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62 And still the Church's children raise That strain so lofty and so strong, Which makes their matin hymn of praise As jubilant as evensong.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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Rh And His the glory still revealed When lovers plight and keep their vows; Himself the Bridegroom Who has sealed The Church to be His Spouse.


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Rh Soft whisperings and laughters loud, The song of birds, the insects' hum, Storm-music of the thunder-cloud— And be no longer dumb.


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Usage
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title: Template:-/doc


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Rh


text: Page:The Mirror of the Sea.djvu/9

title: Page:The Mirror of the Sea.djvu/9


text:
Page:Devil-Worship in France 1896.djvu/2

 {|
 * Demy 8vo, about 450 pages, cloth

BY

ELIPHAS LEVI

A COMPLETE TRANSLATION OF"DOGME ET RITUEL DE LA HAUTE

MAGICMAGIE [sic]"

by

ARTHUR EDWARD WAITE

With all the original engravings and a portrait of the Author.

GEORGE REDWAY

9HART STREET, BLOOMSBURY

 * }


text: Talk:A Study of Mexico

title: Talk:A Study of Mexico


text:
Page:The letters of William Blake (1906).djvu/203

 Rh or sorrow.—I have the honour to remain, your ever affectionate and much obliged,

I thank kindly for the remembrance of I. Denman: he is not my nephew, he is whole brother to Maria and half-brother to my wife.

30.

To William Hayley.

London, 14th January 1804,

,—I write immediately on my arrival, not merely to inform you that in a conversation with an old soldier, who came in the coach with me, I learned that no one, not even the most expert horseman, ought ever to mount a trooper s horse. They are taught so many tricks, such as stopping short, falling down on their knees, running sideways, and in various and innumerable ways endeavouring to throw the rider, that it is a miracle if a stranger escape with his life. All this I learned with some alarm, and heard also what the soldier said confirmed by another person in the coach. I therefore, as it is my duty, beg and entreat you never to mount that wretched horse again, nor again trust to one who has been so


text:
Executive Order 13560

By the authority vested in me as President by the Constitution and the laws of the United States of America, and in order to encourage the growth and maximize the impact of innovative community solutions and civic participation by all Americans, it is hereby ordered as follows:

, December 14, 2010. [FR Doc. 2010–31878 Filed 12–16–10; 8:45 am] Billing Code 3195–W1–P

title: Executive Order 13560


text: 1911 Encyclopædia Britannica/Capillary Action


text: 1911 Encyclopædia Britannica/Cap Haitien


text:
Page:BicentennialBellPresentation Part1.jpg

 HERE ON JULY 6, 1976, HER MAJESTY QUEEN ELIZABETH II OF GREAT BRITAIN PRESENTED THE BICENTENNIAL BELL TO THE PEOPLE OF THE UNITED STATES, IN THESE WORDS

I speak to you as the direct descendant of King George III. He was the last Crowned Sovereign to rule in this country, and it is therefore with a particular personal interest that I view those events which took place 200 years ago.

It seems to me that Independence Day, the Fourth of July, should be celebrated as much in Britain as in America. Not in rejoicing at the separation of the American Colonies from the British Crown, but in sincere gratitude to the Founding Fathers of this great Republic for having taught Britain a very valuable lesson.

We lost the American Colonies because we lacked that statesmanship "to know the right time, and the manner of yielding, what is impossible to keep".


text:
In a Balcony/II

 * enter the.

Queen.

Constance? She is here as he said. Speak quick! Is it so? is it true or false? One word!

Constance.

True.

Queen.

Mercifullest Mother, thanks to thee!

Constance.

Madam?

Queen.

I love you, Constance, from my soul. Now say once more, with any words you will, 'Tis true, all true, as true as that I speak,

Constance.

Why should you doubt it?

Queen.

Ah, why doubt? why doubt? Dear, make me see it. Do you see it so? None see themselves—another sees them best. You say "why doubt it?"—you see him and me. It is because the Mother has such grace That if we had but faith—wherein we fail— Whate'er we yearn for would be granted us; Yet still we let our whims prescribe despair, Our fancies thwart and cramp our will and power, And, while accepting life, abjure its use. Constance, I had abjured the hope of love And being loved, as truly as yon palm The hope of seeing Egypt from that plot.

Constance.

Heaven!

Queen.

But it was so, Constance, it was so! Men say—or do men say it? fancies say— "Stop here, your life is set, you are grown old. Too late—no love for you, too late for love— Leave love to girls. Be queen: let Constance love." One takes the hint—half meets it like a child, Ashamed at any feelings that oppose. "Oh love, true, never think of love again! I am a queen—I rule, not love forsooth." So it goes on; so a face grows like this, Hair like this hair, poor arms as lean as these, Till,—nay, it does not end so, I thank God!

Constance.

I cannot understand——

Queen.

The happier you! Constance, I know not how it is with men: For women (I am a woman now like you) There is no good of life but love—but love! What else looks good, is some shade flung from love; Love gilds it, gives it worth. Be warned by me, Never you cheat yourself one instant! Love, Give love, ask only love, and leave the rest! O Constance, how I love you!

Constance.

I love you.

Queen.

I do believe that all is come through you. I took you to my heart to keep it warm When the last chance of love seemed dead in me; I thought your fresh youth warmed my withered heart. Oh, I am very old now, am I not? Not so! it is true and it shall be true!

Constance.

Tell it me: let me judge if true or false.

Queen.

Ah, but I fear you! you will look at me And say, "she's old, she's grown unlovely quite Who ne'er was beauteous: men want beauty still." Well, so I feared—the curse! so I felt sure!

Constance.

Be calm. And now you feel not sure, you say?

Queen.

Constance, he came,—the coming was not strange— Do not I stand and see men come and go? I turned a half-look from my pedestal Where I grow marble—"one young man the more! He will love some one; that is nought to me: What would he with my marble stateliness?" Yet this seemed somewhat worse than heretofore; The man more gracious, youthful, like a god, And I still older, with less flesh to change— We two those dear extremes that long to touch. It seemed still harder when he first began To labour at those state-affairs, absorbed The old way for the old end—interest. Oh, to live with a thousand beating hearts Around you, swift eyes, serviceable hands, Professing they've no care but for your cause, Thought but to help you, love but for yourself,— And you the marble statue all the time They praise and point at as preferred to life, Yet leave for the first breathing woman's smile, First dancer's, gipsy's or street baladine's! Why, how I have ground my teeth to hear men's speech Stifled for fear it should alarm my ear, Their gait subdued lest step should startle me, Their eyes declined, such queendom to respect, Their hands alert, such treasure to preserve, While not a man of them broke rank and spoke, Wrote me a vulgar letter all of love, Or caught my hand and pressed it like a hand! There have been moments, if the sentinel Lowering his halbert to salute the queen, Had flung it brutally and clasped my knees, I would have stooped and kissed him with my soul.

Constance.

Who could have comprehended?

Queen.

Ay, who—who? Why, no one, Constance, but this one who did. Not they, not you, not I. Even now perhaps It comes too late—would you but tell the truth.

Constance.

I wait to tell it.

Queen.

Well, you see, he came, Outfaced the others, did a work this year Exceeds in value all was ever done You know—it is not I who say it—all Say it. And so (a second pang and worse) I grew aware not only of what he did, But why so wondrously. Oh, never work Like his was done for work's ignoble sake— Souls need a finer aim to light and lure! I felt, I saw, he loved—loved somebody. And Constance, my dear Constance, do you know, I did believe this while 'twas you he loved.

Constance.

Me, madam?

Queen.

It did seem to me your face Met him—where'er he looked: and whom but you Was such a man to love? It seemed to me You saw he loved you, and approved his love, And both of you were in intelligence. You could not loiter in that garden, step Into this balcony, but I straight was stung And forced to understand. It seemed so true, So right, so beautiful, so like you both That all his work should have been done by him Not for the vulgar hope of recompense, But that at last—suppose, some night like this— Borne on to claim his due reward of me, He might say "Give her hand and pay me so." And I (O Constance, you shall love me now!) I thought, surmounting all the bitterness, —"And he shall have it. I will make her blest, My flower of youth, my woman's self that was, My happiest woman's self that might have been! These two shall have their joy and leave me here." Yes—yes!

Constance.

Thanks!

Queen.

And the word was on my lips When he burst in upon me. I looked to hear A mere calm statement of his just desire In payment of his labour. When—O heaven, How can I tell you? lightning on my eyes And thunder in my ears proved that first word Which told 'twas love of me, of me, did all— He loved me—from the first step to the last, Loved me!

Constance.

You hardly saw, scarce heard him speak Of love: what if you should mistake?

Queen.

No, no— No mistake! Ha, there shall be no mistake! He had not dared to hint the love he felt— You were my reflex—(how I understood!) He said you were the ribbon I had worn, He kissed my hand, he looked into my eyes, And love, love came at end of every phrase. Love is begun; this much is come to pass: The rest is easy. Constance, I am yours! I will learn, I will place my life on you, Teach me but how to keep what I have won! Am I so old? This hair was early grey; But joy ere now has brought hair brown again, And joy will bring the cheek's red back, I feel. I could sing once too; that was in my youth. Still, when men paint me, they declare me. . . yes, Beautiful—for the last French painter did! I know they flatter somewhat; you are frank— I trust you. How I loved you from the first! Some queens would hardly seek a cousin out And set her by their side to take the eye: I must have felt that good would come from you. I am not generous—like him—like you! But he is not your lover after all: It was not you he looked at. Saw you him? You have not been mistaking words or looks? He said you were the reflex of myself. And yet he is not such a paragon To you, to younger women who may choose Among a thousand Norberts. Speak the truth! You know you never named his name to me: You know, I cannot give him up—ah God, Not up now, even to you!

Constance.

Then calm yourself.

Queen.

See, I am old—look here, you happy girl! I will not play the fool, deceive—ah, whom? 'Tis all gone: put your cheek beside my cheek And what a contrast does the moon behold! But then I set my life upon one chance, The last chance and the best—am I not left, My soul, myself? All women love great men If young or old; it is in all the tales: Young beauties love old poets who can love— Why should not he, the poems in my soul, The passionate faith, the pride of sacrifice, Life-long, death-long? I throw them at his feet. Who cares to see the fountain's very shape Whether it be a Triton's or a Nymph's That pours the foam, makes rainbows all around? You could not praise indeed the empty conch; But I'll pour floods of love and hide myself. How I will love him! Cannot men love love? Who was a queen and loved a poet once Humpbacked, a dwarf? ah, women can do that! Well, but men too; at least, they tell you so. They love so many women in their youth, And even in age they all love whom they please; And yet the best of them confide to friends That 'tis not beauty makes the lasting love— They spend a day with such and tire the next: They like soul,—well then, they like phantasy, Novelty even. Let us confess the truth, Horrible though it be, that prejudice, Prescription. . . curses! they will love a queen. They will, they do: and will not, does not—he?

Constance.

How can he? You are wedded: 'tis a name We know, but still a bond. Your rank remains, His rank remains. How can he, nobly souled As you believe and I incline to think, Aspire to be your favourite, shame and all?

Queen.

Hear her! There, there now—could she love like me? What did I say of smooth-cheeked youth and grace? See all it does or could do! so youth loves! Oh, tell him, Constance, you could never do What I will—you, it was not born in! I Will drive these difficulties far and fast As yonder mists curdling before the moon. I'll use my light too, gloriously retrieve My youth from its enforced calamity, Dissolve that hateful marriage, and be his, His own in the eyes alike of God and man.

Constance.

You will do—dare do. . . pause on what you say!

Queen.

Hear her! I thank you, sweet, for that surprise. You have the fair face: for the soul, see mine! I have the strong soul: let me teach you, here. I think I have borne enough and long enough, And patiently enough, the world remarks, To have my own way now, unblamed by all. It does so happen (I rejoice for it) This most unhoped-for issue cuts the knot. There's not a better way of settling claims Than this; God sends the accident express: And were it for my subjects' good, no more, 'Twere best thus ordered. I am thankful now, Mute, passive, acquiescent. I receive, And bless God simply, or should almost fear To walk so smoothly to my ends at last. Why, how I baffle obstacles, spurn fate! How strong I am! Could Norbert see me now!

Constance.

Let me consider. It is all too strange.

Queen.

You, Constance, learn of me; do you, like me! You are young, beautiful: my own, best girl, You will have many lovers, and love one— Light hair, not hair like Norbert's, to suit yours: Taller than he is, since yourself are tall. Love him, like me! give all away to him; Think never of yourself; throw by your pride, Hope, fear,—your own good as you saw it once, And love him simply for his very self. Remember, I (and what am I to you?) Would give up all for one, leave throne, lose life, Do all but just unlove him! He loves me.

Constance.

He shall.

Queen.

You, step inside my inmost heart! Give me your own heart: let us have one heart! I'll come to you for counsel; "this he says, This he does; what should this amount to, pray? Beseech you, change it into current coin! Is that worth kisses? Shall I please him there?" And then we'll speak in turn of you—what else? Your love, according to your beauty's worth, For you shall have some noble love, all gold: Whom choose you? we will get him at your choice. —Constance, I leave you. Just a minute since, I felt as I must die or be alone Breathing my soul into an ear like yours: Now, I would face the world with my new life, Wear my new crown. I'll walk around the rooms, And then come back and tell you how it feels. How soon a smile of God can change the world! How we are made for happiness—how work Grows play, adversity a winning fight! True, I have lost so many years: what then! Many remain: God has been very good. You, stay here! 'Tis as different from dreams, From the mind's cold calm estimate of bliss, As these stone statues from the flesh and blood. The comfort thou hast caused mankind, God's moon!

[''She goes out, leaving Constance. Dance-music from within.''

title: In a Balcony/II


text: Index:Lorentz Simplified1899.djvu

title: Index:Lorentz Simplified1899.djvu


text:
Page:BicentennialBellPresentation Part2.jpg

 But the lesson was learnt. In the next century and a half we kept more closely to the principles of Magna Carta which have been the common heritage of both countries.

We learnt to respect the right of others to govern themselves in their own ways. This was the outcome of experience learned the hard way in 1776. Without that great act in the cause of liberty performed in Independence Hall two hundred years ago, we could never have transformed an Empire into a Commonwealth!

Ultimately peace brought a renewal of friendship which has continued and grown over the years and has played a vital part in world affairs. Together we have fought in two world wars in the defence of our common heritage of freedom. Together we have striven to keep the peace so dearly won. Together, as friends and allies, we can face the uncertainties of the future, and this is something for which we in Britain can also celebrate the Fourth of July.


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Page:Lorentz Simplified1899.djvu/1

By. Prof.

§ 1. In former investigations I have assumed that, in all electrical and optical phenomena, taking place in ponderable matter, we have to do with small charged particles or ions, having determinate positions of equilibrium in dielectrics, but free to move in conductors except in so far as there is a resistance, depending on their velocities. According to these views an electric current in a conductor is to be considered as a progressive motion of the ions, and a dielectric polarization in a non-conductor as a displacement of the ions from their positions of equilibrium. The ions were supposed to be perfectly permeable to the aether, so that they can move while the aether remains at rest. I applied to the aether the ordinary electromagnetic equations, and to the ions certain other equations which seemed to present themselves rather naturally. In this way I arrived at a system of formulae which were found sufficient to account for a number of phenomena.

In the course of the investigation some artifices served to shorten the mathematical treatment. I shall now show that the theory may be still further simplified if the fundamental equations are immediately transformed in an appropriate manner.

§ 2. I shall start from the same hypotheses and introduce the same notations as in my "Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern". Thus, $$\mathfrak{d}$$ and $$\mathfrak{H}$$ will represent the dielectric displacement and the magnetic force, $$\varrho$$ the density to which the ponderable matter is charged, $$\mathfrak{v}$$ the velocity of this matter, and $$\mathfrak{E}$$ the force acting on it per unit charge (electric force). It is only in the interior of the ions that the density $$\varrho$$ differs from 0; for simplicity's sake I shall take it to be a continuous function of the coordinates, even at the surface of the ions. Finally, I suppose that each element of an ion retains its charge while it moves.

If, now, V be the velocity of light in the aether, the fundamental equations will be

title: Page:Lorentz Simplified1899.djvu/1


text:
Page:Lorentz Simplified1899.djvu/2

 § 3. We shall apply these equations to a system of bodies, having a common velocity of translation $$\mathfrak{p}$$, of constant direction and magnitude, the aether remaining at rest, and we shall henceforth denote by $$\mathfrak{v}$$, not the whole velocity of a material element, but the velocity it may have in addition to $$\mathfrak{p}$$.

Now it is natural to use a system of axes of coordinates, which partakes of the translation $$\mathfrak{p}$$. If we give to the axis of x the direction of the translation, so that $$\mathfrak{p}_{y}$$ and $$\mathfrak{p}_{z}$$ are 0, the equations (Ia)— (Va) will have to be replaced by

{{MathForm2|(IIIb)|$$\left.\begin{array}{l} \frac{\partial\mathfrak{H}_{z}}{\partial y}-\frac{\partial\mathfrak{H}_{y}}{\partial z}=4\pi\varrho(\mathfrak{p}_{x}+\mathfrak{v}_{x})+4\pi\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}\right)\mathfrak{d}_{x,}\\ \\\frac{\partial\mathfrak{H}_{x}}{\partial z}-\frac{\partial\mathfrak{H}_{z}}{\partial x}=4\pi\varrho\mathfrak{v}_{y}+4\pi\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}\right)\mathfrak{d}_{y},\\ \\\frac{\partial\mathfrak{H}_{y}}{\partial x}-\frac{\partial\mathfrak{H}_{x}}{\partial y}=4\pi\varrho\mathfrak{v}_{z}+4\pi\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}\right)\mathfrak{d}_{z},\end{array}\right\} $$}}

{{MathForm2|(IV{{sub|b}})|$$\left.\begin{array}{l} 4\pi V^{2}\left(\frac{\partial\mathfrak{d}_{z}}{\partial y}-\frac{\partial\mathfrak{d}_{y}}{\partial z}\right)=-\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}\right)\mathfrak{H}_{x},\\ \\4\pi V^{2}\left(\frac{\partial\mathfrak{d}_{x}}{\partial z}-\frac{\partial\mathfrak{d}_{z}}{\partial x}\right)=-\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}\right)\mathfrak{H}_{y},\\ \\4\pi V^{2}\left(\frac{\partial\mathfrak{d}_{y}}{\partial x}-\frac{\partial\mathfrak{d}_{x}}{\partial y}\right)=-\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}\right)\mathfrak{H}_{z},\end{array}\right\} $$}}

{{MathForm2|(V{{sub|b}})|$$\mathfrak{E}=4\pi V^{2}\mathfrak{d}+[\mathfrak{p.H}]+[\mathfrak{v.H}]$$}}

In these formulae the sign Div, applied to a vector $$\mathfrak{A}$$, has still the meaning defined by

$$Div\ \mathit{\mathfrak{A}}=\frac{\partial\mathfrak{A}_{x}}{\partial x}+\frac{\partial\mathfrak{A}_{y}}{\partial y}+\frac{\partial\mathfrak{A}_{z}}{\partial z}$$.

As has already been said, $$\mathfrak{v}$$ is the relative velocity with regard to the moving axes of coordinates. If $$\mathfrak{v}=0$$, we shall speak of a system at rest; this expression therefore means relative rest with regard to the moving axes.

title: Page:Lorentz Simplified1899.djvu/2


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 In most applications $$\mathfrak{p}$$ would be the velocity of the earth in its yearly motion.

§ 4. Now, in order to simplify the equations, the following quantities may be taken as independent variables

The last of these is the time, reckoned from an instant that is not the same for all points of space, but depends on the place we wish to consider. We may call it the local time, to distinguish it from the universal time t.

If we put

$$\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}=k$$,

we shall have

$$\frac{\partial}{\partial x}=k\frac{\partial}{\partial x'}-k^{2}\frac{\mathfrak{p}_{x}}{V^{2}}\frac{\partial}{\partial t'},\ \frac{\partial}{\partial y}=\frac{\partial}{\partial y'},\ \frac{\partial}{\partial z}=\frac{\partial}{\partial z'},\ \frac{\partial}{\partial t}=\frac{\partial}{\partial t'}$$.

The expression

$$\frac{\partial\mathfrak{A}_{x}}{\partial x'}+\frac{\partial\mathfrak{A}_{y}}{\partial y'}+\frac{\partial\mathfrak{A}_{z}}{\partial z'}$$

will be denoted by

$$Div'\ \mathfrak{A}$$.

We shall also introduce, as new dependent variables instead of the components of $$\mathfrak{d}$$ and $$\mathfrak{H}$$, those of two other vectors $$\mathfrak{F}'$$ and $$\mathfrak{H}$$, which we define as follows

In this way I find by transformation and mutual combination of the equations (Ib)—(Vb):

title: Page:Lorentz Simplified1899.djvu/3


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Page:BicentennialBellPresentation Part3.jpg

 This morning I saw the famous Liberty Bell. It came here over 200 years ago when Philadelphia, after London, was the largest English speaking city in the world. It was cast to commemorate the Pennsylvania Charter of Privileges, but is better known for its association with the Declaration of Independence.

Today, to mark the 200th anniversary of that declaration, it gives me the greatest pleasure, on behalf of the British people, to present a new bell to the people of the United States of America. It comes from the same foundry as the Liberty Bell, but written on the side of this Bicentennial Bell are the words "Let Freedom Ring".

It is a message in which both our people can join and which I hope will be heard around the world for centuries to come.


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{{MathForm2|(III{{sub|c}})|$$\left.\begin{array}{l} \frac{\partial\mathfrak{H}'_{z}}{\partial y'}-\frac{\partial\mathfrak{H}'_{y}}{\partial z'}=4\pi k^{2}\varrho\mathfrak{v}_{x}+\frac{k^{2}}{V^{2}}\frac{\partial\mathfrak{F}'_{x}}{\partial t'}\\ \\\frac{\partial\mathfrak{H}'_{x}}{\partial z'}-\frac{\partial\mathfrak{H}'_{z}}{\partial x'}=4\pi k\varrho\mathfrak{v}_{y}+\frac{k^{2}}{V^{2}}\frac{\partial\mathfrak{F}'_{y}}{\partial t'}\\ \\\frac{\partial\mathfrak{H}'_{y}}{\partial x'}-\frac{\partial\mathfrak{H}'_{x}}{\partial y'}=4\pi k\varrho\mathfrak{v}_{z}+\frac{k^{2}}{V^{2}}\frac{\partial\mathfrak{F}'_{z}}{\partial t'}\end{array}\right\} $$}}

{{MathForm2|(IV{{sub|c}})|$$\left.\begin{array}{l} \frac{\partial\mathfrak{F}'_{z}}{\partial y'}-\frac{\partial\mathfrak{F}'_{y}}{\partial z'}=-\frac{\partial\mathfrak{H}'_{x}}{\partial t'}\\ \\\frac{\partial\mathfrak{F}'_{x}}{\partial z'}-\frac{\partial\mathfrak{F}'_{z}}{\partial x'}=-\frac{\partial\mathfrak{H}'_{y}}{\partial t'}\\ \\\frac{\partial\mathfrak{F}'_{y}}{\partial x'}-\frac{\partial\mathfrak{F}'_{x}}{\partial y'}=-\frac{\partial\mathfrak{H}'_{z}}{\partial t'}\end{array}\right\} $$}}

{{MathForm2|(V{{sub|c}})|$$\left.\begin{array}{l} \mathfrak{E}_{x}=\mathfrak{F}'_{x}+k\frac{\mathfrak{p}_{x}}{V^{2}}(\mathfrak{v}_{y}\mathfrak{F}'_{y}+\mathfrak{v}_{z}\mathfrak{F}'_{z})+(\mathfrak{v}_{y}\mathfrak{H}'_{z}-\mathfrak{v}_{z}\mathfrak{H}'_{y})\\ \\\mathfrak{E}_{y}=\frac{1}{k}\mathfrak{F}'_{y}-k\frac{\mathfrak{p}_{x}}{V^{2}}\mathfrak{v}_{x}\mathfrak{F}'_{y}+(\frac{1}{k}\mathfrak{v}_{z}\mathfrak{H}'_{x}-\mathfrak{v}_{x}\mathfrak{H}'_{z})\\ \\\mathfrak{E}_{z}=\frac{1}{k}\mathfrak{F}'_{z}-k\frac{\mathfrak{p}_{x}}{V^{2}}\mathfrak{v}_{x}\mathfrak{F}'_{z}+(\mathfrak{v}_{x}\mathfrak{H}'_{y}-\frac{1}{k}\mathfrak{v}_{y}\mathfrak{H}'_{x})\end{array}\right\}$$.}}

Putting $$\mathfrak{v}=0$$ in the three last equations we see that

$$\mathfrak{F}'_{x},\ \frac{1}{k}\mathfrak{F}'_{y},\ \frac{1}{k}\mathfrak{F}'_{z}$$

are the components of the electric force that would act on a particle at rest.

§ 5. We shall begin with an application of the equations to

title: Page:Lorentz Simplified1899.djvu/4


text:
Page:Lorentz Simplified1899.djvu/5

 electrostatic phenomena. In these we have $$\mathfrak{v}=0$$ and $$\mathfrak{F}'$$ independent of the time. Hence, by (IIc) and (IIIc)

$$\mathfrak{H}'=0$$,

and by (IVc) and (Ic)

These equations show that $$\mathfrak{F}'$$ depends on a potential ω, so that

$$\mathfrak{F}'_{x}=-\frac{\partial\omega}{\partial x'},\ \mathfrak{F}'_{y}=-\frac{\partial\omega}{\partial y'},\ \mathfrak{F}'_{z}=-\frac{\partial\omega}{\partial z'}$$

and

Let S be the system of ions with the translation $$\mathfrak{p}_{x}$$, to which the above formulae are applied. We can conceive a second system S0 with no translation and consequently no motion at all; we shall suppose that S is changed into S0 by a dilatation in which the dimensions parallel to OX are changed in ratio of 1 to k, the dimensions perpendicular to OX remaining what they were. Moreover we shall attribute equal charges to corresponding volume-elements in S and S0; if then $$\varrho_{0}$$ be the density in a point P of S, the density in the corresponding point P0 of S0 will be

$$\varrho_{0}=\frac{1}{k}\varrho$$.

If x, y, z are the coordinates of P, the quantities x', y', z', determined by (1), may be considered as the coordinates of P0.

In the system S0, the electric force, which we shall call $$\mathfrak{E}_{0}$$ may evidently be derived from a potential ω0, by means of the equations

$$\mathfrak{E}_{0x}=-\frac{\partial\omega_{0}}{\partial x'},\ \mathfrak{E}_{0y}=-\frac{\partial\omega_{0}}{\partial y'},\ \mathfrak{E}_{0z}=-\frac{\partial\omega_{0}}{\partial z'}$$,

title: Page:Lorentz Simplified1899.djvu/5


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Page:Lorentz Simplified1899.djvu/6

 and the function ω0 itself will satisfy the condition

Comparing this with (2), we see that in corresponding points

and consequently

In virtue of what has been remarked at the end of § 4, the components of the electric force in the system S will therefore be

Parallel to OX we have the same electric force in S and S0, but in a direction perpendicular to OX the electric force in S will be $$\frac{1}{k}$$ times the electric force in S0.

By means of this result every electrostatic problem for a moving system may be reduced to a similar problem for a system at rest; only the dimensions in the direction of translation must be slightly different in the two systems. If, e.g., we wish to determine in what way innumerable ions will distribute themselves over a moving conductor C, we have to solve the same problem for a conductor C0, having no translation. It is easy to show that if the dimensions of C0 and C differ from each other in the way that has been indicated, the electric force in one case will be perpendicular to the surface of C, as soon as, in the other case, the force $$\mathfrak{E}_{0}$$ is normal to the surface of C0.

Since

exceeds unity only by a quantity of the second order — if we call $$\frac{\mathfrak{p}_{x}}{V}$$ of the first order — the influence of the earth's yearly motion on electrostatic phenomena will likewise be of the second order.

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 § 6. We shall now shew how our general equations (Ic) — (Vc) may be applied to optical phenomena. For this purpose we consider a system of ponderable bodies, the ions in which are capable of vibrating about determinate positions of equilibrium. If the system be traversed by waves of light, there will be oscillations of the ions, accompanied by electric vibrations in the aether. For convenience of treatment we shall suppose that, in the absence of lightwaves, there is no motion at all; this amounts to ignoring all molecular motion.

Our first step will be to omit all terms of the second order. Thus, we shall put k=1, and the electric force acting on ions at rest will become $$\mathfrak{F}'$$ itself.

We shall further introduce certain restrictions, by means of which we get rid of the last term in (Ic) and of the terms containing $$\mathfrak{v}_{x},\mathfrak{v}_{y},\mathfrak{v}_{z}$$ in (Vc).

The first of these restrictions relates to the magnitude of the displacements $$\mathfrak{a}$$ from the positions of equilibrium. We shall suppose them to be exceedingly small, even relatively to the dimensions of the ions and we shall on this ground neglect all quantities which are of the second order with respect to $$\mathfrak{a}$$.

It is easily seen that, in consequence of the displacements, the electric density in a fixed point will no longer have its original value $$\varrho_{0}$$ but will have become

$$\varrho=\varrho_{0}-\frac{\partial}{\partial x}(\varrho_{0}\mathfrak{a}_{x})-\frac{\partial}{\partial y}(\varrho_{0}\mathfrak{a}_{y})-\frac{\partial}{\partial z}(\varrho_{0}\mathfrak{a}_{z})$$.

Here, the last terns, which evidently must be taken into account, have the order of magnitude $$\frac{c\ \varrho_{0}}{a}$$, if c denotes the amplitude of the vibrations; consequently, the first term of the right-hand member of (Ic) will contain quantities of the order

On the other hand, if T is the time of vibration, the last term in (Ic) will be of the order

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 Dividing this by (3), we get

$$\frac{\mathfrak{p}_{x}}{V}\cdot\frac{a}{VT}$$,

an extremely small quantity, because the diameter of the ions is a very small fraction of the wave-length. This is the reason why we may omit the last term in (Ic).

As to the equations (Vc), it must be remarked that, if the displacements are infinitely small, the same will be true of the velocities and, in general, of all quantities which do not exist as long as the system is at rest and are entirely produced by the motion. Such are $$\mathfrak{H}'_{x},\mathfrak{H}'_{y},\mathfrak{H}'_{z}$$. We may therefore omit the last terms in (Vc), as being of the second order.

The same reasoning would apply to the terms containing $$\frac{\mathfrak{p}_{x}}{V^{2}}$$, if we could be sure that in the state of equilibrium there are no electric forces at all. If, however, in the absence of any vibrations, the vector $$\mathfrak{F}'$$ has already a certain value $$\mathfrak{F}_{0}'$$, it will only be the difference $$\mathfrak{F}'-\mathfrak{F}_{0}'$$, that may be called infinitely small; it will then be permitted to replace $$\mathfrak{F}'_{y}$$ and $$\mathfrak{F}'_{z}$$ by $$\mathfrak{F}'_{0y}$$ and $$\mathfrak{F}'_{0y}$$.

Another restriction consists in supposing that an ion is incapable of any motion but a translation as a whole, and that, in the position of equilibrium, though its parts may be acted on by electric forces, as has just been said, yet the whole ion does not experience a resultant electric force. Then, if $$d\tau$$ is an element of volume, and the integrations are extended all over the ion,

Again, in the case of vibrations, the equations (Vc) will only serve to calculate the resultant force acting on an ion. In tho direction of the axis of y e. g. this force will be

$$\int\varrho\mathfrak{E}_{y}d\tau$$.

Its value may be found, if we begin by applying the second of the three equations to each point of the ion, always for the same universal time t, and then integrate. From the second term on the right-hand side we find

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text:
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 $$-\frac{\mathfrak{p}_{x}}{V^{2}}\int\varrho\mathfrak{v}_{x}\mathfrak{F}'_{y}d\tau$$,

or, since we may replace $$\mathfrak{F}'_{y}$$ by $$\mathfrak{F}'_{0y}$$ and $$\varrho$$ by $$\varrho_{0}$$,

$$-\frac{\mathfrak{p}_{x}}{V^{2}}\mathfrak{v}_{x}\int\varrho_{0}\mathfrak{F}'_{0y}d\tau$$,

which vanishes on account of (5).

Hence, as far as regards the resultant force, we may put $$\mathfrak{E}=\mathfrak{F}'$$, that is to say, we may take $$\mathfrak{F}'$$ as the electric force, acting not only on ions at rest, but also on moving ions.

The equations will be somewhat simplified, if, instead of $$\mathfrak{F}'$$, we introduce the already mentioned difference $$\mathfrak{F}'-\mathfrak{F}'_{0}$$. In order to do this, we have only twice to write down the equations (Ic)—(IVc), once for the vibrating system and a second time for the same system in a state of rest; and then to subtract the equations of the second system from those of the first. In the resulting equations, I shall, for the sake of brevity, write $$\mathfrak{F}'$$ instead of $$\mathfrak{F}'-\mathfrak{F}'_{0}$$, so that henceforth $$\mathfrak{F}'$$ will denote not the total electric force, but only the part of it that is due to the vibrations. At the same time we shall replace the value of $$\varrho$$, given above, by

$$\varrho_{0}-\mathfrak{a}_{x}\frac{\partial\varrho_{0}}{\partial x'}-\mathfrak{a}_{y}\frac{\partial\varrho_{0}}{\partial y'}-\mathfrak{a}_{z}\frac{\partial\varrho_{0}}{\partial z'}$$.

We may do so, because we have supposed $$\mathfrak{a}_{x},\mathfrak{a}_{y},\mathfrak{a}_{z}$$ to have the same values all over an ion, and because $$\varrho_{0}$$ is independent of the time, so that

$$\frac{\partial\varrho_{0}}{\partial x}=\frac{\partial\varrho_{0}}{\partial x'}$$.

Finally we have

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text:
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 Since these equations do no longer explicitly contain the velocity $$\mathfrak{p}_{x}$$, they will hold, without any change of form, for a system that has no translation, in which case, of course, t' would be the same thing as the universal time t.

Yet, strictly speaking, there would be a slight difference in the formulae, when applied to the two cases. In the system without a translation $$\mathfrak{a}_{x},\mathfrak{a}_{y},\mathfrak{a}_{z}$$ would be, in all points of an ion, the same functions of t', i. e. of the universal time, whereas, in the moving system, these components would not depend in the same way on t' in different parts of the ion, just because they must everywhere be the same functions of t.

However, we may ignore this difference, of the ions are so small, that we may assign to each of them a single local time, applicable to all its parts.

The equality of form of the electromagnetic equations for the two cases of which we have spoken will serve to simplify to a large extent our investigation. However, it should be kept in mind, that, to the equations (Id)—(IVd), we must add the equations of motion for the ions themselves. In establishing these, we have to take into account, not only the electric forces, but also all other forces acting on the ions. We shall call these latter the molecular forces and we shall begin by supposing them to be sensible only at such small distances, that two particles of matter, acting on each other, may be said to have the same local time.

§ 7. Let us now imagine two systems of ponderable bodies, the one S with a translation, and the other one S0 without such a motion, but equal to each other in all other respects. Since we neglect quantities of the order $$\mathfrak{p}_{x}^{2}/V^{2}$$, the electric force will, by §5 be the same in both systems, as long as there are no vibrations.

After these have been excited, we shall have for both systems the equations (Id)—(IVd).

Further we shall imagine motions of such a kind, that, if in a point (x',y',z') of S0 we find a certain quantity of matter or a certain electric charge at the universal time t, an equal quantity of matter or an equal charge will be found in the corresponding point of S at the local time t'. Of course, this involves that at these corresponding times we shall have, in the point (x', y', z') of both systems, the same electric density, the same displacement $$\mathfrak{a}$$, and equal velocities and accelerations.

Thus, some of the dependent variables in our equations (Id)—(IVd) will be represented in S0 and S by the same functions of x', y', z', t',

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 whence we conclude that the equations will be satisfied by values of $$\mathfrak{H}'_{x},\mathfrak{H}'_{y},\mathfrak{H}'_{z},\mathfrak{F}'_{x},\mathfrak{F}'_{y},\mathfrak{F}'_{z}$$, which are likewise in both cases the same functions of x', y', z', t'. By what has been said at the beginning of this §, not only $$\mathfrak{F}'$$, but also the total electric force will be the same in S0 and S, always provided that corresponding ions at corresponding times (i. e. for equal values of t') be considered.

As to the molecular forces, acting on an ion, they are confined to a certain small space surrounding it, and by what has been said in § 6, the difference of local times within this space may be neglected. Moreover, if equal spaces of this kind are considered in S0 and S, there will be, at corresponding times, in both the same distribution of matter. This is a consequence of what has been supposed concerning the two motions.

Now, the simplest assumption we can make on the molecular forces is this, that they are not changed by the translation of the system. If this be admitted, it appears from the above considerations that corresponding ions in S0 and S will be acted on by the same molecular forces, as well as by the same electric forces. Therefore, since the masses and accelerations are the same, the supposed motion in S will be possible as soon as the corresponding motion in S0 can really exist. In this way we are led to the following theorem.

If, in a body or a system of bodies, without a translation, a system of vibrations be given, in which the displacements of the ions and the components of $$\mathfrak{F}'$$ and $$\mathfrak{H}'$$ are certain functions of the coordinates and the time, then, if a translation be given to the system, there can exist vibrations, in which the displacements and the components of $$\mathfrak{F}'$$ and $$\mathfrak{H}'$$ are the same functions of the coordinates and the local time. This is the theorem, to which I have been led in a much more troublesome way in my Versueh einer Theorie, etc., and by which most of the phenomena, belonging to the theory of aberration may be explained.

§ 8. In what precedes, the molecular forces have been supposed to be confined to excessively small distances. If two particles of matter were to act upon each other at such a distance that the difference of their local times might not be neglected, the theorem would no longer be true in the case of molecular forces that are not altered at all by the translation. However, one soon perceives that the theorem would again hold good, if these forces were changed by the translation in a definite way, in such a way namely that the action between two quantities of matter were determined, not by the simultaneous values of their coordinates, but by their values

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 at equal local times. If therefore, we should meet with phenomena, in which the difference of the local times for mutually acting particles might have a sensible influence, and in which yet observation showed the above theorem to be true, this would indicate a modification, like the one we have just specified, of the molecular forces by the influence of a translation. Of course, such a modification would only be possible, if the molecular forces were no direct actions at a distance, but were propagated by the aether in a similar way as the electromagnetic actions. Perhaps the rotation of the plane of polarization in the so-called active bodies will be found to be a phenomenon of the kind, just mentioned.

§ 9. Hitherto all quantities of the order $$\mathfrak{p}_{x}^{2}/V^{2}$$ have been neglected. As is well known, these must be taken into account in the discussion of 's experiment, in which two rays of light interfered after having traversed rather long paths, the one parallel to the direction of the earth's motion, and the other perpendicular to it. In order to explain the negative result of this experiment and myself have supposed that, in consequence of the translation, the dimensions of the solid bodies serving to support the optical apparatus, are altered in a certain ratio.

Some time ago, M. has emitted the opinion that, according to my theory, the experiment should have a positive result, if it were modified in so far, that the rays had to pass through a solid or a liquid dielectric.

It is impossible to say with certainty what would be observed in such a case, for, if the explication of 's result which I have proposed is accepted, we must also assume that the mutual distances of the molecules of transparent media are altered by the translation.

Besides, we must keep in view the possibility of an influence, be it of the second order, of the translation on the molecular forces.

In what follows I shall shew, not that the result of the experiment must necessarily be negative, but that this might very well be the case. At the same time it will appear what would be the theoretical meaning of such a result.

Let us return again to the equations (Ic)—(Vc). This time we shall not put in them k = 1, but the other simplifications of which we have spoken in § 6 will again be introduced. We shall now have to distinguish between the vectors $$\mathfrak{E}$$ and $$\mathfrak{F}'$$, the former alone

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 being the electric force. By both signs I shall now denote, not the whole vector, but the part that is due to the vibrations.

The equations may again be written in a form in which the velocity of translation does not explicitly appear. For this purpose, it is necessary to replace the variables x', y', z', t', $$\mathfrak{F}',\mathfrak{H}',\mathfrak{a}$$ and $$\varrho_{0}$$ by new ones, differing from the original quantities by certain constant factors.

For the sake of uniformity of notation all these new variables will be distinguished by double accents. Let $$\varepsilon$$ be an indeterminate coefficient, differing from unity by a quantity of the order $$\mathfrak{p}_{x}^{2}/V^{2}$$ and let us put

so that t" is a modified local time; then we find

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 These formulae will also hold for a system without translation; only, in this case we must take k=1, and we shall likewise take $$\varepsilon=1$$, though this is not necessary. Thus, x", y", z" will then be the coordinates, t" the same thing as t, i. e. the universal time, $$\mathfrak{a}$$ the displacement, $$\varrho_{0}$$ the electric density, $$\mathfrak{H}$$ and $$\mathfrak{F}$$ the magnetic and electric forces, the last in so far as it is due to the vibrations.

Our next object will be to ascertain under what conditions, now that we retain the terms with $$\mathfrak{p}_{x}^{2}/V^{2}$$, two systems S and S0, the first having a translation, and the second having none, may be in vibratory states that are related to each other in some definite way. This investigation resembles much the one that has been given in §7; it may therefore be expressed in somewhat shorter terms.

To begin with, we shall agree upon the degree of similarity there shall be between the two systems in their states of equilibrium. In this respect we define S by saying that the system S0 may be changed into it by means of the dilatations indicated by (6); we shall suppose that, in undergoing these dilatations, each element of volume retains its ponderable matter, as well as its charge. It is easily seen that this agrees with the relation (8).

We shall not only suppose that the system S0 may be changed in this way into an imaginary system S, but that, as soon as the translation is given to it, the transformation really takes place, of itself, i. e. by the action of the forces acting between the particles of the system, and the aether. Thus, after all, S will be the same material system as S.

The transformation of which I have now spoken, is precisely such a one as is required in my explication of 's experiment. In this explication the factor $$\varepsilon$$ may be left indeterminate. We need hardly remark that for the real transformation produced by a translatory motion, the factor should have a definite value. I see, however, no means to determine it.

Before we proceed further, a word on the electric forces in S and S0 in their states of equilibrium. If $$\varepsilon=1$$, the relation between these forces will be given by the equations of §5. Now $$\varepsilon$$ indicates an alteration of all dimensions in the same ratio, and it is very easy to see what influence this will have on the electric forces. Thus, it will be found that, in passing from S0, to S, the electric force in the direction of OX will be changed in the ratio of 1 to $$\tfrac{1}{\varepsilon^{2}}$$, and that the corresponding ratio for the other components will be as 1 to $$\tfrac{1}{k\varepsilon^{2}}$$.

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 As to the corresponding vibratory motions, we shall require that at corresponding times, i.e. for equal values of t", the configuration of S may always be got from that of S0 by the above mentioned dilatations. Then, it appears from (7) that $$\mathfrak{a}_{x},\mathfrak{a}_{y},\mathfrak{a}_{z}$$ will be, in both systems, the same functions of x", y", z", t", whence we conclude that the equations (Ie)—(IVe) can be satisfied by values of $$\mathfrak{F}_{x},\mathfrak{H}_{x}$$, etc., which are likewise, in S0 and in S, the same functions of x", y", z", t"''.

Always provided that we start from a vibratory motion in S0 that can really exist, we have now arrived at a motion in S, that is possible in so far as it satisfies the electromagnetic equations. The last stage of our reasoning will be to attend to the molecular forces. In S0 we imagine again, around one of the ions, the same small space, we have considered in § 7 and to which the molecular forces acting on the ion are confined; in the other system we shall now conceive the corresponding small space, i.e. the space that may be derived from the first one by applying to it the dilatations (6). As before, we shall suppose these spaces to be so small that in the second of them there is no necessity to distinguish the local times in its different parts; then we may say that in the two spaces there will be, at corresponding times, corresponding distributions of matter.

We have already seen that, in the states of equilibrium, the electric forces parallel to OX, OY, OZ, existing in S differ from the corresponding forces in S0 by the factors

$$\frac{1}{\varepsilon^{2}},\ \frac{1}{k\varepsilon^{2}}$$ and $$\frac{1}{k\varepsilon^{2}}$$.

From (Ve) it appears that the same factors come into play when we consider the part of the electric forces that is due to the vibrations. If, now, we suppose that the molecular forces are modified in quite the same way in consequence of the translation, we may apply the just mentioned factors to the components of the total force acting on an ion. Then, the imagined motion in S will be a possible one, provided that these same factors to which we have been led in examining the forces present themselves again, when we treat of the product of the masses and the accelerations.

According to our suppositions, the accelerations in the directions of OX, OY, OZ in S are resp. $$\tfrac{1}{k^{3}\varepsilon}, \tfrac{1}{k^{2}\varepsilon}$$ and $$\frac{1}{k^{2}\varepsilon}$$ times what they Proceedings Royal Acad. Amsterdam. Vol. I.

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 are in S0. If therefore the required agreement is to exist with regard to the vibrations parallel to OX, the ratio of the masses of the ions in S and S0 should be $$\tfrac{k^{3}}{\varepsilon}$$; on the contrary we find for this ratio $$\tfrac{k}{\varepsilon}$$, if we consider in the same way the forces and the accelerations in the directions of OY and OZ.

Since k is different from unity, these values cannot both be 1; consequently, states of motion, related to each other in the way we have indicated, will only be possible, if in the transformation of S0 into S the masses of the ions change; even, this must take place in such a way that the same ion will have different masses for vibrations parallel and perpendicular to the velocity of translation.

Such a hypothesis seems very startling at first sight. Nevertheless we need not wholly reject it. Indeed, as is well known, the effective mass of an ion depends on what goes on in the aether; it may therefore very well be altered by a translation and even to different degrees for vibrations of different directions.

If the hypothesis might be taken for granted, 's experiment should always give a negative result, whatever transparent media wore placed on the path of the rays of light, and even if one of these went through air, and the other, say through glass. This is seen by remarking that the correspondence between the two motions we have examined is such that, if in S0 we had a certain distribution of light and dark (interference-bands) we should have in S a similar distribution, which might be got from that in S0 by the dilatations (6), provided however that in S the time of vibration be $$k\varepsilon$$ times as great as in S0. The necessity of this last difference follows from (9). Now the number $$k\varepsilon$$ would be the same in all positions we can give to the apparatus; therefore, if we continue to use the same sort of light, while rotating the instruments, the interference-bands will never leave the parts of the ponderable system, e. g. the lines of a micrometer, with which they coincided at first.

We shall conclude by remarking that the alteration of the molecular forces that has been spoken of in this § would be one of the second order, so that we have not come into contradiction with what has been said in § 7.

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text: 1911 Encyclopædia Britannica/Diagoras

title: 1911 Encyclopædia Britannica/Diagoras


text: 1911 Encyclopædia Britannica/Diagonal

title: 1911 Encyclopædia Britannica/Diagonal


text:
User talk:Adeshram

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title: User talk:Adeshram


text: 1911 Encyclopædia Britannica/Diagram

title: 1911 Encyclopædia Britannica/Diagram


text: Index:Lorentz Grav1900.djvu

title: Index:Lorentz Grav1900.djvu


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Page:Lorentz Grav1900.djvu/1

By. Prof.

§ 1. After all we have learned in the last twenty or thirty years about the mechanism of electric and magnetic phenomena, it is natural to examine in how far it is possible to account for the force of gravitation by ascribing it to a certain state of the aether. A theory of universal attraction, founded on such an assumption, would take the simplest form if new hypotheses about the aether could be avoided, i.e. if the two states which exist in an electric and a magnetic field, and whose mutual connection is expressed by the well known electromagnetic equations were found sufficient for the purpose.

If further it be taken for granted that only electrically charged particles or ions, are directly acted on by the aether, one is led to the idea that every particle of ponderable matter might consist of two ions with equal opposite charges — or at least might contain two such ions — and that gravitation might be the result of the forces experienced by these ions. Now that so many phenomena have been explained by a theory of ions, this idea seems to be more admissible than it was ever before.

As to the electromagnetic disturbances in the aether which might possibly be the cause of gravitation, they must at all events be of such a nature, that they are capable of penetrating all ponderable bodies without appreciably diminishing in intensity. Now, electric vibrations of extremely small wave-length possess this property; hence the question arises what action there would be between two ions if the aether were traversed in all directions by trains of electric waves of small wave-length.

The above ideas are not new. Every physicist knows 's theory in which innumerable small corpuscula are supposed to move with great velocities, producing gravitation by their impact against the coarser particles of ordinary ponderable matter. I shall not here discuss this theory which is not in harmony with modern physical views. But, when it had been found that a pressure against a body may be produced as well by trains of electric waves, by rays of light e.g., as by moving projectiles and when the -rays with their remarkable penetrating power had been discovered, it was natural to replace 's corpuscula by vibratory motions. Why should there not exist radiations, far more penetrating than even the X-rays, and which might therefore serve to account for a force which as far as we know, is independent of all intervening ponderable matter?

I have deemed it worth while to put this idea to the test. In

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 what follows, before passing to considerations of a different order (§ 5), I shall explain the reasons for which this theory of rapid vibrations as a cause of gravitation can not be accepted.

§ 2. Let an ion carrying a charge $$e_{1}$$, and having a certain mass, be situated at the point P(x,y,z); it may be subject or not to an elastic force, proportional to the displacement and driving it back to P, as soon as it has left this position. Next, let the aether be traversed by electromagnetic vibrations, the dielectric displacement being denoted by $$\mathfrak{d}$$, and the magnetic force by $$\mathfrak{H}$$, then the ion will be acted on by a force

$4\pi V^{2}e\mathfrak{d}$,

whose direction changes continually, and whose components are

In these formulae V means the velocity of light.

By the action of the force (1) the ion will be made to vibrate about its original position P, the displacement (x, y,z) being determined by well known differential equations.

For the sake of simplicity we shall confine ourselves to simple harmonic vibrations with frequency n. All our formulae will then contain the factor $$\cos nt$$ or $$\sin nt$$, and the forced vibrations of the ion may be represented by expressions of the form

{{MathForm2|(2)|$$\left.\begin{array}{l} \mathsf{x}=ae\mathfrak{d}_{\mathsf{x}}-be\mathfrak{\dot{d}}_{\mathsf{x}},\\ \mathsf{y}=ae\mathfrak{d}_{\mathsf{y}}-be\mathfrak{\dot{d}}_{\mathsf{y}},\\ \mathsf{z}=ae\mathfrak{d}_{\mathsf{z}}-be\mathfrak{\dot{d}}_{\mathsf{z}},\end{array}\right\} $$}}

with certain constant coefficients a and b. The terms with $$\mathfrak{\dot{d}}_{\mathsf{x}}, \mathfrak{\dot{d}}_{\mathsf{y}}$$ and $$\mathfrak{\dot{d}}_{\mathsf{z}}$$ have been introduced in order to indicate that the phase of the forced vibration differs from that of the force (X,Y,Z); this will be the case as soon as there is a resistance, proportional to the velocity, and the coefficient b may then be shown to be positive. One cause of a resistance lies in the reaction of the aether, called forth by the radiation of which the vibrating ion itself becomes the centre, a reaction which determines at the same time an apparent increase of the mass of the particle. We shall suppose however that we have kept in view this reaction in establishing the equations of motion, and in assigning their values to the coefficients a and b.

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 Then, in what follows, we need only consider the forces due to the state of the aether, in so far as it not directly produced by the ion itself.

Since the formulae (2) contain e as a factor, the coefficients a and b will be independent of the charge; their sign will be the same for a negative ion and for a positive one. Now, as soon as the ion has shifted from its position of equilibrium, new forces come into play. In the first place, the force $$4\pi V^{2}e\mathfrak{d}$$ will have changed a little, because, for the new position, $$\mathfrak{d}$$ will be somewhat different from what it was at the point P. We may express this by saying that, in addition to the force (1), there will be a new one with the components,

In the second place, in consequence of the velocity of vibration, there will be an electromagnetic force with the components

If, as we shall suppose, the displacement of the ion be very small, compared with the wave-length, the forces (3) and (4) are much smaller than the force (1); since they are periodic — with the frequency 2n, — they will give rise to new vibrations of the particle. We shall however omit the consideration of these slight vibrations, and examine only the mean values of the forces (3) and (4), calculated for a rather long lapse of time, or, what amounts to the same thing, for a full period $$\tfrac{2\pi}{n}$$.

§ 3. It is immediately clear that this mean force will be 0 if the ion is alone in a field in which the propagation of waves takes place equally in all directions. It will be otherwise, as soon as a second ion Q has been placed in the neighbourhood of P; then, in consequence of the vibrations emitted by Q after it has been itself put in motion, there may be a force on P, of course in the direction of the line QP. In computing the value of this force, one finds a great number of terms, which depend in different ways on the distance r. We shall retain those which are inversely proportional to r or r2, but we shall neglect all terms varying inversely as the higher powers of r; indeed, the influence of these, compared with that of the first mentioned terms will be of the order $$\tfrac{\lambda}{r}$$, if $$\lambda$$ is the

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 wave-length, and we shall suppose this to be a very small fraction.

We shall also omit all terms containing such factors as $$\cos\ 2\pi k\tfrac{r}{\lambda}$$ or $$\sin\ 2\pi k\tfrac{r}{\lambda}$$ (k a moderate number). These reverse their signs by a very small change in r; they will therefore disappear from the resultant force, as soon as, instead of single particles P and Q, we come to consider systems of particles with dimensions many times greater than the wave-length.

From what has been said, we may deduce in the first place that, in applying the above formulae to the ion P, it is sufficient, to take for $$\mathfrak{d}$$ and $$\mathfrak{H}$$ the vectors that would exist if P were removed from the field. In each of these vectors two parts are to be distinguished. We shall denote by $$\mathfrak{d}_{1}$$ and $$\mathfrak{H}_{1}$$ the parts existing independently of Q, and by $$\mathfrak{d}_{2}$$ and $$\mathfrak{H}_{2}$$ the parts due to the vibrations of this ion.

Let Q be taken as origin of coordinates, QP as axis of x, and let us begin with the terms in (2) having the coefficient a.

To these corresponds a force on P, whose first component is

Since we have only to deal with the mean values for a full period, we may write for the last term

$-e^{2}a\left(\mathfrak{d}_{y}\mathfrak{\dot{H}}_{z}-\mathfrak{d}{}_{z}\mathfrak{\dot{H}}_{y}\right)$,|undefined

and if, in this expression, $$\mathfrak{\dot{H}}_{y}$$ and $$\mathfrak{\dot{H}}_{z}$$ be replaced by

$4\pi V^{2}\left(\frac{\partial\mathfrak{d}_{\mathsf{z}}}{\partial x}-\frac{\partial\mathfrak{d}_{\mathsf{x}}}{\partial z}\right)$ and $4\pi V^{2}\left(\frac{\partial\mathfrak{d}_{\mathsf{x}}}{\partial y}-\frac{\partial\mathfrak{d}_{\mathsf{y}}}{\partial x}\right)$,|undefined

(5) becomes

where $$\mathfrak{d}$$ is the numerical value of the dielectric displacement.

Now, $$\mathfrak{d}^{2}$$ will consist of three parts, the first being $$\mathfrak{d}_{1}^{2}$$, the second $$\mathfrak{d}_{2}^{2}$$ and the third depending on the combination of $$\mathfrak{d}_{1}$$ and $$\mathfrak{d}_{2}$$.

Evidently, the value of (6), corresponding to the first part, will be 0.

As to the second part, it is to be remarked that the dielectric displacement, produced by Q, is a periodic function of the time. At distant points the amplitude takes the form $$\tfrac{c}{r}$$, where c is independent

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 of r. The mean value of $$\mathfrak{d}^{2}$$ for a full period is $$\tfrac{1}{2}\tfrac{c^{2}}{r^{2}}$$ and differentiating this with regard to x or to r, we should get $$r^{3}$$ in the denominator.

The terms in (6) which correspond to the part

$2\left(\mathfrak{d}_{1\mathsf{x}}\mathfrak{d}_{2\mathsf{x}}+\mathfrak{d}_{1\mathsf{y}}\mathfrak{d}_{2\mathsf{y}}+\mathfrak{d}_{1\mathsf{z}}\mathfrak{d}_{2\mathsf{z}}\right)$ |undefined

in $$\mathfrak{d}^{2}$$, may likewise be neglected. Indeed, if these terms are to contain no factors such as $$\cos\ 2\pi k\tfrac{r}{\lambda}$$ or $$\sin\ 2\pi k\tfrac{r}{\lambda}$$, there must be between $$\mathfrak{d}_{1}$$ and $$\mathfrak{d}_{2}$$, either no phase-difference at all, or a difference which is independent of r. This condition can only be fulfilled, if a system of waves, proceeding in the direction of QP, is combined with the vibrations excited by Q, in so far as this ion is put in motion by that system itself. Then, the two vectors $$\mathfrak{d}_{1}$$ and $$\mathfrak{d}_{2}$$ will have a common direction perpendicular to QP, say that of the axis of y, and they will be of the form

The mean value of $$\mathfrak{d}_{1\mathsf{y}}\mathfrak{d}_{2\mathsf{y}}$$ is

$\frac{1}{2}\frac{qc}{r}\cos n\ \left(\varepsilon_{1}-\varepsilon_{2}\right)$,

and its differential coefficient with regard to x has $$r^{2}$$ in the denominator. It ought therefore to be retained, were it not for the extremely small intensity of the systems of waves which give rise to such a result. In fact, by the restriction imposed on them as to their direction, these waves form no more than a very minute part of the whole motion.

§ 4. So, it is only the terms in (2), with the coefficient b, with which we are concerned. The corresponding forces are

and

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 If Q were removed, these forces together would be 0, as has already been remarked. On the other hand, the force (8) taken by itself, would then likewise be 0. Indeed, its value is

or, by theorem $$\tfrac{n^{2}e^{2}b}{V^{2}}S_{\mathsf{x}}$$, if $$S_{\mathsf{x}}$$ be the flow of energy in a direction parallel to the axis of x. Now, it is clear that, in the absence of Q, any plane must be traversed in the two directions by equal amounts of energy.

In this way we come to the conclusion that the force (7), in so far as it depends on the part, ($$\mathfrak{d}_{1}$$), is 0, and from this it follows that the total value of (7) will vanish, because the part arising from the combination of ($$\mathfrak{d}_{1}$$) and ($$\mathfrak{d}_{2}$$), as well as that which is solely due to tho vibrations of Q, are 0. As to the first part, this may be shown by a reasoning similar to that used at the end of, the preceding §. For the second part, the proof is as follows.

The vibrations excited by Q in any point A of the surrounding aether are represented by expressions of the form

$\frac{1}{r}\vartheta\cos n\left(t-\frac{r}{V}+\varepsilon\right)$,

where $$\vartheta$$ depends on the direction of the line QA, and r denotes the length of this line. If, in differentiating such expressions, we wish to avoid in the denominator powers of r, higher than the first — and this is necessary, in order that (7) may remain free from powers higher than the second — $$\tfrac{1}{r}$$ and $$\vartheta$$ have to be treated as constants. Moreover, the factors $$\vartheta$$ are such, that the vibrations are perpendicular to the line QA. If, now, A coincides with P, and QA with the axis of x, in the expression for bx we shall have $$\vartheta=0$$, and since this factor is not to be differentiated, all terms in (7) will vanish.

Thus, the question reduces itself to (8) or (9). If, in this last expression, we take for $$\mathfrak{d}$$ and $$\mathfrak{H}$$ their real values, modified as they are by the motion of Q, we may again write for the force

$\frac{n^{2}e^{2}b}{V^{2}}S_{\mathsf{x}}$;|undefined

this time, however, we have to understand by $$S_{\mathsf{x}}$$ the flow of energy as it is in the actual case.

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 Now, it is clear that, by our assumptions, the flow of energy must be symmetrical all around Q; hence, if an amount E of energy traverses, in the outward direction, a spherical surface described around Q as centre with radius r, we shall have

$S_{\mathsf{x}}=\frac{E}{4\pi r^{2}}$,|undefined

and the force on P will be

$K=\frac{n^{2}e^{2}b\ E}{4\pi V^{2}r^{2}}$.|undefined

It will have the direction of QP prolonged.

In the space surrounding Q the state of the aether will be stationary; hence, two spherical surfaces enclosing this particle must be traversed by equal quantities of energy. The quantity E will be independent of r, and the force K inversely proportional to the square of the distance.

If the vibrations of Q were opposed by no other resistance but that which results from radiation, the total amount of electro-magnetic energy enclosed by a surface surrounding Q would remain constant; E and K would then both be 0. If, on the contrary, in addition to the just mentioned resistance, there were a resistance of a different kind, the vibrations of Q would be accompanied by a continual loss of electro-magnetic energy; less energy would leave the space within one of the spherical surfaces than would enter that space. E would be negative, and, since b is positive, there would be attraction. It would be independent of the signs of the charges of P and Q.

The circumstance however, that this attraction could only exist, if in some way or other electromagnetic energy were continually disappearing, is so serious a difficulty, that what has been said cannot be considered as furnishing an explanation of gravitation. Nor is this the only objection that can be raised. If the mechanism of gravitation consisted in vibrations which cross the aether with the velocity of light, the attraction ought to be modified by the motion of the celestial bodies to a much larger extent than astronomical observations make it possible to admit.

§ 5. Though the states of the aether, the existence and the laws of which have been deduced from electromagnetic phenomena, are found insufficient to account for universal attraction, yet one may try to establish a theory which is not wholly different from that of

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 electricity, but has some features in common with it. In order to obtain a theory of this kind, I shall start from an idea that has been suggested long ago by Ottaviano-Fabrizio Mossotti and has been afterwards accepted by Wilhelm Eduard Weber and Johann Karl Friedrich Zöllner.

According to these physicists, every particle of ponderable matter consists of two oppositely electrified particles. Th us, between two particles of matter, there will be four electric forces, two attractions between the charges of different, and two repulsions between those of equal signs. supposes the attractions to be somewhat greater than the repulsions, the difference between the two being precisely what we call gravitation. It is easily seen that such a difference might exist in cases where an action of a specific electric nature is not exerted.

Now, if the form of this theory is to be brought into harmony with the present state of electrical science, we must regard the four forces of as the effect of certain states in the aether which are called forth by the positive and negative ions.

A positive ion, as well as a negative one, is the centre of a dielectric displacement, and, in treating of electrical phenomena, these two displacements are considered as being of the same nature, so that, if in opposite directions and of equal magnitude, they wholly destroy each other.

If gravitation is to be included in the theory, this view must be modified. Indeed, if the actions exerted by positive and negative ions depended on vector-quantities of the same kind, in such a way that all phenomena in the neighbourhood of a pair of ions with opposite charges were determined by the resulting vector, then electric actions could only be absent, if this resulting vector were 0, but, if such were the case, no other actions could exist; a gravitation, i.e. a force in the absence of an electric field, would be impossible.

I shall therefore suppose that the two disturbances in the aether, produced by positive and negative ions, are of a somewhat different nature, so that, even if they are represented in a diagram by equal and opposite vectors, the state of the aether is not the natural one. This corresponds in a sense to idea that positive and negative charges differ from each other to a larger extent, than may be expressed by the signs + and –.

After having attributed to each of the two states an independent and separate existence, we may assume that, though both able to act on positive and negative ions, the one has more power over the positive particles and the other over the negative ones. This difference

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 will lead us to the same result that attained by means of the supposed inequality of the attractive and the repulsive forces.

§ 6. I shall suppose that each of the two disturbances of the aether is propagated with the velocity of light, and, taken by itself, obeys the ordinary laws of the electromagnetic field. These laws are expressed in the simplest form if, besides the dielectric displacement $$\mathfrak{d}$$, we consider the magnetic force $$\mathfrak{H}$$, both together determining, as we shall now say, one state of the aether or one field. In accordance with this, I shall introduce two pairs of vectors, the one $$\mathfrak{d}$$, $$\mathfrak{H}$$ belonging to the field that is produced by the positive ions, whereas the other pair $$\mathfrak{d}'$$, $$\mathfrak{H}'$$ serve to indicate the state of the aether which is called into existence by the negative ions. I shall write down two sets of equations, one for $$\mathfrak{d}$$, $$\mathfrak{H}$$, the other for $$\mathfrak{d}', \mathfrak{H}'$$, and having the form which I have used in former papers for the equations of the electromagnetic field, and which is founded on the assumption that the ions are perfectly permeable to the aether and that they can be displaced without dragging the aether along with them.

I shall immediately take this general case of moving particles.

Let us further suppose the charges to be distributed with finite volume-density, and let the units in which these are expressed be chosen in such a way that, in a body which exerts no electrical actions, the total amount of the positive charges has the same numerical value as that of the negative charges.

Let $$\varrho$$ be the density of the positive, and $$\varrho'$$ that of the negative charges, the first number being positive and the second negative.

Let $$\mathfrak{v}$$ (or $$\mathfrak{v}'$$) be the velocity of an ion.

Then the equations for the state $$(\mathfrak{d}, \mathfrak{H})$$ are

{{MathForm2|(I)|$$\left.\begin{array}{c} Div\ \mathfrak{d}=\varrho\\ Div\ \mathfrak{H}=0\\ Rot\ \mathfrak{H}=4\pi\varrho\mathfrak{v}+ 4\pi\mathfrak{\dot{d}}\\ 4\pi V^{2}Rot\ \mathfrak{d}=-\mathfrak{\dot{H}};\end{array}\right\} $$}}

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 and those for the state $$(\mathfrak{d}', \mathfrak{H}')$$

{{MathForm2|(II)|$$\left.\begin{array}{c} Div\ \mathfrak{d}'=\varrho'\\ Div\ \mathfrak{H}'=0\\ Rot\ \mathfrak{H}'=4\pi\varrho'\mathfrak{v}'+4\pi\mathfrak{\dot{d}}'\\ 4\pi V^{2}Rot\ \mathfrak{d}'=-\mathfrak{\dot{H}}';\end{array}\right\} $$.}}

In the ordinary theory of electromagnetism, the force acting on a particle, moving with velocity $$\mathfrak{v}$$, is

$4\pi V^{2}\mathfrak{d}+\left[\mathfrak{v\ }.\ \mathfrak{H}\right]$

per unit charge.

In the modified theory, we shall suppose that a positively electrified particle with charge e experiences a force

on account of the field $$(\mathfrak{d}, \mathfrak{H})$$, and a force

on account of the field $$(\mathfrak{d}', \mathfrak{H}')$$, the positive coefficients $$\alpha$$ and $$\beta$$ having slightly different values.

For the forces, exerted on a negatively charged particle I shall write

and

expressing by these formulae that e is acted on by ($$\mathfrak{d}, \mathfrak{H}$$) in the same way as e'  by ($$\mathfrak{d}', \mathfrak{H}'$$), and vice versa.

§ 7. Let us next consider the actions exerted by a pair of oppositely charged ions, placed close to each other, and remaining so during their motion. For convenience of mathematical treatment, we may even reason as if the two charges penetrated each other, so that, if they are equal, $$\varrho'=-\varrho$$.

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 On the other hand $$\mathfrak{v}'=\mathfrak{v}$$ ; hence, by (I) and (II),

$\mathfrak{d}'=-\mathfrak{d}$ and $\mathfrak{H}'=-\mathfrak{H}$.

Now let us put in the field, produced by the pair of ious, a similar pair with charges e and $$e'=-e$$, and moving with the common velocity $$\mathfrak{v}$$. Then, by (10) — (13),

$k_{2}=-\frac{\beta}{\alpha}k_{1},\ k_{3}=-\frac{\beta}{\alpha}k_{1},\ k_{4}=k_{1}.$

The total force on the positive particle will be

$k_{1}+k_{2}=k_{1}\left(1-\frac{\beta}{\alpha}\right)$

and that on the negative ion

$k_{3}+k_{4}=k_{1}\left(1-\frac{\beta}{\alpha}\right)$.

These forces being equal and having the same direction, there is no force tending to separate the two ions, as would be the case in an electric field. Nevertheless, the pair is acted on by a resultant force

$2k_{1}\left(1-\frac{\beta}{\alpha}\right)$.

If now β be somewhat larger than $$\alpha$$, the factor $$2\left(1-\frac{\beta}{\alpha}\right)$$ have a certain negative value –ε, and our result may be expressed as follows:

If we wish to determine the action between two ponderable bodies, we may first consider the forces existing between the positive ions in the one and the positive ions in the other. We then have to reverse the direction of these forces, and to multiply them by the factor ε. Of course, we are led in this way to law of gravitation.

The assumption that all ponderable matter is composed of positive and negative ions is no essential part of the above theory. We might have confined ourselves to the supposition that the state of the aether which is the cause of gravitation is propagated in a similar way as that which exists in the electromagnetic field. Proceedings Royal Acad. Amsterdam. Vol. II.

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 Instead of introducing two pairs of vectors ($$\mathfrak{d,\ H}$$) and ($$\mathfrak{d',H'}$$), both of which come into play in the electromagnetic actions, as well as in the phenomenon of gravitation, we might have assumed one pair for the electromagnetic field and one for universal attraction.

For these latter vectors, say $$\mathfrak{d,\ H}$$, we should then have established the equations (I), $$\varrho$$ being the density of ponderable matter, and for the force acting on unit mass, we should have put

$-\eta\left\{ 4\pi V^{2}\mathfrak{d}+\mathfrak{\left[v.\ H\right]}\right\}$,

where $$\eta$$ is a certain positive coefficient.

§ 8. Every theory of gravitation has to deal with the problem of the influence, exerted on this force by the motion of the heavenly bodies. The solution is easily deduced from our equations; it takes the same form as the corresponding solution for the electromagnetic actions between charged particles.

I shall only treat the case of a body A, revolving around a central body M, this latter having a given constant velocity p. Let r be the line MA, taken in the direction from M towards A, x, y, z the relative coordinates of A with respect to M, w the velocity of A's motion relatively to M, $$\vartheta$$ the angle between w and p, finally $$p_{r}$$ the component of p in the direction of r.

Then, besides the attraction

which would exist if the bodies were both at rest, A will be subject to the following actions.

1st. A force

in the direction of r.

2nd. A force whose components are

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 3rd. A force

parallel to the velocity p.

4th. A force

in the direction of r.

Of these, (15) and (16) depend only on the common velocity p, ( 17) and (18) on the contrary, on p and w conjointly.

It is further to be remarked that the additional forces (15) — (18) are all of the second order with respect to the small quantities

$\frac{p}{V}$ and $\frac{w}{V}$.

In so far, the law expressed by the above formulae presents a certain analog)- with the laws proposed by, Bernhard Riemann and Rudolf Clausius for the electromagnetic actions, and applied by some astronomers to the motions of the planets. Like the formulae of , our equations contain the absolute velocities, i. e. the velocities, relatively to the aether.

There is no doubt but that, in the present state of science, if we wish to try for gravitation a similar law as for electromagnetic forces, the law contained in (15) — (18) is to be preferred to the three other just mentioned laws.

§ 9. The forces (15) — (18) will give rise to small inequalities in the elements of a planetary orbit ; in computing these, we have to take for p the velocity of the Sun's motion through space. I have calculated the secular variations, using the formulae communicated by w:f:Félix Tisserand in his Mécanique céleste.

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 $\varkappa'+\int\limits_{0}^{t}\ n\ dt$.

Further, let λ, μ, and ν be the direction-cosines of the velocity p with respect to: 1st. the radius vector of the perihelion, 2nd. a direction which is got by giving to that radius vector a rotation of 90°, in the direction of the planet's revolution, 3rd. the normal to the plane of the orbit, drawn towards the side whence the planet is seen to revolve in the same direction as the hands of a watch.

Put $$\omega =\varpi - \theta$$, $$\frac{p}{V}=\delta$$ and $$\frac{na}{V}=\delta'$$ (na is the velocity in a circular orbit of radius a).

Then I find for the variations during one revolution

§ 10. I have worked out the case of the planet Mercury, taking 276° and + 34° for the right ascension and declination of the apex of the Sun's motion. I have got the following results:

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Now, $$\delta'=1,6\ \times\ 10^{-4}$$ and, if we put $$\delta =5,3\ \times\ 10^{-5}$$, we get

The changes that take place in a century are found from these numbers, if we multiply them by 415, and, if the variations of φ, θ, $$\varpi$$, and $$\varkappa'$$ are to be expressed in seconds, we have to introduce the factor $$2,06 \times 10^{5}$$. The result is, that the changes in φ, θ, $$\varpi$$, and $$\varkappa'$$ amount to a few seconds, and that in e to 0,000005.

Hence we conclude that our modification of law cannot account for the observed inequality in the longitude of the perihelion — as  law can to some extent do — but that, if we do not pretend to explain this inequality by an alteration of the law of attraction, there is nothing against the proposed formulae. Of course it will be necessary to apply them to other heavenly bodies, though it seems scarcely probable that there will be found any case in which the additional terms have an appreciable influence.

The special form of these terms may perhaps be modified. Yet, what has been said is sufficient to show that gravitation may be attributed to actions which are propagated with no greater velocity than that of light.

As is well known, Pierre-Simon Laplace has been the first to discuss this question of the velocity of propagation of universal attraction, and later astronomers have often treated the same problem. Let a body B be attracted by a body A, moving with the velocity p. Then, if the action is propagated with a finite velocity V, the influence which reaches B at time t, will have been emitted by A at an anterior moment, say t—τ. Let A1 be the position of the acting body at this moment, A2 that at time t. It is an easy matter to calculate the distance between these positions. Now, if the action at time

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( 574 ) t is calculated, as if A had continued to occupy the position A1, one is led to an influence on the astronomical motions of the order $$\tfrac{p}{V}$$; if V were equal to the velocity of light, this influence would be much greater than observations permit us to suppose. If, on the contrary, the terms with $$\tfrac{p}{V}$$ are to have admissible values, V ought to be many millions of times as great as the velocity of light.

From the considerations in this paper, it appears that this conclusion can be avoided. Changes of state in the aether, satisfying equations of the form (I), are propagated with the velocity V; yet, no quantities of the first order $$\tfrac{p}{V}$$ or $$\tfrac{w}{V}$$ (§ 8), but only terms containing $$\tfrac{p^{2}}{V^{2}}$$ and $$\tfrac{pw}{V^{2}}$$ appear in the results. This is brought about by the peculiar way—determined by the equations—in which moving matter changes the state of the aether; in the above mentioned case the condition of the aether will not be what it would have been, if the acting body were at rest in the position A1.

(April 25, 1900)

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text:
Author:Charles Henry Timperley

Works

 * A Dictionary of Printers and Printing (1839)

title: Author:Charles Henry Timperley


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