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 We replace in (9), by (4) and (III)

by

and transform the parts of the integral, that contain derivatives of $$\mathfrak{H}_{x},\mathfrak{H}_{y},\mathfrak{H}_{z}$$, by partial integration.

By consideration of equation (IV) we will find

where

At first is should be assumed, that the electric motions are restricted to a certain finite space, and that surface $$\sigma$$ is entirely outside of that space. Then at the surfaces it will be $$\mathfrak{d}=0,\mathfrak{H}=0$$, and

Therefore the magnitude L + U really applies, whose increase is equal to the work of the external forces, and which therefore is denoted by the expression "energy". It is composed of the ordinary mechanical energy L and the "electrical" energy U, and as regards the latter we find again the value given by.

The theorem of Pointing.
§ 14. Even if we abandon the previously made assumption about $$\sigma$$, formula (11) allows of a simple interpretation. With we not only assume that the electric force would have the value (12), but also, that it is really distributed over the space as it is expressed by the formula, i.e. that it amounts for unit volume


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 In equations (11), L + U thus means the whole energy within surface $$\sigma$$, and therefore the view is near at hand, that a quantity of energy

has traveled through the surface into the outside. It is most simple, if we put for the "energy flow" related to unit time and area

By that we come to the known theorem formulated by. Here, we don't discuss the subtle, related question concerning the localization of the energy. We can restrict ourselves with the fact, that the entire energy located in an arbitrary space — the "electric" portion calculated by formula (12) — always varies, as if the energy would travel according to the way determined by (13).

Tensions in the aether.
§ 15. The forces determined by our formula (V), not only require the motion of ions in ponderable bodies, but also in some circumstances can unify themselves to an action, that tends to set the body into motion. In this way all "ponderomotive" forces emerge, as for example the ordinary electrostatic and electrodynamic ones, as well as the pressure that is exerted by light rays on a body.

We want to consider the body as rigid, and calculate (by simple addition) all the forces that were exerted by the aether in the direction of the x-axis, i.e. the total force $$\Xi$$ in this direction. The investigation should be based on the things said at the beginning of § 13.

We immediately obtain

$$\begin{array}{cl} \Xi & =4\pi V^{2}\int\mathfrak{d}_{x}\rho\ d\ \tau+\int\rho[\mathfrak{v.H}]_{x}\ d\ \tau=\\ & =4\pi V^{2}\int\mathfrak{d}_{x}\rho\ d\ \tau+\int\rho\left(\mathfrak{v}_{y}\mathfrak{H}_{z}-\mathfrak{v}_{z}\mathfrak{H}_{y}\right)\ d\ \tau{,} \end{array}$$

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


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 where the integrals only have to be extended over the ponderable body, but like in § 13, it should taken for the entire space enclosed by $$\sigma$$.

At first, we replace $$4\pi\rho\mathfrak{v}_{x}$$, etc. by the expressions (10), and, because of (I), $$\rho$$ by

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

thus

Furthermore, a partial integration and application of (IV) and (II) gives (when we denote the direction constants of the perpendicular to $$\sigma$$ by $$\chi,\beta,\gamma$$)

$$\begin{array}{cl} \int\mathfrak{d}_{x}\frac{\partial\mathfrak{d}_{y}}{\partial y}\ d\ \tau & =\int\beta\mathfrak{d}_{x}\mathfrak{d}_{y}\ d\ \sigma-\int\mathfrak{d}_{y}\frac{\partial\mathfrak{d}_{x}}{\partial y}\ d\ \tau=\\ & =\int\beta\mathfrak{d}_{x}\mathfrak{d}_{y}\ d\ \sigma-\int\mathfrak{d}_{y}\frac{\partial\mathfrak{d}_{y}}{\partial x}\ d\ \tau-\frac{1}{4\pi V^{2}}\int\mathfrak{d}_{y}\frac{\partial\mathfrak{H}_{z}}{\partial t}\ d\ \tau{,}\\ \int\mathfrak{d}_{x}\frac{\partial\mathfrak{d}_{z}}{\partial z}\ d\ \tau & =\int\gamma\mathfrak{d}_{x}\mathfrak{d}_{z}\ d\ \sigma-\int\mathfrak{d}_{z}\frac{\partial\mathfrak{d}_{x}}{\partial z}\ d\ \tau=\\ & =\int\gamma\mathfrak{d}_{x}\mathfrak{d}_{z}\ d\ \sigma-\int\mathfrak{d}_{z}\frac{\partial\mathfrak{d}_{z}}{\partial x}\ d\ \tau+\frac{1}{4\pi V^{2}}\int\mathfrak{d}_{z}\frac{\partial\mathfrak{H}_{y}}{\partial t}\ d\ \tau{,} \end{array}$$

$$\int\left(\mathfrak{H}_{y}\frac{\partial\mathfrak{H}_{x}}{\partial y}+\mathfrak{H}_{z}\frac{\partial\mathfrak{H}_{x}}{\partial z}\right)\ d\ \tau=\int\left(\beta\mathfrak{H}_{x}\mathfrak{H}_{y}+\gamma\mathfrak{H}_{x}\mathfrak{H}_{z}\right)\ d\ \sigma-$$

$$-\int\mathfrak{H}_{x}\left(\frac{\partial\mathfrak{H}_{y}}{\partial y}+\frac{\partial\mathfrak{H}_{z}}{\partial z}\right)\ d\ \tau=\int\left(\beta\mathfrak{H}_{x}\mathfrak{H}_{y}+\gamma\mathfrak{H}_{x}\mathfrak{H}_{z}\right)\ d\ \sigma+$$

$$+\int\mathfrak{H}_{x}\frac{\partial\mathfrak{H}_{x}}{\partial x}\ d\ \tau.$$

If we substitute this value into (14), then several terms occur, that can be completely integrated, and eventually by a simple transformation


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 we have

Two similar equations serve for the determination of the other components $$\Eta$$ and $$\Zeta$$ of the ponderomotive action.

Besides it is to be noticed, that $$\Xi$$, $$\Eta$$ and $$\Zeta$$ must vanish, as soon space $$\tau$$ doesn't contain ponderable matter. Then it would be

§ 16. In some cases the space integral the remained in (15), will become independent of t, and if the last member vanishes, namely as soon as we have to deal with a stationary state, may it be with an electric charge, or may it be with a system of constant currents. Then, at least concerning the resultant force, the ponderomotive action can be calculated by integration over an arbitrary surface that encloses the body, and it is near at hand, to view them in a way, so that we (like Maxwell did) attribute to the aether a certain state of tension, and consider the tensions as the cause of the ponderomotive actions. If we as usual understand by $$\left(X_{n}, Y_{n}, Z_{n}\right)$$ the force related to unit area, that the aether exerts at the side (given by n) of an element $$d\sigma$$ upon the opposite aether, then by (15) we would have to put

From that, it is easy to derive the values of $$X_{x}$$, $$X_{y}$$, $$X_{z}$$, $$Y_{x}$$;


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 then we exactly obtain the system of tensions that was given by.

§ 17. Since in (15) the space integral doesn't vanish, the assumption of tensions (17) doesn't generally lead to the action stated by us. If we would reject equation (V) as the basis of the calculation of the ponderomotive forces, and employ the tensions, then the case would in no way be finished by formulas (I)—(IV) and (17). One wouldn't even obtain the same values for $$\Xi$$, when we would apply the equation

on one area and then on the other area, that encloses the considered body. It is connected with the fact, that the tensions (17) wouldn't let the aether to be at rest.

Above we have found formulas (16) for a space which is free of ponderable matter. That it's correct, as long as the aether is at rest, can hardly be doubted, since for the derivation only generally taken equations come into play. From the formulas

and

it is given, namely, that the right-hand side of equation (14) is zero for the free aether; the application of (IV) and (II) then leads to the first of formulas (16).

Now, in those formulas, the forces (that follow from the tensions at surface $$\sigma$$) are on the left side, and thus the formulas say that the considered part of the aether cannot remain at rest under the influence of these forces. All who consider equations (17) as generally valid, must conclude that in all cases, where 's energy flow is variable with time, the aether as a whole will be set into motion.


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 Thus it would be necessary to study the from of the emerging aether flows, and under consideration of them to again tackle the question after the ponderomotive actions.

The basics of a theory of the mentioned aether flows was already sketched by the masterhand of in one of his last papers, which he was able to complete.

We cannot discuss the considered questions, because the fundamental assumption by which we started, gives another view. Indeed, why should we, since we assumed that the aether is not in motion, ever speak about a force acting on that medium? The most simple would by, to assume that on a volume element of the aether, considered as a whole, never acts a force, or even refuse to apply the concept of force on such an element that of course never moves from its place. Of course this view violates the equality of action and reaction —, since we have reason to say that the aether exerts forces on ponderable matter —; but, as far I can see, nothing forces us to elevate this theorem to an unrestricted fundamental law.

Once we have decided ourselves in favor of the previously discusses view, then we must refuse from the outset, to reduce the ponderomotive forces (that follow from (V)) to tensions of the aether.

Nevertheless we may apply equation (15) to simplify the calculation, and it won't cause a misunderstanding, when we express ourselves for brevities sake, as if the elements of the two first integrals would mean real tensions in the aether.


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 From these merely fictitious "tensions" we can, as we saw, directly derive the interaction between charged bodies and electrodynamic actions. It is also to be recommended, to operate with them, when the phenomena are periodic and when we only wish to know the averages of the ponderomotive forces during a full period; the last member of (15) namely doesn't contribute anything to these values.

In this way we come to 's theorem on the pressure generated by motion of light.

The reversibility of motions and the mirror image of motion.
§ 18. For subsequent applications we include the following considerations at this place.

Let a system of moving ions be given, and $$\rho_{1}$$, $$\mathfrak{v}_{1}$$, $$\mathfrak{d}_{1}$$ and $$\mathfrak{H}_{1}$$ are the various relevant magnitudes within. We may denote the corresponding magnitudes for a second system by $$\rho_{2}$$, $$\mathfrak{v}_{2}$$, $$\mathfrak{d}_{2}$$ and $$\mathfrak{H}_{2}$$, and we want to imagine that in an arbitrary point, these magnitudes are at time +t in agreement with the magnitudes $$\rho_{1}$$, $$-\mathfrak{v}_{1}$$, $$\mathfrak{d}_{1}$$ and $$-\mathfrak{H}_{1}$$ at time -t.

We can easily see that, as regards $$\rho_{2}$$ and $$\mathfrak{v}_{2}$$, those conditions can be satisfied by a real motion of the ions, and namely the system of these ions must completely be in agreement with the first system; the same configurations with the same interval must occur one after the other, as in that first system, but in opposite order; in other words, we obtain the motions of the ions in the second system, when we reverse the motions given at first.

Furthermore, since $$\mathfrak{d}_{2}$$ and $$\mathfrak{H}_{2}$$ satisfy the conditions (I), (II), (III) and (IV), thus the condition of the aether as determined by these vectors, is in agreement with the motion of the ions.

Eventually it follows from equation (V), that in the second


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 system at time +t, the forces exerted upon the ions have the same direction and magnitude, as the corresponding forces in the first system at time -t. Now, if also the remaining forces that act on the ions in both cases — and in the same instances — are the same, then we can conclude, that the second state of motion is realizable in any way.

By means of similar considerations the possibility of motion can be demonstrated, which is the "mirror image" of a given motion with respect to a fixed plane.

We call $$P_{2}$$ the mirror image of a point $$P_{1}$$ and denote the magnitudes that are valid for two system — namely for the first in $$P_{1}$$ and for the second in $$P_{2}$$ — by $$\rho_{1}$$, $$\mathfrak{v}_{1}$$, $$\mathfrak{d}_{1}$$, $$\mathfrak{H}_{1}$$ and $$\rho_{2}$$, $$\mathfrak{v}_{2}$$, $$\mathfrak{d}_{2}$$, $$\mathfrak{H}_{2}$$. There it should constantly be $$\rho_{2} = \rho_{1}$$, and the vectors $$\mathfrak{v}_{2}$$, $$\mathfrak{d}_{2}$$, $$\mathfrak{H}_{2}$$ should be the mirror images of the vectors $$\mathfrak{v}_{1}$$, $$\mathfrak{d}_{1}$$ and $$-\mathfrak{H}_{1}$$.

That the second state of motion can now conveniently be called "mirror image", requires no explanation. If the forces of non-electric origin are of such a manner, so that the vectors by which they can be represented in both cases behave like objects and their mirror images, then the second motion will be possible as soon as the first one is possible.


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 CHAPTER III

To George F. Babbitt, as to most prosperous citizens of Zenith, his motor car was poetry and tragedy, love and heroism. The office was his pirate ship but the car his perilous excursion ashore.

Among the tremendous crises of each day none was more dramatic than starting the engine. It was slow on cold mornings; there was the long, anxious whirr of the starter; and sometimes he had to drip ether into the cocks of the cylinders, which was so very interesting that at lunch he would chronicle it drop by drop, and orally calculate how much each drop had cost him.

This morning he was darkly prepared to find something wrong, and he felt belittled when the mixture exploded sweet and strong, and the car didn't even brush the door-jamb, gouged and splintery with many bruisings by fenders, as he backed out of the garage. He was confused. He shouted "Morning!" to Sam Doppelbrau with more cordiality than he had intended.

Babbitt's green and white Dutch Colonial house was one of three in that block on Chatham Road. To the left of it was the residence of Mr. Samuel Doppelbrau, secretary of an excellent firm of bathroom-fixture jobbers. His was a comfortable house with no architectural manners whatever; a large wooden box with a squat tower, a broad porch, and glossy paint yellow as a yolk. Babbitt disapproved of Mr. and Mrs. Doppelbrau as "Bohemian." From their house came midnight music and obscene laughter; there were neighborhood rumors of bootlegged whisky and fast motor

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Rh rides. They furnished Babbitt with many happy evenings of discussion, during which he announced firmly, "I'm not strait-laced, and I don't mind seeing a fellow throw in a drink once in a while, but when it comes to deliberately trying to get away with a lot of hell-raising all the while like the Doppelbraus do, it's too rich for my blood!"

On the other side of Babbitt lived Howard Littlefield, Ph.D., in a strictly modern house whereof the lower part was dark red tapestry brick, with a leaded oriel, the upper part of pale stucco like spattered clay, and the roof red-tiled. Littlefield was the Great Scholar of the neighborhood; the authority on everything in the world except babies, cooking, and motors. He was a Bachelor of Arts of Blodgett College, and a Doctor of Philosophy in economics of Yale. He was the employment-manager and publicity-counsel of the Zenith Street Traction Company. He could, on ten hours' notice, appear before the board of aldermen or the state legislature and prove, absolutely, with figures all in rows and with precedents from Poland and New Zealand, that the street-car company loved the Public and yearned over its employees; that all its stock was owned by Widows and Orphans; and that whatever it desired to do would benefit property-owners by increasing rental values, and help the poor by lowering rents. All his acquaintances turned to Littlefield when they desired to know the date of the battle of Saragossa, the definition of the word "sabotage," the future of the German mark, the translation of "hinc illæ lachrimæ," or the number of products of coal tar. He awed Babbitt by confessing that he often sat up till midnight reading the figures and footnotes in Government reports, or skimming (with amusement at the author's mistakes) the latest volumes of chemistry, archeology, and ichthyology.

But Littlefield's great value was as a spiritual example. Despite his strange learnings he was as strict a Presbyterian and as firm a Republican as George F. Babbitt. He confirmed

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26 the business men in the faith. Where they knew only by passionate instinct that their system of industry and manners was perfect, Dr. Howard Littlefield proved it to them, out of history, economics, and the confessions of reformed radicals.

Babbitt had a good deal of honest pride in being the neighbor of such a savant, and in Ted's intimacy with Eunice Littlefield. At sixteen Eunice was interested in no statistics save those regarding the ages and salaries of motion-picture stars, but—as Babbitt definitively put it—"she was her father's daughter."

The difference between a light man like Sam Doppelbrau and a really fine character like Littlefield was revealed in their appearances. Doppelbrau was disturbingly young for a man of forty-eight. He wore his derby on the back of his head, and his red face was wrinkled with meaningless laughter. But Littlefield was old for a man of forty-two. He was tall, broad, thick; his gold-rimmed spectacles were engulfed in the folds of his long face; his hair was a tossed mass of greasy blackness; he puffed and rumbled as he talked; his Phi Beta Kappa key shone against a spotty black vest; he smelled of old pipes; he was altogether funereal and archidiaconal; and to real-estate brokerage and the jobbing of bathroom-fixtures he added an aroma of sanctity.

This morning he was in front of his house, inspecting the grass parking between the curb and the broad cement sidewalk. Babbitt stopped his car and leaned out to shout "Mornin'!" Littlefield lumbered over and stood with one foot up on the running-board.

"Fine morning," said Babbitt, lighting—illegally early—his second cigar of the day.

"Yes, it's a mighty fine morning," said Littlefield.

"Spring coming along fast now."

"Yes, it's real spring now, all right," said Littlefield.

"Still cold nights, though. Had to have a couple blankets, on the sleeping-porch last night."

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Rh "Yes, it wasn't any too warm last night," said Littlefield.

"But I don't anticipate we'll have any more real cold weather now."

"No, but still, there was snow at Tiflis, Montana, yesterday," said the Scholar, "and you remember the blizzard they had out West three days ago—thirty inches of snow at Greeley, Colorado—and two years ago we had a snow-squall right here in Zenith on the twenty-fifth of April."

"Is that a fact! Say, old man, what do you think about the Republican candidate? Who'll they nominate for president? Don't you think it's about time we had a real business administration?"

"In my opinion, what the country needs, first and foremost, is a good, sound, business-like conduct of its affairs. What we need is—a business administration!" said Littlefield.

"I'm glad to hear you say that! I certainly am glad to hear you say that! I didn't know how you'd feel about it, with all your associations with colleges and so on, and I'm glad you feel that way. What the country needs—just at this present juncture—is neither a college president nor a lot of monkeying with foreign affairs, but a good—sound economical—business—administration, that will give us a chance to have something like a decent turnover."

"Yes. It isn't generally realized that even in China the schoolmen are giving way to more practical men, and of course you can see what that implies."

"Is that a fact! Well, well!" breathed Babbitt, feeling much calmer, and much happier about the way things were going in the world. "Well, it's been nice to stop and parleyvoo a second. Guess I'll have to get down to the office now and sting a few clients. Well, so long, old man. See you tonight. So long."

They had labored, these solid citizens. Twenty years before, the hill

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28 on which Floral Heights was spread, with its bright roofs and immaculate turf and amazing comfort, had been a wilderness of rank second-growth elms and oaks and maples. Along the precise streets were still a few wooded vacant lots, and the fragment of an old orchard. It was brilliant to-day; the apple boughs were lit with fresh leaves like torches of green fire. The first white of cherry blossoms flickered down a gully, and robins clamored.

Babbitt sniffed the earth, chuckled at the hysteric robins as he would have chuckled at kittens or at a comic movie. He was, to the eye, the perfect office-going executive—a well-fed man in a correct brown soft hat and frameless spectacles, smoking a large cigar, driving a good motor along a semi-suburban parkway. But in him was some genius of authentic love for his neighborhood, his city, his clan. The winter was over; the time was come for the building, the visible growth, which to him was glory. He lost his dawn depression; he was ruddily cheerful when he stopped on Smith Street to leave the brown trousers, and to have the gasoline-tank filled.

The familiarity of the rite fortified him: the sight of the tall red iron gasoline-pump, the hollow-tile and terra-cotta garage, the window full of the most agreeable accessories—shiny casings, spark-plugs with immaculate porcelain jackets tire-chains of gold and silver. He was flattered by the friendliness with which Sylvester Moon, dirtiest and most skilled of motor mechanics, came out to serve him. "Mornin', Mr. Babbitt!" said Moon, and Babbitt felt himself a person of importance, one whose name even busy garagemen remembered—not one of these cheap-sports flying around in flivvers. He admired the ingenuity of the automatic dial, clicking off gallon by gallon; admired the smartness of the sign: "A fill in time saves getting stuck—gas to-day 31 cents"; admired the rhythmic gurgle of the gasoline as it flowed into the tank, and the mechanical regularity with which Moon turned the handle.

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Rh "How much we takin' to-day?" asked Moon, in a manner which combined the independence of the great specialist, the friendliness of a familiar gossip, and respect for a man of weight in the community, like George F. Babbitt.

"Fill 'er up."

"Who you rootin' for for Republican candidate, Mr. Babbitt?"

"It's too early to make any predictions yet. After all, there's still a good month and two weeks—no, three weeks—must be almost three weeks—well, there's more than six weeks in all before the Republican convention, and I feel a fellow ought to keep an open mind and give all the candidates a show—look 'em all over and size 'em up, and then decide carefully."

"That's a fact, Mr. Babbitt."

"But I'll tell you—and my stand on this is just the same as it was four years ago, and eight years ago, and it'll be my stand four years from now—yes, and eight years from now! What I tell everybody, and it can't be too generally understood, is that what we need first, last, and all the time is a good, sound business administration!"

"By golly, that's right!"

"How do those front tires look to you?"

"Fine! Fine! Wouldn't be much work for garages if everybody looked after their car the way you do."

"Well, I do try and have some sense about it." Babbitt paid his bill, said adequately, "Oh, keep the change," and drove off in an ecstasy of honest self-appreciation. It was with the manner of a Good Samaritan that he shouted at a respectable-looking man who was waiting for a trolley car, "Have a lift?" As the man climbed in Babbitt condescended, "Going clear down-town? Whenever I see a fellow waiting for a trolley, I always make it a practice to give him a lift—unless, of course, he looks like a bum."

"Wish there were more folks that were so generous with

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30 their machines," dutifully said the victim of benevolence.

"Oh, no, 'tain't a question of generosity, hardly. Fact, I always feel—I was saying to my son just the other night—it's a fellow's duty to share the good things of this world with his neighbors, and it gets my goat when a fellow gets stuck on himself and goes around tooting his horn merely because he's charitable."

The victim seemed unable to find the right answer. Babbitt boomed on:

"Pretty punk service the Company giving us on these car-lines. Nonsense to only run the Portland Road cars once every seven minutes. Fellow gets mighty cold on a winter morning, waiting on a street corner with the wind nipping at his ankles."

"That's right. The Street Car Company don't care a damn what kind of a deal they give us. Something ought to happen to 'em."

Babbitt was alarmed. "But still, of course it won't do to just keep knocking the Traction Company and not realize the difficulties they're operating under, like these cranks that want municipal ownership. The way these workmen hold up the Company for high wages is simply a crime, and of course the burden falls on you and me that have to pay a seven-cent fare! Fact, there's remarkable service on all their lines—considering."

"Well—" uneasily.

"Darn fine morning," Babbitt explained. "Spring coming along fast."

"Yes, it's real spring now."

The victim had no originality, no wit, and Babbitt fell into a great silence and devoted himself to the game of beating trolley cars to the corner: a spurt, a tail-chase, nervous speeding between the huge yellow side of the trolley and the jagged row of parked motors, shooting past just as the trolley stopped—a rare game and valiant.

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Rh And all the while he was conscious of the loveliness of Zenith. For weeks together he noticed nothing but clients and the vexing To Rent signs of rival brokers. To-day, in mysterious malaise, he raged or rejoiced with equal nervous swiftness, and to-day the light of spring was so winsome that he lifted his head and saw.

He admired each district along his familiar route to the office: The bungalows and shrubs and winding irregular drive ways of Floral Heights. The one-story shops on Smith Street, a glare of plate-glass and new yellow brick; groceries and laundries and drug-stores to supply the more immediate needs of East Side housewives. The market gardens in Dutch Hollow, their shanties patched with corrugated iron and stolen doors. Billboards with crimson goddesses nine feet tall advertising cinema films, pipe tobacco, and talcum powder. The old "mansions" along Ninth Street, S. E., like aged dandies in filthy linen; wooden castles turned into boarding-houses, with muddy walks and rusty hedges, jostled by fast-intruding garages, cheap apartment-houses, and fruit-stands conducted by bland, sleek Athenians. Across the belt of railroad-tracks, factories with high-perched water-tanks and tall stacks-factories producing condensed milk, paper boxes, lighting-fixtures, motor cars. Then the business center, the thickening darting traffic, the crammed trolleys unloading, and high doorways of marble and polished granite.

It was big—and Babbitt respected bigness in anything; in mountains, jewels, muscles, wealth, or words. He was, for a spring-enchanted moment, the lyric and almost unselfish lover of Zenith. He thought of the outlying factory suburbs; of the Chaloosa River with its strangely eroded banks; of the orchard-dappled Tonawanda Hills to the North, and all the fat dairy land and big barns and comfortable herds. As he dropped his passenger he cried, "Gosh, I feel pretty good this morning!"

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32

Epochal as starting the car was the drama of parking it before he entered his office. As he turned from Oberlin Avenue round the corner into Third Street, N.E., he peered ahead for a space in the line of parked cars. He angrily just missed a space as a rival driver slid into it. Ahead, another car was leaving the curb, and Babbitt slowed up, holding out his hand to the cars pressing on him from behind, agitatedly motioning an old woman to go ahead, avoiding a truck which bore down on him from one side. With front wheels nicking the wrought-steel bumper of the car in front, he stopped, feverishly cramped his steering-wheel, slid back into the vacant space and, with eighteen inches of room, manoeuvered to bring the car level with the curb. It was a virile adventure masterfully executed. With satisfaction he locked a thief-proof steel wedge on the front wheel, and crossed the street to his real-estate office on the ground floor of the Reeves Building.

The Reeves Building was as fireproof as a rock and as efficient as a typewriter; fourteen stories of yellow pressed brick, with clean, upright, unornamented lines. It was filled with the offices of lawyers, doctors, agents for machinery, for emery wheels, for wire fencing, for mining-stock. Their gold signs shone on the windows. The entrance was too modern to be flamboyant with pillars; it was quiet, shrewd, neat. Along the Third Street side were a Western Union Telegraph Office, the Blue Delft Candy Shop, Shotwell's Stationery Shop, and the Babbitt-Thompson Realty Company.

Babbitt could have entered his office from the street, as customers did, but it made him feel an insider to go through the corridor of the building and enter by the back door. Thus he was greeted by the villagers.

The little unknown people who inhabited the Reeves Building corridors—elevator-runners, starter, engineers, superintendent, and the doubtful-looking lame man who conducted the

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Rh news and cigar stand—were in no way city-dwellers. They were rustics, living in a constricted valley, interested only in one another and in The Building. Their Main Street was the entrance hall, with its stone floor, severe marble ceiling, and the inner windows of the shops. The liveliest place on the street was the Reeves Building Barber Shop, but this was also Babbitt's one embarrassment. Himself, he patronized the glittering Pompeian Barber Shop in the Hotel Thornleigh, and every time he passed the Reeves shop—ten times a day, a hundred times—he felt untrue to his own village.

Now, as one of the squirearchy, greeted with honorable salutations by the villagers, he marched into his office, and peace and dignity were upon him, and the morning's dissonances all unheard.

They were heard again, immediately.

Stanley Graff, the outside salesman, was talking on the telephone with tragic lack of that firm manner which disciplines clients: "Say, uh, I think I got just the house that would suit you—the Percival House, in Linton.... Oh, you've seen it. Well, how'd it strike you? ... Huh? ... Oh," irresolutely, "oh, I see."

As Babbitt marched into his private room, a coop with semi-partition of oak and frosted glass, at the back of the office, he reflected how hard it was to find employees who had his own faith that he was going to make sales.

There were nine members of the staff, besides Babbitt and his partner and father-in-law, Henry Thompson, who rarely came to the office. The nine were Stanley Graff, the outside salesman—a youngish man given to cigarettes and the playing of pool; old Mat Penniman, general utility man, collector of rents and salesman of insurance—broken, silent, gray; a mystery, reputed to have been a "crack" real-estate man with a firm of his own in haughty Brooklyn; Chester Kirby Laylock, resident salesman out at the Glen Oriole acreage development—an enthusiastic person with a silky mustache

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Transformation of the fundamental equations.
§ 19. From now on it will be assumed that the bodies to be considered are moving at a steady velocity of translation $$\mathfrak{p}$$, under which we will have to understand in almost all applications, the speed of the earth in its motion around the sun. It would be interesting at first to further develop the theory for stationary bodies, but for brevity's sake let us immediately turn to the more general case. Besides, it may be still set $$\mathfrak{p}=0$$.

The treatment of the problems that are now coming into play is most simple, when instead of the co-ordinate system used above, we introduce another one which is rigidly connected with ponderable matter and therefore shares its displacement.

While the coordinates of a point with respect to the fixed system were called x, y, z, let those, which refer to the moving system and which I call the relative coordinates, denoted by (x), (y), (z) for the time being. Until now, all the variable parameters were seen as functions of x, y, z, t; furthermore $$\mathfrak{d}_{x}$$, $$\mathfrak{d}_{y}$$, etc. shall be seen as functions of (x) (y), (z) and t.

Under a fixed point, we now understand one point,


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34 and much family; Miss Theresa McGoun, the swift and rather pretty stenographer; Miss Wilberta Bannigan, the thick, slow, laborious accountant and file-clerk; and four freelance part-time commission salesmen.

As he looked from his own cage into the main room Babbitt mourned, "McGoun's a good stenog., smart's a whip, but Stan Graff and all those bums—" The zest of the spring morning was smothered in the stale office air.

Normally he admired the office, with a pleased surprise that he should have created this sure lovely thing; normally he was stimulated by the clean newness of it and the air of bustle; but to-day it seemed flat—the tiled floor, like a bathroom, the ocher-colored metal ceiling, the faded maps on the hard plaster walls, the chairs of varnished pale oak, the desks and filing-cabinets of steel painted in olive drab. It was a vault, a steel chapel where loafing and laughter were raw sin.

He hadn't even any satisfaction in the new water-cooler! And it was the very best of water-coolers, up-to-date, scientific, and right-thinking. It had cost a great deal of money (in itself a virtue). It possessed a non-conducting fiber ice-container, a porcelain water-jar (guaranteed hygienic), a drip-less non-clogging sanitary faucet, and machine-painted decorations in two tones of gold. He looked down the relentless stretch of tiled floor at the water-cooler, and assured himself that no tenant of the Reeves Building had a more expensive one, but he could not recapture the feeling of social superiority it had given him. He astoundingly grunted, "I'd like to beat it off to the woods right now. And loaf all day. And go to Gunch's again to-night, and play poker, and cuss as much as I feel like, and drink a hundred and nine-thousand bottles of beer."

He sighed; he read through his mail; he shouted "Msgoun," which meant "Miss McGoun"; and began to dictate.

This was his own version of his first letter:

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Rh "Omar Gribble, send it to his office, Miss McGoun, yours of twentieth to hand and in reply would say look here, Gribble, I'm awfully afraid if we go on shilly-shallying like this we'll just naturally lose the Allen sale, I had Allen up on carpet day before yesterday and got right down to cases and think I can assure you—uh, uh, no, change that: all my experience indicates he is all right, means to do business, looked into his financial record which is fine—that sentence seems to be a little balled up, Miss McGoun; make a couple sentences out of it if you have to, period, new paragraph.

"He is perfectly willing to pro rate the special assessment and strikes me, am dead sure there will be no difficulty in getting him to pay for title insurance, so now for heaven's sake let's get busy—no, make that: so now let's go to it and get down—no, that's enough—you can tie those sentences up a little better when you type 'em, Miss McGoun—your sincerely, etcetera."

This is the version of his letter which he received, typed, from Miss McGoun that afternoon:

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36 As he read and signed it, in his correct flowing business-college hand, Babbitt reflected, "Now that's a good, strong letter, and clear's a bell. Now what the— I never told McGoun to make a third paragraph there! Wish she'd quit trying to improve on my dictation! But what I can't understand is: why can't Stan Graff or Chet Laylock write a letter like that? With punch! With a kick!"

The most important thing he dictated that morning was the fortnightly form-letter, to be mimeographed and sent out to a thousand "prospects." It was diligently imitative of the best literary models of the day; of heart-to-heart-talk advertisements, "sales-pulling" letters, discourses on the "development of Will-power," and hand-shaking house-organs, as richly poured forth by the new school of Poets of Business. He had painfully written out a first draft, and he intoned it now like a poet delicate and distrait:

I just want to know can I do you a whaleuva favor? Honest! No kidding! I know you're interested in getting a house, not merely a place where you hang up the old bonnet but a love-nest for the wife and kiddies—and maybe for the flivver out beyant (be sure and spell that b-e-y-a-n-t, Miss McGoun) the spud garden. Say, did you ever stop to think that we're here to save you trouble? That's how we make a living—folks don't pay us for our lovely beauty! Now take a look:

Sit right down at the handsome carved mahogany escritoire and shoot us in a line telling us just what you want, and if we can find it we'll come hopping down your lane with the good tidings, and if we can't, we won't bother you. To save your time, just fill out the blank enclosed. On request will also send blank regarding store properties in Floral Heights, Silver Grove, Linton, Bellevue, and all East Side residential districts.

P.S.—Just a hint of some plums we can pick for you—some genuine bargains that came in to-day:

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Rh —Cute four-room California bungalow, a.m.i., garage, dandy shade tree, swell neighborhood, handy car line. $3700, $780 down and balance liberal, Babbitt-Thompson terms, cheaper than rent.

.—A corker! Artistic two-family house, all oak trim, parquet floors, lovely gas log, big porches, colonial,, a bargain at $11,250.

Dictation over, with its need of sitting and thinking instead of bustling around and making a noise and really doing something, Babbitt sat creakily back in his revolving desk-chair and beamed on Miss McGoun. He was conscious of her as a girl, of black bobbed hair against demure cheeks. A longing which was indistinguishable from loneliness enfeebled him. While she waited, tapping a long, precise pencil-point on the desk-tablet, he half identified her with the fairy girl of his dreams. He imagined their eyes meeting with terrifying recognition; imagined touching her lips with frightened reverence and— She was chirping, "Any more, Mist' Babbitt?" He grunted, "That winds it up, I guess," and turned heavily away.

For all his wandering thoughts, they had never been more intimate than this. He often reflected, "Nev' forget how old Jake Offutt said a wise bird never goes love-making in his own office or his own home. Start trouble. Sure. But—"

In twenty-three years of married life he had peered uneasily at every graceful ankle, every soft shoulder; in thought he had treasured them; but not once had he hazarded respectability by adventuring. Now, as he calculated the cost of repapering the Styles house, he was restless again, discontented about nothing and everything, ashamed of his discontentment, and lonely for the fairy girl.

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 that has a steady position with respect to the new axis; in the same way, by rest or motion of a physical particle we shall mean the relative rest or the relative motion in relation to ponderable matter. With ions, which move in this sense of the word, we will have to do as soon as the displaced matter is the seat of electric motions.

By $$\mathfrak{v}$$ we shall not represent the real velocity, but the velocity of the previously mentioned relative motion. The real velocity is thus

and hereby $$\mathfrak{v}$$ is to be replaced in equations (4) and (V).

In addition, we have, instead of the derivatives with respect to x, y, z and t, to establish such with respect to (x), (y), (z) and t.

The first mentioned derivative I denote by

however, the latter by

Now we have, by application to an arbitrary function,

$$\frac{\partial}{\partial x}=\frac{\partial}{\partial(x)},\ \frac{\partial}{\partial y}=\frac{\partial}{\partial(y)},\ \frac{\partial}{\partial z}=\frac{\partial}{\partial(z)}{,}$$

$$\left(\frac{\partial}{\partial t}\right)_{1}=\left(\frac{\partial}{\partial t}\right)_{2}-\mathfrak{p}_{x}\frac{\partial}{\partial(x)}-\mathfrak{p}_{y}\frac{\partial}{\partial(y)}-\mathfrak{p}_{z}\frac{\partial}{\partial(z)}.$$

By that it follows, that we can write for $$Div\ \mathfrak{A}$$ the expression

and for the components of $$Rot\ \mathfrak{A}$$

The expressions $$Div\ \mathfrak{A}$$ and $$Rot\ \mathfrak{A}$$ have still the meaning given in § 4, g and h, if, after having abandoned the old coordinates one and for all, for simplification we don't indicate the new ones with (x), (y), (z), but with x, y, z.


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 We also want, after we have passed to the new coordinates, use the sign $$\tfrac{\partial}{\partial t}$$ instead of $$\left(\tfrac{\partial}{\partial t}\right)_{2}$$ for a differentiation with respect to time at constant relative coordinates, so that

The derivative with respect to time, which occurs in the basic equations (I) - (V), are all of the kind indicated by $$\left(\tfrac{\partial}{\partial t}\right)_{1}$$. We will maintain this sign as an abbreviation for the longer term (18).

In contrast, a point over a letter shall henceforth — such as $$\partial/\partial t$$ - indicate a differentiation with respect to time at constant relative coordinates. Thus the terms $$\dot{\mathfrak{d}}$$ and $$\dot{\mathfrak{H}}$$ in (4) and (IV) may not be left unaltered. By $$\mathfrak{d}$$, for example, we understood a vector with components

$$\left(\frac{\partial\mathfrak{d}_{x}}{\partial t}\right)_{1}\text{, etc.,}$$

or

$$\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}-\mathfrak{p}_{y}\frac{\partial}{\partial y}-\mathfrak{p}_{z}\frac{\partial}{\partial z}\right)\mathfrak{d}_{x}\text{, etc.}$$

We can suitably write this vector

$$\left(\frac{\partial\mathfrak{d}}{\partial t}\right)_{1}{,}$$

while

$$\dot{\mathfrak{d}}$$ or $$\frac{\partial\mathfrak{d}}{\partial t}$$

will mean the vector with components

$$\frac{\partial\mathfrak{d}_{x}}{\partial t}\text{, etc.}$$

Based on the system of axes associated with ponderable matter, eventually the fundamental equations become


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§ 20. For some purposes, a different form of some equations is more appropriate.

The first of the three (IV) summarized relations is namely

where, by equation (IIa), we can write for the last three members

which is nothing else than the first component of

Accordingly, we obtain instead of (IVa)

Furthermore, the current $$\mathfrak{S}$$ can be entirely eliminated. The first of equations (IIIa) becomes, when we consider (4a) and (Ia),

$$\frac{\partial\mathfrak{H}_{z}}{\partial y}-\frac{\partial\mathfrak{H}_{y}}{\partial z}=4\pi\rho\left(\mathfrak{p}_{x}+\mathfrak{v}_{x}\right)+4\pi\left(\frac{\partial\mathfrak{d}_{x}}{\partial t}-\mathfrak{p}_{x}\frac{\partial\mathfrak{d}_{x}}{\partial x}-\mathfrak{p}_{y}\frac{\partial\mathfrak{d}_{x}}{\partial y}-\right.$$

$$\left.-\mathfrak{p}_{z}\frac{\partial\mathfrak{d}_{x}}{\partial z}\right)=4\pi\rho\mathfrak{v}_{x}+4\pi\left\{ \left(\mathfrak{p}_{x}\frac{\partial\mathfrak{d}_{y}}{\partial y}-\mathfrak{p}_{y}\frac{\partial\mathfrak{d}_{x}}{\partial y}\right)-\left(\mathfrak{p}_{z}\frac{\partial\mathfrak{d}_{x}}{\partial z}-\right.\right.$$

$$\left.\left.-\mathfrak{p}_{x}\frac{\partial\mathfrak{d}_{z}}{\partial z}\right)\right\}+4\pi\frac{\partial\mathfrak{d}_{x}}{\partial t}.$$

By that it follows, if we define a new vector $$\mathfrak{H}'$$ by means of the equation

thus

If we now introduce the sign $$\mathfrak{F}$$ for the electric force-action on stationary ions, we obtain the following set of formulas


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User talk:Closeapple

— billinghurst  sDrewth  11:03, 13 January 2011 (UTC)

title: User talk:Closeapple


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§ 21. From equations (Ia)—(Va) (§ 19) also some formulas can be derived, any of them contains only one of the magnitudes $$\mathfrak{d}_{x}$$, $$\mathfrak{d}_{y}$$, $$\mathfrak{d}_{z}$$, $$\mathfrak{H}_{x}$$, $$\mathfrak{H}_{y}$$, $$\mathfrak{H}_{z}$$.

At first, it follows from (IVa)

If we consider here what has been said in § 4, h, as well as the relations (Ia), (IIIa) and (4a), we arrive at the three formulas

Similarly, we find

The last members of these six equations are completely known once we know how the ions are moving.

Application to electrostatics.
§ 22. We want to calculate by which forces the ions act on one another, when all of them are at rest with respect to ponderable matter. In this case a state occurs, where at every point $$\mathfrak{d}$$ and $$\mathfrak{H}$$ are independent of time. We have

and equations (A) and (B) will be reduced, when for brevity's sake the operation


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 is indicated by $$\Delta'$$, to

and

To fulfill these conditions, we determine a function $$\omega$$ by

and put

i.e., values that really satisfy the fundamental equations (Ia) - (IVa).

From (Va) it also follows

so that the sought forces are found.

Without prejudice to generality, we may assume that the translation happens in the direction of the x-axis. It is then $$\mathfrak{p}_y= \mathfrak{p}_z = 0$$, and the formula for the determination of $$\omega$$ will be transformed into

§ 23. To clearly define the meaning of the above formulas, we will compare the considered system $$S_{1}$$ with a second one $$S_{2}$$. The latter should not be moved, and it arises from $$S_{1}$$ by increasing all the dimensions that have the direction of the x-axis (therefore the relevant dimensions of the ions as well), in the ratio $$\textstyle\sqrt{V^{2}-\mathfrak{p}^{2}}$$ to $$V$$, or: between the coordinates x, y, z of a point of $$S_{1}$$ and the coordinates $$x', y', z'$$ of the same corresponding point of $$S_{2}$$, we let remain the relations


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In addition, the mutually corresponding volume elements, and therefore also the ions, shall have the same charges in $$S_{1}$$ and $$S_{2}$$.

If we apply to all magnitudes, which are related to the second system, a prime so they can be distinguished, then

$$\rho'=\rho\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}{,}$$

and

$$\frac{\partial^{2}\omega'}{\partial x'^{2}}+\frac{\partial^{2}\omega'}{\partial y'^{2}}+\frac{\partial^{2}\omega'}{\partial z'^{2}}=\rho'=\rho\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}.$$

Then the equation (23) can be written in the form

$$\frac{\partial^{2}\omega}{\partial x'^{2}}+\frac{\partial^{2}\omega}{\partial y'^{2}}+\frac{\partial^{2}\omega}{\partial z'^{2}}=\rho$$

then

$$\omega=\frac{\omega'}{\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}}{,}$$

and since in the second system

$$\mathfrak{E'}_{x}=4\pi V^{2}\frac{\partial\omega'}{\partial x'}\text{, etc.}$$

thus

$$\mathfrak{E}_{x}=\mathfrak{E'}_{x},\ \mathfrak{E}_{y}=\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}\mathfrak{E'}_{y},\ \mathfrak{E}_{z}=\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}\mathfrak{E'}_{z}.$$

The same relations, as they exist between the components of $$\mathfrak{E}$$ and $$\mathfrak{E}'$$, also exist, since the charges in $$S_{1}$$ and $$S_{2}$$ are equal, between the force components acting on an ion.

If in the second system at certain places $$\mathfrak{E}'=0$$, then $$\mathfrak{E}$$ vanishes at the corresponding points of the first system.

§ 24. Several implications of this theorem are obvious. From ordinary electrostatics, we know for example that an excess of positive (or negative) ions can be distributed over a


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 conductor, namely over its surface $$\Sigma$$, so that in the interior no electric force is acting. If we take this distribution for the system $$S_2$$ and derive from it a system $$S_1$$ by the above-discussed transformation, then also in this one an excess of positive ions only exists at a certain surface $$\Sigma$$, while in all interior points the electric force $$\mathfrak{E}$$ vanishes. The fact that an electric charge is located at the surface of a conductor, won't be changed by the translation of ponderable matter.

Similar considerations apply to two or more bodies. If a conductor C is confronted with a charged body K, then there exists, according to a known theorem, always a certain amount of charge on the surface of C, which together with K exerts no action on the ions in the interior of the conductor. This theorem remains valid, if the ponderable matter is moving, and it is even still allowed to assume that, under the influence of K, an "induced" charge is formed by itself upon C, which just cancels the effect of K on the interior points.

Since by (22) the components of $$\mathfrak{E}$$ are proportional to the derivative of ω, we can also say that inducing and inducted charges together cause a constant ω at all points of C. It follows then by means of equations (20), (21) and (Va), that also a moving ion in the interior of C does not experience any force-action from the two charges.

Finally, it should be noted that by our formulas, the distribution of a charge over a given conductor, as well as the attraction or repulsion of charged bodies by the motion of the earth, must be changed. But this influence is limited to the second order, namely if the fraction $$\mathfrak{p}/V$$ is called a magnitude of first order, and thus the fraction $$\mathfrak{p}^{2}/V^{2}$$ is called a magnitude of second order.

Since $$\mathfrak{p}/V=1/10000$$, we may not hope, neglecting some very special cases, to find with respect to electrical and optical phenomena an influence of earth's motion that depends on $$\mathfrak{p}^{2}/V^{2}$$. The only thing that could be


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 observed in relation to bodies at rest on earth, is the magnetic force (21). At first glance, we might expect a corresponding effect on the current elements. We will return to this question in § 26.

Values of $$\mathfrak{d}$$ and $$\mathfrak{H}$$ at a stationary current.
§ 25. On the basis of equations (A) and (B) we again tackle the problem treated in § 11. We consider, as there, the mean values and take into account that for them the simplification (19) is permitted in stationary states; moreover, we assume at first that the conductors do not have a significant charge, so that $$\bar{\rho}=0$$.

It is near at hand to interpret the vector $$\overline{\rho\mathfrak{v}}$$ as being a "current". We think of it as solenoidally distributed and denote it by $$\bar{\mathfrak{S}}$$, where it remains, however, temporarily undecided whether this is also the mean value of the vector occurring in (4a).

We now derive from (A) and (B)

$$V^{2}\Delta'\bar{\mathfrak{d}}_{x}=-\left(\mathfrak{p}_{x}\frac{\partial}{\partial x}+\mathfrak{p}_{y}\frac{\partial}{\partial y}+\mathfrak{p}_{z}\frac{\partial}{\partial z}\right)\bar{\mathfrak{S}}_{x}\text{, etc.,}$$

$$\Delta'\overline{\mathfrak{H}_{x}}=4\pi\left(\frac{\partial\overline{\mathfrak{S}}_{y}}{\partial z}-\frac{\partial\overline{\mathfrak{S}}_{z}}{\partial y}\right)\text{, etc.}$$

If we determine thus the three auxiliary magnitudes $$\chi_{x}$$, $$\chi_{y}$$, $$\chi_{z}$$ by means of the equations

$$\Delta'\chi_{x}=\overline{\mathfrak{S}}_{x},\ \Delta'\chi_{y}=\overline{\mathfrak{S}}_{y},\ \Delta'\chi_{z}=\overline{\mathfrak{S}}_{z}{,}$$

so everywhere we have


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 and by (Va) the electric force acting on stationary ions,

At first glance, it therefore seems as if a current that streams through a conductor, is acting on a stationary ion with a force of first order. However, on closer reflection we find that the force (27) is just being compensated by another force.

The values (27) are in fact in perfect agreement with the expressions (22), if we substitute

By § 22, ω would belong to an electric charge, its density is

or by the given formulas

Let us imagine for a moment that the current does not exist, but there is a charge with the average density ρ. This would of course exist only in the conductor, and the total sum would be zero, as it follows from (29) and

Obviously this ion distribution would completely vanish, if it is left alone. This can also be expressed by saying that the charge will set them in motion by virtue of its action on resting ions, and that therefore eventually another charge with the average density -ρ occurs besides it, or

Since the current that we considered initially, exactly acts on resting ions as the charge (29), it will also generate the charge A after a short time; this eliminates the effects on stationary ions, namely


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 not only in the outer points, but also, at least with respect to the averages of the forces, in the interior of the conductor.

I want to call this charge A the compensation charge. Once generated, the conductor does not cause any electricity motion in a neighboring body. A stationary current in a wire moving with the Earth therefore exerts no inductive action on a circuit which is also at rest with respect to Earth, regardless of Earth's motion.

It should be noted now that in the finally occurring state of the system, ρ and $$\mathfrak{d}$$ have certain values of order $$\mathfrak{p}$$. Neglecting the magnitudes of second order, then it really follows from (4a)

Interaction between a charged body K and a conductor.
§ 26. After the foregoing, we have to assume that in the conductor next to the current $$\overline{\mathfrak{S}}$$, a compensation charge does exist, and also (at the surface of the conductor) the electrostatic induction-charge B caused by K. For simplicity, we imagine that $$\overline{\mathfrak{S}}$$, $$A$$ and $$B$$ co-exist as independent ion systems This mode of imagination, however, is in no way necessary. To show that the considerations communicated in the texts are correct, we don't need to assume, that the ions which form the charges A and B, were remaining at rest and were altogether uninfluenced by the adjacent existing current. We can also imagine that all ions are moving, similar to an electrolyte, in a most irregular manner. But a constant, non-zero mean value $$\bar{\ ho}$$ is very well possible; because this constitutes the charges designated by A and B (i.e., $$\bar{\rho}$$ is composed of two terms of a sum $$\bar{\rho_{A}}$$ and $$\bar{\rho_{B}}$$), while the current $$\overline{\mathfrak{S}}$$ is determined by $$\overline{\rho\mathfrak{v}}$$.

If in (A) and (B) all members are replaced by the mean values, one easily sees that each of the vectors $$\overline{\mathfrak{d}}$$ and $$\overline{\mathfrak{H}}$$ consists of two parts, where one of them only depends on $$\bar{\rho}$$ and the other one only depends on $$\overline{\rho\mathfrak{v}}$$. Now, as the actions to the outside were determined by those vectors, then they are just so, as if the charge and the current were not connected with each other at all.

The same is true for the actions exerted on the conductor. Namely, if $$\mathfrak{d}$$ and $$\mathfrak{H}$$ are the variations caused by external causes in the aether, then by (Va) the force acting on a volume element is given by

The action, to which a noticeable part of the body is subjected, can thus be calculated in a manner, by which we put as unit volume

which again decomposes into two parts $$\bar{\rho}$$ and $$\overline{\rho\mathfrak{v}}$$.

Strictly taken, also a third charge would have to be taken into account. The current can not exist without a potential gradient, and this cannot exist without electric charges of the parts of the conductor. These charges, however, play in the considered questions no essential role, and could even more be left out, as we can think of them as vanishingly small if we assume a very high conductivity. .


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 Each of the four systems $$\overline{\mathfrak{S}}$$, $$A$$, $$B$$ and $$K$$ now forces a special state to the aether, and thus acts on any of the others. To shortly indicate these actions, we want to put $$\left(\overline{\mathfrak{S}}, K\right)$$ for those actions, which for example were exerted by $$\overline{\mathfrak{S}}$$ on $$K$$, where we have to notice that perhaps $$\left(\overline{\mathfrak{S}}, K\right)$$ and $$\left(K, \overline{\mathfrak{S}}\right)$$ are not equal and opposite, and that also actions such as $$\left(\overline{\mathfrak{S}}, \overline{\mathfrak{S}}\right)$$ may exist, namely forces which act on one of the ion systems due to condition changes in the aether, which were caused by itself.

In easily understandable symbols we can now write for the total action on K

which, however, due to § 25

is reduced to the first two terms and thus becomes independent of the current.

On the other hand, the forces which act on the conductor, can be represented by a expression consisting of 12 members,


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 since the action of $$K$$, $$\overline{\mathfrak{S}}$$, $$A$$ and $$B$$ on $$\overline{\mathfrak{S}}$$, $$A$$ and $$B$$, has to be considered each time. It is now

so that by the aforementioned expression it only remains

Those forces represented by the first two members would also exist, when $$\overline{\mathfrak{S}}=0$$, and the last two members are independent of the charged body K. An action of K exerted on the conductor as such, doesn't exist.

Besides, in each of the four members (30), the part that depends on $$\mathfrak{p}$$ is of second order. We already know this from $$(K, B) + (B, B)$$, since this represents an electrostatic effect. $$(A, \overline{\mathfrak{S}})$$ and $$(\overline{\mathfrak{S}}, \overline{\mathfrak{S}})$$, however, represent forces acting on a current, in which the mean electric density is zero. As it can be seen from (Va), such forces are determined by the value of $$\mathfrak{H}$$, which belongs to the acting system. Inasmuch as $$\mathfrak{H}$$ (that belongs to $$\overline{\mathfrak{S}}$$) depends on $$\mathfrak{p}$$, it is of second order (§ 25), and the compensation charge A only produces by its velocity $$\mathfrak{p}$$ a magnetic force of second order, since its density already contains the factor $$\mathfrak{p}/V$$.

Electrodynamic actions.
§ 27. The question as to how these effects are influenced by earth's motion, can now easily be answered. If we denote the currents in two conductors by $$\overline{\mathfrak{S}}$$ and $$\overline{\mathfrak{S}'}$$, and the corresponding compensation charges by A and $$A'$$, then the action exerted on the second conductor is

in which the last two terms are mutually canceled. That $$(A, \overline{\mathfrak{S}'})$$ and the $$\mathfrak{p}$$-dependent part $$(\overline{\mathfrak{S}}, \overline{\mathfrak{S}'})$$ are of order $$\mathfrak{p}^{2}/V^{2}$$, follows from considerations such as those communicated above.


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Induction in a linear conductor.
§ 28. A closed secondary wire from B will be displaced from $$B_1$$ into position $$B_2$$, while a primary conductor A at the same time passes from position $$A_1$$ to $$A_2$$, and the intensity of the primary current increases from $$i_1$$ to $$i_2$$. At the beginning and the end of time T, in which these processes take place, the two conductors shall be at rest and the primary current shall be constant; if no other electromotive forces acts on B, then this wire will eventually be, as before, without current. We want to determine the quantity of electricity, which has passed in time T through a cross section of the wire, and namely we will only consider the convection current at this place.

After the expiry of the whole process, the surface of B has nowhere a electric charge. It follows that the quantity of electricity that streamed through is the same for all cross sections, and that the conductor can be decomposed into infinitely thin current tubes, so that in each of them and equally through all cross-sections, the same quantity of electricity is streaming.

We consider in detail one of these tubes, and call ds an element of their length, ω is a vertical cross-section, Ndt the number of positive ions which pass through it during the time dt in the assumed positive direction s, N'dt the number of negative ions which move in the opposite direction, e is the charge of a positive and $$-e'$$ the charge of a negative ion. The total current through ω is then

Furthermore, $$\mathfrak{E}_{s}$$ and $$\mathfrak{E}'_{s}$$ are the electric forces acting in the direction of ds, which come into consideration for a positive or a negative ion. By 's law we shall assume, that the motion of ions by these forces is thus determined, so that N and $$N'$$ are proportional to its mean value; this and the proportionality to ω, we express by

where p and q are constant factors.


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 It is now necessary to distinguish between the velocity of the considered conductor element and the relative velocity of an ion in the wire. The former shall be called $$\mathfrak{v}$$ and the latter $$\mathfrak{w}$$. From (Va) it is given

$$\mathfrak{E}=4\pi V^{2}\mathfrak{d}+[\mathfrak{p.H}]+[\mathfrak{v.H}]+[\mathfrak{w.H}].$$

Yet, the velocity $$\mathfrak{w}$$ has the direction of ds; consequently we have $$[\mathfrak{w.H}]_{s}=0$$, and for positive as well as for negative ions

$$\mathfrak{E}_{s}=\mathfrak{E}'_{s}=4\pi V^{2}\mathfrak{d}_{s}+[\mathfrak{p.H}]_{s}+[\mathfrak{v.H}]_{s}.$$

Finally, equation (31) transforms into

$$i=c\omega\int\left\{ 4\pi V^{2}\bar{\mathfrak{d}_{s}}+[\mathfrak{p.\bar{H}}]_{s}+[\mathfrak{v.\bar{H}}]_{s}\right\}\ d\ t{,}$$

$$c = pe + qe'.$$

Let us divide by $$c\ \omega$$, multiply by ds, and integrate over the whole current-line. If we consider here, that i has everywhere the same value in the current-line, and if we put

$$\int\frac{d\ s}{c\ \omega}=\frac{1}{C}{,}$$

we shall find

§ 29. The following discussion is intended to derive the known fundamental law of induction from this formula. Imagine an area σ on which the current-line constantly is located during its motion, and consider the integral

for the part that is cut by the line.

This quantity, which is usually called "the number of magnetic force-lines covered by s", changes over time, namely for two reasons. First, $$\bar{\mathfrak{H}}$$ varies at each point, and second, the area of integration changes.

During time dt, the first cause produces the following increase of P

$$d\ t\int\dot{\overline{\mathfrak{H}_{n}}}\ d\ \sigma.$$


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Template talk:Lc

subst:
I would think that we should be encouraging people to subst: this template, or looking to make it do it internally. The issue for our text is not just the presentation of the text, it should also be looking reasonable when it is found through search engines and the like. Taking the time to get the case right should be part of the mission, not just the look. — billinghurst  sDrewth  11:19, 13 January 2011 (UTC)

title: Template talk:Lc


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 As to the second variation, it should be noted that each element ds describes an infinitely small parallelogram on the surface, and that the value of the surface integral $$\textstyle{\int\dot{\overline{\mathfrak{H}_{n}}}\ d\ \sigma}$$ of this parallelogram, by suitably chosen signs, goes into dP. This value is determined by the area of the parallelepiped, with $$d\ s$$, $$\mathfrak{H}$$ as its sides, and the distance $$\mathfrak{v}\ d\ t$$ in the direction of $$\mathfrak{v}$$. We will find for it

$$-d\ t[\mathfrak{v.\bar{H}}]_{s}\ d\ s{,}$$

and for the whole increase of (33)

$$d\ P=d\ t\int\dot{\overline{\mathfrak{H}_{n}}}\ d\ \sigma-d\ t\int[\mathfrak{v.\bar{H}}]_{s}\ d\ s{,}$$

or, if the relations (IVb) and (Vb), as well as the theorem stated in (1) (§ 4, h), were considered,

$$-d\ t\int\left\{ 4\pi V^{2}\bar{\mathfrak{d}_{s}}+[\mathfrak{p.\bar{H}}]_{s}\right\}\ d\ s-d\ t\int[\mathfrak{v.\bar{H}}]_{s}\ d\ s.$$

Consequently, (32) transforms into

$$i=-C\int d\ P=C\left(P_{1}+P_{2}\right){,}$$

where $$P_1$$ and $$P_2$$ belong to the beginning and the end of the considered time.

The magnitude P depends on the different parts of $$\mathfrak{H}$$. Since an induced current neither exists at the beginning nor at the end of time T, we commit no mistake when we substitute into (33) for $$\mathfrak{H}$$ only the magnetic force generated by the primary current. The prime above the letter can be omitted here, and if the induced wire is very thin, we may calculate for all current-lines with the same P. Finally, if $$C_1$$ is the sum of all numbers C (i.e., the conductivity of the induced electrical circuit), then the integral-current which we wished to calculate, becomes

$$I=C_{1}\left(P_{1}-P_{2}\right){,}$$

which is consistent with a known theorem.

The motion of Earth was never overlooked during the given derivation; consequently the formula admits of a conclusion about the influence of this motion on the phenomena of induction. There, only magnitudes of second order


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 come into account. $$\mathfrak{H}$$, which should serve to determine the magnitude P, is indeed composed of the vector specified by (26) and the magnetic force which is generated by the compensation charge. The latter magnetic force is of order $$\mathfrak{p}^{2}/V^{2}$$, and since in equations (§ 25) that serve to determine $$\chi_{x}$$, $$\chi_{y}$$, $$\chi_{z}$$, also just the square of $$\mathfrak{p}$$ is included, then the values (26) differ only to second order from the expressions that apply to a stationary earth.

By proving, that no first-order influence may be expected from the phenomena of induction, we have achieved the explanation for the negative result of.


text: Author talk:Chandler Rathfon Post

title: Author talk:Chandler Rathfon Post


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General formulas.
§ 30. Once the motion of the ions is given, known functions of x, y, z and t appear on the right-hand side of equations (A) and (B) (§ 21); with respect to the last variable, these are periodic functions if the ions carry out oscillations with constant amplitude and a common oscillation interval T. It is easy to see, that in this case the equations are satisfied by values of $$\mathfrak{d}_{x}$$, $$\mathfrak{d}_{y}$$, $$\mathfrak{d}_{z}$$, $$\mathfrak{H}_{x}$$, $$\mathfrak{H}_{y}$$, $$\mathfrak{H}_{z}$$, which also have the period T. Therefore, the important and almost self-evident theorem is given:

If ion oscillations of period T take place in a light source, then $$\mathfrak{d}$$ and $$\mathfrak{H}$$ indicate the same periodicity at each point that shares the translation of the source.

The resolution of the equations leads to quite complicated expressions. For simplicity, it is advisable to calculate the components of the vector $$\mathfrak{H}'$$ (§ 20) at first.

According to (VIb)

Accordingly, we want to multiply the second and third of equations (A) by $$4\pi\mathfrak{p}_{z}$$ and $$-4\pi\mathfrak{p}_{y}$$ respectively, and then add them to the first of equations (B). We obtain in this way, under consideration of the importance of $$\left(\tfrac{\partial}{\partial t}\right)_{1}$$ (§ 19),


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 $$V^{2}\Delta\mathfrak{H'}_{x}-\left(\frac{\partial^{2}\mathfrak{H'}_{x}}{\partial t^{2}}\right)_{1}=4\pi V^{2}\left\{ \frac{\partial(\rho\mathfrak{v}_{y})}{\partial z}-\frac{\partial(\rho\mathfrak{v}_{z})}{\partial y}\right\} +$$

$$+4\pi\mathfrak{p}_{z}\left\{ \frac{\partial(\rho\mathfrak{v}_{y})}{\partial t}-\mathfrak{p}_{x}\frac{\partial(\rho\mathfrak{v}_{y})}{\partial x}-\mathfrak{p}_{y}\frac{\partial(\rho\mathfrak{v}_{y})}{\partial y}-\mathfrak{p}_{z}\frac{\partial(\rho\mathfrak{v}_{y})}{\partial z}\right\} -$$

$$-4\pi\mathfrak{p}_{y}\left\{ \frac{\partial(\rho\mathfrak{v}_{z})}{\partial t}-\mathfrak{p}_{x}\frac{\partial(\rho\mathfrak{v}_{z})}{\partial x}-\mathfrak{p}_{y}\frac{\partial(\rho\mathfrak{v}_{z})}{\partial y}-\mathfrak{p}_{z}\frac{\partial(\rho\mathfrak{v}_{z})}{\partial z}\right\}.$$

§ 31. In the following calculation, magnitudes of order $$\mathfrak{p}^{2}/V^{2}$$ should be neglected. First, we neglect on the right-hand side the terms with two factors $$\mathfrak{p}_{x}$$, $$\mathfrak{p}_{y}$$ or $$\mathfrak{p}_{z}$$, since we find a similar term in $$V^2$$; and we therefore retain only

Second, we write for the operation that has to be applied to $$\mathfrak{H'}_{x}$$,

$$V^{2}\Delta-\left(\frac{\partial}{\partial t}-\mathfrak{p}_{x}\frac{\partial}{\partial x}-\mathfrak{p}_{y}\frac{\partial}{\partial y}-\mathfrak{p}_{z}\frac{\partial}{\partial z}\right)^{2}=\left(V^{2}\frac{\partial^{2}}{\partial x^{2}}+2\mathfrak{p}_{x}\frac{\partial^{2}}{\partial x\ \partial t}\right)+$$

$$+\left(V^{2}\frac{\partial^{2}}{\partial y^{2}}+2\mathfrak{p}_{y}\frac{\partial^{2}}{\partial y\ \partial t}\right)+\left(V^{2}\frac{\partial^{2}}{\partial z^{2}}+2\mathfrak{p}_{x}\frac{\partial^{2}}{\partial z\ \partial t}\right)-\frac{\partial^{2}}{\partial t^{2}}=$$

$$=V^{2}\left(\frac{\partial}{\partial x}+\frac{\mathfrak{p}_{x}}{V^{2}}\frac{\partial}{\partial t}\right)^{2}+V^{2}\left(\frac{\partial}{\partial y}+\frac{\mathfrak{p}_{y}}{V^{2}}\frac{\partial}{\partial t}\right)^{2}+$$

$$+V^{2}\left(\frac{\partial}{\partial z}+\frac{\mathfrak{p}_{z}}{V^{2}}\frac{\partial}{\partial t}\right)^{2}-\frac{\partial^{2}}{\partial t^{2}}.$$

The form of this expression suggests the introduction of a new independent variable instead of t

and to consider $$\mathfrak{H'}_{x}$$, as well as $$\rho\mathfrak{v}_{y}$$ and $$\rho\mathfrak{v}_{z}$$, as functions of x, y, z and $$t'$$. We denote the derivative that corresponds to this view by

and give to the sign $$\Delta'$$ the meaning


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 It is now

and

$$\frac{\partial}{\partial t}=\frac{\partial}{\partial t'}{,}$$

so that we find for the determination of $$\mathfrak{H}'$$

$$V^{2}\Delta'\mathfrak{H'}_{x}-\frac{\partial^{2}\mathfrak{H'}_{x}}{\partial t'^{2}}=4\pi V^{2}\left[\left\{ \frac{\partial(\rho\mathfrak{v}_{y})}{\partial z}\right\}'-\left\{ \frac{\partial(\rho\mathfrak{v}_{z})}{\partial y}\right\}'\right]\text{, etc.}$$

A solution of these equations is easy to give. Namely, imagine three functions $$\psi_{x}$$, $$\psi_{y}$$, $$\psi_{z}$$ that satisfy the conditions

and put

Once $$\mathfrak{H}'$$ is found by that, equation (IIIb) provides us with the value of $$\dot{\mathfrak{d}}$$ and thus also, as far as we don't use additive constants, the value of $$\mathfrak{d}$$. From (VIb) it also follows $$\mathfrak{H}$$; from (Vb) and (VIIb) it follows $$\mathfrak{F}$$ and $$\mathfrak{E}$$. That in this way really all the equations are satisfied, can be proven, but should not be discussed here for brevity.

In contrast, in the next section the value of $$\psi_{x}$$ shall be given, and in § 33 the solution for a special case shall be further developed.

It should also be remarked before, that the variable $$t'$$ can be regarded as a time, counting from an instant that depends on the location of the point. We can therefore call this variable the local time of this point, in contrast to the general time t. The transition from one time to another is provided by equation (34).

§ 32. The product $$\rho\mathfrak{v}_{x}$$ in the first of equations (36), as noted already, is a known function of x, y, z and $$t'$$. We accordingly set


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and thus have

a solution of (36). By that we have to imagine two points; first, the fixed point (x, y, z), for which we want to calculate $$\psi_{x}$$ and which we call P; second, a moving point Q, which has to traverse the whole space, where $$\rho\mathfrak{v}_{x}$$ is different from zero. r represents the distance QP, and $$t'$$ the local time of P at the instant for which we wish to calculate $$\psi_{x}$$; furthermore we have to understand by ξ, η, ζ, the coordinates of Q, and by dτ an element of the just mentioned space. The function $$\textstyle{f\left(\xi,\eta,\zeta,t'-\frac{r}{V}\right)}$$ is the value of $$\rho\mathfrak{v}_{x}$$ in this element, namely, if the local time that is valid at this place, is $$\textstyle{t'-\frac{r}{V}}$$.

A single luminous molecule.
§ 33. To excite electric oscillations, a single molecule with oscillating ions shall serve; let $$Q_0$$ be an arbitrary fixed point in it — for brevity, we say, "the molecule is present in $$Q_0$$" —, and for P a place is chosen, whose distance from $$Q_0$$ is much larger than the dimensions of the molecules. For distinction, $$Q_{0}P=r_{0}$$.

We now want to replace the various distances r, that are present in formula (38), by $$r_0$$ and also neglect the differences of local times at the various points of the molecule. In this way,


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 where all occurring $$\rho\mathfrak{v}_{x}$$ are related to the same instant, namely to the instant when

is the local time of $$Q_0$$.

Since $$\mathfrak{v}_{x}$$ is equal for all points of an ion, then, if we write e for the charge of such a particle, the last integral transforms into

The sum is extending over all ions of the molecule.

Furthermore, if $$\mathfrak{q}$$ is now the displacement of an ion from its equilibrium position, then

and

This has a simple meaning. We can conveniently call the vector $$\Sigma e\mathfrak{q}$$ the electric moment of the molecule and denote it by $$\mathfrak{m}$$. Then it is

$$\Sigma e\mathfrak{q}_{x}=\mathfrak{m}_{x}{,}$$

$$\psi_{x}=-\frac{1}{r_{0}}\frac{d\mathfrak{m}_{x}}{dt}=-\frac{\partial}{\partial t}\left(\frac{\mathfrak{m}_{x}}{r_{0}}\right);$$

after the things said here, we have to take the value of the derivative for the instant when the local time in $$Q_0$$ is $$t'=\tfrac{r_{0}}{V}$$. Obviously we can also write

where $$\mathfrak{m}_{x}$$ means the first component of the electric moment in that very instant. After (by that and by two equations of the same from) we have found $$\psi_{x}$$, $$\psi_{y}$$, $$\psi_{z}$$ for the point (x, y, z) and the local time $$t'$$ at this place, the study of the propagating oscillations is very simple. The equations (37) give


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 and because we seek the value of $$\mathfrak{d}$$ outside the molecule, (IIIb) is transformed into

or, due to (35), it is transformed into

If we bring the last two terms on the left side, then we just obtain $$\textstyle{\frac{1}{V^{2}}\dot{\mathfrak{F}}_{x}}$$ or $$\textstyle{\frac{1}{V^{2}}\frac{\partial\mathfrak{F}_{x}}{\partial t'}}$$, as it can be seen by (Vb); since $$\mathfrak{H}$$ and $$\mathfrak{H}'$$ only differ by magnitudes of order $$\mathfrak{p}$$, we may replace the vector product (Vb) by $$[\mathfrak{p.H'}]$$.

From

we obtain $$\mathfrak{F}$$ by integration; constants were omitted by us, since we are only dealing with vibrations.

We substitute the values (39) and put

It is then

and namely, $$\mathfrak{m}_{x}$$, $$\mathfrak{m}_{y}$$, $$\mathfrak{m}_{z}$$ are still related to the instant given above.

As to how the other magnitudes occurring in (Ib)-(VIIb) can be determined, can immediately be seen.

§ 34. Just some words on the error committed in the above calculation. That in (38) the factor $$\tfrac{1}{r}$$ was replaced by $$\tfrac{1}{r_{0}}$$, needs surely no justification. But we also haven't taken the values of $$\rho\mathfrak{v}$$ for the function f at the the correct times. Once we have replaced $$t'-\tfrac{r}{V}$$ by $$t'-\tfrac{r_{0}}{V}$$ in (38), then in the time when l is one of the


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 dimensions of the molecule, we have committed an error of order $$\textstyle{\frac{l}{V}}$$, secondly, the inequality of the local times at the various locations of the molecule were not considered, and in this lies an error of order $$\textstyle{\frac{l\mathfrak{p}}{V^{2}}}$$ by (34). But even then, if we want to keep magnitudes of the order $$\textstyle{\frac{\mathfrak{p}}{V}}$$, we don't need to care about this second error, when already the first may be neglected. Now this is indeed the case when the dimensions of the molecule are much smaller than the wavelength of TV. Then also l/V is considerably smaller than T, and the state in the molecule will not noticeably change in the time l/V.

§ 35. The formulas for the propagation of oscillations is obtained, if goniometric functions of time are substituted into the equations (39) and (40) for $$\mathfrak{m}_{x}$$, $$\mathfrak{m}_{y}$$ ,$$\mathfrak{m}_{z}$$. If, for example,

$$\mathfrak{m}_{y}=0,\ \mathfrak{m}_{z}=0{,}$$

and, as a function of local time which is valid for the location of the molecule,

$$\mathfrak{m}_{x}=a\cos 2\pi\frac{t'}{T}\text{, (a constant),}$$

thus at an external point in the distance r and for the local time $$t'$$ that belongs to it

$$\mathfrak{H'}_{x}=0,\ \mathfrak{H'}_{y}=\frac{\partial}{\partial t'}\left(\frac{\partial\chi}{\partial z}\right)',\ \mathfrak{H'}_{z}=-\frac{\partial}{\partial t'}\left(\frac{\partial\chi}{\partial y}\right)'{,}$$

$$\mathfrak{F}_{z}=-V^{2}\left(\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}\right)'\chi,\ \mathfrak{F}_{y}=V^{2}\left(\frac{\partial^{2}\chi}{\partial x\ \partial y}\right)',\ \mathfrak{F}=V^{2}\left(\frac{\partial^{2}\chi}{\partial x\ \partial z}\right)'.$$

$$\chi=\frac{a}{r}\cos\frac{2\pi}{T}\left(t'-\frac{r}{V}\right).$$

If we eventually want to consider a stationary light source once, so we simply have to omit all accents. The formulas then are in accordance with the expressions, by which represented the oscillations in the vicinity of his vibrator.


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The direction of the wave normal.
§ 36. Now we shall examine the oscillations in such distances from the luminous molecules, which are considerably larger than the wavelength. It should be noted that in (39) and (40), $$\mathfrak{m}_{x}$$, $$\mathfrak{m}_{y}$$, $$\mathfrak{m}_{z}$$ are goniometric functions of

we namely want to write from now on r instead of $$r_0$$. The assumption made about the length of this line justifies to consider only the variability of the argument of any goniometric function for all differentiations with respect to x, y, z, but to consider as constant all factors such as $$\tfrac{1}{r}$$, or $$\cos(r, x)$$, by which these functions are multiplied.

For any of the magnitudes $$\mathfrak{H'}_{x}$$, $$\mathfrak{H'}_{y}$$, $$\mathfrak{H'}_{z}$$, $$\mathfrak{F}_{x}$$, $$\mathfrak{F}_{y}$$, $$\mathfrak{F}_{z}$$ - we will call them φ — it can therefore be found an expression of the form

where A and B are indeed dependent on the length and the direction of line $$Q_0 P$$ — $$Q_0$$ is the location of the molecule, and P is the considered external point —, but, if r were just big enough, it may be regarded as constant in a space that comprises many wavelengths. While x, y, z are the coordinates of P, we denote by ξ, η, ζ the coordinates of $$Q_0$$, and by $$b_{x}$$, $$b_{y}$$, $$b_{z}$$ the direction constants of the connection-line $$Q_0 P$$. If we now replace in the formula (41) r by

and $$t'$$ by the value (34), we obtain


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 In an area that isn't too extended, we may also regard $$b_{x}$$, $$b_{y}$$, $$b_{z}$$ as constant, and thus regard the motion as a system of plane waves. The direction constants $$b'_{x}$$, $$b'_{y}$$, $$b'_{z}$$ of the wave normal are obviously to be determined from the condition

For $$\mathfrak{p}=0$$, $$b'_{x}$$, $$b'_{y}$$, $$b'_{z}$$ fall into $$b_{x}$$, $$b_{y}$$, $$b_{z}$$, and the waves are perpendicular to $$Q_0 P$$. This is not the case if the light source is moving. From (43) follows, that the waves are perpendicular to the line that connects P with that point at which the light source was at the moment, when the light was sent that reaches P at time t.

The law of Doppler.
§ 37. In a point that moves together with the luminous molecule — and thus also for an observer who shares the translation — the values of $$\mathfrak{d}_{x},...\mathfrak{H}_{x},...$$ are changing, as we have seen (§ 30), as often in unit time as it corresponds to the actual period of oscillation T of the ions.

We can also examine, with which frequency these values in a stationary point are changing their sign. This frequency causes the oscillation period for a stationary observer. The question can be solved immediately, if instead of x, y, z we introduce new coordinates $$\mathbf{x}$$, $$\mathbf{y}$$, $$\mathbf{z}$$, which refer to a stationary system of axes. If the two systems have the same directions of axes and the same origin at time t = 0, then

and by (42) for $$\mathfrak{d}_{x},...\mathfrak{H}_{x}, ...$$ we obtain expressions of the form

where


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 is the component of $$\mathfrak{p}$$ with respect to the connection line $$Q_0 P$$.

The "observed" period of oscillation is thus

what is in agreement with the known law of The derivation given here can easily be generalized so that it can be applied to all similar cases, for example also to sounding bodies. An arbitrary body A move with constant velocity $$\mathfrak{p}$$ in a medium that either remains at rest, or comes into a stationary state of motion. In this latter case (which also encloses the former one) we find at any point P, which translates with the body A, always the same state of motion, and it can be said, that the whole figure representing the distribution of velocities in the vicinity of A, shares the translation $$\mathfrak{p}$$.

Furthermore, imagine now that the parts of the body perform simple oscillations of period T and of constant amplitude. It seems clear without further ado, when a sufficiently long time has elapsed since the beginning of this motion, that in the just-mentioned point P, the deviation from equilibrium or rather from the stationary state of flow, must necessarily have the period T. If we now introduce the co-ordinates x, y, z with respect to a system of axes progressing with the body (relative coordinates), and if we restrict ourselves to a space, that is so far from A and so small that we can speak of plane waves in it, then the above deviation can be represented by expressions of the form

Here, $$a_{x}$$, $$a_{y}$$, $$a_{z}$$ are the direction constants of the wave normal, while V is the velocity of propagation.

If we now want to know, by which frequency φ (in a stationary point) its sign is varying, then we have to introduce coordinates $$\mathbf{x}$$, $$\mathbf{y}$$, $$\mathbf{z}$$ with respect to stationary axes. By using the relations (44), (45) transforms into

where

are the components of $$\mathfrak{p}$$ with respect to the wave normal

For the observed oscillation period we now obtain

What we have already stated without proof, namely that the period T exists throughout in the medium, is nothing else than what, in his attacks against 's theory, called the law of the immutability of the oscillation period (Wiener Sitz.-Ber., vol 8, p. 134, 1852). He only forgot to notice, that this law only would apply, if we consider the phenomena as a function of t and the relative coordinates.

The proof of the theorem is, by the way, easy to give, when the oscillations are infinitely small, and when we have to do with homogeneous linear differential equations.

As regards the acoustic phenomena, the problem was discussed in detail by (Het beginsel van  in de geluidsleer, Leiden, Engels, 1881). .


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 If the law, as it is usually applied, should be given, it must of course still be assumed, that the translation does not change the actual period of oscillation of the luminous particles. I must abstain from giving an account of this hypothesis, since we know nothing about nature of the molecular forces that determine the oscillation period.

§ 38. The case that the light source is at rest and the observer progresses, allows of a similar treatment. If namely, as above, $$\mathbf{x}$$, $$\mathbf{y}$$, $$\mathbf{z}$$ are the coordinates based on stationary axes, then in a distant point P, any of the magnitudes $$\mathfrak{d}_{x},...\mathfrak{H}_{x},...$$ shall now be represented by

We most conveniently describe the perception of motion by means of a co-ordinate system, which shares the translation $$\mathfrak{p}$$ of the observer. Here, again the relations (44) are applicable, and (46) transforms into

from which it is given for the "observed" period of oscillation


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Equations for the aether enclosed in ponderable bodies.
§ 39. Let us now turn to the motion of light in ponderable, dielectric, and completely transparent bodies. It shall be assumed, that they are moving with velocity $$\mathfrak{p}$$ in an arbitrary direction, and that, as already said, the molecules contain ions that are connected with certain equilibrium position.

For one of these particles we again denote the charge by e, and the displacement from the equilibrium position by $$\mathfrak{q}$$. The components $$\mathfrak{q}_{x}$$,$$\mathfrak{q}_{y}$$,$$\mathfrak{q}_{z}$$, as well as the velocities $$\dot{\mathfrak{q}}_{x}$$,$$\dot{\mathfrak{q}}_{y}$$,$$\dot{\mathfrak{q}}_{z}$$ we consider as infinitely small; i.e. besides magnitudes that only contain one of these components as factor, we neglect terms in which two such factors occur.

Any of the considered bodies shall be homogeneous. However, for that the cases of reflexion and refraction are not excluded, we imagine two different bodies, they may (Fig. 1) either sharply mutually separated at a surface $$\Sigma$$, or steadily go into one another at a thin limiting layer, such as


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 between the surfaces $$\Sigma_1$$, and $$\Sigma_2$$, (Fig. 2). If we speak in the latter case about a "limiting surface", then we shall mean by that, for example, a surface $$\Sigma$$ halfway between $$\Sigma_1$$ and $$\Sigma_2$$.

We will always calculate by averages, and namely not only by those defined in §4 l, but sometimes also by others, that come into consideration when the relevant magnitude only exist in a single point Q, for example in one point of the various molecules, or if we have reason to consider only the values of a function in such points. Such a average of second kind we distinguish from the averages of first kind by a double horizontal prime, and besides we follow a similar calculation rule as during the last calculation. Namely we understand under the value of $$\overline{\overline{\phi}}$$ in a point P the arithmetic average of the values of $$\varphi$$ in the points Q, so far as they are present within the sphere I around P (as mentioned in §4, l).

By the assumption made about the radius R (§4), all "rapid" variations are vanished from the averages; however, concerning the velocity of the remaining variations, we have to distinguish between the interior of the body and the border. If we are positioning in Figures 1 and 2 the surfaces $$\sigma_1$$ and $$\sigma_2$$ in such a way, so that in the first figure they are both distant from $$\Sigma$$ by R, while in the second this distance exists, first, between $$\Sigma_1$$ and $$\sigma_1$$, and second, between $$\Sigma_2$$ and $$\sigma_2$$, then for the calculation of $$\overline{\overline{\phi}}$$ or $$\overline{\phi}$$ in the points, that are outside of the layer ($$\sigma_1$$, $$\sigma_2$$), only the values of $$\varphi$$ come into play. While the averages, although in a completely steady way, can be considerably different from $$\sigma_1$$ to $$\sigma_2$$, we want to assume, that the variations from point to point are much slower in the interior of the body. This will be indeed satisfied in the problems to be considered, when only the wavelength $$\lambda$$ is many times greater than the distance a of $$\sigma_1$$ and $$\sigma_2$$

We even want to assume, that between $$\lambda$$ and a,


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 we can also introduce the distance l, so that $$\lambda/l$$ and $$l/a$$ become very great. The purpose of this assumption will become clear soon.

If the limiting surface $$\Sigma$$ is curved, then the radii of curvature shall be greater than $$\lambda$$, or at least of the same order.

§ 40. We already have spoken about the electric moment of a molecule in § 33. We want also now to retain the definition given there, and in similar manner call the vector

where the sum is extended over all ions in the interior of sphere I, the moment of unit volume. More precisely we say, that (47) may indicate the value of this moment in the center of the sphere. If we choose for this new vector the sign $$\mathfrak{M}$$, then

With this $$\mathfrak{M}$$, another magnitude is most closely connected. During the displacement of the ions from the equilibrium positions, a fixed surface will namely be interspersed, which we may call a "convection current through the surface". If $$d\sigma$$ is an element of surface, with P as its center and n as its perpendicular, then the charge $$\epsilon$$ that passed through it into the side designated by n, will depend on the location of P, if we specify the magnitude $$d\sigma$$ and the direction of n once and for all. Let $$d\sigma$$ be very small in relation to the molecular distances, but as great, so that we don't have to consider the cases in which an ion just is in contact with the borderline. Obviously some locations of P will exist, where the element won't intercept any ion at all, and others where it will intersect the path $$\mathfrak{q}$$ of an ion. In the first case $$\epsilon = 0$$, in the latter it is equal to the positive and negative calculated charge of the ion.

Since $$\epsilon$$ depends on the location of P, we can form the average $$\overline{\epsilon}$$ in the ordinary way; it is now, as it shall be shown in the next §,

$$\mathfrak{M}_{n}d\sigma$$.


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 § 41. The rule contained in the formula

$$\overline{\epsilon}=\frac{1}{l}\int\epsilon d\mathfrak{r}$$

can be express somewhat differently. Namely we shall choose for the point P an infinite amount, we want to say k, locations that are uniformly distributed over the sphere I, and take the arithmetic mean of the values of $$\epsilon$$ that are valid for these locations, i.e. we put

Any ion, that has its equilibrium position in the interior of I, will (during its displacement) now pass through some positions that are connected with the element $$d\sigma$$ and thus add some terms to the sum $$\Sigma\epsilon$$. We obtain the whole sum, if we at first add to one another the terms that stem from a certain ion, and than sum over all ions.

Let Q be the equilibrium position of the considered ions, and Q the new position; so $$QQ'=\mathfrak{q}$$. The length and the direction of this line are given, as well as the direction and magnitude of $$d\sigma$$. Whether the particle hits the surface element and provides the part e for the sought sum, only depends on the relative positions of P and Q. Thus we can, instead of giving k positions in the sphere l to P, also let remain the point at its position and lead point Q around a sphere I. As $$QQ'$$ now hits the fixed element $$d\sigma$$, when Q lies in a certain, easily specifiable cylinder of area $$\mathfrak{q}_{n}d\sigma$$, then the number of "relevant" positions is related to the integer k, as the area of that cylinder is related to the area of sphere I. This number is thus

$$\frac{k}{I}\mathfrak{q}_{n}d\sigma$$,

and the sum $$\Sigma\epsilon$$, as far it is caused by the ion Q,

$$\frac{k}{I}e\mathfrak{q}_{n}d\sigma$$.


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 Eventually in formula (49) we obtain

where the sum is extended over all ions of sphere I, and

or by (48)

§ 42. Equations ($$I_b$$) — ($$VII_b$$) (§ 20) may form the initial point for the subsequent considerations. At first we notice, that the first of them is equivalent to

for an arbitrary closed surface (n is to be drawn into the outside), when E is the electric charge enclosed by it. If now in one element $$d\mathfrak{r}$$ of the inner space in equilibrium state, a density $$\rho_0$$ exists, and if (for an element of the surface) $$\epsilon$$ has the meaning given above, then

where the sum is related to all elements $$d\sigma$$.

By that

From the definition of the average we easily find now

Since now

and

it is eventually given


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 We now want to define a new vector $$\mathfrak{D}$$ by the equation

$$\mathfrak{D}=\overline{\mathfrak{d}}+\mathfrak{M}$$,

and call it the dielectric polarization.

This vector, that goes over for the free aether (where $$\mathfrak{M}=0$$) into $$\mathfrak{d}$$, is exactly that, what Maxwell calls "dielectric displacement". Its basic property is according to the above, that for any closed surface

and also in the interior of any body

§ 43. Formula (50) leads to an important limiting-condition, if we apply it to a surface, that lies partly in the first, and partly in the second body. Around a certain point P of the limiting-surface $$\Sigma$$(Fig. 1 and 2) we shall lay a cylinder-surface C that is parallel to the perpendicular in P, and choose for the mentioned area the surface of the space that is cut from layer ($$\sigma_1$$, $$\sigma_2$$). If now the dimensions of the parts limited in $$\sigma_1$$ and $$\sigma_2$$ are of order l (§ 39), then we may consider the parts as elements that are equal, parallel and plane, and as they are much greater than the part of C that lies between $$\sigma_1$$ and $$\sigma_2$$, we can omit the integral taken over the latter

$$\int\mathfrak{D}_{n}d\sigma$$

Thus we find, if we mutually distinguish the values that are valid in $$\sigma_1$$ and $$\sigma_2$$ by the indices 1 and 2, and draw either at $$\sigma_1$$ as well as at $$\sigma_2$$ the perpendicular n from the first to the second body,

In relation to this, we have to notice one thing. In any medium, $$\mathfrak{D}_{x},\mathfrak{D}_{y},\mathfrak{D}_{z}$$ can be represented as slowly (§ 39) varying functions of coordinates, and we would have to substitute in these functions the coordinates of a point of $$\sigma_1$$ or $$\sigma_2$$, to obtain $$\mathfrak{D}_{n(1)}$$ and $$\mathfrak{D}_{n(2)}$$. Instead of this we can without noticeable error — due to the small distance of


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 the surfaces — introduce the coordinates of the point P that lies in $$\Sigma$$. Thus it is allowed to say, that $$\mathfrak{D}_{n(1)}$$ and $$\mathfrak{D}_{n(2)}$$ are the values at the limiting-surface and that the previous formula expresses the continuity of $$\mathfrak{D}_{n}$$.

Similar formulas as equations (Ic) and (51) are emerging from ($$II_b$$); namely for the interior of a body

and for the limiting-surface

§ 44. From fundamental equation ($$III_b$$) we derive

or, be means of the definition

This derivation is true for the interior of a body. To arrive at the limiting condition, we note at first, that (§ 4, h) (by the equation ($$III_b$$) for an arbitrary surface $$\sigma$$, with the borderline s)

and thus also

Now we lay through the point p (Fig. 1 and 2) a plane, that contains the perpendicular of the borderline and the arbitrary direction h tangential to $$\Sigma$$, and choose as surface $$\sigma$$ the part of this plane, that lies between $$\sigma_1$$ and $$\sigma_2$$ and which is limited by two lines parallel to that perpendicular. If the length of this layer in the direction h is of order l (§ 39), then we may neglect all magnitudes of order a and we obtain from (52)

where the indices 1 and 2 have the same meaning as above. For the two components of $$\overline{\mathfrak{H}^{'}}$$ we may take at this place the values in point P again, and thus the equations says, that the tangential components of vector $$\overline{\mathfrak{H}^{'}}$$ were steady.