text:
A Christmas Eve in Spain

I.
The last gleam of the sun is dying away, and the air is terribly sharp. It seems as if the barren fields were shivering with cold. The neighboring chimneys send out puffs of smoke, which are swiftly whirled away by the icy north wind that leaves some snow-flakes in exchange. The startled black-bird chirps complainingly from her perch on the wall, or calls anxiously to her mate from the top of an apple-tree. From time to time is heard the melancholy and monotonous cry of the laborer calling his cattle; occasionally the scurrying of wooden shoes over the stones in the alley is perceived, and you need listen no longer, for no other sound indicates that there is any life in that pale and somber landscape.

In the broad porch of one of the houses that adorn it are two little chaps stretched out upon a heap of dried rushes. They are lying on their stomachs, their faces rest upon their hands, and they are looking into each other's eyes. As they have passed the afternoon frolicking over the yielding material upon which they are now resting, to overcome the cold which they scarcely feel, poorly clad as they are, they need only to blow upon their fingers from time to time.

One of the boys belongs in the house; the other, next door. It is the first one who suddenly exclaims:

"I'm going to have fritters to eat, aha!"

"So am I, too," replies the other.

"But I'm going to have honey with mine."

"I'll have something better than that—sugar!"

"Well, in our house we're going to have a meat stew and some wheat bread to go with it."

"My father brought yesterday two fishes—tremendous big ones!"

"My mother is down in the town now to get some lard, some wheat bread, and some cake, and this noon my father brought two jars of white wine, the very best kind. And all the eggs we've had this week are kept for to-day, more than fifteen, as big as that! I tell you what, we're going to spend to-night all of a dollar and thirty-five cents which are saved up!"

"What does all that amount to! Why, my father brought up from the harbor four dollars, and forty cents besides, and we're going to lay out the whole of it to-night. I say, will you keep a taste of the stew for me, if I'll give you a piece of the fish?"

"You needn't think I care anything about that! But you have not got a brother who is a student and who is coming home on a vacation this very afternoon, the way I have."

"But I have got a splendid young bull, and a cow that gives three quarts of milk. Oh, what a lot of it we've got for to-night!"

"Milk, did you say! Do you want to see two jars of milk right away? I'll show you some worth looking at."

Up springs the eager boy, with the other one after him, and together they go into the kitchen, looking cautiously about lest Uncle Jeromo, the master of the establishment, should be anywhere about.

The evening is already coming on, and the boy of the house catches up a brand from the hearth. By dint of much blowing upon it he produces a flickering flame, and by its light they go up to a smoke-begrimed chest that stands underneath a shelf, also blackened by smoke. They raise the cover, and there on the bottom, between heaps of flour, bran, and half of a ham, are seen two large jars filled with milk. The boy of the house looks at his friend with an air of triumph. Both of them fasten their eager eyes upon the jars. Both of them stretch out their right hand, and both moisten their forefinger in the milk, each in his own jar. Continuing to move as if by a common impulse, they draw their arms out of the chest, and stand face to face, sucking their fingers.

"That's good milk!" says the boy of the house.

"It's better cream, though," replies his comrade.

"Did you get the cream?"

"What do you suppose! I got up every bit of it on my finger."

At that moment the former recalls with alarm the fearful disturbance his father makes whenever the cream is missing from the daily supply of milk, and remembers the witness to such troubles left on his own ribs by his father's hand. For that reason, fearing a fresh drubbing, and in order to make his own innocence perfectly clear, he flings the brand into the fire, grasps his friend by the trousers, and begins to call out in a tone of the greatest anguish:

"Father! Father!"

But the sybaritic prisoner, seized with mortal terror, gives a terrific twist to his jailer's hands, and scuttles out through the yard, licking his chops.

Uncle Jeromo, who is tiring a wheel out behind the house, hurries in at the sound of the cries, and, not believing his boy's story about the theft of the cream, thinks that the fellow himself has skimmed it, and gives him a couple of resounding blows. The boy shrieks, and the father threatens him with more of the same. In the midst of the cries and warnings the voice of Aunt Simona is heard in the doorway.

"Oh, was there ever such folks as you! This is the way I always find you."

"Oh dear, mother, mother!" exclaims the boy, running to cling to the good woman's skirts.

"What are you crying for, my boy? Who has been beating you?"

"Mo-mo-mother! It was fa-fa-father!"

"And you'll get some more of it," murmured the latter, going out to the stable to look after the cattle. "I will teach you to guzzle the cream!"

"I didn't take it, I tell you! It was Tonu, Tonu Zancuda—oh, dear!"

"Very likely it was, little angel! He is the sneakingest creature! Come, here's some chestnuts. Now don't cry any more. Your father is too quick. Has the student come?"

"No, ma'am."

"God grant that a wolf may not eat him up on the way! But where is your sister?"

"She went to the fountain."

"The gadabout! I'll go out and settle her accounts. No, though, I won't, it's no fun going out into the cold at this time of day. Oh that boy of Lambiona's, what a thing he is! Worse luck to him!"

Muttering words like these, Aunt Simona leaves her sabots at the door of the closet, takes off the baize skirt which she had thrown over her shoulders, hangs it upon a plow-handle which depends from one of the attic floor-beams, and goes into the kitchen, closely followed by the little fellow. The basket which she had brought in under the folds of the skirt she sets down by the fire, throwing on some dried thistles to add to the flame. Then she lights a candle, and takes from the basket a little lard, a jar of honey, and two cents' worth of cinnamon. All these she places upon the shelf within easy reach of her hand, and sets about preparing the Christmas supper. In that operation her daughter soon comes in to aid her, bringing home her two jars of water, and protesting that she has not stopped to talk with a living being, she vows she had not, and no one ever found a lie in her mouth!

Uncle Jeromo comes in a little later, and takes his seat near the fire, so as to help the family in the important affair; for the country-people of this part of Spain, of simple habits both by necessity and by custom, take as much delight in witnessing the preparation of the Christmas supper as in eating it. The sputtering of the lard in the frying-pan, the cutting out of the fritters, the breaking and beating of the eggs, and the soaking of the bread in them, the sprinkling of the sugar over the fritters as they come piping hot out of the frying-pan, the pouring of milk or honey over them—all these processes, and the thought, furthermore, that all this fine food, together with the jar of wine saved up sacredly for the occasion, is to be luxuriously enjoyed by the poor family of laboring people looking on, arouse in them such delightful feelings that—well, you would only need to look at the faces of Uncle Jeromo's family! The affair of the cream is entirely forgotten now. What demonstrative faces! What evident happiness! Aunt Simona, grasping the handle of the frying-pan in one hand and a wooden spoon in the other, and crouching down upon the bare floor, feels more important than the emperor of China, and regards her task as more difficult and critical than that of an ambassador of peace between two powerful nations engaged in desperate fighting. A great pity it was that the student had not arrived, in order fittingly to celebrate Christmas Eve. For there are many peculiar features of that festival in the country. After the pleasure of preparing and eating the supper, comes that of the arrival of the marzantes, for whom the little fellow of the beating has already inquired many times. If the truth must be told, he is much more concerned about them than about the delay of his brother in getting home. This is because the unlucky boy has never heard them, either at Christmas Eve or at New Year or at Epiphany, as he has always fallen asleep before they reach the door; thus it is that he believes in the marzantes just as he does in the other world, simply from what he has been told.

II.
You must not imagine that Uncle Jeromo is a rich man, taking the word in its literal sense, simply because he has a son who is away as a student; Aunt Simona's husband, for a laboring man, is middling well off, as he would say. But a chaplaincy belongs to the family, and Uncle Jeromo gladly parted with some of his property in order to meet the expense of his oldest son's education for the priesthood, in the hope of his obtaining the living. He did this, but at the same time he said plainly to his son:

"If you spend this pretty penny I got from the sale of my land, without making yourself a priest, Heaven help you, for, as far as I am concerned, I will cut you in two pieces."

The boy's entrance into the Seminary was hastened by the advice of the Lord of the Manor. This individual had studied a bit of Latin in his youth, and was so great a pedant that, simply in order to have some one with whom to show off his knowledge, he kept at Uncle Jeromo day after day, until at last he got him to decide that the boy should learn Latin. And so proud is the great man of his learning, and such is his pedantry, that since Uncle Jeromo's eldest born has entered the Seminary, he has several times wanted to give up his vacations so as not to have to face his neighbor at home, who lies in wait for him with Latin talk that is a perfect riddle, as the student says.

Thus it is that the Seminarist, dreading to meet him in some alley or seated on the bench before his father's door, has left the Seminary late so as to reach the village in the evening. This explains the delay which is causing Aunt Simona so much anxiety.

But what she did not know, and what the student himself could not suspect, was that, when he stepped into the tavern hard by to spend sixpence in quenching his thirst, the first person he was to meet, after the tavern-keeper, would be none other than the Lord of the Manor. No sooner did the latter catch sight of the student than he launched at him an "amice, quo modo vales," which bereft the poor fellow even of his thirst.

"The deuce take the man!" he muttered, hastening to pick up again the bundle of clothes which he had laid down on a bench as he came in.

"Unde venis? Quorsum tendis!"

"Oh, see here, I say! I've come four leagues on foot, and have no mind for joking of that kind. Goodby."

"Wait a moment, man! Why are you always so unsociable?"

"You're such a Paul Pry! One of these fine days I'll set you a sticker in Latin that you won't see into in a year. For, as far as that goes, I know a thing or two. Why, if I get you into theology once, we'll see what a figure you'll cut."

"Parce mihi, incipiens sacerdo."

"Be careful what you say, I tell you, for, though you may think I do not understand, I know how to translate. I can't stand everything."

"You are a weak creature, without a bit of the virum fortem. Oh, don't hurry off so! Why, I want to walk along with you to your home."

It was all of no avail. The Seminarist quickened his pace, cursing his evil star; he left the importunate great man in the middle of the road, and did not stop until safe inside his father's kitchen, where he presented himself in the worst possible temper.

"The deuce take that man!" was his only greeting as he found himself in the bosom of his family.

"Why, what has happened?" asks Uncle Jeromo.

"The same old thing! That crack-brained fellow—with his everlasting Latin!"

"But what trouble is that to you? Haven't you studied three years now? Why didn't you answer him?"

"Because I am not such a braggart as he. And besides, he has studied in a different way. My Latin doesn't have all those catches that he knows. I tell you, I wish he'd try me in psychology once! I'm perfectly at home in that."

"But when are you going to say your first mass?" asks Aunt Simona, lost in wonder, while the two children are gazing at the student with open eyes. "Why, the place is going to ruin—the priest is so old."

"And he doesn't know a thing, mother. If it was only one of us students now! We are learning such a lot! You ought to see the sermons I deliver when it comes my turn!"

III.
As it is by no means our main object to draw the portrait of Uncle Jeromo's eldest son, we may leave out entirely the dialogue brought on by his outburst against the Lord of the Manor, and go on again with the matter we had in hand, witnessing the supper of this worthy family on Christmas Eve. The student takes from the fire the pot containing the stew, in order that the flame may dry the mud upon his trousers. His sister has carefully gathered up his cloak and shirts. Then he takes the jar of milk, his father attending to the sugar, and is ready to help his mother and sister in making the toast, enlivening the work with tales of his achievements and adventures as a student.

When every delicacy is "fit to be eaten by an angel," as Aunt Simona avers, so perfectly prepared is it, and while all are carefully keeping close to the fire for the sake of the warmth, another operation is got under way, no less important than the supper itself: namely, setting the lazy table.

This table is only a rectangular board, fastened to the kitchen wall by a hinge at one end; the other end is held up to the same wall by a strap. When this is loosened, the table comes down like the portcullis of a fortress, and is kept in a horizontal position by means of a foot, or prop, hanging down from the board itself. This kind of table is used in the country only on occasions of festival importance, or when there is a wedding in the house. Therefore we should not be surprised at the tumult caused in Uncle Jeromo's kitchen by the operation of lowering it.

"Don't let it fall on you!"

"Help me with this side!"

"Take away that bench!"

"Pick up that spoon!"

"There it goes!"

"It's twisted!"

"Fasten it there!"

"Look out for your foot!"

There are little less outcry and greater precautions than if it were a question of launching a three-decker!

The table once set, and the luxuries all on it at last, the student says grace. But we will leave the family to sup in perfect freedom; it is an operation which, save for a few slight differences in the dishes and the vigor of mastication, we all perform in the same manner. Besides, our presence might prevent the good Jeromo from scraping out the gravy from the pot in which the stew was made, or his wife from putting her finger into the dish of toast to mix the sugar well, or the Seminarist from so completely draining his glass of white wine.

Let us return to the same kitchen an hour later. All are more talkative than before, and even the old laboring man has smoothed out his customary wrinkles. The small boy is stretched out on a bench, soundly snoring, and the student is talking Latin, and vowing that, if at that moment the Lord of the Manor were to quiz him, there would be a terrible time. Aunt Simona is humming in a low voice:

"To-night is Christmas Eve, To-morrow's Christmas holy; At twelve o'clock the Virgin's babe Is laid in cradle holy."

Her daughter is preparing to join in the song, when there is heard in the yard a chorus of neighs and a sound on the cobble stones as if twenty horses were coming in.

"Robbers are there!" a frightened city man would say in such circumstances.

Not at all, sir; these are the marzantes—that is to say, a couple of dozen of the young men of the place who go running about from house to house. The noise over the stones and the neighing are produced by the wooden shoes and the lungs of the youths.

This event has a most pleasing effect upon the persons in the kitchen; they are silent as statues, and listen intently.

"Well, Señor Don Jeromo," says a voice from the doorway, speaking in falsetto so as to disguise its true tones, "give us a sight of those shoulders curing there on the pole, and of those rich eggs of the speckled hen that was cackling out in the yard. Come, do it!"

The chorus neighs vigorously in answer to this first onset of algarabía, as the introduction of the marzantes is technically called, and then the voice begs again for a morsel of this and of that, until at last it stops, saying,

"Which will you have? Shall we sing or pray?"

"Pray," says Jeromo.

"Sing, of course!" breaks in the student. "I like these things very much. Come, sing!" he adds, opening the window the merest crack.

This command is received outside with another chorus of neighs, and the marzantes at once begin to sing, in a doleful and unvarying tone, a very long narrative poem, beginning,

"In Bethlehem's manger rude, The Virgin's child is born, A babe more rich than gold, Bright as the sun at morn,"

and ending,

"To the people of this place, May victory be given; In the present life much grace, And glory then in heaven."

The last verse has a variant which the marzantes are accustomed to use when nothing is given to them, or when they are tricked with food filled with ashes:

"For all in this house, I wish at this hour That even their bones The mange may devour."

The terrific encounters of wit to which this ejaculation usually gives rise, and the by no means despicable ones which are provoked when the marzantes go to the tavern, perhaps meeting others from another neighborhood, perhaps getting into a dispute with some passerby, are doubtless the reason for the leader's disguising his voice, and for his companions keeping in the background.

But in Jeromo's house no tricks are tried upon anybody, and Aunt Simona passes a bit of refreshment to the marzantes; and they, after performing the former verse, march away neighing with delight.

The family gives the finishing blows to the supper, drinking the last drops of the wine. The little fellow wakes up and asks for the marzantes. When he learns that they have come and gone, he fills the kitchen with his howls. His father begins to cuff him, but his mother and the student interpose. The fire goes out, the light of the candle flickers, the girl nods, the cat lazily mews, Jeromo's pipe falls from his mouth more than once, the Seminarist talks stupidly about the phenomena of light; and when the neighing of the marzantes is heard far away, with slow and reluctant steps the family of Uncle Jeromo go to seek in repose the end of the lively and pleasant evening.

Aunt Simona is the last to go. While she is lamenting that prayers had been omitted on account of the extra doings, and vowing that the deficiency shall be made up the next day, she puts away in the chest that we know of the remains of the bread, the sugar, and the lard, so that on Easter day the family would have something to remind them, provided the materials were skillfully handled, of Christmas Eve.

La noche de navidad

title: A Christmas Eve in Spain


text: Talk:A Christmas Eve in Spain

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title: Talk:A Christmas Eve in Spain


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INTRODUCTION.
At the present time, different opinions are being held about the fundamental equations of Electrodynamics for moving bodies. The ian forms must be given up, for it has appeared that they are contrary to many experimental results.

In 1895 H. A. published his theory of


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 optical and electrical phenomena in moving bodies; this theory was based upon the atomistic conception (vorstellung) of electricity, and on account of its great success appears to have justified the bold hypotheses, by which it has been ushered into existence. In his theory, proceeds from certain equations, which must hold at every point of "Æther"; then by forming the average values over "physically infinitely small" regions, which however contain large numbers of electrons, the equations for electro-magnetic processes in moving bodies can be successfully built up.

In particular, 's theory gives a good account of the non-existence of relative motion of the earth and the luminiferous "Æther"; it shows that this fact is connected with the covariance of the original equation, at certain simultaneous transformations of the space and time co-ordinates; these transformations have obtained from H. the name of -transformations. The covariance of these fundamental equations, when subjected to the -transformation, is a purely mathematical fact; I will call this the Theorem of Relativity; this theorem rests essentially on the form of the differential equations for the propagation of waves with the velocity of light.

Now without recognizing any hypothesis about the connection between "Æther" and matter, we can expect these mathematically evident theorems to have their consequences so far extended — that thereby even those laws of ponderable media which are yet unknown may anyhow possess this covariance when subjected to a -transformation; by saying this, we do not indeed express an opinion, but rather a conviction, — and this conviction I may be permitted to call the Postulate of Relativity. The position of affairs here is almost the same as when the Principle of Conservation of Energy was postulated in cases, where the corresponding forms of energy were unknown.

Now if hereafter, we succeed in maintaining this covariance as a definite connection between pure and simple observable phenomena in moving bodies, the definite connection may be styled the Principle of Relativity.


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 These differentiations seem to me to be useful for enabling us to characterise the present day position of the electro-dynamics for moving bodies.

H. A. has found out the Relativity theorem and has created the Relativity postulate as a hypothesis that electrons and matter suffer contractions in consequence of their motion according to a certain law.

has brought out the point very clearly, that this postulate is not an artificial hypothesis but is rather a new way of comprehending the time-concept which is forced upon us by observation of natural phenomena.

The Principle of Relativity has not yet been formulated for electro-dynamics of moving bodies in the sense characterized by me. ''In the present essay, while formulating this principle, I shall obtain the fundamental equations for moving bodies in a sense which is uniquely determined by this principle. But it will be shown that none of the forms hitherto assumed for these equations can exactly fit in with this principle.''

We would at first expect that the fundamental equations which are assumed by for moving bodies would correspond to the Relativity Principle. But it will be shown that this is not the case for the general equations which has for any possible, and also for magnetic bodies; but this is approximately the case (if we neglect the square of the velocity of matter in comparison to the velocity of light) for those equations which  hereafter infers for non-magnetic bodies. But this latter accordance with the relativity principle is due to the fact that the condition of non-magnetisation has been formulated in a way not corresponding to the relativity principle; therefore the accordance is due to the fortuitous compensation of two contradictions to the relativity postulate. But meanwhile enunciation of the Principle in a rigid manner does not signify any contradiction to the hypotheses of 's molecular theory, but it shall become clear that the assumption of the contraction of the electron in 's theory must be introduced at an earlier stage than has actually done.


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 In an appendix, I have gone into discussion of the position of Classical Mechanics with respect to the relativity postulate. Any easily perceivable modification of mechanics for satisfying the requirements of the Relativity theory would hardly afford any noticeable difference in observable processes; but would lead to very surprising consequences. By laying down the relativity postulate from the outset, sufficient means have been created for deducing henceforth the complete series of Laws of Mechanics from the principle of conservation of energy (and statements concerning the form of the energy) alone.

§ 1. NOTATIONS.
Let a rectangular system (x, y, z, t,) of reference be given in space and time. The unit of time shall be chosen in such a manner with reference to the unit of length that the velocity of light in space becomes unity.

Although I would prefer not to change the notations used by, it appears important to me to use a different selection of symbols, for thereby certain homogeneity will appear from the very beginning. I shall denote the vector

 * electric force by $$\mathfrak{E}$$, the magnetic induction by $$\mathfrak{M}$$, the electric induction by $$\mathfrak{e}$$ and the magnetic force by $$\mathfrak{m}$$,

so that $$\mathfrak{E,M,e,m}$$ are used instead of 's $$\mathfrak{E,B,D,H}$$ respectively.

I shall further make use of complex magnitudes in a way which is not yet current in physical investigations, i.e., instead of operating with t, I shall operate with it, where i denotes $$\sqrt{-1}$$. If now instead of (x, y, z, it), I use the method of writing with indices, certain essential circumstances will come into evidence; on this will be based a general use of the suffixes (1, 2, 3, 4). The advantage of this method will be, as I expressly emphasize here, that we shall have to handle symbols which have a purely real appearance; we can however at any moment pass to real equations


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 if it is understood that of the symbols with indices, such ones as have the suffix 4 only once, denote imaginary quantities, while those which have not at all the suffix 4, or have it twice denote real quantities.

An individual system of values of x, y, z t, i. e., of $$x_{1},\ x_{2},\ x_{3},\ x_{4}$$ shall be called a space-time point.

Further let $$\mathfrak{w}$$ denote the velocity vector of matter, $$\epsilon$$ the dielectric constant, $$\mu$$, the magnetic permeability, $$\sigma$$ the conductivity of matter, while $$\varrho$$ denotes the density of electricity in space, and $$\mathfrak{s}$$ the vector of "Electric Current" which we shall come across in §7 and §8.

§ 2. The Fundamental Equations for Æther.
By using the electron theory, in his above mentioned essay traces the Laws of Electrodynamics of Ponderable Bodies to still simpler laws. Let us now adhere to these simpler laws, whereby we require that for the limiting case $$\epsilon = 1,\ \mu = 1,\ \sigma = 0$$, they should constitute the laws for ponderable bodies. In this ideal limiting case $$\epsilon = 1,\ \mu = 1,\ \sigma = 0$$, we shall have $$\mathfrak{E}=\mathfrak{e},\mathfrak{M}=\mathfrak{m}$$. At every space time point x, y, z, t we shall have the equations:

I shall now write $$x_{1},\ x_{2},\ x_{3},\ x_{4}$$ for x, y, z, it $$\left(i=\sqrt{-1}\right)$$ and

$$\varrho_{1},\ \varrho_{2},\ \varrho_{3},\ \varrho_{4}$$

for

$$\varrho\mathfrak{w}_{x},\ \varrho\mathfrak{w}_{y},\ \varrho\mathfrak{w}_{z},\ i\varrho$$


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 i.e. the components of the convection current $$\varrho\mathfrak{w}$$, and the electric density multiplied by $$\sqrt{-1}$$.

Further I shall write

$$f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}$$

for

$$\mathfrak{m}_{x},\ \mathfrak{m}_{y},\ \mathfrak{m}_{z},\ -i\mathfrak{e}_{x},\ -i\mathfrak{e}_{y},\ -i\mathfrak{e}_{z}$$,

i.e., the components of $$\mathfrak{m}$$ and $$-i\mathfrak{e}$$ along the three axes; now if we take any two indices h, k out of the series

$$f_{kh} = -f_{hk}$$,

therefore

Then the three equations comprised in (I), and the equation (II) multiplied by i becomes

On the other hand, the three equations comprised in (III) multiplied by -i, and equation (IV) multiplied by -1, become

By means of this method of writing we at once notice the perfect symmetry of the 1st as well as the 2nd system of equations


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 as regards permutation with the indices (1,2,3,4).

§ 3. The Theorem of Relativity of Lorentz.
It is well-known that by writing the equations I) to IV) in the symbol of vector calculus, we at once set in evidence an invariance (or rather a (covariance) of the system of equations A) as well as of B), when the co-ordinate system is rotated through a certain amount round the null-point. For example, if we take a rotation of the axes round the z-axis. through an amount $$\varphi$$, keeping $$\mathfrak{e,m,w}$$ fixed in space, and introduce new variables $$x'_{1},\ x'_{2},\ x'_{3},\ x'_{4}$$ instead of $$x_{1},\ x_{2},\ x_{3},\ x_{4}$$, where

and introduce magnitudes

where

and $$f'_{12},\dots f'_{34}$$, where

then out of the equations (A) would follow a corresponding system of dashed equations (A') composed of the newly introduced dashed magnitudes.

So upon the ground of symmetry alone of the equations (A) and (B) concerning the suffixes (1, 2, 3, 4), the theorem of Relativity, which was found out by, follows without any calculation at all.

I will denote by $$i\psi$$ a purely imaginary magnitude, and consider the substitution

Putting


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 We shall have

$$\cos\ i\psi=\frac{1}{\sqrt{1-q^{2}}},\ \sin\ i\psi=\frac{iq}{\sqrt{1-q^{2}}}$$,

where $$-1 < q < 1$$, and $$\sqrt{1-q^{2}}$$ is always to be taken with the positive sign.

Let us now write

then the substitution 1) takes the form

the coefficients being essentially real.

If now in the above-mentioned rotation round the z-axis, we replace 1, 2, 3, 4 throughout by 3, 4, 1, 2, and $$\varphi$$ by $$i\psi$$, we at once perceive that simultaneously, new magnitudes $$\varrho'_{1},\ \varrho'_{2},\ \varrho'_{3},\ \varrho'_{4}$$, where

$$\begin{array}{ccc} & \varrho'_{1}=\varrho_{1},\ \varrho'_{2}=\varrho_{2},\\ \varrho'_{3}=x_{3}\cos\ i\psi+\varrho_{4}\sin\ i\psi, & & \varrho'_{4}=-\varrho_{3}\sin\ i\psi+\varrho_{4}\cos\ i\psi\end{array}$$

and $$f'_{12},\dots f'_{34}$$, where

must be introduced. Then the systems of equations in (A) and (B) are transformed into equations (A'), and (B'), the new equations being obtained by simply dashing the old set.

All these equations can be written in purely real figures, and we can then formulate the last result as follows.

If the real transformations 4) are taken, and x', y', z', t' be taken as a new frame of reference, then we shall have

furthermore


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and

Then we have for these newly introduced vectors $$\mathfrak{w',e',m'}$$ with components $$\mathfrak{w}'_{x},\mathfrak{w}'_{y},\mathfrak{w}'_{z}; \mathfrak{e}'_{x},\mathfrak{e}'_{y},\mathfrak{e}'_{z}$$; $$\mathfrak{m}'_{x},\mathfrak{m}'_{y},\mathfrak{m}'_{z}$$ and the quantity $$\varrho'$$ a series of equations I'), II'), III'), IV) which are obtained from I), II), III), IV) by simply dashing the symbols.

We remark here that e$$\mathfrak{e}_{x}-q\mathfrak{m}_{y},\ \mathfrak{e}_{y}+q\mathfrak{m}_{x},\ \mathfrak{e}_{z}$$ are components of the vector $$\mathfrak{e}+[\mathfrak{vm}]$$, where $$\mathfrak{v}$$ is a vector in the direction of the positive z-axis, and $$\left|\mathfrak{v}\right|=q$$, and $$[\mathfrak{vm}]$$ is the vector product of $$\mathfrak{v}$$ and $$\mathfrak{m}$$; similarly $$\mathfrak{m}_{x}+q\mathfrak{e}_{y},\ \mathfrak{m}_{y}-q\mathfrak{e}_{x},\ \mathfrak{m}_{z}$$ are the components of the vector $$\mathfrak{m}-[\mathfrak{ve}]$$.

The equations 6) and 7), as they stand in pairs, can be expressed as.

If $$\varphi$$ denotes any other real angle, we can form the following combinations : —

§ 4. Special Lorentz-Transformation.
The role which is played by the z-axis in the transformation (4)


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 can easily be transferred to any other axis when the system of axes are subjected to a transformation about this last axis. So we came to a more general law: —

Let $$\mathfrak{v}$$ be a vector with the components $$\mathfrak{v}_{x},\ \mathfrak{v}_{y},\ \mathfrak{v}_{z}$$, and let $$\left|\mathfrak{v}\right|=q<1$$. By $$\mathfrak{\bar{v}}$$ we shall denote any vector which is perpendicular to $$\mathfrak{v}$$, and by $$\mathfrak{r_{v}}$$, $$\mathfrak{r_{\bar{v}}}$$ we shall denote components of $$\mathfrak{r}$$ in direction of $$\mathfrak{\bar{v}}$$ and $$\left|\mathfrak{v}\right|$$.

Instead of x, y, z, t, new magnetudes x,' y,' z,' t'  will be introduced in the following way. If for the sake of shortness, $$\mathfrak{r}$$ is written for the vector with the components x, y, z in the first system of reference, $$\mathfrak{r}'$$ for the same vector with the components x', y', z'  in the second system of reference, then for the direction of $$\mathfrak{v}$$ we have

and for every perpendicular direction $$\mathfrak{\bar{v}}$$

and further

The notations $$\mathfrak{r'_{v}}$$ and $$\mathfrak{r'_{\bar{v}}}$$ are to be understood in the sense that with the directions $$\mathfrak{v}$$, and every direction $$\mathfrak{v}$$ perpendicular to $$\mathfrak{\bar{v}}$$ in the system x, y, z are always associated the directions with the same direction cosines in the system x', y', z' ,

A transformation which is accomplished by means of (10), (11), (12) with the condition $$0< q < 1$$ will be called a special -transformation. We shall call $$\mathfrak{v}$$ the vector, the direction of $$\mathfrak{v}$$ the axis, and the magnitude of $$\mathfrak{v}$$ the moment of this transformation.

If further $$\varrho'$$ and the vectors $$\mathfrak{w}',\ \mathfrak{e}',\ \mathfrak{m}'$$, in the system x', y', z' are so defined that,


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further

Then it follows that the equations I), II), III), IV) are transformed into the corresponding system with dashes.

The solution of the equations (10), (11), (12) leads to

Now we shall make a very important observation about the vectors $$\mathfrak{w}$$ and $$\mathfrak{w}'$$. We can again introduce the indices 1, 2, 3, 4, so that we write $$x'_{1},\ x'_{2},\ x'_{3},\ x'_{4}$$ instead of x,' y,' z,' it' , and $$\varrho'_{1},\ \varrho'_{2},\ \varrho'_{3},\ \varrho'_{4}$$ instead of $$\varrho'\mathfrak{w}'_{x'}\ \varrho'\mathfrak{w}'_{y'}\ \varrho'\mathfrak{w}'_{z'}\ i\varrho'$$. Like the rotation round the z-axis, the transformation (4), and more generally the transformations (10), (11), (12), are also linear transformations with the determinant +1, so that

is transformed into

On the basis of the equations (13), (14), we shall have

transformed into $$\varrho'(1-\mathfrak{w}'^{2})$$ or in other words,

is an invariant in a -transformation.


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 If we divide $$\varrho_{1},\ \varrho_{2},\ \varrho_{3},\ \varrho_{4}$$ by this magnitude, we obtain the four values

$$w_{1}=\frac{\mathfrak{w}_{x}}{\sqrt{1-\mathfrak{w}^{2}}},\ w_{2}=\frac{\mathfrak{w}_{y}}{\sqrt{1-\mathfrak{w}^{2}}},\ w_{3}=\frac{\mathfrak{w}_{z}}{\sqrt{1-\mathfrak{w}^{2}}},\ w_{4}=\frac{i}{\sqrt{1-\mathfrak{w}^{2}}}$$,

so that

It is apparent that these four values, are determined by the vector $$\mathfrak{w}$$ and inversely the vector $$\mathfrak{w}$$ of magnitude $$< 1$$ follows from the 4 values $$w_{1},\ w_{2},\ w_{3},\ w_{4}$$, where $$w_{1},\ w_{2},\ w_{3}$$ are real, $$-iw_{4}$$ real and positive and condition (19) is fulfilled.

The meaning of $$w_{1},\ w_{2},\ w_{3},\ w_{4}$$ here is, that they are the ratios of $$dx_{1},\ dx_{2},\ dx_{3},\ dx_{4}$$ to

The differentials denoting the displacements of matter occupying the spacetime point $$x_{1},\ x_{2},\ x_{3},\ x_{4}$$ to the adjacent space-time point.

After the -transfornation is accomplished the velocity of matter in the new system of reference for the same space-time point x', y', z', t'  is the vector $$\mathfrak{w}'$$ with the ratios $$\frac{dx'}{dt'},\ \frac{dy'}{dt'},\ \frac{dz'}{dt'}$$ as components.

Now it is quite apparent that the system of values

$$x_{1} = w_{1},\ x_{2} = w_{2},\ x_{3} = w_{3},\ x_{4} = w_{4}$$

is transformed into the values

$$x'_{1} = w'_{1},\ x'_{2} = w'_{2},\ x'_{3} = w'_{3},\ x'_{4} = w'_{4}$$

in virtue of the -transformation (10), (11), (12).

The dashed system has got the same meaning for the velocity $$\mathfrak{w}'$$ after the transformation as the first system of values has got for $$\mathfrak{w}$$ before transformation.


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 If in particular the vector $$\mathfrak{v}$$ of the special -transformation be equal to the velocity vector $$\mathfrak{w}$$ of matter at the space-time point $$x_{1},\ x_{2},\ x_{3},\ x_{4}$$, then it follows out of (10), (11), (12) that

$$w'_{1} = 0,\ w'_{2} = 0,\ w'_{3} = 0,\ w'_{4} = i$$,

Under these circumstances therefore, the corresponding space-time point has the velocity $$\mathfrak{w}' = 0$$ after the transformation, it is as if we transform to rest. We may call the invariant $$\varrho\sqrt{1-\mathfrak{w}^{2}}$$ as the rest-density of Electricity.

§ 5. Space-time Vectors. Of the 1st and 2nd kind.
If we take the principal result of the transformation together with the fact that the system (A) as well as the system (B) is covariant with respect to a rotation of the coordinate-system round the null point, we obtain the general relativity theorem. In order to make the facts easily comprehensible, it may be more convenient to define a series of expressions, for the purpose of expressing the ideas in a concise form, while on the other hand I shall adhere to the practice of using complex magnitudes, in order to render certain symmetries quite evident.

Let us take a linear homogeneous transformation,

the Determinant of the matrix is +1, all co-efficients without the index 4 occurring once are real, while $$\alpha_{14},\ \alpha_{24},\ \alpha_{34}$$, are purely imaginary, but $$\alpha_{44}$$ is real and $$>0$$, and $$x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4}$$ transforms into $$x^{'2}_{1} + x^{'2}_{2} + x^{'2}_{3} + x^{'2}_{4}$$. The operation shall be called a general  transformation.

If we put $$x'_{1} = x',\ x'_{2} = y',\ x'_{3} = z',\ x'_{4} = it'$$ then immediately there occurs a homogeneous linear transformation


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 of x, y, z, t in x', y', z', t'  with essentially real co-efficients, whereby the aggregrate $$-x^{2} - y^{2} - z^{2} + t^{2}$$ transforms into $$-x'^{2} - y'^{2} - z'^{2} + t'^{2}$$, and to every such system of values x, y, z, t with a positive t, for which this aggregate $$>0$$, there always corresponds a positive t'; this last is quite evident from the continuity of the aggregate x, y, z, t.

The last vertical column of co-efficients has to fulfill, the condition

If $$\alpha_{14}=0,\ \alpha_{24}=0,\ \alpha_{34}=0$$ then $$\alpha_{44}=1$$, and the transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point.

If $$\alpha_{14},\ \alpha_{24},\ \alpha_{34}$$ are not all zero, and if we put

$$\alpha_{14} : \alpha_{24} : \alpha_{34} : \alpha_{44} = \mathfrak{v}_{x}:\mathfrak{v}_{y}:\mathfrak{v}_{z}:i$$,

$$q=\sqrt{\mathfrak{v}_{x}^{2}+\mathfrak{v}_{y}^{2}+\mathfrak{v}_{z}^{2}}<1$$.

On the other hand, with every set of value of $$\alpha_{14},\ \alpha_{24},\ \alpha_{34},\ \alpha_{44}$$ which in this way fulfill the condition 22) with real values of $$\mathfrak{v}_{x}+\mathfrak{v}_{y}+\mathfrak{v}_{z}$$, we can construct the special -transformation (16) with $$\alpha_{14},\ \alpha_{24},\ \alpha_{34},\ \alpha_{44}$$ as the last vertical column, — and then every -transformation with the same last vertical column $$\alpha_{14},\ \alpha_{24},\ \alpha_{34},\ \alpha_{44}$$ supposed to be composed of the special -transformation, and a rotation of the spatial co-ordinate system round the null-point.

The totality of all -Transformations forms a group.

Under a space-time vector of the 1st kind shall be understood a system of four magnitudes $$\varrho_{1},\ \varrho_{2},\ \varrho_{3},\ \varrho_{4}$$ with the condition that in case of a -transformation it is to be replaced by the set $$\varrho'_{1},\ \varrho'_{2},\ \varrho'_{3},\ \varrho'_{4}$$, where these are the values $$x'_{1},\ x'_{2},\ x'_{3},\ x'_{4}$$ obtained by substituting $$\varrho_{1},\ \varrho_{2},\ \varrho_{3},\ \varrho_{4}$$ for $$x_{1},\ x_{2},\ x_{3},\ x_{4}$$ in the expression (21).

Besides the time-space vector of the 1st kind $$x_{1},\ x_{2},\ x_{3},\ x_{4}$$ we shall also make use of another spacetime vector of the first kind $$u_{1},\ u_{2},\ u_{3},\ u_{4}$$, and let us form the linear combination


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with six coefficients $$f_{23},\dots f_{34}$$. Let us remark that in the vectorial method of writing, this can be constructed out of the four vectors

and the constants $$x_{4}$$ and $$u_{4}$$ at the same time it is symmetrical with regard the indices (1, 2, 3, 4).

If we subject $$x_{1},\ x_{2},\ x_{3},\ x_{4}$$ and $$u_{1},\ u_{2},\ u_{3},\ u_{4}$$ simultaneously to the transformation (21), the combination (23) is changed to.

where the coefficients $$f'_{23},\dots f'_{34}$$ depend solely on $$f_{23},\dots f_{34}$$ and the coefficients $$\alpha_{11},\ \alpha_{12},\dots \alpha_{44}$$.

We shall define a space-time Vector of the 2nd kind as a system of six-magnitudes $$f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}$$ with the condition that when subjected to a transformation, it is changed to a new system $$f'_{23},\ f'_{31},\ f'_{12},\ f'_{14},\ f'_{24},\ f'_{34}$$ which satisfies the connection between (23) and (24).

I enunciate in the following manner the general theorem of relativity corresponding to the equations (I) — (IV), — which are the fundamental equations for Æther.

If x, y, z, it (space co-ordinates, and time it) is subjected to a transformation, and at the same time $$\varrho\mathfrak{w}_{x},\ \varrho\mathfrak{w}_{y},\ \varrho\mathfrak{w}_{z},\ i\varrho$$ (convection-current, and charge density × i) is transformed as a space time vector of the 1st kind, further $$\mathfrak{m}_{x},\ \mathfrak{m}_{y},\ \mathfrak{m}_{z},\ -i\mathfrak{e}_{x}\ -i\mathfrak{e}_{y},\ -i\mathfrak{e}_{z}$$ (magnetic force, and electric induction × i) is transformed as a space time vector of the 2nd kind, then the system of equations (I), (II), and the system of equations (III), (IV) transforms into essentially corresponding relations between the corresponding magnitudes newly introduced info the system.

These facts can be more concisely expressed in these words: the system of equations (I, and II) as well as the system of equations (III) (IV) are covariant in all cases of -transformation, where $$\varrho\mathfrak{w},\ i\varrho$$ is to be transformed as a space time vector of the 1st kind, $$\mathfrak{m},\ -i\mathfrak{e}$$ is to be treated as a vector of the 2nd kind, or more significantly, —

$$\varrho\mathfrak{w},\ i\varrho$$ is a space time vector of the 1st kind, $$\mathfrak{m},\ -i\mathfrak{e}$$ is a space-time vector of the 2nd kind.


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 I shall add a few more remarks here in order to elucidate the conception of space-time vector of the 2nd kind. Clearly, the following are invariants for such a vector when subjected to a group of transformation.

A space-time vector of the second kind $$\mathfrak{m},\ -i\mathfrak{e}$$, where $$\mathfrak{m}$$ and $$\mathfrak{e}$$ are real magnitudes, may be called singular, when the scalar square $$(\mathfrak{m}-i\mathfrak{e})^{2}=0$$, i.e. $$\mathfrak{m}^{2}-\mathfrak{e}^{2}=0$$, and at the same time $$(\mathfrak{me})=0$$, i.e. the vector $$\mathfrak{m}$$ and $$\mathfrak{e}$$ are equal and perpendicular to each other; when such is the case, these two properties remain conserved for the space-time vector of the 2nd kind in every -transformation.

If the space-time vector of the 2nd kind is not singular, we rotate the spacial co-ordinate system in such a manner that the vector-product $$[\mathfrak{me}]$$ coincides with the z-axis, i.e. $$\mathfrak{m}_{z}=0, \mathfrak{e}_{z}=0$$. Then

Therefore $$\frac{\mathfrak{e}_{y}+i\mathfrak{m}_{y}}{\mathfrak{e}_{x}+i\mathfrak{m}_{x}}$$ is different from $$\pm i$$, and we can therefore define a complex argument $$\varphi+i\psi$$ in such a manner that

If then, by referring back to equations (9), we carry out the transformation (1) through the angle $$\psi$$ and a subsequent rotation round the z-axis through the angle $$\varphi$$, we perform a -transformation at the end of which $$\mathfrak{m}_{y}=0,\ \mathfrak{e}_{y}=0$$, and therefore $$\mathfrak{m}$$ and $$\mathfrak{e}$$ shall both coincide with the new x-axis. Then by means of the invariants $$\mathfrak{m}^{2}+\mathfrak{e}^{2}$$ and $$(\mathfrak{me})$$ the final values of these vectors, whether they are of the same or of opposite directions, or whether one of them is equal to zero, would be at once settled.

§ 6. Concept of Time.
By the transformation, we are allowed to effect certain changes of the time parameter. In consequence of this fact, it is no longer permissible to speak of the absolute simultaneity of two events. The ordinary idea of simultaneity rather presupposes that six independent parameters, which are evidently required for defining a system of space and time axes, are somehow reduced to three.


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 Since we are accustomed to consider that these limitations represent in a unique way the actual facts very approximately, we maintain that the simultaneity of two events exists of themselves. In fact, the following considerations will prove conclusive.

Let a reference system x, y, z, t for space time points (events) be somehow known. Now if a space point $$A(x_{0},\ y_{0},\ z_{0})$$ at the time $$t_{0}=0$$ be compared with a space point P(x, y, z) at the time t and if the difference of time $$t - t_{0}$$, (let $$t > t_{0}$$) be less than the length AP, i.e. less than the time required for the propagation of light from A to P, and if $$q=\frac{t-t_{0}}{AP}<1$$, then by a special transformation, in which AP is taken as the axis, and which has the moment q, we can introduce a time parameter t which (see equation 11, 12, § 4) has got the same value $$t'= 0$$ for both space-time points $$A, t_{0}$$, and P, t. So the two events can now be comprehended to be simultaneous.

Further, let us take at the same time $$t_{0} = 0$$, two different space-points A, B, or three space-points A, B, C which are not in the same space-line, and compare therewith a space point P, which is outside the line AB, or the plane ABC at another time t, and let the time difference $$t - t_{0}$$ be less than the time which light requires for propagation from the line AB, or the plane ABC to P. Let q be the ratio of $$t - t_{0}$$ by the second time. Then if a transformation is taken in which the perpendicular from P on AB, or from P on the plane ABC is the axis, and q is the moment, then all the three (or four) events $$A,\ t_{0};\ B,\ t_{0};\ (C,\ t_{0})$$ and P, t are simultaneous.

If four space-points, which do not lie in one plane are conceived to be at the same time to, then it is no longer permissible to make a change of the time parameter by a -transformation, without at the same time destroying the character of the simultaneity of these four space points.

To the mathematician, accustomed on the one hand to the methods of treatment of the poly-dimensional manifold, and on the other hand to the conceptual figures of the so-called non-Euclidean Geometry, there can be no difficulty in adopting this concept


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 of time to the application of the -transformation. The paper of which has been cited in the Introduction, has succeeded to some extent in presenting the nature of the transformation from the physical standpoint.

§ 7. Fundamental Equations for bodies at rest.
After these preparatory works, which have been first developed on account of the small amount of mathematics involved in the limitting case $$\epsilon=1,\ \mu=1,\ \sigma=1$$, let us turn to the electro-magnetic phenomena in matter. We look for those relations which make it possible for us — when proper fundamental data are given — to obtain the following quantities at every place and time, and therefore at every spacetime point as functions of x, y, z, t: — the vector of the electric force $$\mathfrak{E}$$, the magnetic induction $$\mathfrak{M}$$, the electrical induction $$\mathfrak{e}$$, the magnetic force $$\mathfrak{m}$$, the electrical space-density $$\varrho$$, the electric current $$\mathfrak{s}$$ (whose relation hereafter to the conduction current is known by the manner in which conductivity occurs in the process), and lastly the vector $$\mathfrak{w}$$, the velocity of matter.

The relations in question can be divided into two classes.

Firstly — those equations, which, — when $$\mathfrak{w}$$, the velocity of matter is given as a function of x, y, z, t, — lead us to a knowledge of other magnitude as functions of x, y, z, t — I shall call this first class of equations the fundamental equations —

Secondly, the expressions for the ponderomotive force, which, by the application of the Laws of Mechanics, gives us further information about the vector $$\mathfrak{w}$$ as functions of x, y, z, t.

For the case of bodies at rest, i.e. when $$\mathfrak{w}(x,\ y,\ z,\ t) = 0$$, the theories of (Heaviside, ) and  lead to the same fundamental equations. They are ; —

(1) The Differential Equations: — which contain no constant referring to matter: —


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 (2) Further relations, which characterise the influence of existing matter for the most important case to which we limit ourselves, i.e. for isotopic bodies; — they are comprised in the equations

where $$\epsilon$$ = dielectric constant, $$\mu$$ = magnetic permeability, $$\sigma$$ = the conductivity of matter, all given as function of x, y, z, t. $$\mathfrak{s}$$ is here the conduction current.

By employing a modified form of writing, I shall now cause a latent symmetry in these equations to appear. I put, as in the previous work,

$$x_{1} = x,\ x_{2} = y,\ x_{3} =z,\ x_{4} = it$$

and write $$s_{1},\ s_{2},\ s_{3},\ s_{4}$$ for $$\mathfrak{s}_{x},\ \mathfrak{s}_{y},\ \mathfrak{s}_{z},\ i\varrho$$,

further $$f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}$$

for $$\mathfrak{m}_{x},\ \mathfrak{m}_{y},\ \mathfrak{m}_{z},\ -i\mathfrak{e}_{x},\ -i\mathfrak{e}_{y},\ -i\mathfrak{e}_{z}$$,

and $$F_{23},\ F_{31},\ F_{12},\ F_{14},\ F_{24},\ F_{34}$$

for $$\mathfrak{M}_{x},\ \mathfrak{M}_{y},\ \mathfrak{M}_{z},\ -i\mathfrak{E}_{x},\ i\mathfrak{E}_{y},\ i\mathfrak{E}_{z}$$;

lastly we shall have the relation $$f_{kh} = -f_{hk},\ F_{kh} = -F_{hk}$$, (the letter f, F shall denote the field, s the (i.e. current).

Then the fundamental Equations can be written as


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 and the equations (III) and (IV), are

§ 8. The Fundamental Equations for Moving Bodies.
We are now in a position to establish in a unique way the fundamental equations for bodies moving in any manner by means of these three axioms exclusively.

The first Axion shall be, —

When a detached region of matter is at rest at any moment, therefore the vector $$\mathfrak{w}$$ is zero, for a system x, y, z, t — the neighbourhood may be supposed to be in motion in any possible manner, then for the spacetime point x, y, z, t the same relations (A) (B) (V) which hold in the case when all matter is at rest, snail also hold between $$\varrho$$, the vectors $$\mathfrak{s,e,m,E,M}$$ and their differentials with respect to x, y, z, t.

The second axiom shall be : —

Every velocity of matter is < 1, smaller than the velocity of propagation of light.

The third axiom shall be : —

The fundamental equations are of such a kind that when x, y, z, it are subjected to a transformation and thereby $$\mathfrak{m},\ -i\mathfrak{e}$$ and $$\mathfrak{M},\ -i\mathfrak{E}$$ are transformed into space-time vectors of the second kind, $$\mathfrak{s},\ i\varrho$$ as a space-time vector of the 1st kind, the equations are transformed into essentially identical forms involving the transformed magnitudes.

Shortly I can signify the third axiom as ; —

$$\mathfrak{m},\ -i\mathfrak{e}$$ and $$\mathfrak{M},\ -i\mathfrak{E}$$ are space-time vectors of the second kind, $$\mathfrak{s},\ i\varrho$$ is a space-time vector of the first kind.

This axiom I call the Principle of Relativity.


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 In fact these three axioms lead us from the previously mentioned fundamental equations for bodies at rest to the equations for moving bodies in an unambiguous way.

According to the second axiom, the magnitude of the velocity vector $$\left|\mathfrak{w}\right|$$ is < 1 at any space-time point. In consequence, we can always write, instead of the vector $$\mathfrak{w}$$, the following set of four allied quantities

$$w_{1}=\frac{\mathfrak{w}_{x}}{\sqrt{1-\mathfrak{w}^{2}}},\ w_{2}=\frac{\mathfrak{w}_{y}}{\sqrt{1-\mathfrak{w}^{2}}},\ w_{3}=\frac{\mathfrak{w}_{z}}{\sqrt{1-\mathfrak{w}^{2}}},\ w_{4}=\frac{i}{\sqrt{1-\mathfrak{w}^{2}}}$$,

with the relation

From what has been said at the end of § 4, it is clear that in the case of a -transformation, this set behaves like a space-time vector of the 1st kind, and we want to call it space-time vector velocity.

Let us now fix our attention on a certain point x, y, z of matter at a certain time t. If at this space-time point $$\mathfrak{w}=0$$, then we have at once for this point the equations (A), (B) (V) of § 7. If $$\mathfrak{w}\ne0$$, then there exists according to 16), in case $$\left|\mathfrak{w}\right|<1$$, a special -transformation, whose vector $$\mathfrak{v}$$ is equal to this vector $$\mathfrak{w}(x,\ y,\ z,\ t)$$ and we pass on to a new system of reference x', y', z', t' in accordance with this transformation. Therefore for the space-time point considered, there arises as in § 4, the new values

therefore the new velocity vector $$\mathfrak{w}'=0$$, the space-time point is as if transformed to rest. Now according to the third axiom the system of equations for the transformed point x, y, z, t involves the newly introduced magnitude $$\mathfrak{w}',\varrho',\mathfrak{s',e',m',E',M'}$$ and their differential quotients with respect to x', y', z, t' in the same manner as the original equations for the point x, y, z, t. But according to the first axiom, when $$\mathfrak{w}'=0$$ these equations must be exactly equivalent to

(1) the differential equations (A'), (B'), which are obtained from the equations (A), (B) by simply dashing the symbols in (A) and (B).

(2) and the equations


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where $$\epsilon,\ \mu,\ \sigma$$ are the dielectric constant, magnetic permeability, and conductivity for the system x', y', z', t', i.e. in the space-time point x, y, z, t of matter.

Now let us return, by means of the reciprocal -transformation to the original variables x, y, z, t, and the magnitudes $$\mathfrak{w},\varrho,\mathfrak{s,e,m,E,M}$$ and the equations, which we then obtain from the last mentioned, will be the fundamentil equations sought by us for the moving bodies.

Now from § 4, and § 6, it is to be seen that the equations A), as well as the equations B) are covariant for a -transformation, i.e. the equations, which we obtain backwards from A') B'), must be exactly of the same form as the equations A) and B), as we take them for bodies at rest. We have therefore as the first result: —

The differential equations expressing the fundamental equations of electrodynamics for moving bodies, when written in $$\varrho$$ and the vectors $$\mathfrak{s,\ e,\ m,\ E,\ M}$$ are exactly of the same form as the equations for moving bodies. The velocity of matter does not enter in these equations. In the vectorial way of writing, we have

The velocity of matter occurs only in the auxilliary equations which characterise the influence of matter on the basis of their characteristic constants $$\epsilon,\ \mu,\ \sigma$$. Let us now transform these auxilliary equations x, y, z into the original co-ordinates x, y, z, and t.)

According to formula 15) in § 4, the component of $$\mathfrak{e}'$$ in the direction of the vector $$\mathfrak{w}$$ is the same us that of $$\mathfrak{e}+[\mathfrak{wm}]$$, the component of $$\mathfrak{m}'$$ is the same as that of $$\mathfrak{m}-[\mathfrak{we}]$$, but for the perpendicular direction $$\mathfrak{\bar{w}}$$, the components of $$\mathfrak{e}'$$ and $$\mathfrak{m}'$$


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 are the same as those of $$\mathfrak{e}+[\mathfrak{wm}]$$ and $$\mathfrak{m}-[\mathfrak{we}]$$, multiplied by $$\frac{1}{\sqrt{1-\mathfrak{w}^{2}}}$$. On the other hand $$\mathfrak{E}'$$ and $$\mathfrak{M}'$$ shall stand to $$\mathfrak{E}+[\mathfrak{wM}]$$, and $$\mathfrak{M}-[\mathfrak{wE}]$$ in the same relation us $$\mathfrak{e}'$$ and $$\mathfrak{m}'$$ to $$\mathfrak{e}+[\mathfrak{wm}]$$ and $$\mathfrak{m}+[\mathfrak{we}]$$. From the relation $$\mathfrak{e}'=\epsilon\mathfrak{E}'$$, the following equations follow

and from the relation $$\mathfrak{M}'=\mu\mathfrak{m}'$$ we have

For the components in the directions perpendicular to $$\mathfrak{w}$$, and to each other, the equations are to be multiplied by $$\sqrt{1-\mathfrak{w}^{2}}$$.

Then the following equations follow from the transfermation equations (12), 10), (11) in § 4, when we replace $$q,\ \mathfrak{r_{v},\ r_{\bar{v}}},t,\mathfrak{r'_{v},\ r'_{\bar{v}}},t'$$ by $$\left|\mathfrak{w}\right|,\ \mathfrak{s_{w},s_{\bar{w}}},\varrho,\mathfrak{s'_{w},s'_{\bar{w}}},\varrho'$$.

In consideration of the manner in which $$\sigma$$ enters into these relations, it will be convenient to call the vector $$\mathfrak{s}-\varrho\mathfrak{w}$$ with the components $$\mathfrak{s_{w}}-\varrho\mathfrak{\left|w\right|}$$ in the direction of $$\mathfrak{w}$$ and $$\mathfrak{s_{\bar{w}}}$$ in the directions $$\mathfrak{w}$$ perpendicular to $$\mathfrak{\bar{w}}$$ the Convection current. This last vanishes for $$\sigma = 0$$.

We remark that for $$\epsilon = 1,\ \mu = 1$$ the equations $$\mathfrak{e'=E',\ m'=M'}$$ immediately lead to the equations $$\mathfrak{e=E,\ m=M}$$ by means of a reciprocal -transformation with $$-\mathfrak{w}$$ as vector; and for $$\sigma = 0$$, the equation $$\mathfrak{s}'=0$$ leads to $$\mathfrak{s}=\varrho\mathfrak{w}$$, that the "fundamental equations of Æther" discussed in § 2 becomes in fact the limitting case of the equations obtained here with $$\epsilon = 1,\ \mu = 1,\ \sigma = 0$$.


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§ 9. The Fundamental Equations in Lorentz's Theory.
Let us now see how far the fundamental equations assumed by correspond to the Relativity postulate, as defined in §8. In the article on Electron-theory (Ency, Math., Wiss., Bd. V. 2, Art 14) has given the fundamental equations for any possible, even magnetised bodies (see there page 209, Eq. XXX', formula (14) on page 78 of the same (part).

Then for moving non-magnetised bodies, puts (page 223, 3) $$\mu = 1$$, $$\mathfrak{B}=\mathfrak{H}$$, and in addition to that takes account of the occurrence of the di-electric constant $$\epsilon$$, and conductivity $$\sigma$$ according to equations

's $$\mathfrak{E,B,D,H}$$ are here denoted by $$\mathfrak{E,M,e,m}$$ while $$\mathfrak{F}$$ denotes the conduction current.

The three last equations which have been just cited here coincide with eq. (II), (III), (IV), the first equation would be, if $$\mathfrak{F}$$ is identified with $$\mathfrak{s}-\mathfrak{w}\sigma$$ (the current being zero for $$\sigma = 0$$),

and thus comes out to in in a different form than (1) here. Therefore for magnetised bodies, 's equations do not correspond to the Relativity Principle.

On the other hand, the form corresponding to the relativity principle, for the condition of non-magnetisation is to be taken out of (D) in §8, with $$\mu=1$$, not as $$\mathfrak{B}=\mathfrak{H}$$, as takes, but as


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 Now by putting $$\mathfrak{H}=\mathfrak{B}$$, the differential equation (29) is transformed into the same form as eq. (1) here when $$\mathfrak{m}-[\mathfrak{we}]=\mathfrak{M}-[\mathfrak{wE}]$$. Therefore it so happens that by a compensation of two contradictions to the relativity principle, the differential equations of for moving non-magnetised bodies at last agree with the relativity postulate.

If we make use of (30) for non-magnetic bodies, and put accordingly $$\mathfrak{H}=\mathfrak{B}+[\mathfrak{w},\ \mathfrak{D}-\mathfrak{E}]$$, then in consequence of (C) in §8,

$$(\epsilon-1)(\mathfrak{E}+(\mathfrak{wB}])=\mathfrak{D}-\mathfrak{E}+(\mathfrak{w}[\mathfrak{w},\ \mathfrak{D}-\mathfrak{E}])$$,

i.e. for the direction of $$\mathfrak{w}$$

$$(\epsilon-1)(\mathfrak{E}+(\mathfrak{wB}])_{\mathfrak{w}}=(\mathfrak{D}-\mathfrak{E})_{\mathfrak{w}}$$,

and for a perpendicular direction $$\mathfrak{\bar{w}}$$,

$$(\epsilon-1)(\mathfrak{E}+(\mathfrak{wB}])_{\mathfrak{\bar{w}}}=(1-\mathfrak{w}^{2})(\mathfrak{D}-\mathfrak{E})_{\mathfrak{\bar{w}}}$$,

i.e. it coincides with 's assumption, if we neglect $$\mathfrak{w}^2$$ in comparison to 1.

Also to the same order of approximation, 's form for $$\mathfrak{F}$$ corresponds to the conditions imposed by the relativity principle [comp. (E) § 8] — that the components of $$\mathfrak{F_{w}}$$, $$\mathfrak{F_{\bar{w}}}$$ are equal to the components of $$\sigma(\mathfrak{E}+(\mathfrak{wB}])$$ multiplied by $$\sqrt{1-\mathfrak{w}^{2}}$$ or $$\frac{1}{\sqrt{1-\mathfrak{w}^{2}}}$$ respectively.

§ 10. Fundamental Equations of E. Cohn.
E. assumes the following fundamental equations

where E, M are the electric and magnetic field intensities (forces), $$\mathfrak{E,M}$$ are the electric and magnetic polarisation (induction).


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 The equations also permit the existence of true magnetism; if we do not take into account this consideration, $$div\ \mathfrak{M}$$ is to be put = 0.

An objection to this system of equations, is that according to these, for $$\epsilon = 1,\ \mu = 1$$, the vectors force and induction do not coincide. If in the equations, we conceive E and M and not $$E-[\mathfrak{wM}]$$ and $$M+[\mathfrak{wE}]$$ as electric and magnetic forces, and with a glance to this we substitute for $$\mathfrak{E,M},E,M,div\ \mathfrak{E}$$ the symbols $$\mathfrak{e,M,\ E}+[\mathfrak{wM}],\ \mathfrak{m}-[\mathfrak{we}],\ \varrho$$, then the differential equations transform to our equations, and the conditions (32) transform into

then in fact the equations of become the same as those required by the relativity principle, if errors of the order $$\mathfrak{w}^{2}$$ are neglected in comparison to 1.

It may be mentioned here that the equations of become the same as those of, if the auxilliary conditions are

§ 11. Typical Representations of the fundamental equations.
In the statement of the fundamental equations, our leading idea had been that they should retain a covariance of form, when subjected to a group of -transformations. Now we have to deal with ponderomotive reactions and energy in the electro-magnetic field. Here from the very first there can be no doubt that the settlement of this question is in some way connected with the simplest forms which can be given to the fundamental equations, satisfying the conditions of covariance. In order to arrive at such forms, I shall first of all put the fundamental equations in a typical form which brings out clearly their covariance in case of a -transformation.


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 Here I am using a method of calculation, which enables us to deal in a simple manner with the space-time vectors of the 1st, and 2nd kind, and of which the rules, as far as required are given below.

1°. A system of magnitudes $$a_{hk}$$, formed into the matrix

$$\left|\begin{array}{ccc} a_{11}, & \dots & a_{1q}\\ \vdots & & \vdots\\ a_{p1}, & \dots & a_{pq}\end{array}\right|$$

arranged in p horizontal rows, and q vertical columns is called a $$p \times q$$ series-matrix, and will be denoted by the letter A.

If all the quantities $$a_{hk}$$ are multiplied by c, the resulting matrix will be denoted by $$cA$$.

If the roles of the horizontal rows and vertical columns be intercharged, we obtain a $$q \times p$$ series matrix, which will be known as the transposed matrix of A, and will be denoted by A.

$$\bar{A}=\left|\begin{array}{ccc} a_{11}, & \dots & a_{q1}\\ \vdots & & \vdots\\ a_{1p}, & \dots & a_{pq}\end{array}\right|$$.

If we have a second $$p \times q$$ series matrix B.

$$B=\left|\begin{array}{ccc} b_{11}, & \dots & b_{1q}\\ \vdots & & \vdots\\ b_{p1}, & \dots & b_{pq}\end{array}\right|$$,

then A+B shall denote the $$p \times q$$ series matrix whose members are $$a_{hk}+b_{hk}$$.

2° If we have two matrices

$$A=\left|\begin{array}{ccc} a_{11}, & \dots & a_{1q}\\ \vdots & & \vdots\\ a_{p1}, & \dots & a_{pq}\end{array}\right|,\ B=\left|\begin{array}{ccc} b_{11}, & \dots & b_{1r}\\ \vdots & & \vdots\\ b_{p1}, & \dots & b_{qr}\end{array}\right|$$

where the number of horizontal rows of B, is equal to the number of vertical columns of A,


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 then by AB, the product of the matrices A and B, will be denoted the matrix

$$C=\left|\begin{array}{ccc} c_{11}, & \dots & c_{1r}\\ \vdots & & \vdots\\ c_{p1}, & \dots & c_{pr}\end{array}\right|$$

where

$$c_{hk}=a_{h1}b_{1k}+a_{h2}b_{2k}+\dots+a_{hq}b_{qk}\quad\left({h=1,2,\dots p\atop k=1,2,\dots r}\right)$$

these elements being formed by combination of the horizontal rows of A with the vertical columns of B. For such a point, the associative law $$(AB)S = A(BS)$$ holds, where S is a third matrix which has got as many horizontal rows as B (or AB) has got vertical columns.

For the transposed matrix of $$ C = AB$$, we have $$\bar{C}=\bar{B}\bar{A}$$.

3°. We shall have principally to deal with matrices with at most four vertical columns and for horizontal rows.

As a unit matrix (in equations they will be known for the sake of shortness as the matrix 1) will be denoted the following matrix (4 ✕ 4 series) with the elements.

For a 4✕4 series-matrix, Det A shall denote the determinant formed of the 4✕4 elements of the matrix. If $$Det A \ne 0$$, then corresponding to A there is a reciprocal matrix, which we may denote by $$A^{-1}$$ so that $$A^{-1} A = 1$$

A matrix

$$f=\left|\begin{array}{cccc} 0, & f_{12}, & f_{13}, & f_{14}\\ f_{21}, & 0, & f_{23}, & f_{24}\\ f_{31}, & f_{32}, & 0, & f_{34}\\ f_{41}, & f_{42}, & f_{43}, & 0\end{array}\right|$$,


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 in which the elements fulfill the relation $$f_{kh} = -f_{hk}$$, is called an alternating matrix. These relations say that the transposed matrix $$\bar{f}=-f$$. Then by $$f^{*}$$ will be the dual, alternating matrix

Then

i.e. We shall have a 4✕4 series matrix in which all the elements except those on the diagonal from left up to right down are zero, and the elements in this diagonal agree with each other, and are each equal to the above mentioned combination in (36).

The determinant of f is therefore the square of the combination, by $$Det^{\frac{1}{2}}f$$ we shall denote the expression

4°. A linear transformation

which is accomplished by the matrix

$$\mathsf{A}=\left|\begin{array}{cccc} \alpha_{11}, & \alpha_{12}, & \alpha_{13}, & \alpha_{14}\\ \alpha_{21}, & \alpha_{22}, & \alpha_{23}, & \alpha_{24}\\ \alpha_{31}, & \alpha_{32}, & \alpha_{33}, & \alpha_{34}\\ \alpha_{41}, & \alpha_{42}, & \alpha_{43}, & \alpha_{44}\end{array}\right|$$,

will be denoted as the transformation $$\mathsf{A}$$

By the transformation $$\mathsf{A}$$, the expression

$$x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4}$$

is changed into the quadratic form

$$\Sigma a_{hk}x'_{h}x'_{k}\qquad (h,k = 1, 2, 3, 4)$$


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 where

$$a_{hk} = \alpha_{1h}\alpha_{1k} + \alpha_{2h}\alpha_{2k} + \alpha_{3h}\alpha_{3k} + \alpha_{4h}\alpha_{4k}$$

are the members of a 4✕4 series matrix which is the product of $$\mathsf{\bar{A}A}$$, the transposed matrix of $$\mathsf{A}$$ into $$\mathsf{A}$$. If by the transformation, the expression is changed to

$$x^{'2}_{1} + x^{'2}_{2} + x^{'2}_{3} + x^{'2}_{4}$$

we must have

$$\mathsf{A}$$ has to correspond to the following relation, if transformation (38) is to be a -transformation. For the determinant of $$\mathsf{A}$$ it follows out of (39) that $$(Det \mathsf{A})^{2} = 1, Det \mathsf{A} = \pm 1$$.

From the condition (39) we obtain

i.e. the reciprocal matrix of $$\mathsf{A}$$ is equivalent to the transposed matrix of $$\mathsf{A}$$.

For $$\mathsf{A}$$ as transformation, we have further $$Det \mathsf{A} = + 1$$, the quantities involving the index 4 once in the subscript are purely imaginary, the other co-efficients are real, and $$\alpha_{44}>0$$.

5°. A space time vector of the first kind which is represented by the 1✕4 series matrix,

is to be replaced by $$s\mathsf{A}$$ in case of a transformation

A space-time vector of the 2nd kind with components $$f_{23},\ f_{31},\ f_{12},\ f_{14},\ f_{24},\ f_{34}$$ shall be represented by the alternating matrix

and is to be replaced by $$\mathsf{\overline{A}}f\mathsf{A}=\mathsf{A}^{-1}f\mathsf{A}$$ in case of a transformation [see the rules in § 5 (23) (24)]. Therefore referring to the expression


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 (37), we have the identity $$Det^{\frac{1}{2}}(\mathsf{\overline{A}}f\mathsf{A})=Det\ \mathsf{A}\ Det^{\frac{1}{2}}f$$. Therefore $$Det^{\frac{1}{2}}f$$ becomes an invariant in the case of a transformation [see eq. (26) Sec. § 5].

Looking back to (36), we have for the dual matrix

from which it is to be seen that the dual matrix $$f^{*}$$ behaves exactly like the primary matrix f, and is therefore a space time vector of the II kind; $$f^{*}$$ is therefore known as the dual space-time vector of f with components $$f_{14},\ f_{24},\ f_{34},\ f_{23},\ f_{31},\ f_{12}$$.

6°.If w and s are two space-time vectors of the 1st kind then by $$w\bar{s}$$ (as well as by $$s\bar{w})$$) will be understood the combination

In case of a transformation $$\mathsf{A}$$, since $$(w\mathsf{A})(\mathsf{\bar{A}}\bar{s})=w\bar{s}$$ this expression is invariant. — If $$w\bar{s}=0$$, then w and s are perpendicular to each other.

Two space-time rectors of the first kind w, s gives us a 2✕4 series matrix

$$\left|\begin{array}{cccc} w_{1}, & w_{2}, & w_{3}, & w_{4}\\ s_{1}, & s_{2}, & s_{3}, & s_{4}\end{array}\right|$$

Then it follows immediately that the system of six magnitudes

behaves in case of a -transformation as a space-time vector of the II. kind. The vector of the second kind with the components (44) are denoted by [w,s]. We see easily that $$Det^{\frac{1}{2}}[w,s] =0$$. The dual vector of [w,s] shall be written as [w,s]*.

If w is a space-time vector of the 1st kind, f of the second kind, wf signifies a 1✕4 series matrix. In case of a -transformation $$\mathsf{A}$$, w is changed into $$w'=w\mathsf{A}$$, f into $$f'=\mathsf{A}^{-1}f\mathsf{A}$$, therefore $$w'f'=w\mathsf{A}\ \mathsf{A}^{-1}f\mathsf{A}=(wf)\mathsf{A}$$, i.e., wf is transformed as a space-time vector of the 1st kind. We can verify, when w is a space-time vector of the 1st kind, f of the 2nd kind, the important identity


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 Margaret (, pp. 673, 721, 772, 778-780;, No. 2602).

Forman was rewarded by permission to hold benefice in England (24 May 1498), and with a pension of a thousand merks 'till he "be promovit to a bishoprik or abbasy' (13 Oct.) (, Pref. p. clviii); and by the grant of the wardship of the Rutherford heiress (12 Nov. 1502), who ultimately married his brother, Sir John Forman (Reg. of Great Seal, Nos. 2677, 3612). By 8 Oct. 1501 he was postulate of Moray, and by 12 Nov. 1502 full bishop of this see (ib. No. 2677;, p. 778). In 1502 he was also commendator of Pittenweem in Fife and of Cottinghame in England (Reg. of Great Seal, No. 2677). On 30 July 1509 Forman was appointed ambassador to Henry VIII. Early in 1511 (January?) James IV commissioned him to bring about a general peace among Christian princes with a view to a great crusade. For the next few years he was occupied in this work. The pope, Julius II, determined to make him a cardinal (, i. 1459, 1461, 1643, &c.) Forman succeeded in making a truce between Julius and Louis XII (id. ii. 776), but not in securing universal peace. James IV made an alliance with Louis for an attack on England, and Louis made the ambassador archbishop of Bourges, for which see, after a contested election, he did homage on 12 Sept. 1513 (, i. 318-21; Gallia Christiana, ii. 93-4). Henry, suspecting the king of France's intentions, refused the bishop a safe-conduct through his country (12 Nov. 1512); but Forman was abroad by April 1513, and sent news of Julius II's death to Scotland. In these days he was reckoned omnipotent with James (, No. 3651). Leo X, who succeeded Julius II in the papacy, had promoted the Bishop of Moray to St. Andrews (by 27 Jan. 1514), then vacant by the death of Alexander Stewart, James IV's son, who was slain at Flodden (No. 4682,, p. 95). His election to this see was contested by [q. v.] and John Hepburn. It was generally believed that Forman was supported by the new regent, the Duke of Albany, whom, however, the bishop did not accompany to Scotland. In March 1515 the bishop was at Lyons, and about 3 June he left Bruges for Scotland. Leo had already appointed the new archbishop his legate in Scotland, but promised to revoke the commission on hearing of Henry VIII's disapproval (2 March 1515) (, ii. Nos. 210, 291, 365, 576, 593).

The archbishop was so unpopular in Scotland that in January 1515 it was reported that the lords would league against him, and that 'the duke will be the werr ressavit if he tak his part.' His great offence seems to have been the accumulation of ecclesiastical benefices which the lords thought would be better in the hands of members of their own family. Besides the offices already noticed he had held the monasteries of Dryburgh, Dunfermline, Kilwinning, and Arbroath, and was accused of aiming at the see of Glasgow also (ib. ii. Nos. 27, 50, 776;, p. 101). He appears, however, to have very soon resigned everything, except St. Andrews and Dunfermline (No. 776); and in February 1516 the three competitors for St. Andrews consented to abide by Albany's decision. Albany gave St. Andrews to Forman, and promoted James Hepburn to the see of Moray (, p. 106). In May 1516 Albany was still urging his claims to the cardinalate (No. 1869); and it appears that, notwithstanding Henry VIII's opposition, he was ' legatus natus cum potestate legati a latere' (regni Scotise) (Great Seal, ii. No. 389). As bishop of Moray he had procured for this see an exemption from the authority of St. Andrews, much to the displeasure of James IV and his son. As archbishop of St. Andrews he sought to limit, though he could not at once annul, the exemption and authority of Glasgow (, pp. ccxxvi-ccxxviii). As primate of Scotland he issued an important series of constitutions in 1515-16, which are printed in the 'Scotiæ Concilia ' (pp. cclxx, &c.) He died in 1522, and was buried at Dunfermline (Scotichron. p. 245).

Forman is praised for his generosity, his politicalcapacity, and his scholarship. Demster makes Forman the author of three works: Robertson, in the notes to his 'Scotiæ Concilia,' prints some interesting documents showing the debts Forman incurred in his candidature for the cardinalate, and how the bishop laid his ill-success to the charge of Henry VIII, who would not suffer him to pass through England (i. p. cxxvi).
 * 1) 'Contra Lutherum.'
 * 2) 'De Stoica Philosophia.'
 * 3) 'Collectanea Decretalium ' (ib. p. 243).


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 The sum of the two space time vectors of the second kind on the left side is to be understood in the sense of the addition of two alternating matrices.

For example, for $$w_{1} = 0,\ w_{2} = 0,\ w_{3} =0,\ w_{4} = i$$

The fact that in this special case, the relation is satisfied, suffices to establish the theorem (45) generally, for this relation has a covariant character in case of a transformation, and is homogeneous in $$w_{1},\ w_{2},\ w_{3},\ w_{4}$$.

After these preparatory works let us engage ourselves with the equations (C,) (D,) (E) by means which the constants $$\epsilon,\ \mu,\ \sigma$$ will be introduced.

Instead of the space vector $$\mathfrak{w}$$, the velocity of matter, we shall introduce the space-time vector of the first kind w with the components.

$$w_{1}=\frac{\mathfrak{w}_{x}}{\sqrt{1-\mathfrak{w}^{2}}},\ w_{2}=\frac{\mathfrak{w}_{y}}{\sqrt{1-\mathfrak{w}^{2}}},\ w_{3}=\frac{\mathfrak{w}_{z}}{\sqrt{1-\mathfrak{w}^{2}}},\ w_{4}=\frac{i}{\sqrt{1-\mathfrak{w}^{2}}}$$

where

and $$-iw_{4} > 0$$..

By F and f shall be understood the space time vectors of the second kind $$\mathfrak{M},\ -i\mathfrak{E}$$, $$\mathfrak{m},\ -i\mathfrak{e}$$.

In $$\Phi = -wF$$, we have a space time vector of the first kind with components

$$\begin{array}{ccccccccc} \Phi_{1} & = & &  & w_{2}F_{12} & + & w_{3}F_{13} & + & w_{4}F_{14},\\ \Phi_{2} & = & w_{1}F_{21} & &  & + & w_{3}F_{23} & + & w_{4}F_{24},\\ \Phi_{3} & = & w_{1}F_{31} & + & w_{2}F_{32} & &  & + & w_{4}F_{34},\\ \Phi_{4} & = & w_{1}F_{41} & + & w_{2}F_{42} & + & w_{3}F_{43}.\end{array}$$.

The first three quantities $$\Phi_{1},\ \Phi_{2},\ \Phi_{3}$$ are the components of the space-vector


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 and further

Because F is an alternating matrix,

i.e. $$\Phi$$ is perpendicular to the vector to w; we can also write

I shall call the space-time vector $$\Phi$$ of the first kind as the Electric Rest Force.

Relations analogous to those holding between $$-wF,\ \mathfrak{E,\ M,\ w}$$, hold amongst $$-wf,\ \mathfrak{e,\ m,\ w}$$, and in particular -wf is normal to w. The relation (C) can be written as

The expression (wf) gives four components, but the fourth can be derived from the first three.

Let us now form the time-space vector 1st kind $$\Psi=iwf^{*}$$, whose components are

$$\begin{array}{cccccccccc} \Psi_{1} & = & -i( & &  & w_{2}f_{34} & + & w_{3}f_{42} & + & w_{4}f_{23}),\\ \Psi_{2} & = & -i( & w_{1}f_{43} & &  & + & w_{3}f_{14} & + & w_{4}f_{31}),\\ \Psi_{3} & = & -i( & w_{1}f_{24} & + & w_{2}f_{41} & &  & + & w_{4}f_{12}),\\ \Psi_{4} & = & -i( & w_{1}f_{32} & + & w_{2}f_{13} & + & w_{3}f_{21} & & ).\end{array}$$

Of these, the first three $$\Psi_{1},\ \Psi_{2},\ \Psi_{3}$$ are the x-, y-, z-components of the space-vector

and further

Among these there is the relation

which can also be written as


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The vector $$\Psi$$ is perpendicular to w; we can call it the Magnetic rest-force.

Relations analogous to these hold among the quantities $$iwF^{*},\mathfrak{M,E,w}$$ and Relation (D) can be replaced by the formula

We can use the relations (C) and (D) to calculate F and f from $$\Phi$$ and $$\Psi$$, we have

and applying the relation (45) and (46), we have

i.e.

Let us now consider the space-time vector of the second kind $$[\Phi \Psi]$$, with the components

Then the corresponding space-time vector of the first kind

vanishes identically owing to equations 49) and 53).

Let us now take the vector of the 1st kind

with the components

$$\Omega_{1}=-i\left|\begin{array}{ccc} w_{2}, & w_{3}, & w_{4}\\ \Phi_{2}, & \Phi_{3}, & \Phi_{4}\\ \Psi_{2}, & \Psi_{3}, & \Psi_{4}\end{array}\right|$$, etc.

Then by applying rule (45), we have


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i,e.

$$\Phi_{1}\Psi_{2} - \Phi_{2}\Psi_{1} = i(w_{3}\Omega_{4} - w_{4}\Omega_{3})$$, etc..

The vector $$\Omega$$ fulfills the relation

which we can write as

$$\Omega_{4}=i(\mathfrak{w}_{x}\Omega_{1}+\mathfrak{w}_{y}\Omega_{2}+\mathfrak{w}_{z}\Omega_{3})$$

and $$\Omega$$ is also normal to w. In case $$\mathfrak{w} =0$$, we have $$\Phi_{4} = 0,\ \Psi_{4} = 0,\ \Omega_{4} = 0$$, and

I shall call $$\Omega$$, which is a space-time vector 1st kind the Rest-Ray.

As for the relation E), which introduces the conductivity $$\sigma$$, we have

$$-w\bar{s}=-(w_{1}s_{1}+w_{2}s_{2}+w_{3}s_{3}+w_{4}s_{4})=\frac{-\left|\mathfrak{w}\right|s_{\mathfrak{w}}+\varrho}{\sqrt{1-\mathfrak{w}^{2}}}=\varrho'$$

This expression gives us the rest-density of electricity (see §8 and §4). Then

represents a space-time vector of the 1st kind, which since $$w\bar{w}=1$$, is normal to w, and which I may call the rest-current. Let us now conceive of the first three component of this vector as the x-, y-, z co-ordinates of the space-vector, then the component in the direction of $$\mathfrak{w}$$ is

$$\mathfrak{s_{w}}-\frac{\left|\mathfrak{w}\right|\varrho'}{\sqrt{1-\mathfrak{w}^{2}}}=\frac{\mathfrak{s_{w}}-\left|\mathfrak{w}\right|\varrho}{\sqrt{1-\mathfrak{w}^{2}}}=\frac{\mathfrak{F_{w}}}{1-\mathfrak{w}^{2}}$$

and the component in a perpendicular direction is $$\mathfrak{s_{\bar{w}}}=\mathfrak{F_{\bar{w}}}$$.

This space-vector is connected with the space-vector $$\mathfrak{F}=\mathfrak{s}-\varrho\mathfrak{w}$$, which we denoted in § 8 as the conduction-current.

Now by comparing with $$\Phi = -wF$$, the relation (E) can be brought into the form


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 This formula contains four equations, of which the fourth follows from the first three, since this is a space-time vector which is perpendicular to w.

Lastly, we shall transform the differential equations (A) and (B) into a typical form.

§ 12. The Differential Operator Lor.
A 4✕4 series matrix

with the condition that in case of a transformation it is to be replaced by $$\mathsf{\bar{A}}S\mathsf{A}$$, may be called a space-time matrix of the II. kind. We have examples of this in : —

 * 1) the alternating matrix f, which corresponds to the space-time vector of the II. kind, —

 * 2) the product fF of two such matrices, for by a transformation $$\mathsf{A}$$, it is replaced by $$(\mathsf{A}^{-1}f\mathsf{A})(\mathsf{A}^{-1}F\mathsf{A})=\mathsf{A}^{-1}fF\mathsf{A}$$,

 * 3) further when $$w_{1},\ w_{2},\ w_{3},\ w_{4}$$ and $$\Omega_{1},\ \Omega_{2},\ \Omega_{3},\ \Omega_{4}$$ are two space-time vectors of the 1st kind, the 4✕4 matrix with the $$S_{hk}=w_{h}\Omega_{k}$$,

 * lastly in a multiple L of the unit matrix of 4✕4 series in which all the elements in the principal diagonal are equal to L, and the rest are zero.

We shall have to do constantly with functions of the space-time point x, y, z, it, and we may with advantage employ the 1✕4 series matrix, formed of differential symbols, —

$$\left|\frac{\partial}{\partial x},\ \frac{\partial}{\partial y},\ \frac{\partial}{\partial z},\ \frac{\partial}{i\partial t}\right|$$,

or


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 For this matrix I shall use the shortened from lor.

Then if S is, as in (62), a space-time matrix of the II. kind, by lor S' will be understood the 1✕4 series matrix

$$\left|K_{1},\ K_{2},\ K_{3},\ K_{4}\right|$$

where

When by a transformation $$\mathsf{A}$$, a new reference system $$x'_{1},\ x'_{2},\ x'_{3},\ x'_{4}$$ is introduced, we can use the operator

$$lor'=\left|\frac{\partial}{\partial x'_{1}},\ \frac{\partial}{\partial x'_{2}},\ \frac{\partial}{\partial x'_{3}},\ \frac{\partial}{x'_{4}}\right|$$

Then S is transformed to $$S'=\bar{\mathsf{A}}S\mathsf{A}=\left|S'_{hk}\right|$$, so by lor' Sis meant the 1✕4 series matrix, whose element are

$$K'_{k}=\frac{\partial S'_{1k}}{\partial x'_{1}}+\frac{\partial S'_{2k}}{\partial x'_{2}}+\frac{\partial S'_{3k}}{\partial x'_{3}}+\frac{\partial S'_{4k}}{\partial x'_{4}}\qquad (k=1,2,3,4)$$

Now for the differentiation of any function of (x y z t) we have the rule

so that, we have symbolically

$$lor'=lor\ (\mathsf{A}$$

Therefore it follows that

i.e., lor S behaves like a space-time vector of the first kind.

If L is a multiple of the unit matrix, then by lor L will be denoted the matrix with the elements


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If $$s=\left|s_{1},\ s_{2},\ s_{3},\ s_{4}\right|$$ is a space-time vector of the 1st kind, then

In case of a transformation $$\mathsf{A}$$, we have

i.e., lor s is an invariant in a {sc|Lorentz}}-transformation.

In all these operations the operator lor plays the part of a space-time vector of the first kind.

If f represents a space-time vector of the second kind, -lor f denotes a space-time vector of the first kind with the components

$$\begin{array}{ccccccc} & & \frac{\partial f_{12}}{\partial x_{2}} & + & \frac{\partial f_{13}}{\partial x_{3}} & + & \frac{\partial f_{14}}{\partial x_{4}},\\ \\\frac{\partial f_{21}}{\partial x_{1}} & &  & + & \frac{\partial_{23}}{\partial x_{3}} & + & \frac{\partial_{24}}{\partial x_{4}},\\ \\\frac{\partial f_{31}}{\partial x_{1}} & + & \frac{\partial_{32}}{\partial x_{2}} & &  & + & \frac{\partial_{34}}{\partial x_{4}},\\ \\\frac{\partial f_{41}}{\partial x_{1}} & + & \frac{\partial_{42}}{\partial x_{2}} & + & \frac{\partial_{43}}{\partial x_{3}},\end{array}$$

So the system o£ differential equations (A) can be expressed in the concise form

and the system (B) can be expressed in the form

Referring back to the definition (67) for $$lor\ \bar{s}$$, we find that the combinations $$lor (\overline{lor\ f})$$ and $$lor (\overline{lor\ F^{*}})$$ vanish identically, when f and F* are alternating matrices. Accordingly it follows out of (A), that


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 while the relation

signifies that of the four equations in (B), only three represent independent conditions.

I shall now collect the results.

Let w denote the space-time vector of the first kind

$$\frac{\mathfrak{w}}{\sqrt{1-\mathfrak{w}^{2}}}, \frac{i}{\sqrt{1-\mathfrak{w}^{2}}}$$

($$\mathfrak{w}$$ = velocity of matter),

F the space-time vector of the second kind $$\mathfrak{M},\ -i\mathfrak{E}$$ ($$\mathfrak{M}$$ = magnetic induction, $$\mathfrak{E}$$ = Electric force),

f the space-time vector of the second kind $$\mathfrak{m},\ -i\mathfrak{e}$$ ($$\mathfrak{m}$$ = magnetic force,)

$$\mathfrak{e}$$ = Electric Induction.

s the space-time vector of the first kind $$\mathfrak{s}, i\varrho$$ ($$\varrho$$ = electrical space-density,)

$$\mathfrak{s}-\varrho\mathfrak{w}$$ = conductivity current,

$$\epsilon$$ = dielectric constant,

$$\mu$$ = magnetic permeability,

$$\sigma$$ = conductivity.

then the fundamental equations for electromagnetic processes in moving bodies are

$$w\bar{w}=-1$$, and wF, wf, wF*, wf*, $$s+(w\bar{s})w$$ which are space-time vectors of the first kind are all normal to w, and for the system {B}, we have

$$lor\ (\overline{lor\ F^{*}})=0$$

Bearing in mind this last relation, we see that we have as many independent equations at our disposal as are necessary for determining the processes when proper fundamental data are given, where the motion of matter, thus the vector $$\mathfrak{w}$$ as a function of x, y, z, t are given.


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§ 13. The Product of the Field-vectors fF.
Finally let us enquire about the laws which lead to the determination of the vector w as a function of x, y, z, t. In these investigations, the expressions which are obtained by the multiplication of two alternating matrices

$$f=\left|\begin{array}{cccc} 0, & f_{12}, & f_{13}, & f_{14}\\ f_{21}, & 0, & f_{23}, & f_{24}\\ f_{31}, & f_{32}, & 0, & f_{34}\\ f_{41}, & f_{42}, & f_{43}, & 0\end{array}\right|,\ F=\left|\begin{array}{cccc} 0, & F_{12}, & F_{13}, & F_{14}\\ F_{21}, & 0, & F_{23}, & F_{24}\\ F_{31}, & F_{32}, & 0, & F_{34}\\ F_{41}, & F_{42}, & F_{43}, & 0\end{array}\right|$$

are of much importance. Let us write.

Then (71)

Let L now denote the symmetrical combination of the indices 1, 2, 3, 4, given by

Then we shall have

In order to express in a real form, we write

Now


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Page:Dictionary of National Biography volume 19.djvu/444

 FORMAN, SIMON (1552–1611), astrologer and quack-doctor, was fifth son of the eight children of William Forman and his wife Mary, daughter of John Foster, by Marianna Hallam. Simon's grandfather, Richard Forman, was governor of Wilton Abbey before the suppression of the monasteries, and when the abbey was made over to William Herbert, earl of Pembroke, held some office about the park. Dying in 1556 Richard was buried at Foulson, Devonshire. Simon's father, William, born at Quidhampton, Wiltshire, in 1524, served as page to Lady Willoughby; married in 1544 Mary Foster, who came from the neighbourhood of Andover; was deprived of property which he should have inherited from his father, and died 1 Jan. 1564, being buried at Foulson. Simon's mother lived to the age of ninety-seven, dying in 1602, and being buried with her husband. She was vigorous to the last, walking two miles within a fortnight of her death. Simon, who paid much attention to the genealogy of his family, claimed descent from some apocryphal Richard Forman, earl of Devonshire in the time of William I, who is said to have built the church of St. James at Exeter. A Sir George Forman was created K.B. in 1485, and Sir William Forman, haberdasher, was lord mayor of London in 1538-9. With both of these Simon declared that he was connected.

Simon was born at Quidhampton, 30 Dec. 1552. Lilly's statement that he was son of a chandler, and was born in Westminster, is untrue. He suffered as a child from bad dreams, presaging 'the troubles of his riper years.' A clergyman of Salisbury, named Riddout, who had formerly been a cobbler, and who removed to Quidhampton, when the plague raged in Salisbury, first taught Simon his accidence. Afterwards he went for two years to a free school in the Close at Salisbury, under a master named Boole or Bowie, 'a severe and furious man,' and was thence removed to the care of one Minterne, prebendary of the cathedral, a person of unpleasantly frugal habits. The death of Simon's father in January 1563-4 left him destitute. His mother neglected him, and made him do menial work. On 8 Feb. 1567 he apprenticed himself to Matthew Comin, a general dealer, of Salisbury. His master treated him kindly, but his mistress had a violent temper, and he left after a serious quarrel with her (29 June 1572). He had Kept up his studies by getting a schoolboy who lodged with his master to teach him at night all he learned by day. He went through the Isle of Wight on his way home to Quidhampton. His mother still declined to maintain him; he became a schoolmaster near his native place, and received 40s. for half a year's work. On 20 May 1573 Simon made his way to Oxford with a friend, Thomas Ridear. He entered Magdalen College as a poor scholar, and studied at the school attached to the college. John Thornborough, a demy of the college (afterwards bishop of Limerick), and his friend Robert Pinkney of St. Mary's Hall, two pleasure-loving young gentlemen, took him into their service. He had to attend them on hunting expeditions to Shotover, and to walk to Cowley almost every day to assist them in the courtship of a young lady for whose hand they were both suitors. Forman left Oxford 12 Sept. 1574, and until midsummer 1578 found employment as an usher in several small schools at Wilton, Ashmore, and Salisbury. Early in 1579 he was lodging in the parsonage of Fisherton, and it was about that date that he discovered what he claimed to be his miraculous powers. 'I did prophesy,' he records in his diary, 'the truth of many things which afterwards came to pass, and the very spirits were subject to me.' In June he was robbed of his goods and books, and, on the information of one William Estcourt, was sent to gaol for sixty weeks, apparently on the ground of practising magic. This proved the first of a long series of similar experiences. He was set free 14 July 1580, begged his way to London, and obtained work as a carpenter at Greenwich. On 14 Aug. he first practised his healing arts, which cured one Henry Jonson of London of a pulmonary complaint. In September he accompanied his patient to Holland; stayed for a fortnight at the Hague, and largely increased his knowledge of astrology and medicine. He was home again in October, and went to Quidhampton for a year, 'curing sick and lame folk,' but the justices at the Lent assizes bound him over to abstain from his quackery, and he had often to 'thresh and dig and hedge' for his living. In the autumn of 1581 he hired a house at Salisbury, and renewed his practice of physic and surgery. In August 1582 he went to sea, and landed in Studland. On his return he travelled much, but finally set up in the next year (1583) in London as a doctor and astrologer. There he remained till the end of his life. He lived at different times in New Street, St. Thomas's Churchyard, Philpot Street, and elsewhere. The authorities invariably condemned his methods of gaining a livelihood, and he repeatedly suffered imprisonment, but gradually he acquired a lucrative practice, although for the most part a disreputable one. The Bishop of London summoned him in 1583; he was


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User talk:NotFromUtrecht

— billinghurst  sDrewth  10:57, 28 January 2011 (UTC)

Did you stop to think...
... that whole point of Wikisource is to faithfully reproduce works as they were intially published?

You recently changed the title of a Wordsworth poem that no longer matches the source it was taken from as given on in its infobox on it's talkpage. If you had a "scholarlly edition" of the poem you should have added it separately, cited that source on that talk page and disambiguate both titles on a separate disambiguation page. All you've done is defiled the credibility and integrity of the work here on Wikisource. Why and what do you suppose we should do about this now? -- George Orwell III (talk) 10:51, 19 May 2012 (UTC)
 * The text of the source has already deviated fairly substantially from the source text linked on the talk page -- look at the formatting of the first lines in each paragraph or the use of em dashes for instance. As I said in my edit summary, my page move and edits reflect the titles given in the Lyrical Ballads (both the 1798 and 1800 editions) which is where they were originally published, and in modern scholarly editions -- such as WJB Owen's edition of the 1798 Lyrical Ballads or in Stephen Gill's edition of Wordsworth's poems. Since the text as it appears now closely matches the 1798 edition (see Gutenberg version), surely we could just change the source to that? After all, the formatting which has subsequently been applied to the poetry-online.org source text has apparently taken this edition as its model. NotFromUtrecht (talk) 12:00, 19 May 2012 (UTC)
 * Yes. Please do those changes to reflect the corresponding source for current content. I will delete the redirects created by the move (title change). Thanks for following up. -- George Orwell III (talk) 13:08, 19 May 2012 (UTC)

title: User talk:NotFromUtrecht


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These quantities are all real. In the theory for bodies at rest, the combinations ($$X_{x},\ X_{y},\ X_{z},\ Y_{x},\ Y_{y},\ Y_{z},\ Z_{x},\ Z_{y},\ Z_{z}$$ are known as 's Stresses", $$T_{x},\ T_{y},\ T_{z}$$ are known as the Poynting's Vector, $$T_{t}$$ as the electromagnetic energy-density, and L as the Langrangian function.

On the other hand, by multiplying the alternating matrices of f and F, we obtain

and hence, we can put

where by L, we mean L-times the unit matrix, i.e. the matrix with elements

$$\left|Le_{hk}\right|\ \left(\begin{array}{c} e_{hh}=1,\ e_{hk}=0,\ h\gtrless k\\ h,k=1,2,3,4\end{array}\right)$$

Since here $$SL = LS$$, we deduce that,

$$F^{*}f^{*}fF = (-S-L)(S-L) = -SS + L^{2}$$,

and find, since $$f^{*}f = Det^{\frac{1}{2}}f,\ F^{*}F = Det\frac{1}{2}F$$, we arrive at the interesting conclusion

i.e. the product of the matrix S into itself can be expressed as the multiple of a unit matrix — a matrix in which all the elements except those in the principal diagonal are zero, the elements in the principal diagonal are all equal


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User talk:Liumx

— billinghurst  sDrewth  10:58, 28 January 2011 (UTC)

title: User talk:Liumx


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 and have the value given on the right-hand side of (79). Therefore the general relations

h, k being unequal indices in the series 1, 2, 3, 4, and

for h = 1,2,3,4.

Now if instead of F and f in the combinations (72) and (73), we introduce the electrical rest-force $$\Phi$$, the magnetic rest-force $$\Psi$$ and the rest-ray $$\Omega$$ [(55), (56) and (57)], we can pass over to the expressions, —

Here we have

The right side of (82) as well as L is an invariant in a transformation, and the 4✕4 element on the right side of (83) as well as $$S_{hk}$$, represent a space time vector of the second kind. Remembering this fact, it suffices, for establishing the theorems (82) and (83) generally, to prove it for the special case $$w_{1} = 0,\ w_{2} = 0,\ w_{3} = 0,\ w_{4} = i$$. But for this case $$\mathfrak{w} = 0$$, we immediately arrive at the equations (82) and (83) by means (45), (51), (60) on the one hand, and $$\mathfrak{e}=\epsilon\mathfrak{E},\ \mathfrak{M}=\mu\mathfrak{m}$$ on the other hand.

The expression on the right-hand side of (81), which equals

$$=\left(\frac{1}{2}(\mathfrak{mM}-\mathfrak{eE})\right)^{2}+(\mathfrak{em})(\mathfrak{EM})$$

is $$\geqq0$$, because $$(\mathfrak{em})=\epsilon\Phi\bar{\Psi},\ (\mathfrak{EM})=\mu\Phi\bar{\Psi}$$, now referring back to 79), we can denote the positive square root of this expression as $$Det^{\frac{1}{2}}S$$.

Since $$\bar{f}=-f,\ \bar{F}=-F$$, we obtain for $$\bar{S}$$, the transposed matrix of S, the following relations from (78),

Then is