of forces can be said to have been held by investigators, that it is necessary to lay a secure foundation for the superstructure of the argument to foUow. Thus we have to develop a line of reasoning of some length; yet in the end we shall be compensated for our labors by a clearly defined view of the invisible mechanism underlying the processes of the physical Universe. Heretofore this view has been hazy and so securely hidden from our imagination as to be utterly bewildering to both the geometer and the natural philosopher. (i) The simple case of the reflection of Light from a plane mirror. We shall first illustrate these operations in the case of reflected light, because this case is the simplest, and has the advantage that it clearly illuminates some of the deepest mysteries of Nature. Fig. 2. Illustration of the reflection of a ray of light from a plane mirror, in the path of Least Time, along the perimeter of an Isosceles Triangle. It was an early discovery of the Greek natural philosophers, implied in the Geometrical Theorems of Euclid, about 300 B.C., but on physical grounds ascribed to Ptolemy by Laplace, that in the case of a plane mirror the bent line formed by the incident and reflected rays of light are shorter than any other THEOEY OF THE TRANSMISSION OP PHYSICAL FORCES 63 bent line with the same extremities, having its point of bending in the plane of reflection in the mirror. On account of the geometrical reasoning to follow this important observed fact is illustrated by the accompanying figure. Now what is the real cause of this apparent propagation of the light in the yath of Least Time? We shall try to answer this question briefly, in accordance with valid geometrical and physical principles. 1. In Euclid's Elements, Book I, Propositions 10-12, (Todhunter's edition, MacMillan & Co., 1903) we see how the Greek mathematicians reasoned in regard to the bisection of a line AB by means of a circle described from the centre with radius CF = CG, and the perpendicular CH at the mid-point of the circular arc FDG. 2. One of Euclid's celebrated axioms is that a straight line is the shortest distance between two points. He also shows that a perpendicular is the shortest distance from any point as C to a right line, as AB; and proves that any right line drawn from C to AB but not meeting it in the perpendicular at H is longer than CH. He does this by the use of the simple rule and compass; and his conclusion is so obvious that no elaborate demonstration is required. Fig. 3. Euclid's Geometrical Theorem of the perpendicular as the shortest distance from a point to given line, and the isosceles triangle which may be erected at equal distances from the base of the perpendicular. 3. Now in the above figure of the ray of light reflected from the mirror, AMB corresponds to Euclid's isosceles triangle FCG. Therefore AMB is an isosceles triangle, and MP is the perpendicular from the reflecting point M in the plane of the mirror. Consequently by geometrical construction the angles of incidence and of reflection necessarily are equal. 64 THEORY OF THE TRANSMISSION OF PHYSICAL FORCES 4. In regard to the physical cause of this equaUty of angles, it is obvious that it can only rest on the elastic rebound of the waves, at the surface of the mirror, due to the physical properties of the aether when the motion is retarded. This medium is elastic, and disturbances in it are therefore reflected according to the recognized law of action and reaction in ponderable bodies of elastic constitution. 5. A few familiar illustrations will suffice: (a) When a boy bounces a rubber ball on the smooth pavement, the descending and ascending angle of the ball's motion are equal, (b) When a glass, stone, or metal marble is thrown against a smooth, highly rigid slab, the angle of the rebound is noticed to be equal to that of the descent, and the same law holds for the reaction of all elastic solids in collision, (c) So also for arrested fluid movements, as when waves in water are reflected from a smooth wall. Here we may easily observe that the angle of incidence is equal to the angle of reflection. This law of Nature is indeed uni- versal, and rests on the physical property of the more or less perfect elasticity of form and volume. 6. Now since the propagation of light involves the transmission of waves in the aether, this medium evidently will react if it is elastic, and the forward motion is obstructed, as at the surface of the mirror. 7. It is shown geometrically, in the reasoning above cited from Euclid, that the shortest distance from M to AB is the perpendicular MP; and as the points A and B are equidistant from the base of the perpendicular P they too are equally distant from M, so that AM — MB. 8. Of all the triangles erected on AB as a base, and with vertices in the parallel line MD, in the plane of the mirror, the isosceles triangle has the shortest perimeter.* Consequently any triangle not isosceles, as AEB wiU have the sum of the sides AE + EB longer than AM + MB. Hence AMB is the path of the shortest time. 9. This result was known to the Greek mathematicians in the time of Euclid, 300 B.C., but the cause of it has not been fully agreed upon even in our own time. Hence we have entered with some detail into the circumstances of the motion of light. *The celebrated problem of perimeters was first treated by the Greek Geometer Zeno- DORUS, in the second century B.C. In general the perimeter is less the nearer the approach of the figure to that of a regular polygon, whether made up of 3 or n sides, so that an Isosceles triangle has the same property. THEOBY OP THE TRANSMISSION OP PHYSICAL FORCES 65 10. With these geometrical principles before us, we now ask what, then, is the physical cause of so wonderful a result? It was long thought by philos- ophers to be an indication of a tendency to simpUcity and economy in Nature. But from the above reasoning we conclude that it rests on two distinct grounds, as follows: (a) The very perfect elasticity of the aetherial medium. This gives a physical basis for the reflection of wave disturbances which cannot proceed in a right line, and necessarily makes the angle r = i. (b) As the reflection, on the same physical grounds, requires the reflected waves to follow the other equal side of an isosceles triangle, the geometrical path AMB nec- essarily is the shortest possible. And since light is confined to this isosceles triangle, by the physical properties of the elastic aether, it must on purely geometrical grounds be propagated in the Least Time. {ii) The case of Light bent out of its rectilinear path by Refraction. The above discussion is sufficient for the simple case of aether waves under- going reflection from a plane mirror; but in the case of refraction, more complex phenomena arise, and it is necessary to consider the path of the light as it goes through media which bend the rays from a rectilinear course. This problem was first critically examined by the French mathematician Fermat (1601 — 1665) whose researches on maxima and minima are considered by many modern geometers to be the earliest germs of the Differential Calculus. When Fermat tried to trace out this supposed economy of Nature, in the more complex problem of refraction, he encountered considerable difficulty, because in his time the relative velocities in the refracting media were not yet established by observation. About 1620, Snellius had discovered the law of refraction, which DesCartes, 1637, put in the form now used: sin i = n sin r. It was soon found that if the two lengths or indices be measured along the in- cident ray prolonged and the refracted ray, they have a common projection in the plane of refraction. In order to reconcile these phenomena with the acknowledged propagation in least time, supposed to follow from natural laws since the time of the Greek 66 THEORY OF THE TRANSMISSION OF PHYSICAL FORCES philosophers, Fermat was led to conclude from considerations of simplicity that the relative velocities are inversely as the refractive indices, and therefore that the velocity o£ light is diminished on entering the denser medium, in which the path was found by observation to approach the perpendicular.* Fermat believed that the total time of propagation would be found by taking the sum of the products of the length of path of the incident light by its velocity, in the first medium, kvi, and of the length of path of the refracted light by its velocity in the second medium, I2V2. Indeed, Fermat's mathematical treatment showed this compound sum, liVi + I2V2 to be less, for a path in a plane refraction, than that which would result if the light went by any other than its actual path. Hence resulted Fermat's condition of minimum path, which has since become so celebrated in physical science. Thus he was enabled to reconcile his mathematical discovery that in the actual path the sum r = liVi + I2V2 is a minimum, with his cosmological no- tion of propagation in Least Time, recognized in simple reflection since the age of the Greek geometers. For a time these results of Fermat's researches were attacked by DesCartes and his followers, but after the discovery of the Calculus they were zealously and effectively defended by Leibnitz. Huyghens was among the first to recognize Roemer's discovery of the velocity of light (1675), and thus he too was led to adopt Fermat's conclusions of a velocity inversely as the index of the medium, and of minimum time of propagation. But as Newton's widely current emission theory was not reconcilable with Fermat's results, the theorem of shortest time was abandoned by many. It was at this stage that Maupertuis proposed in its stead the celebrated Principle of Least Action. This new principle was afterwards much elaborated by the researches of Euler and Lagrange. (Hi) Maupertuis' Principle of Least Action. Maupertuis' principle of Least Action as formulated by Lagrange, is stated thus: (1) A = / / ^ niiVidSi *Cf. Tait, Article, Light, Encyc. Brit, 9th edition, Vol. XIV, p. 598. THEOEY OF THE TRANSMISSION OF PHYSICAL FORCES 67 This means that in any independent system, the integral of the actions between t = 0, t = t, is found by summing up the products of mass by velocity, by element of curvilinear space traversed. Maupertuis tried to reconcile the results of Fermat and Newton, by show- ing that the course chosen by light corresponds to the least possible action, though not always to the least possible time. The introduction of the product of masses by velocity, in the above formula, was required to compose the difficulties of Newton's corpuscular theory of light, and first occurs in Lagrange's researches on Dynamics. Among the several great mathematicians of the preceding age Euler was pre-eminently Newton's intellectual successor. And as the Swiss geometer became attached to Maupertuis during his residence at St. Petersburg and Ber- lin, and was agreeably impressed with the new principle of Least Action, he em- ployed his own great mathematical powers to show that the law of Least Action applies to motion in all curves described by points under the influence of central forces, such as planets and comets revolving about the Sun, and of heavy bodies falling to the Earth in the curve of least time (Brachistochrone), which had also long engaged the attention of Newton, Leibnitz, and the Bernoulis. In Euler's method of proof he takes the curve actually described and com- pares it with another having the same extremities, but differing from it indefinitely little in shape and position, yet such as may be imagined described by a neighbor- ing point with the same law of velocity. Then, applying the term action to the integral of the product of the velocity and element of the curve, he finds that the difference of the two neighboring values of this action will be indefinitely less than the greatest linear distance (itseff indefinitely small) between the two adjacent curves. Sir Wm. R. Hamilton, whose discussion we have here followed, calls this unchanging result Stationary Action. Lagrange extended this theorem of Euler to the motion of a system of bodies which act in any manner on each other, the action being formed of the sum of the masses by the velocity-space integrals employed by Euler, so as to give the Lagrange-Hamilton form: A =/ ^m,V,f:dt (2) dt In tridimensional space the velocity along the arc of any curve of double curvature becomes 68 THEORY OF THE TRANSMISSION OF PHYSICAL FORCES And therefore <^' ^:/l(:-^'m-m-m)-'M'-'^ " <=o This formula therefore represents the integral for the preservation of the Kinetic Energy or Vis-Viva. Ill The Principles of Least Action and of Varying Action Deducible PROM the Conservation of the Kinetic Energy under Physical Forces Due to Waves We have just seen that in any conservative system, such as an isolated star cluster, in steady motion but without collisions and in space free of a resisting medium, the changes of the squared velocities, in a given time t — to, mutually bal- ance up, and within the whole system entirely disappear; so that (5) 2m.y..^=/(^m.F.|')d« In other words: Given the system oi i + 1 bodies moving with velocities Vi at the epoch to, they will at the epoch t also be moving with other velocities v, of such a nature that although individual increases and decreases occur, yet in the whole system the squares multiplied by the masses mutually compensate, so that the action becomes (6) ^=/(2-'-^'S)'^^ ^1=0 And therefore the variation of the Action vanishes: (7) oA = o/ (S^ vu Yi ^) dt = 8(y^ m, vA = (=0 THEORY OF THE TRANSMISSION OF PHYSICAL FORCES 69 Now whilst the variation of the Action vanishes in a conservative system, it does not do so in a non-conservative one. Yet when work is done the change in the Kinetic Energy of the system in any time, to to t, depends only on the relative distances of the masses at the beginning and end of that time. The changes in the mutual distances is purely a problem of mutual action, under the attractive forces exerted by the bodies, along right lines joining their centres, and hence the absurdity of the claim made by Einstein and his followers that Gravity is not a "force," but a "property of space." For it is obvious that the paths of the motions follow from the initial co-ordinates Xi, yi, Zi, and the velocities -jj ' -^ > -ttS and the accelerations produced by the mutual action of the bodies of the system in the interval to to t. The conservation of areas holds even with internal collisions, but the vis-viva only for a conservative system. In the force function introduced by Lagrange the mutual potential of the I + 1 bodies taken in pairs mt mj is formulated thus: ^, momi , mowiz , m,,mi , , moin„ \ ■ „ • i U = -r 1- -r h -r h . . . + \ t = 0, J = 1 . Ao,2 Ao,3 A- Ai,2 Ai,3 Ai,„ A2 , 3 "211. + . / I = n - 1, ] = n^ 1 = 1 j-n = 5; S 'S ^^> (=0 J=l \,J And since A?,, = (e. - i)' + im - vjY + (r< - 1,)' (10) we have 9Ao,i ^ go - gi e)Ao,2 ^ go- ^2 ^^o„^ ^ go - g.. ago ~ Ao,i ' ago Ao,2 ' ■ ■ ■ ago Ao,„ (11) And the corresponding differential equations for the actions of the other bodies upon mo become: 70 THEORY OF THE TRANSMISSION OF PHYSICAL FORCES '^0 ~17T = ^0^1 1i — T + '^o'^ —A — T + ■ • • + mom,, dP Ao,i Ao,2' (12) ^^'?0 ™ ™ '?! ~ '?!' I '?2 — I70 Ao, 3 1 n Vn - - JJn Ao: 3 1 n fn- -fo Similarly for the action of the other bodies on the mass Wi, etc. These familiar equations (12) show that the actions of the bodies always are mutual; and therefore no change can be made in the component velocities of one body without corresponding reactions on the component velocities of the other bodies. It is obvious therefore that changes of velocities due to inter-actions necessarily are mutual. It is only under interpenetration of waves that stress of the medium is developed, and the magnitudes of the changes produced in the energy of these masses depend upon the /orces at work, under the stresses operating in the action of the attracting bodies to which any single mass may be subjected within the system. Accordingly the changes within the system, depending on the mutual actions of its component masses, necessarily will be such that the Action is in accordance with the Lagrange-Hamilton Equation (13) ^ =/(S'"'"'^>^^ = And therefore, in a conservative system, free of collisions and devoid of friction (14) 5.4 = s/(y^w,vAdt = Thus any system necessarily pursues the path of Least Action under its own forces; for this merely conserves the vis viva. If the bodies be started with the velocities v,- and co-ordinates x,-, y,, Zi, the changes in the velocities due to the mutual wave action of the bodies of the system necessarily will be such as to con- form to the forces which are appropriate to the given co-ordinates. No other changes are dynamically possible. Consequently the system always conserves the vis viva, and hence automatically pursues the path of Least Action; and therefore for any interval, to to t, we have the above integrals (13) and (14). But starting at any epoch to it is obvious that THEORY OF THE TRANSMISSION OF PHYSICAL FORCES 71 ^ = /(y^^niVA dt = / Cdt , C = a constant, (15) and therefore for any given interval of time t - U there is no change in the Action. From the point of view of the conservation of energy, we may write T for the kinetic energy, V for the potential energy; and then for the total energy of the sj^stem we have And as the total energy is constant for any system subjected only to the mutual actions of its parts, it is obvious that we shall have BT = —ZV; therefore Accordingly this equation defines any variation in the potential energy which may occur as due to variations in the co-ordinates. The integral therefore 8A = hi ^ {m,Y,^dt = (18) expresses Hamilton's Stationary Condition. Hamilton calls the changes of the Action due to variation of path, with the same terminal co-ordinates, Varying Action. Let W denote the work function, and T the kinetic and V the potential energy; then we have L = T + W = T - V . (19) But W and V are functions of the co-ordinates, not of the velocities; and therefore if AB be the actual trajectory, CD the neighboring path, with the same terminal points and times of description, we obviously have identical action in the two paths Cbdi - fldt = (20) which shows that the Action is Stationary. 72 THEORY OF THE TRANSMISSION OF PHYSICAL FORCES This result is an extension of the researches of Euler and Lagrange, and known as Hamilton's Principle. It implies that the motion in the same time is one of Least Action, along paths appropriate to the conservation of the kinetic energy, and thus combines Fermat's Principle of Least Time with Maupertuis' Principle of Least Action. All these results really depend on the conservation of the kinetic energy, and thus are referable to wave action propagated in least time. In any conservative system, the conservation of the Kinetic Energy is obvious. This is all that Least Action implies; and it is traceable to physical forces directed to the masses mu- tually, and due to the interaction of infinite systems of waves in the aethereal medium, everywhere under tension and therefore propagated in the least time, and thus more rapidly in free space than through ponderable bodies, involving resistance to the progress of the disturbances. Accordingly, if such an attractive force as Universal Gravitation be due to electrodynamic wave action, as shown by the writer's researches of 1914-17, it will follow that the disturbances due to such waves, pursuing paths of least action, in least time, — but with the adjacent path offering unequal resistance — will necessarily suffer refraction and dispersion, perhaps absorption, while traversing the solid matter of the heavenly bodies. . The Sun's gravitational attraction on the Moon, therefore, will be decreased in and near the shadow of the Earth, giving that very "expenditure or absorption of energy somewhere in the solar system" postulated by Newcomb for explaining the Moon's Fluctuations, in his posthumous memoir of 1912. Accordingly whilst there is conservation of energy in the Universe, this con- servation is not perfect in the temporary operations of a single system involving the mutual interaction of parts under forces propagated through solid masses, in which the electrodynamic waves may be refracted and dispersed, when bent from a rectilinear path in transmission. It was in the hope of unfolding this view of the mutual actions of the heavenly bodies, when these actions are somewhat obstructed, by the interposition of some of these masses in the rectilinear paths of the mutual electrodynamic wave action, that we have dwelt at length on the laws of Dynamics. We cannot assume the ordinary formulae for the dynamics of particles, with- out considering the interposition of ponderable masses of finite dimensions in the line of mutual action for wave disturbances traveling with the velocity of Light, and therefore suffering both refraction and dispersion, perhaps absorption, of part of the wave energy. But as the dimensions of the heavenly bodies are very small compared to the immense distances which separate them, there is reason THEORY OF THE TRANSMISSION OF PHYSICAL FORCES 73 to anticipate that the modification of the gravitational forces in propagation through the masses should be very small. It is not wonderful therefore that this refraction, dispersion and perhaps absorption of gravitational energy long escaped attention; and is only now be- ginning to be placed on a sound mathematical and physical basis, and verified by observations drawn from the motion of the Moon. IV Application of Hamilton's Principle to the Wave-Theory of Light, AND Thence to- Electrodynamic Waves in Electrodynamics and Universal Gravitation In the Wave Theory of Light the time of passage from one point to another, as was found by Fermat, rests on the principle of least time. Accordingly this time of transmission should thus fulfiU Hamilton's Stationary Condition. We restrict our present considerations to the case of single refraction in a non-homo- geneous medium; for if it applies to this case perfectly, it will obviously apply to the more complicated phenomena of light which occur in Nature, and also to the Electrodynamic Waves which give rise to the attractions observed in Elec- trodynamic Action and Universal Gravitation. In the problems of refractions, we have to consider the curved path of the light; and the measure of curvature is defined by the equations: s = acj) , \ curvature = ^- = - ' I as a ) where cj) is the angle between the osculating tangent planes, and ds is the element of the curve, and a the radius of curvature, for the osculating circle passing through three consecutive points. For any path in space, the curvature is 1 ^ I f'^dd = K ■ V ■ p , h dA = vds — - ds , V where p is the perpendicular distance from the focus upon the tangent to the ellipse, and h is constant. Now we have also (of. Williamson and Tarlton's Dynamics, 1885, §166, p. 174) for the radius vector r' and perpendicular p' from the upper focus V p' r' (64) P = — ' ^ = - ■ V ^ ' p' p r Therefore we have dt = Kvp , (6^^ ) dA = \,ds- p' THEORY OF THE TEANSMISSION OF PHYSICAL FORCES 83 And as ^' = p » we see that while dt depends on the area described about the lower focus, the element of the action dA on the other hand, depends on the area described about the upper focus.* This is a very curious result in the plan- etary theory, and shows that the Time and the Action are in a reciprocal rela- tion as to the origins of their respective areas used as the basis of the measure- ments. Fig. 4. Illustration of the Areas afc and a'fb' described about the occupied focus /, which measure the times t' and t, according to Kepler's second Law; and of the Actions a'fb' and afc described about the empty focus/' in the intervals t and t'. The Actions are very unequal, as might be expected from the very unequal intensities of the central forces at perihelion and aphelion respectively, the product vds magnifying the difference in the ratio of the square of the forces. The integrals between the instants to and t are respectively: r = r r'dO = C{t - to) , A = r (vds) dt . *Cf. Tait, Article Mechanics, Ency. Brit., 9th edition. (66) 84 THEORY OF THE TRANSMISSION OF PHYSICAL FORCES Here C is the constant of areas, according to Kepler's second law; and A may be calculated from the areas described about the empty focus as origin. (m) Hamilton's method of Varying Action applied to Light of uniform wave length propagated through a medium arranged in concentric spherical shells of uni- form density. We may assume that at a distance r from the centre of the sphere, the light will be traveling in the spherical shell of mass (67) d7n = Awar^dr It thus appears that the density varies inversely as r^, for we have cbu, 1 rr = (68) ■■ - A ^ But the velocity decreases in denser media, so that it varies in the inverse ratio of the density, or directly as r^. We may therefore take b' + r' (69) V = —^ where h and c are absolute constants, such that at the centre of the sphere, where r = , b' (70) Vo= - On account of symmetry, the path of any ray necessarily will be in a plane through the centre. CalUng this plane that of xy we have ^^'^> \dx) ^ [dyj v' ib' + rY In the more convenient polar co-ordinates this expression becomes (72) [Tr)+7^ drj ' r\d9j (b'^ + ry THEORY OP THE TRANSMISSION OF PHYSICAL FORCES 85 Of this equation see seek the general sokition. If now we assume, as before, dr then (72) gives 36 = ^' (^^) dr ~ \( a i{b^ + rY r' And as (73) yields the first solution t = a6, the equation (74) leads to (74) . =y d.^ (75) f (-52 _,_ ^y ^2 Hence the complete solution from (74) becomes the general solution: The equation of the path of the light therefore is , dr „ f dr a '■•{i da I I e 2 (7y) (6" + ry r" = 6 — cos^ r^t,-v »« V This gives at once = y ^] - 46^ • COS (9 - a') (79) 6^ -r' (iv) The Paths of the Rays in Diametral Planes through the Concentric Spherical Shells are Circles. Accordingly, we conclude that the above equation represents a series of circles, with the following common property. Let ¥ be the square of the least half chord, through any point as cut by any one of the concentric spherical shells. Then it is shown in Geometry (cf. Chauvenet's Geometry, Book III, Prop. XVII, 86 THEORY OF THE TRANSMISSION OF PHYSICAL FORCES p. 114) that this is equal to the product of the two segments AO-OA' in any other direction whatever, (80) AO • OA' = b' The points A and A' are conjugate foci, in respect to 0, the common centre of the concentric spherical shells; for light going from either focus, will return to the other through the paths of the circle specified. In the theory of Light this remarkable result is of great interest in connection with the phenomena of the Mirage. In Electrodynamics it opens up a physical theory of the decrease of the Sun's gravitation in the shadows of the planets, and thus will throw new light on the Unexplained Fluctuations of the Moon's Mean Motion first suspected by New- comb in 1869, and definitely established by his last Researches in 1909. Fig. 5. Maxwell's Theory of the Circular Refraction of rays of light in the eye of a fish, which also applies to the Refraction of Electrodynamic Waves in traversing the Earth made up of concentric spherical shells with density increasing towards the centre. From the above reasoning we see that every ray through a sphere made up of concentric spherical shells of uniform density, but with the density increas- ing towards the centre, will be confined to a diametral plane, and describe a cir- cle; and a pair of conjugate foci will always lie on a line through the centre of the shells. Maxwell discovered this remarkable result from the arrangement of the eye of a fish (Cambridge and Dublin Mathematical Journal, Vol. XI), but THEORY OF THE TRANSMISSION OF PHYSICAL FORCES 87 the present discussion follows the analysis given by Tait, in the Article, Light, Encyc. Brit., 9th edition. In the figure let be the common centre of the spherical shells of equal re- fractive index; then obviously, every ray from A describes a circle which passes through A', and AOA' is a straight fine through the centre. A and A', B and B' are conjugate foci; and by Geometry the common property is : AO ■ OA' = b' , BO ■ OB' = b' , etc. ) ^^^^ If we suppose an eye situated in the layer at Ei, in the small circle of density cTi, about the centre, it will receive the light from A along the tangent to the larger circle through AEiA', and similarly for any other point B, seen along the tangent to the circle BE^B'. These images wiU be erect, for they have not yet passed through the con- jugate foci, A', B' ; but if the eye now be turned in the opposite direction, towards A', B', the light will have traversed the longer paths AE2A', BEzB', and enter the eye after turning its course through 180° and passing through the conjugate foci A' , B' , so that the image will he inverted and moreover represent the hack of the object. It is obvious from the figure why this results — namely, the circuit- ous course to the opposite focus produces inversion, and the light so proceeding is from the hack of the objects at AB. (v) Description of the paths of Gravitational Waves propagated through a Heterogeneous Sphere made up of concentric spherical shells of uniform density. If the gravitation of the Sun pass through the Earth, in the form of Elec- trodynamic Waves, it will obviously suffer some decrease in and near the shadow of the Earth; and thus operate to disturb the Moon during echpses, as pointed out by Dr. K. F. Bottlinger in his crowned Prize Inaugural Dissertation, Munich, 1912 (c. Troemer's Universitats Buchhandlung, Freiburg, 1912). The above outline of the Hamiltonian theory of refraction in a sphere made up of concentric spherical shells of uniform density, but increasing in density towards the centre will enable us to see that in passing through the globe the Elec- trodynamic Waves must necessarily be somewhat diverted from a rectilinear course, and therefore the effect would be to decrease the Sun's gravity on the Moon when near the Earth's shadow. This new physical theory thus confirms Bottlinger's researches, indicating that the Sun's gravitation is less near the anti-solar point; and the subject there- fore becomes one of such high interest in connection with the Lunar Theory that it deserves the most careful investigation. 88 THEORY OP THE TRANSMISSION OF PHYSICAL FORCES As the waves are refracted through the Earth, they are partially turned aside in circles of various radii. Within a curved conical area having its vertex at the Earth's centre and extending beyond the Moon's orbit, the gravity of the Sun is decreased. This physical refraction of part of the Gravitational Waves ,h. ! Illustration of the Refraction of the Electrodynamic Waves from the Sun as they traverse the globe of the Earth, and are partially refracted to either side, before reach mg the Moon, so that the Sun's effective gravity is decreased near the shadow of the Ear t and the result .s periodic Fluctuations in the Moon's motion depending on the ^aland otW cycles connected with the movements of the Perigee and Nodes of the Lunar Orbit is iUustrated roughly by the accompanying figure, which for the sake of simplicity IS restricted to uniform wave length, and thus omits the effects of dispersion. THEORY OF THE TRANSMISSION OP PHYSICAL FORCES 89 VI Analysis of the Secular Effects of the Circular Refraction, Dis- persion AND perhaps Absorption of Gravitational Waves Propagated Through the Concentric Homogeneous Spherical Shells Composing THE Earth. (i) Cause of the circular refraction, dispersion and perhaps absorption of rays. From the foregoing mathematical theory and the accompanying figure of the law of density in the Earth {A.N. 3992) . 3e sin v , And since the integral J always has the same positive fore sign, we find that when 0° < «; < 180°, or the Moon in appulse is going from perigee to apogee, the mean motion n is diminished; but when 180° the angle about the axis of symmetry, and the other quantities as illustrated in the figure. M Fig. 3. Bottlingee's Theory of a slight absorption of a ray of gravitation in traversing the Earth. Accordingly, the attraction becomes (9) z = i B cos adm = 'k'^ \ l J (Tdpdad