text:
United Nations General Assembly Resolution 3314

29th Session  Agenda item: 86

Committee: 6

Vote: Adopted without a vote

Ref: 3105(XXVIII), 2967(XXVII), 2781(XXVII), 2644(XXV), 2549(XXIV), 2420(XXIII), 2330(XXII), 1181(XII), 895(IX), 688(VII), 599(VI)

The General Assembly,

Having considered the report of the Special Committee on the Question of Defining Aggression, established pursuant to its resolution 2330(XXII) of 18 December 1967, covering the work of its seventh session held from 11 March to 12 April 1974, including the draft Definition of Aggression adopted by the Special Committee by consensus and recommended for adoption by the General Assembly,[FN1]

Deeply, convinced that the adoption of the Definition of Aggression would contribute to the strengthening of international peace and security,

1. Approves the Definition of Aggression, the text of which is annexed to the present resolution;

2. Expresses its appreciation to the Special Committee on the Question of Defining Aggression for its work which resulted in the elaboration of the Definition of Aggression;

3. Calls upon all States to refrain from all acts of aggression and other uses of force contrary to the Charter of the United Nations and the Declaration on Principles of International Law concerning Friendly Relations and Cooperation among States in accordance with the Charter of the United Nations;[FN2]

4. Calls the attention of the Security Council to the Definition of Aggression, as set out below, and recommends that it should, as appropriate, take account of that Definition as guidance in determination, in accordance with the Charter, the existence of an act of aggression.

2319th plenary meeting

14 December 1974

Annex

Definition of Aggression

The General Assembly,

Basing itself on the fact that one of the fundamental purposes of the United Nations is to maintain international peace and security and to take effective collective measures for the prevention and removal of threats to the peace, and for the suppression of acts of aggression or other breaches of the peace,

Recalling that the Security Council, in accordance with Article 39 of the Charter of the United Nations, shall determine the existence of any threat to the peace, breach of the peace or act of aggression and shall make recommendations, or decide what measures shall be taken in accordance with Articles 41 and 42, to maintain or restore international peace and security,

Recalling also the duty of States under the Charter to settle their international disputes by peaceful means in order not to endanger international peace, security and justice,

Bearing in mind that nothing in this Definition shall be interpreted as in any way affecting the scope of the provisions of the Charter with respect to the functions and powers of the organs of the United Nations,

Considering also that, since aggression is the most serious and dangerous form of the illegal use of force, being fraught, in the conditions created by the existence of all types of weapons of mass destruction, with the possible threat of a world conflict and all its catastrophic consequences, aggression should be defined at the present stage,

Reaffirming the duty of States not to use armed force to deprive peoples of their right to self-determination, freedom and independence, or to disrupt territorial Integrity,

Reaffirming also that the territory of a State shall not be violated by being the object, even temporarily, of military occupation or of other measures of force taken by another State in contravention of the Charter, and that it shall not be the object of acquisition by another State resulting from such measures or the threat thereof,

Reaffirming also the provisions of the Declaration on Principles of International Law concerning Friendly Relations and Cooperation among States in accordance with the Charter of the United Nations,

Convinced that the adoption of a definition of aggression ought to have the effect of deterring a potential aggressor, would simplify the determination of acts of aggression and the implementation of measures to suppress them and would also facilitate the protection of the rights and lawful interests of, and the rendering of assistance to, the victim,

Believing that, although the question whether an act of aggression has been committed must be considered in the light of all the circumstances of each particular case, it is nevertheless desirable to formulate basic principles as guidance for such determination,

Adopts the following Definition of Aggression:[FN3]

Article I

Aggression is the use of armed force by a State against the sovereignty, territorial integrity or political independence of another State, or in any other manner inconsistent with the Charter of the United Nations, as set out in this Definition.

Explanatory note: In this Definition the term "State":

(a) Is used without prejudice to questions of recognition or to whether a State is a member of the United Nations;

(b) Includes the concept of a "group of States" where appropriate.

Article 2

The First use of armed force by a State in contravention of the Charter shall constitute prima facie evidence of an act of aggression although the Security Council may, in conformity with the Charter, conclude that a determination that an act of aggression has been committed would not be justified in the light of other relevant circumstances, including the fact that the acts concerned or their consequences are not of sufficient gravity.

Article 3

Any of the following acts, regardless of a declaration of war, shall, subject to and in accordance with the provisions of article 2, qualify as an act of aggression:

(a) The invasion or attack by the armed forces of a State of the territory of another State, or any military occupation, however temporary, resulting from such invasion or attack, or any annexation by the use of force of the territory of another State or part thereof;

(b) Bombardment by the armed forces of a State against the territory of another State or the use of any weapons by a State against the territory of another State;

(c) The blockade of the ports or coasts of a State by the armed forces of another State;

(d) An attack by the armed forces of a State on the land, sea or air forces, or marine and air fleets of another State;

(e) The use of armed forces of one State which are within the territory of another State with the agreement of the receiving State, in contravention of the conditions provided for in the agreement or any extension of their presence in such territory beyond the termination of the agreement;

(f) The action of a State in allowing its territory, which it has placed at the disposal of another State, to be used by that other State for perpetrating an act of aggression against a third State;

(g) The sending by or on behalf of a State of armed bands, groups, irregulars or mercenaries, which carry out acts of armed force against another State of such gravity as to amount to the acts listed above, or its substantial involvement therein.

Article 4

The acts enumerated above are not exhaustive and the Security Council may determine that other acts constitute aggression under the provisions of the Charter.

Article 5

1. No consideration of whatever nature, whether political, economic, military or otherwise, may serve as a justification for aggression.

2. A war of aggression is a crime against international peace. Aggression gives rise to international responsibility.

3. No territorial acquisition or special advantage resulting from aggression is or shall be recognized as lawful.

Article 6

Nothing in this Definition shall be construed as in any way enlarging or diminishing the scope of the Charter, including its provisions concerning cases in which the use of force is lawful.

Article 7

Nothing in this Definition, and in particular article 3, could in any way prejudice the right to self-determination, freedom and independence, as derived from the Charter, of peoples forcibly deprived of that right and referred to in the Declaration on Principles of International Law concerning Friendly Relations and Cooperation among States in accordance with the Charter of the United Nations, particularly peoples under colonial and racist regimes or other forms of alien domination: nor the right of these peoples to struggle to that end and to seek and receive support, in accordance with the principles of the Charter and in conformity with the above-mentioned Declaration.

Article 8

In their interpretation and application the above provisions are interrelated and each provision should be construed in the context of the other provisions.

FOOTNOTES

1. Official Records of the General Assembly, Twenty-ninth Session, Supplement No. 19 (A/9619 and Corr. 1).

2. Resolution 2625 (XXV), annex.

3. Explanatory notes on articles 3 and 5 are to be found in paragraph 20 of the Report of the Special Committee on the Question of Defining Aggression (Official Records of the General Assembly, Twenty-ninth Session, Supplement No. 19 (A/9619 and Corr. 1). Statements on the Definition are contained in paragraphs 9 and 10 of the report of the Sixth Committee (A/9890).

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text: Index:Little Red Riding Hood.djvu

title: Index:Little Red Riding Hood.djvu


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Page:Flora Australiensis Volume 5.djvu/330

 318

—Fruit an indehiscent nut or drupe, either 1-seeded or if 2-seeded the seeds separated by a complete dissepiment continuous with the endocarp. Flowers usually solitary within each bract, in cones or spikes or solitary, very rarely racemose, the spikes often shortened into heads.

—Anthers all perfect, or very rarely the upper one abortive, with 2 parallel cells adnate to the connectivum, inserted at the base of the short spreading lamin&aelig of the perianth. Ovule 1 or in a very few species a second one more or less developed. Stigma at the point of the straight style end. Fruit a dry nut.


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Rh "I have had no little sorrow in my day," he would say; "and goodness knows I have not lived long. God has not given me happiness, but what He may give, so will it be. That's so, friend Vasily Stepanych."

Vasily Stepanych knocked the ashes out of his pipe against a rail, stood up, and said: "It is not luck which follows us in life, but human beings. There is no crueller beast on this earth than man. Wolf does not eat wolf, but man will readily devour man."

"Come, friend, don't say that; a wolf eats wolf."

"The words came into my mind and I said it. All the same, there is nothing crueller than man. If it were not for his wickedness and greed, it would be possible to live. Everybody tries to sting you to the quick, to bite and eat you up."

Semyon pondered a bit. "I don't know, brother," he said; "perhaps it is as you say, and perhaps it is God's will."

"And perhaps," said Vasily, "it is waste of time for me to talk to you. To put everything unpleasant on God, and sit and suffer, means, brother, being not a man but an animal. That's what I have to say." And he turned and went off without saying good-bye.

Semyon also got up. "Neighbour," he called, "why do you lose your temper?" But his neighbour did not look round, and kept on his way.

Semyon gazed after him until he was lost to sight in the cutting at the turn. He went home and said to his wife: "Arina, our neighbour is a wicked person, not a man."

However, they did not quarrel. They met again and discussed the same topics.

"Ah, friend, if it were not for men we should not be poking in these huts," said Vasily, on one occasion.

"And what if we are poking in these huts? It's not so bad. You can live in them."

"Live in them, indeed! Bah, you . . . You have lived long and learned little, looked at much and seen little. What sort of life is there for a poor man in a hut here or there? The cannibals are devouring you. They are sucking up all your life-blood, and when you become


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64 Only half satisfied, they stalked away one by one. We watched them at a distance kill and divide the beef. It surprised us to see how they despatched it, and that hardly a vestige of it was left.

Many of the Indians coming from reservations carried papers which they valued and carefully guarded. After burrowing under robe and shirt, something was produced wrapped in layers of soiled cotton cloth. It was a recommendation of them obtained from some officer or Indian agent. This was presented on entering, as their letter of introduction. Most of these papers read very much the same way. Giving the Indian's name, it stated that he had been living on the reservation for a certain length of time, that he was friendly to the whites, etc.

One of our guests that day carried something a little different. He was called "Medicine Jo." Lingering behind the rest, he presented his letter with perfect good faith and great pomposity. Some wag had composed it, and it read something like this:

"Medicine Jo says he is a good Indian, that you can trust him. If he is, he is the first I have ever seen, and in my opinion he, like all the rest, will bear watching."

It was all the general could do to keep his face straight as he handed back to the unconscious owner this little libel on himself.

The interpreter kept constantly before us the fine post that we were approaching, and the last day before we reached there it was visible for a long distance. The atmosphere of Dakota was so deceptive that we

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Rh ourselves within a few miles of the garrison, when, in reality, there was a march of twenty-nine long miles before us. Our road led up from the river valley on the high bluffs, and sometimes followed along the backbone of hills from which on either side we looked down a great distance. There was barely room for the travelling-wagon. Occasionally I had been obliged to take refuge from the cold for a little while and drive. Our lead-mules were tiny, quick-moving little dots, and I soon discovered that they were completely demoralized at the sight of an Indian. They could see one in advance long before the driver could. A sudden shying and quick turning of these agile little brutes, a general tangle of themselves in the harness and legs of the wheelers, loud shouts of the driver, and a quick downfall of his foot on the brake, to keep us from overturning, made an exciting mêlée.

Nothing would get them righted and started again. They would have to be unharnessed, and the rebellious pair tied to the rear of the wagon until we had gone far beyond the object of terror. Part of the day that we were following the wanderings of the road alongside hills and over the narrow, smooth level of the hill-tops, I was compelled to drive, and I watched anxiously the ears of these wretched little beasts to see if they expressed any sentiment of fright. We came to such steep descents, the brake holding the wheels seemed of no use. Looking down from the wagon on to the mules below us, we appeared to be in the position of flies on a wall.

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Rh dish of fried fish, which Tom, "Bos," and I caught a few steps from my tent last evening.

The other day, on our march from Powder River, I shot an antelope. That night, while sitting round the camp-fire, and while Hughes was making our coffee, I roasted some of the ribs Indian fashion, and I must say they were delicious. We all slept in the open air around the fire, Tom and I under a fly, "Bos" and Autie Reed on the opposite side. Tom pelted "Bos" with sticks and clods of earth after we had retired. I don't know what we would do without "Bos" to tease Yesterday Tom and I saw a wild-goose flying over-head quite high in the air. We were in the bushes and could not see each other. Neither knew that the other intended to fire. Both fired simultaneously, and down came the goose, killed. Don't you think that pretty good shooting for rifles?

On our march here we passed through some very extensive Indian villages—rather the remains of villages occupied by them last winter. I was at the head of the column as we rode through one, and suddenly came upon a human skull lying under the remains of an extinct fire. I halted to examine it, and lying near by I found the uniform of a soldier. Evidently it was a cavalry uniform, as the buttons on the overcoat had "C" on them, and the dress-coat had the yellow cord of the cavalry uniform running through it. The skull was weather-beaten, and had evidently been there several months. All the circumstances went to show that the skull was that of some poor mortal who had been a prisoner in the hands of the savages, and who doubtless had been tortured to death, probably burned

We are expecting the Josephine to arrive in a day or two. I hope that it will bring me a good long letter from you, otherwise I do not feel particularly interested in her arrival—unless, by good-luck, you should be on board; you might just as well be here as not I hope to begin another Galaxy article, if the spirit is favorable

1em Look on my map and you will find our present location on the Yellowstone, about midway between Tongue River and the Big Horn.

The scouting-party has returned. They saw the trail and deserted camp of a village of three hundred and eighty (380) lodges. The trail was about one week old. The scouts reported that they could

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66 As we came to one descent more awful than the rest, the general, who was always near, rode up to the carriage and told me not to be afraid, for he would order the wheels manned. The head-quarters escort of over a hundred men, dismounting, attached ropes to the wheels, and held on with all their strength while I went down the steepest declivity I had ever descended. After that I begged to get out, and the general carried me to a bank and set me down where I could watch the repairing of the road.

He took off his coat and joined the soldiers in carrying logs and shovelling earth, for they were obliged to fill up the soft bed of the stream before the command could cross. It took a long time and much patience; but the general enjoyed it all, and often helped when the crossings needed to be prepared. When the logs were all laid, I had to laugh at the energy he showed in cracking a whip he borrowed from a teamster, and shouting to the mules to urge them to pull through where there was danger of their stalling. When the road was completed, I was ready to mount my horse, for it seemed to me preferable to die from accident, surrounded with friends, than to expire alone in the mule-wagon. The ascent was rendered so wet and slippery, the general feared my saddle would turn, and I was once more shut in by myself. The soldiers again manned the wheels to prevent the carriage sliding back, the mules scrambled, and with the aid of language prepared expressly for them, we reached the summit.

The driver had named the lead-mules Bettie and Jane, and when they were over their tempers he petted

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Rh and caressed them. Their repeated rebellion at last wore out even his patience. One morning I noticed new leaders, but the imperturbable face of the driver gave no hint of his successful plotting. Mary told me, however, that he was worn out with his struggles, and had gone after dark into the herd of mules with Bettie and Jane, and, as he expressed it, "lost them." He selected two more from among those belonging to the wagon-train, and returned triumphant over his premeditated exchange. He carefully reclipped their manes and tails, and disguised them still further with blotches of black paint, to give them a mottled appearance. When the other teamster prepared to harness in the morning, of course he discovered the fraud perpetrated on him. There was no redress then, and he had to take out his wrath in language more forcible than elegant, which the teamsters have adapted expressly for extreme occasions. Our driver told Mary, with a chuckle, that with a command of many hundred men waiting for a teamster to harness, he found "no time for swapping horses."

Burkman, the soldier who took care of our horses, was a middle-aged man, so deliberate in speech and slow in his movements, he seemed as incongruous among the spirited cavalrymen as would be an old-time farmer. Early in the march I had heard him coughing as he groomed the horses. When I asked if he had done anything for his cold, he replied, "Bottle after bottle of stuff, mum, but it don't do no good," so I begged the surgeon to look more carefully into his case. He made an examination, and told me, as the result, that the man

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text:
Translation:The Principle of Relativity (Laue 1913)

§ 6. The Lorentz transformation.
The relativity principle claims: ''From the totality of natural phenomena, one can determine with increased approximation and increased exactitude, a reference system x, y, z, t in which the natural laws are valid in certain, mathematically simple forms. This reference system is, however, in no way uniquely defined by the phenomena. There is rather a three-times infinite manifold of equally valid systems, mutually moving with uniform velocity.''

From this principle, we want to derive the transformation equations leading from one valid reference system to another. Therefore, we require the knowledge of an arbitrary natural law; which of them is chosen by us, must principally be irrelevant. If one arrives by a different choice at different results, then all laws wouldn't be invariable with respect to the same transformation, thus the relativity principle would be wrong. Evidently, such laws related to physical processes in empty space, are the most simple ones. Since in the other ones, the velocity of matter relative to the references system is included in its mathematical formulation, and must be co-transformed by a transformation; in vacuum, however, were we know nothing about moving things, this complication vanishes. We know of only two different actions propagating through vacuum, gravitation and electromagnetic processes. About the propagation of the first one we know nothing else than that its velocity is very great against astronomical velocities. That it is greater than the speed of light, although it is often claimed, finds no support in experience. Much better is our knowledge of the law of light propagation in vacuum, since it is proven by 's experiment (with a precision which is hardly attained in other physical measurements) that it is equal in all directions with respect to all systems. We add the assumption, which goes beyond the experimental results but which is required by the relativity principle, that the speed of light has got in all systems the same value $$c=3.10^{10}cm\ sec^{-1}$$. The following consideration will be based on this law.

At first, we consider an example. From a material point $$A$$ at rest in a valid reference system $$K$$ at the origin of coordinates, a short light signal emanates at time $$t=0$$ into all directions. At time $$t>1$$ the points receiving this signal are all located upon a sphere, whose equation reads:

$x^{2}+y^{2}+z^{2}-c^{2}t^{2}=0.\,$

Two material points $$B$$ and $$C$$ (also resting in $$A$$, and located at the same distance from $$A$$ on the same line with $$A$$), consequently are receiving the signal simultaneously, i.e. for the same value of $$t$$.

The same process we relate now to another valid system $$K'$$ moving relative to the first one with velocity $$v$$ in the direction $$\overrightarrow{BC}$$. The material points $$A, B, C$$ thus have in $$K'$$ the velocity $$v$$ parallel to $$\overrightarrow{CB}$$. The origin of coordinates $$x', y', z'$$ as well as time $$t'$$ can evidently be defined by us, so that the light signal is emitted at time $$t'=0$$ at origin $$O$$ (of primed coordinates) by the material point $$A$$. After expiration of time $$t'$$, all points of the sphere

$x'^{2}+y'^{2}+z'^{2}-c^{2}t'^{2}=0\,$

are reached by the signal. However, since $$A$$ has now moved away from $$O$$ to $$A'$$ by the distance $$vt'$$, there is no value $$t'$$ for which the material points $$B$$ and $$C$$ (equally distant from $$A$$) are located on such a sphere; thus they won't be simultaneously reached by the signal, rather (as shown by Fig. 4) this happens earlier in $$C$$ than in $$B$$, since $$C$$ approaches the light sphere, while $$B$$ runs away from it. Two events simultaneous in $$K$$, are in general not simultaneous in $$K'$$. The identical transformation for time, which forms a part of the Galilean transformation, cannot be maintained at this place.

This forces us to subject the concept of time to a critical investigation. We measure the time at a place by clocks, however, we don't need to think of the usual mechanical instruments for which the periods of elastic oscillations serve as the measure of time, but we can read the time in terms of progression of any physical process, as long as we can control its causes so that no unknown or quantitatively uncontrollable reasons can act. Such completely equal clocks are located by us at different places of space, all in one and the same valid system $$K$$. Thus the equality of the measure of time is guaranteed for all those places. However, to arrive at a unitedly defined time for the whole system $$K$$, we must synchronize the clocks, i.e. the zero-points of their time indications must be brought into a suitable relation. This is given to us by the requirement, that in system $$K$$ the speed of light shall have the value $$c$$, and namely is independent of the beam direction. If two clocks are mutually distant by $$l$$, then the time indication of the first one (when a light signal is emanating from here) shall differ from that of the second one (when arriving here) by the amount $$l/c$$. This definition doesn't lead to an inherent contradiction, when the clocks are (in pairs) combined on different ways. Namely, if the clocks in the arbitrarily chosen points $$B$$ and $$C$$ are synchronous with a clock at an arbitrary third point $$A$$, and if we send at time $$O$$ a light signal from $$A$$ to $$B$$, which is sent back without time loss to $$C$$, and from there it is sent back to $$A$$, then it's time of return is $$t_{A}=1/c(AB+BC+CA)$$, since it has traversed the distance $$AB+BC+CA$$. It has reached point $$B$$ by the presupposition of synchronism of the clocks in $$A$$ and $$B$$, when the clock in $$B$$ indicated the time $$t_{B}=AB/c$$, and has left point $$C$$ when the clock in $$C$$ indicated the time $$t_{C}=t_{A}-AC/c$$ because of the synchronism of the clocks in $$A$$ and $$C$$. When the signal started in $$B$$ and when it arrives in $$C$$, the difference between the time indications of the clocks in $$C$$ and $$B$$ is $$t_{C}-t_{B}=BC/c$$, i.e. the clocks in $$B$$ and $$C$$ are mutually synchronous. – Events are simultaneous, when they occur at equal indications of clocks located at the places of the events.

That one achieves a specific time for any valid system, by which it differs from the other valid systems, is already shown by the previous example. Herein lies the boldness and the high philosophical meaning of 's thought, that he removes the traditional prejudice of a time valid for all systems. As enormous this revolution (to which he forces our thinking) may be, there is not the slightest epistemological difficulty in it. Because time is (as space in 's way of expression) a pure form of our intuition; a scheme into which the events must be classified, so that they (contrary to subjective and highly accidental perceptions) gain objective meaning, and thus it is one of the conditions for the possibility of objective facts of experience. This classification can only be done on the basis of empirical knowledge of natural laws. Location and time of the observed change of a celestial body, for example, can only found out on the basis of optical laws. That two differently moving observers, when anyone considers himself as at rest, make this classification on the basis of the same natural laws in a different manner, doesn't contain a logical impossibility. However, both classifications have objective meaning, since from any of them (by means of the transformation formulas to be derived) one can unequivocally derive the valid classification for differently moving observers.

For example, one shall think of two astronomers on different, mutually moving celestial bodies, which are able to exchange thoughts with each other. They would find out, that they would have essentially different time indications of all astronomic events; and all of this (although they apply the same physical foundations as we will assume) because of the single reason that any of them considers himself as at rest. Consequently, the quarrel between them would result in the question, who of them is right with his assumption. According to the relativity principle, this is undecidable, both assumptions are completely equally valid. The time indications of both are just related to different systems. With this insight and with the aid of the transformation equation, the proof must be possible, that the meaning of both kinds of time specification is throughout the same, because they can be completely computed into each other.

To arrive at the transformation formulas, we ask after those linear relations between $$x', y', z', t'$$ and $$x, y, z, t$$ which transform the wave equation

$\Delta\varphi-\frac{1}{c^{2}}\frac{\partial^{2}\varphi}{\partial t^{2}}=0$|undefined

into itself; i.e. to which the identity applies:

where $$\alpha$$ is a proportionality factor, which is different from zero and dependent (in a unknown way at first) on $$v$$. Because equation (22) is the mathematical formulation of the law of light propagation. We choose this substitutions as linear, because otherwise the coordinates and the origin of time (at least in one of both systems) would become preferred points, which cannot be displaced without essential changes. But we consider the equal validity of all spatial points and temporal moments as an empirically verified natural law, equally valid for both systems according to the relativity principle. Furthermore we define, that corresponding coordinate axes in both systems are parallel, namely the $$x$$- and $$x'$$-axis by which the primed system $$K'$$ is moving against the unprimed $$K$$, shall be parallel to the velocity $$v$$; and that eventually the values $$x=0,\ y=0,\ z=0,\ t=0$$ shall correspond to the values $$x'=0,\ y'=0,\ z'=0,\ t'=0$$. The most general equations which satisfy these conditions, read:

{{MathForm2|(26)|$$\left.\begin{array}{c} x'=\varkappa(v)(x-vt)\quad y'=\lambda(v)y\quad z'=\lambda(v)z\\ t'=\mu(v)t-\nu(v)x\end{array}\right\} $$}}

where $$\varkappa(v),\ \lambda(v),\ \mu(v)$$ and $$\nu(v)$$ are functions of $$v$$ still to be determined. Namely, it is clear at first, that there is only one preferred direction, that of the $$x$$-axis. Already because of this reason, $$y$$ and $$z$$ cannot occur in the equation for $$t'$$, since otherwise the direction would be preferred whose direction cosines behave as the coefficients of $$x, y, z$$. Furthermore, plane $$x'=0$$ must be identical with plane $$x=vt$$, plane $$y'=0$$ with that of $$y=0$$, and $$z'=0$$ with that of $$z=0$$. The proportionality factor between $$y'$$ and $$y$$ as well as between $$z'$$ and $$z$$ eventually must be the same because of the equal validity of all directions normal to $$x$$.

However, additionally it can be shown, that it must be

$\lambda=1\,$

Namely, from the start it must be clear, that $$\lambda(v)$$ can only be a function of $$v$$, i.e. $$\lambda(v)=\lambda(-v)$$. Whether the motion of $$K'$$ against $$K$$ happens in the positive or negative $$x$$-direction, it is of course the same process for the $$y$$- and $$z$$-direction. On the other hand, the inversion of

$y'=\lambda(v)y\quad z'=\lambda(v)z$

must read:

$y=\lambda(-v)y'\quad z=\lambda(-v)z',$

since the motions of $$K'$$ against $$K$$ and those of $$K$$ against $$K'$$ only differ by the direction of $$v$$. From that it follows, however, the value 1 for $$\lambda$$ because of $$\lambda(v)=\lambda(-v)$$.

By substitution (26) a function arises now:

$\varphi(x',y',z',t')=\varphi(\varkappa(x-vt),\ y,\ z,\ \mu t-\nu x),$

from which it follows:

$\begin{array}{c} \frac{\partial\varphi}{\partial x}=\varkappa\frac{\partial\varphi}{\partial x'}-\nu\frac{\partial\varphi}{\partial t'},\quad\frac{\partial\varphi}{\partial t}=-\varkappa v\frac{\partial\varphi}{\partial x'}-\mu\frac{\partial\varphi}{\partial t'},\\ \\\frac{\partial^{2}\varphi}{\partial x^{2}}=\varkappa^{2}\frac{\partial^{2}\varphi}{\partial x'^{2}}-2\nu\varkappa\frac{\partial^{2}\varphi}{\partial x'\partial t'}+\nu^{2}\frac{\partial^{2}\varphi}{\partial t'^{2}},\quad\frac{\partial^{2}\varphi}{\partial y^{2}}=\frac{\partial^{2}\varphi}{\partial y'^{2}},\\ \\\frac{1}{c^{2}}\frac{\partial^{2}\varphi}{\partial t^{2}}=\varkappa^{2}\frac{v^{2}}{c^{2}}\frac{\partial^{2}\varphi}{\partial x'^{2}}-2\frac{\varkappa\mu v}{c^{2}}\frac{\partial^{2}\varphi}{\partial x'\partial t'}+\frac{\mu^{2}}{c^{2}}\frac{\partial^{2}\varphi}{\partial t'^{2}},\quad\frac{\partial^{2}\varphi}{\partial z^{2}}=\frac{\partial^{2}\varphi}{\partial z'^{2}}.\end{array}$|undefined

If the equation (25) shall represent an identity, then in the course of summation of the last four equations and the multiplication with $$\alpha$$, the coefficients

$\frac{\partial^{2}\varphi}{\partial x'^{2}},\ \frac{\partial^{2}\varphi}{\partial y'^{2}},\ \frac{\partial^{2}\varphi}{\partial z'^{2}},\ -\frac{1}{c^{2}}\frac{\partial^{2}\varphi}{\partial t'^{2}},$|undefined

must become equal to $$1$$, while that of $$\tfrac{\partial^{2}\varphi}{\partial x'\partial t'}$$ must vanish, i.e. it must be

For $$\varkappa$$ and $$\mu$$ we choose the positive sign, so that for $$v=0$$ the reference system $$K'$$ becomes equal to $$K$$, as it must be by the previous definition concerning the location of both cross-axes. Equations (26) then assume the form;

Due to the equal validity of both systems it must be permitted, to permute the primed and unprimed magnitudes with each other in these equations, as long as one only reverses the signs of $$v$$ simultaneously, because system $$K$$ has (against $$K'$$) the velocity $$-v$$ parallel to the $$x'$$-axis. Thus one finds:

The calculation shows, that (XIa) is actually the inversion of (XI). The totality of equations (XI) or (XIa) we denote as a Lorentz transformation; the relativity principle claims, that all reference systems arising by such transformations from a valid reference system, are not only completely equally valid for the propagation of electromagnetic actions in vacuum, but also for all physical processes.

The first of equations (27) ($$\alpha=1$$) shows by (25), that the expression $$\Delta\varphi-\tfrac{1}{c^{2}}\tfrac{\partial^{2}\varphi}{\partial t^{2}}$$ is invariant with respect to a Lorentz transformation. One easily can convince himself, that instead of this, one can also use the invariance of expression $$x^{2}+y^{2}+z^{2}-c^{2}t^{2}$$ as characteristic for a Lorentz transformation; namely, if one determines the coefficients $$\varkappa,\lambda,\mu,\nu$$ in (26) for the latter requirement, then one finds the exact same values as above.

As well as the totality of all Galilean transformations, also the totality of all Lorentz transformations forms a group; if one applies (one after the other) two of them with arbitrarily directed relative velocities $$v$$, then one obtains a result to be reached by a single transformation as well. Because, when both in the linear passage of a system $$K$$ to $$K'$$ and in the linear passage of $$K'$$ to $$K''$$, the expression

$\Delta\varphi-\frac{1}{c^{2}}\frac{\partial^{2}\varphi}{\partial t^{2}}$|undefined

remains unchanged, then this is also the case in the (also linear) immediate passage from $$K$$ to $$K''$$.

We further notice, that it must be

otherwise the coefficients of (XI) would become imaginary or infinite.

§ 7. The kinematics of Einstein.
In order to discuss formulas (XI) and (XIa), we first investigate the rate of a clock at rest in the primed system (x'=const), as viewed from the unprimed system, by constantly comparing its hand position with the indication of the clock at rest in K at which it just passes by. According to (XIa), the time interval $$\Delta t'$$ which will be read from it by a co-moving observer, corresponds (since $$x'$$ is invariant) to the interval

$\Delta t=\frac{\Delta t'}{\sqrt{1-\frac{v^{2}}{c^{2}} }}$|undefined

which will be detected by an observer in system K by comparison of the times, that were given by two clocks (at rest in the start- and endpoint of the now completed distance) during the start and the arrival of the moving one. This can also be derived from (XI); because in time $$\Delta t$$, the clock in system K travels the distance $$\Delta x=v\Delta t$$, from which it follows

The rate of a clock moving with velocity $$\mathfrak{q}$$ is thus slower in the ratio $$\sqrt{1-\tfrac{q^{2}}{c^{2}}}$$ than for the same clock when at rest. This theorem is of course true, independent to which of the valid system we relate the expressions "at rest" or "in motion". We of course have to consider as a clock, every process whose progression can serve as a measure of time. In a moving hydrogen atom (canal rays), for example, the light-emanating proper oscillations will have a smaller frequency than in a stationary one.

Now we consider two equal clocks. Both are initially at rest in a valid reference system $$K^0$$ at the same location. The second one will be accelerated by us in a quasi-stationary way, i.e. so that its inner condition is always dependent only on the instant velocity, not on the acceleration. We let remain the clock for some time at uniform velocity, then we reverse it by quasi-stationary acceleration, and in this way we bring back the clock to its origin, where we — again quasi-stationary — bring it to rest. So in every moment of this process, the moving clock's rate is slower than the stationary clock, and thus it is retarded with respect to the latter. — This consequence of the theory of relativity, which was further explained in a very interesting manner namely by, initially appears to be so paradox, that it even was subject to the objection, that it disproves the principle of relativity; because it would be possible to consequently decide, which of the two clocks was at rest, and which one was in motion. Indeed we can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. However, among them there is of course a real physical difference. (see § 8.)

Quite similar relations occur with respect to measurements of the length of a moving rod. If it is at rest in $$K'$$ and located parallel to the $$x'$$-axis, then its rest length $$s_0$$ is determined by the difference of the $$x'$$-coordinates of its endpoints, thus:

$s_{0}=x'_{1}-x'_{2};\,$

those can be determined by applying a measuring rod also at rest in $$K'$$. Its length related to $$K$$ is, however, equal to the difference of the $$x$$-coordinates, which are to be attributed to its endpoints for equal values of t; therefore it follows from (XI):

''The length of a body parallel to the direction of motion is considered greater (when viewed from the co-moving system) than in any other system. If we bring a rod under maintenance of its inner condition  (like in empty space and without heat supply) from a state of rest to velocity $$\mathfrak{q}$$, then it contracts in the ratio $$\sqrt{1-\tfrac{q^{2}}{c^{2}}}$$;'' namely if we call the system $$K$$ in which it was at rest at the beginning, and the other system $$K'$$ in which it will be at rest after acceleration, then its length $$s'_{2}$$ in the second state related to $$K'$$, will exactly have the value $$s_0$$, as the length $$s_1$$ which it had in the first state of $$K$$.

This theorem justifies the contraction hypothesis, employed by and  (presented already before the theory of relativity) for the explanation of 's experiment, and it is indeed necessarily required when one wants the give the theory of this experiment for an non-co-moving system. Namely, if in Fig. 1 (p. 14) the distance $$PS_1$$ is contracted (in its motion parallel to it) to length $$l\sqrt{1-\tfrac{q^{2}}{c^{2}}}$$, then the traversing time becomes by (13) and (14):

$t_{1}=\frac{2l}{\sqrt{c^{2}-q^{2}}}=t_{2}.$|undefined

During the rotation of the apparatus, which is followed by the permutation of the roles of $$PS_1$$ and $$PS_1$$, no replacement of the interference fringes takes place. That this time itself is still dependent on $$q$$, lies in the fact that it is related to that system, in which the interferometer has the velocity $$q$$. In the co-moving system $$K'$$, the traversing time would be $$t'=t_{1}\sqrt{1-\tfrac{q^{2}}{c^{2}}}=\tfrac{2l}{c}$$ according to (29), as it must be, since in $$K'$$ the distances $$PS_1$$ and $$PS_1$$ are traversed with velocity $$c$$.

The dimensions normal to motion were not altered by (XI). Consequently the angles, which were formed by a material line with the direction of velocity $$\mathfrak{v}$$ or another line, are in general different in $$K$$ and $$K'$$; as well as the angles between two material planes or between a plane and a line. The volume $$V$$ of a body is to be transformed by the Lorentz transformation because of the same reason as the length parallel to the velocity. Since by that, $$\tfrac{V}{\sqrt{1-\frac{q^{2}}{c^{2}}}}$$ in every system is equal to the volume $$V^0$$ in the co-moving system, if follows that the equation, which serves as the transformation formula for the volume of a body with velocity $$\mathfrak{q}$$ relative to $$K$$ and $$\mathfrak{q}'$$ relative to $$K'$$, is:

Another consequence of contraction is, that the form of a body is differently seen in different systems. For example, a body that has (investigated in a co-moving system) a spherical form, is an oblated ellipsoid of revolution in all other systems.

In its first half (namely as far as it is spoken about the assessment of the lengths of different systems), the theorem on p. 43 is the consequence of different ways of application of the space- and time-scheme by differently moving observers. The two astronomers, about who we spoke on p. 38, despite of the same geometric foundations would be getting into quarrel, as soon as they apply them on real bodies. They just assess their shape in an essentially different way. But the main points of their quarrel would be located in the mutually contradicting assumptions concerning their state of rest. As soon as these assumptions would be recognized, it would be possible to transform both indications into each other by a Lorentz transformation. – In its second half, however, every theorem includes a statement about mechanics; the elastic forces that determine the shape of the body, must be influenced by motion so that they exactly cause this contraction.

As the last kinematic consequence, the transformation formula for the velocity of a material point shall be derived. If its coordinates $$x,y,t$$ in system $$K$$ are functions of time $$t$$, then the velocity components are:

$\mathfrak{q}_{x}=\frac{dx}{dt},\ \mathfrak{q}_{y}=\frac{dy}{dt},\ \mathfrak{q}_{z}=\frac{dz}{dt}.$

The corresponding definition applies to its velocity $$\mathfrak{q}'$$ with respect to $$K'$$. From (XI) it follows, however:

$\begin{array}{c} \frac{dx'}{dt'}=\frac{\frac{dx}{dt}-v}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{dt}{dt'},\ \frac{dy'}{dt'}=\frac{dy}{dt}\frac{dt}{dt'},\ \frac{dz'}{dt'}=\frac{dz}{dt}\frac{dt}{dt'},\\ \\\frac{dt'}{dt}=\frac{1-\frac{v}{c^{2}}\frac{dx}{dt}}{\sqrt{1-\frac{v^{2}}{c^{2}}}};\end{array}$|undefined

thus

or, when one inversely expresses $$\mathfrak{q}$$ by $$\mathfrak{q}'$$:

If one mutually multiplies the equations in (32) and (33) related to the $$y$$- or $$z$$-components, then one finds:

Furthermore, a simple calculation shows, by squaring and adding the three equations (32) and $$v\mathfrak{q}_{x},\ v\mathfrak{q}'_{x}$$ is replaced by the scalar products $$(\mathfrak{vq}),\ (\mathfrak{vq'})$$:

From (31), (34) and (35) it follows:

Eventually, if we denote by $$\vartheta'$$ the angle between velocity $$\mathfrak{q}'$$ and the $$x$$-axis, then it holds according to (33):

since

These equations show, as to how the velocities $$\mathfrak{v}$$ and $$\mathfrak{q}'$$ are composed when viewed from system $$K$$; they contain the famous  addition theorem of velocity.

The most remarkable thing there, is that both velocities that have to be added, are not equally valid. For example, if $$\mathfrak{v}_{2}$$ is parallel to $$y$$, $$\mathfrak{v}_{1}$$ parallel to $$x$$, then by (33) under the assumption, that one adds $$\mathfrak{v}_{2}$$ to $$\mathfrak{v}_{1}$$ by setting $$v_1$$ instead of $$v$$, $$v_2$$ instead of $$\mathfrak{q}'_{y}$$ and $$\mathfrak{q}'_{x}=\mathfrak{q}'_{z}=0$$, the resulting velocity $$\mathfrak{q}_1$$ obtains the components:

$\mathfrak{q}_{1x}=v_{1},\ \mathfrak{q}_{1y}=v_{2}\sqrt{1-\frac{v_{1}^{2}}{c^{2}}};$|undefined

however, when in reverse $$\mathfrak{v}_{1}$$ is added to $$\mathfrak{v}_{2}$$, where the indices 1 and 2 and furthermore the $$x$$- and $$y$$-direction permute their roles, then the velocity $$\mathfrak{q}_{2}$$ is given with components:

$\mathfrak{q}_{2x}=v_{1}\sqrt{1-\frac{v_{2}^{2}}{c^{2}}},\ \mathfrak{q}_{2y}=v_{2}.$|undefined

The absolute value $$q$$ is the same for both cases [see also the symmetric relation (37) in $$q'$$ and $$v$$], however, the directions are different (Fig. 5). If we imagine a material right-angled triangle with velocity $$\mathfrak{v}_{1}$$ displaced parallel to the first side, and introduce a pencil along the second side with velocity $$\mathfrak{v}_{2}$$, then its top follows on the drawing plane the line $$\mathfrak{q}_{1}$$ in unit time. However, if we replace the triangle parallel to the second side with velocity $$\mathfrak{v}_{2}$$ and the pencil along the first one with velocity $$\mathfrak{v}_{1}$$, then its top follows the line $$\mathfrak{q}_{2}$$.

By (28) $$v<c$$. If we let grow $$q'$$ starting from 0, then whatever the direction of $$\mathfrak{q}$$ may be, the denominator $$c^{2}+(\mathfrak{vq}')$$ in (35) cannot vanish, as long as $$q'\leqq c$$. The right-hand side therefore remains positive and finite, until it vanishes for $$q'=c$$.

To the value of $$q'=0$$ corresponds, of course, the value $$\mathfrak{q=v}$$, satisfying (35) as well. If $$q'$$ is steadily increasing starting with this value, then also $$q$$ must be steadily changing, namely in a way, so that the left-hand side of (35), $$\tfrac{c^{2}-q^{2}}{c^{2}-(\mathfrak{vq})}$$, as well as the right-hand side remain positive, until $$q'=c$$. For that, however, to a value of $$q'<c$$ must necessarily correspond a value $$q<c$$, however, to the value $$q'=c$$ must correspond a value $$q=c$$. ''Two added subluminal velocities thus again and again give a subliminal velocity. The addition of the velocity of light to a subluminal velocity gives the velocity of light.'' The latter plays in physics the role of an infinitely great velocity throughout, in so far as it can never be reached by an accumulation of subluminal velocities. There, also the determination $$vc$$ to point $$B(x=x_{B},\ y=z=0)$$, then it would arrive there later in time by

$\Delta t=\frac{x_{B}-x_{A}}{V}.$|undefined

By the last of equations (XI), the duration of this propagation, related to $$K'$$, would be:

$\Delta t'=\frac{\Delta t-\frac{v}{c^{2}}\left(x_{B}-x_{A}\right)}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\Delta t\frac{1-\frac{vV}{c^{2}}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$|undefined

Thus, by the choice of a sufficiently great value for subluminal velocity $$v$$, one can achieve that $$\Delta t'$$ becomes zero or even negative, i.e. there would be a valid system, in which the effect in $$B$$ would chronologically preceding the cause; however, this is impossible, since the natural law in question would remain unchanged in the passage from $$K$$ to $$K'$$ in opposition to the relativity principle, when cause and effect would permute their roles with respect to time and causality.

This consequence from the relativity principle is only apparently refuted by the fact, that in dispersing bodies the refractive index is $$n<1$$ for some spectral areas, thus the "velocity of light" is $$\tfrac{c}{n}>c$$. Because $$\tfrac{c}{n}$$ is namely the velocity, by which the phases of periodic waves propagate in the stationary state, i.e. when they are distributed over the whole space. As to the propagation of a suddenly occurring electromagnetic disturbance, it has no immediate meaning. The first precursor of such a disturbance, the head of the wave, is rather propagating, according to the theory of electrons, steadily with the velocity of light $$c$$. [ ].

Also if one leads (in a valid system) a ruler above a resting second one which is nearly parallel, then the intersection of both can easily travel with superluminal velocity; since its velocity (like that of the first ruler) is divided by the angle between both, and can be increased above any limit by decreasing this angle. This contains no contradiction to the above; since the intersection is no material point, it also cannot be used to transfer actions along the resting ruler. It would be easy to install mechanisms upon it, which cause any physical event (like rings of bells) in the moment when the intersection passes them; and then one can claim in a certain sense, that the rings of bells are propagating with superluminal velocity. However, if one removes any number of these devices again, then the other ones will work as before – a clear sign that these rings of bells are not connected as cause and effect. Their common cause is rather in the motion of the second ruler.

However, it would be in contradiction to the previous theorems, when it would be possible to realize the following case: "At a point $$B$$ of the resting ruler, we again install one of the mentioned devises. At the beginning, also the second ruler is at rest; the intersection of both may coincide at that stage with point $$A$$ of the first one. However, afterwards an arbitrary event takes place in $$A$$, by which the second ruler is practically momentarily brought into uniform motion. The intersection then traverses the resting ruler as the signal of the occurrence of the event in $$A$$, and the event caused by it in $$B$$ becomes the effect of the cause happening in $$A$$. The transfer from $$A$$ to $$B$$ happens with superluminal velocity."

Now, whey isn't it possible to realize the mentioned process between the quotes? Would the second ruler, as previously assumed, really be rigid and incapable of changes of shape, then it would indeed be executable. The transfer of the driving motion starting from $$A$$ would occur momentarily by the elastic forces. And by that we arrive at the reason of this contradiction. The assumption of a rigid body is incompatible with relativity theory. The ruler bulges (under all circumstances) at first in $$A$$ and then it doesn't remain straight; the conclusion to a superluminal velocity of the intersection, however, is essentially based on its straightness.

That any action through vacuum, thus also gravitation, must be propagated at the speed of light $$c$$, when the relativity principle shall be correct, follows from the reasoning, that otherwise one obtains a substitution (by transfer of the consideration leading to equation (XI)), which would be different from (XI) by the value of $$c$$, therefore it would define (together with (XI)) a preferred system again.

§ 8. Minkowski's geometric interpretation of the Lorentz transformation.
The principle of relativity, as it was announced in § 6 and mathematically formulated in the Lorentz-Transformation (XI), completely contains the foundations of relativity theory. Though the calculations partly become inelegant and complicated, because with respect to every vector, one has to transform the component parallel to velocity $$v$$ in a different way than the perpendicular ones. It is the great merit of the so early deceased mathematician of Göttingen,, to have created a geometric interpretation that avoids this difficulty, and due to the elegance that it lends to the calculations of relativity theory, it forms a nearly indispensable tool for the reliable usage of it.

In general, all coordinates are concerned by a Lorentz transformation. Only our special assumption concerning the location of the axis-cross in the systems $$K$$ and $$K'$$ gave rise to the identical transformation in (XI) for the $$y$$- and $$z$$-coordinate. However, since also time must be altered, it can be easily seen that a geometric analogy can only consist in a four-dimensional manifold. That such one is inaccessible to our intuition, shall not frighten us; it is only about a symbolic illustration of certain analytical relations between four variables. We denote this manifold as "the world" and choose as "world coordinates" the space coordinates $$x, y, z$$ and the time coordinate

The $$x, y, z$$-axes shall be mutually perpendicular; the $$u$$-axis preliminarily shall also be perpendicular upon the three mentioned axes, i.e., $$x$$ and $$u$$ shall at first be ordinary rectangular coordinates in the $$xu$$-plane, and corresponding relations shall apply in the $$yu$$- and $$zu$$-plane. Then there are six coordinate planes (pairwise mutually perpendicular), and four (three-dimensional) coordinate spaces (mutually perpendicular). A point $$(x, y, z, u)$$ of this manifold we call a "world point"; it represents an event taking place at location $$x, y, z$$ at time $$t=\tfrac{u}{c}$$.

The space coordinates are functions of time for any material point; thus it determines a curve in the world, a "world-line". At uniform velocity $$q$$ it is a straight line whose angle with respect to the $$u$$-axis is equal to $$\operatorname{arctg}\tfrac{q}{c}$$, thus it is smaller than $$\tfrac{\pi}{4}$$, because $$q<c$$; in general it is an arbitrary curved line, whose inclination with respect to the $$u$$-axis, however, never reaches the given amount.

In the special case of § 6, where the $$x$$- and $$x'$$-axis are parallel to the translatory velocity, the transformation only takes place in the $$xu$$-plane, which we want to use as drawing plane (Fig. 6). Therein we draw hyperbolas of equal sides:

as well as their common pair of asymptotes:

From origin $$O$$ we draw the line in the direction of any point $$U'$$ of that branch of the first hyperbola, for which $$u$$ is positive;

afterwards a second line into a point $$X'$$ of branch $$x>0$$ of the second hyperbola; $$X'$$ is thus determined, so that the asymptote $$x=u$$ becomes the angle bisector, i.e. that $$PU'$$ and $$OX'$$ become conjugate diameters for both hyperbolas (39) and (40). If U$$'$$ falls upon the $$u$$-axis in the direction of $$U$$, than $$X'$$ coincides with $$X$$. The Lorentz transformation (XI) consists in the fact, that instead of line $$OU$$ and $$OX$$, one uses $$OU'$$ and $$OX'$$ as axes; and instead of the distances $$OU$$ and $$OX$$, one uses $$OU'$$ and $$OX'$$ as distances.

Indeed: the lines $$OU'$$ and $$OX'$$ have the equations

$x=\beta u,\ u=\beta x,\ \left

The coordinates of $$U'$$ and $$X'$$ are therefore according to (39) and (40):

{{MathForm2|(42)|$$\left.\begin{array}{ccc} u_{U'}=\frac{1}{\sqrt{1-\beta^{2}}}, & & x_{U'}=\frac{\beta}{\sqrt{1-\beta^{2}}}\\ \\u_{X'}=\frac{\beta}{\sqrt{1-\beta^{2}}}, & & x_{X'}=\frac{1}{\sqrt{1-\beta^{2}}}\end{array}\right\} $$}}

However, now in the primed system it should become:

$u'_{U'}=x'_{X'}=1,\ u'_{X'}=x'_{U'}=0$

The four constants available by a linear transformation from $$x, u$$ to $$x', u'$$, are definitely specified by that; the transformation equations thus must read:

according to (38), they go over into (XI) if one puts

The expression $$x^{2}-u^{2}$$ as well as $$x^{2}+y^{2}+z^{2}-u^{2}$$ therefore remain invariant with substitution (43). The equations of both hyperbolas and their pairs of asymptotes, read the same in the primed system as well as in the unprimed system. Angle $$\psi$$ in Fig. 6 is specified by equation

With respect to every world point $$W$$ of the $$UX$$-plane, for which $$u^{2}>x^{2}$$, one evidently can draw a time-axis starting from $$O$$, with direction $$\overrightarrow{OW}$$ when $$u>0$$, and with direction $$\overrightarrow{WO}$$ when $$u<0$$; because there is alway a conjugate diameter to $$OW$$ as the $$x'$$-axis. Conversely, every world point for which $$u^{2}<x^{2}$$, can be made "simultaneous" to $$O$$, by choosing the line $$OW$$ with appropriate direction as the $$x'$$-axes, and its conjugate diameter as $$u'$$-axis.

To represent the general Lorentz transformation, at which the spatial axis-crosses $$x, y, z$$ and $$x', y', z'$$ are arbitrarily oriented to the relative velocity $$v$$, we have to design this construction in a four-dimensional way. Instead of hyperbolas, the two-sheet hyperbolic space occurs:

and the single-sheet hyperbolic space:

both will be asymptotically contacted by the conic space

The hyperbolas in Fig. 6 and their asymptotes are the intersections of those three-dimensional spaces with the $$ux$$-plane.

Now we choose, upon the positive sheet of the two-sheet hyperbolic space (46), an arbitrary point $$U'$$ with coordinates $$x_{U'},\ y_{U'},\ z_{U'},\ u_{U'}$$, and draw in its direction the line from $$O$$. Then we construct the conjugate diameter-space to this diameter:

which intersects the single-sheet hyperbolic space (47) into an ellipsoid. If $$U'$$ falls into the vertex $$U$$ of the first hyperbolic space, than the conjugate diameter-space becomes the $$x y z$$-space $$(u=0)$$ and the ellipsoid becomes a sphere

and all distances from $$O$$ in the direction of a point of this sphere have the same length 1; it is the measuring surface for distances in space $$x y z$$. The general Lorentz transformation now consists in this, that instead of the line $$OU$$ as $$u$$-axis and the distance $$OU$$ as unity, and instead of the $$xyz$$-space with sphere (50) as measuring surface, we render the line $$OU'$$ (with $$OU'$$ as unity-distance) as the $$u'$$-axis, and its conjugate diameter-space with the mentioned ellipsoid as measuring surface we render as the corresponding $$x'y'z'$$-space. The latter condition expresses, that the equation of the ellipsoid should read

$x'^{2}+y'^{2}+z'^{2}=1\,$

For confirmation, we first change the coordinate system $$x' y' z'$$ by a rotation executed in $$x' y' z'$$-space; $$u'$$ remains totally untouched here, while the sum $$x'^{2}+y'^{2}+z'^{2}$$ is invariant; the same also holds for the expression $$x'^{2}+y'^{2}+z'^{2}-u'^{2}$$. By rotation it can be achieved, that the intersection of $$x' y' z'$$-space becomes the plane (located through the $$u$$ and $$u'$$ axis) to the $$x'$$-axis. By an analogous rotation of coordinate system $$xyz$$, we move the $$x$$-axis into the same plane. However, then we have the same relations as in Fig. 6: $$OX'$$ and $$OU'$$ became the (by construction) conjugate diameter of hyperbola $$x^{2}-u^{2}=-1$$, and since $$x^{2}+y^{2}+z^{2}-u^{2}$$ is invariant with respect to the special transformation illustrated there, then this also holds for the generalized construction now given, by which it proves to be a Lorentz transformation. Thus there exist as many valid systems as points in the positive sheet of hyperbolic space (46), i.e. three-times infinitely many, as already mentioned earlier. The mutual velocity of both systems is specified by the angle $$\psi$$ between the $$u$$- and the $$u'$$-axis according to equation (45).

The conical space (48) separates the world into three parts:

1. $$u^{2}>x^{2}+y^{2}+z^{2},\ u>0$$. One can place a time-axis with direction $$\overrightarrow{OW}$$ after every world point $$W$$ of this area; it contains the world-points "beyond $$O$$", i.e. those arising later than $$O$$ in all valid systems.

2. $$u^{2}>x^{2}+y^{2}+z^{2},\ u<0$$. The world points of this area are arising earlier than $$O$$ in all systems, they are "on this side of $$O$$". One can draw a time-axis with direction $$\overrightarrow{WO}$$ with respect to them.

3. $$u^{2}0$$. The first one encloses those world points representing events, from which a possible effect propagating with the speed of light arrives at point $$x=0,\ y=0,\ z=0$$ at time $$t=0$$. The after-cone on the other hand, represents the totality of events happening simultaneously with the arrival of a possible light-effect emanating from $$O$$. We say, that the former ones "are sending light in the direction of $$O$$", and the latter ones "are receiving light from there".

Due to the invariance of expression $$x^{2}+y^{2}+z^{2}-u^{2}$$, both hyperbolic spaces (46) and (47) as well as the conic space (48) have an absolute meaning; their equation is the same in all reference systems. Therefore, also this arrangement of

the world is independent from the reference system. Its physical meaning can be seen from the following consideration:

The world points $$O$$ and $$R$$ in Fig. 7 may represent two material points $$A$$ and $$B$$ at rest in $$K$$; the first at time $$t(=0)$$ when an arbitrary effect emanates from it; the latter at time $$t_R$$ when this effect reaches $$B$$. Now, if the velocity $$V$$ (by which it propagates) could be greater than $$c$$, then it would be

$u_{R}=ct_{R}=\left

$$R$$ belongs to the intermediary area of $$O$$, as it is also drawn in Fig. 7. Thus valid systems would always exist, in which $$u'_{R}<0$$ as in system $$K'$$ of the figure, i.e. $$R$$ would be earlier than $$O$$; thus the effect would arrive at $$B$$ before its cause would have taken place in $$A$$. This geometric consideration thus again disproves the possibility of superluminal speeds (see p. 48 and 49). ''Thus all world points in a position to causally influence $$O$$, are lying beyond or upon the after-cone, and all points from which $$O$$ can suffer such an influence, are on this side or upon the for-cone of $$O$$. Points of the intermediary space, however, can never be in causal connection with $$O$$.''

For an infinitely small distance on the $$u'$$-axis or a curve-section parallel to it, it is given by the second of equations (43)

$du'=\frac{du-\beta dx}{\sqrt{1-\beta^{2}}}$,|undefined

however, according to the first of these equations and due to $$dx'=0$$,

$dx=\beta du\,;$

thus

The equation includes, like formula (29) with which it is identical, the theorem that a clock at rest in $$K'$$, when related to $$K$$, is retarded. If we denote by dx, dy, dz the projections of the distance (as covered in time dt) upon the coordinate axes, and by q the velocity, then it is given

$\frac{q}{c}dt=\beta dt=\frac{1}{c}\sqrt{dx^{2}+dy^{2}+dz^{2}}$.|undefined

Furthermore, dx, dy, dz, du are the components of the worldline-element covered in time dt. By (51) the time increase is thus given

{{MathForm2|(52)|$$\left.\begin{array}{ll} d\tau & =\frac{1}{c}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}\\ \\ & =dt\sqrt{1-\frac{q^{2}}{c^{2}}}\end{array}\right\} $$}}

which is indicated by the clock while it traverses the element. We denote by $$d\tau$$ the proper time of that worldline-element. For a finite section of a worldline, we find it by integration. The proper time of a worldline-section is an invariant of the transformation. This already becomes clear due to its physical meaning, since the hand positions of the clock are the same for all reference systems at the start and the end of that section; this we can also recognize by the fact, that the expression $$du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)$$ in (52) is transformed the same way, as the invariant expression $$u^{2}-\left(x^{2}+y^{2}+z^{2}\right)$$. However, it is to be noticed, that that equation (52) is initially derived only for uniformly moving clocks, i.e. derived for straight worldlines. In general, the acceleration will have an influence on the rate. Only when the acceleration is sufficiently small, i.e. the curvature of the worldlines is sufficiently weak, the proper time will directly have the given physical meaning. Where the limit for that is located, depends on the constitution of the clock. However, the proper time defined by (52), is always a mathematically useful concept.

Of all worldlines, which connect two given worldpoints 1 and 2, the straight connection has the longest proper time. Because, if we put the z-axis parallel to it — the choice of reference system is totally arbitrary due to the invariance of proper time —, then by (52) the proper time for this line is $$\tfrac{1}{c}\int\limits _{1}^{2}du$$, though for another connecting line it is

$\frac{1}{c}\int\limits _{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}<\frac{1}{c}\int\limits _{1}^{2}du$.

Of the two clocks, of which we have spoken on pp. 42 and 43, one covers the straight worldline from 1 to 2, yet the other one covers a broken line. We confirm at this place, that the second one is retarded at the re-encounter; because its worldline corresponds to a shorter proper time. Simultaneously we also see the physical difference of both motions in the most illustrative way.

Let us come back to Fig. 7 once more. The endpoints of a rod (parallel to the $$x$$-axis and at rest in $$K$$) are following straight (parallel to the $$u$$-axis) world lines $$P_{1}P'_{1}$$ and $$P_{2}P'_{2}$$. The ratio $$\tfrac{P_{1}P_{2}}{OX}$$ denotes its rest length $$s^0$$. If a second rod is at rest in $$K'$$, then the world lines of its endpoints are two lines $$Q_{1}Q'_{1}$$ and $$Q_{2}Q'_{2}$$ parallel to the $$u'$$-axis, and its rest length which also may be equal to $$s^0$$, is represented by the ratio $$\tfrac{Q'_{1}Q'_{2}}{OX'}$$. The length of the first rod in $$K'$$ is, however, $$s'=\tfrac{P'_{1}P'_{2}}{OX'}$$, because $$\tfrac{OP'_{1}}{OX'}$$ and $$\tfrac{OP'_{2}}{OX'}$$ are the $$x'$$-coordinates of its endpoints for the same value (0) of $$t'$$. Analogously, the length of the second rod in $$K$$ is $$s=\tfrac{Q_{1}Q_{2}}{OX}$$. Now, according to (42):

$\frac{OX'}{OX}=\sqrt{x_{X'}^{2}+u_{X'}^{2}}=\sqrt{\frac{1+\beta^{2}}{1-\beta^{2}}}$|undefined

furthermore according to Fig. 7 and (45):

$\begin{array}{ll} \frac{P'_{1}P'_{2}}{P_{1}P_{2}}=\frac{1}{\cos\psi} & =\sqrt{1+\beta^{2}},\ \frac{Q_{1}Q_{2}}{Q'_{2}Q'_{2}}=\frac{OQ_{1}}{OQ'_{1}}=\frac{\sin\left(\sphericalangle OQ'_{1}Q_{1}\right)}{\sin\left(\sphericalangle OQ_{1}Q'_{1}\right)}\\ \\ & =\frac{\sin\left(\frac{\pi}{2}-2\psi\right)}{\sin\left(\frac{\pi}{2}+\psi\right)}=\frac{1-\beta^{2}}{\sqrt{1+\beta^{2}}}\end{array}$|undefined

Thus for the first rod

and for the second

Thus we confirm the theorem already found in §7, that a body when viewed from the co-moving system, appears to be longer than from any other system. Since it is chosen in Fig. 7 that $$Q'_{1}Q'_{2}:OX'=P_{1}P_{2}:OX$$, it thus can be seen that $$Q_{1}Q_{2}<P_{1}P_{2}$$.

§ 9. The Lorentz transformation as an imaginary rotation
Analytically, it is even more elegant to use the imaginary quantity

as time variable, where an imaginary point of the $$x, y, z, l$$-manifold indeed corresponds to any real world-point. Thus it is

As we have seen in § 6, a Lorentz transformation is characterized by the invariance of this expression.

Let us first consider the simple case of the transformation expressed in (XI) and (43) again, by which only $$x$$ and $$t$$ are concerned. If we introduce $$u$$ into (43) instead of $$l$$, then we find:

Now we can define an imaginary angle $$\varphi$$, so that

By (45), it is connected with angle $$\psi$$ of Fig. 6 and 7 by the relation

If one expresses $$\beta$$ by $$\varphi$$ in (57), then one finds the equations:

and as their inversions:

If $$l$$ and $$\varphi$$ were real quantities, this would mean that the coordinate system $$x', l'$$ emerges from system $$x, l$$ by a clockwise rotation around the angle $$\varphi$$. But also here we can maintain the following mode of expression without doubts. The special Lorentz transformation (XI) is a rotation of the axis-cross in the $$xl$$-plane by the imaginary angle $$\varphi$$ defined by (58).

If one successively executes several such transformations in the $$xl$$-plane of rotational angles $$\varphi_{1},\varphi_{2}\dots\varphi_{n}\dots$$, then a single rotation around the angle

$\varphi=\sum\varphi_{n}$

is resulting. The velocity of the $$n$$th with respect to the $$(n-1)$$th system, is according to (59):

$v_{n}=-ic\ \operatorname{tg}\ \varphi_{n}$

that of the latter with respect to the first is accordingly:

$v=-ic\ \operatorname{tg}\ \left(\sum\varphi_{n}\right)$

The function

$\operatorname{tg}\ x=-i\frac{e^{ix}-e^{-ix}}{e^{ix}+e^{-ix}}$|undefined

however, when its argument traverses the purely imaginary value from zero to $$i\infty$$, only increases from zero to value $$i$$. Therefore it is under all circumstances $$\operatorname{tg}\left(\sum\varphi_{n}\right)<i$$ and $$v<c$$. The speed of light remains unreachable, as we have already seen in § 7.

If $$\sum\varphi_{n}$$ is confined to two terms, then it follows:

$v=-ic\ \operatorname{tg}\left(\varphi_{1}+\varphi_{2}\right)=-ic\frac{\operatorname{tg}\ \varphi_{1}+\operatorname{tg}\ \varphi_{2}}{1-\operatorname{tg}\ \varphi_{1}\cdot\operatorname{tg}\ \varphi_{2}}=\frac{v_{1}+v_{2}}{1+\frac{v_{1}v_{2}}{c^{2}}}$|undefined

's addition theorem for velocities of same direction [see the first of relations (33)], thus directly emerges from the addition theorem of the tangent function.

Already in § 8 (p. 55) we saw, that the most general Lorentz transformation can be derived from rotations of the spatial axis-cross in the systems $$K$$ and $$K'$$, as well as from the special transformation (43) being identical with (XI) and (57). All properties of the general one can already be recognized in the latter transformation; for example, the general one has always to be interpreted as an imaginary rotation in the plane determined by the $$l$$- and $$l'$$-axis, to which only the inessential real rotations of the coordinate systems $$x, y, z$$ and $$x', y', z'$$ are supplemented.

Analytically it is more elegant, however, to investigate in a quite general way those linear transformations of the world-coordinate system $$x, y, z, l$$, which represent a Lorentz transformation, i.e., in which the expression (56) $$x^{2}+y^{2}+z^{2}+l^{2}$$ stays invariant. This task indeed corresponds to the question after the substitutions, which represent a rotation of the space-coordinate system $$x, y, z$$. Because these are characterized by the invariance of $$x^{2}+y^{2}+z^{2}$$. (see appendix under b.)

If $$x^{2}+y^{2}+z^{2}+l^{2}=x'^{2}+y'^{2}+z'^{2}+l'^{2}$$ shall follow from the linear substitution

{{MathForm2|(62)|$$\left.\begin{array}{c} x'=\alpha_{1}^{(1)}x+\alpha_{1}^{(2)}y+\alpha_{1}^{(3)}z+\alpha_{1}^{(4)}l\\ \\y'=\alpha_{2}^{(1)}x+\alpha_{2}^{(2)}y+\alpha_{2}^{(3)}z+\alpha_{2}^{(4)}l\\ \\z'=\alpha_{3}^{(1)}x+\alpha_{3}^{(2)}y+\alpha_{3}^{(3)}z+\alpha_{3}^{(4)}l\\ \\l'=\alpha_{4}^{(1)}x+\alpha_{4}^{(2)}y+\alpha_{4}^{(3)}z+\alpha_{4}^{(4)}l\end{array}\right\} $$}}

then (as the calculation shows) the equations

{{MathForm2|(63)|$$\left.\begin{array}{lllcc} \sum\limits _{n}\alpha_{n}^{(m)^{2}}=1 & & m=1,2,3,4\\ \\\sum\limits _{n}\alpha_{n}^{(m)}=0 & & m,o=1,2,3,4 &  & m\lessgtr o\end{array}\right\} $$}}

must be satisfied, which are formed analogously to the condition of orthogonality (p. 255) throughout. Therefore only 6 of the 16 coefficients are arbitrary. If we combine equations (61) for once with $$\alpha_{1}^{(1)},\ \alpha_{2}^{(1)},\ \alpha_{3}^{(1)},\ \alpha_{4}^{(1)}$$, and then with $$\alpha_{1}^{(2)},\ \alpha_{2}^{(2)},\ \alpha_{3}^{(2)},\ \alpha_{4}^{(2)}$$ etc., one finds as there solutions with respect to the unprimed quantities:

{{MathForm2|(64)|$$\left.\begin{array}{c} x=\alpha_{1}^{(1)}x'+\alpha_{2}^{(1)}y'+\alpha_{3}^{(1)}z'+\alpha_{4}^{(1)}l'\\ \\y=\alpha_{1}^{(2)}x'+\alpha_{2}^{(2)}y'+\alpha_{3}^{(2)}z'+\alpha_{4}^{(2)}l'\\ \\z=\alpha_{1}^{(3)}x'+\alpha_{2}^{(3)}y'+\alpha_{3}^{(3)}z'+\alpha_{4}^{(3)}l'\\ \\l=\alpha_{1}^{(4)}x'+\alpha_{2}^{(4)}y'+\alpha_{3}^{(4)}z'+\alpha_{4}^{(4)}l'\end{array}\right\} $$}}

so that by analogy of (63), it is also

We denote the determinant of the coefficient as $$A$$:

their sub-determinants we denote as usually as $$A_{m}^{(n)}$$. Resolving (62) into $$x, y, z, l$$ must give the form now:

$Ax=A_{1}^{(1)}x'+A_{2}^{(1)}y'+A_{3}^{(1)}z'+A_{4}^{(1)}l'$ etc.,

thus it must by according to (64)

If one substitutes these values into (66), if follows for the determinants of the sub-determinants:

$\pm A_{1}^{(1)}A_{2}^{(2)}A_{3}^{(3)}A_{4}^{(4)}=A^{5}$

According to a known theorem, however, we have $$A^{n-1}$$ when the number of the column is $$n$$, thus in our case we have $$A^{3}$$, therefore it must be

because $$A=-1$$ is excluded, since $$A$$ is a steady function of the coefficient, and with an identical transformation $$x'=x$$ etc. it evidently has the value $$+1$$. By (67) it is then

We can combine transformations (62) and (64) into the scheme (being intelligible without further explanation):

{{MathForm2|(XII)|$$\left.\begin{array}{c|c|c|c|c|} & x' & y' & z' & l'\\ \hline x & \alpha_{1}^{(1)} & \alpha_{2}^{(1)} & \alpha_{3}^{(1)} & \alpha_{4}^{(1)}\\ \hline y & \alpha_{1}^{(2)} & \alpha_{2}^{(2)} & \alpha_{3}^{(2)} & \alpha_{4}^{(2)}\\ \hline z & \alpha_{1}^{(3)} & \alpha_{2}^{(3)} & \alpha_{3}^{(3)} & \alpha_{4}^{(3)}\\ \hline l & \alpha_{1}^{(4)} & \alpha_{2}^{(4)} & \alpha_{3}^{(4)} & \alpha_{4}^{(4)}\end{array}\right\} $$}}

''If the substitution (XII) shall represent a Lorentz transformation, the coefficients: 1. must satisfy the conditions of orthogonality (63) and (65); 2. those containing the index 4 once, must be purely imaginary, all others must be real; 3. its determinant and $$\alpha_{4}^{(4)}$$ must be positive.'' We have added conditions 2., so that from the real values $$x, y, z$$ and an imaginary value of $$l$$, real values for $$x', y', z'$$ and imaginary values for $$l'$$ are emerging again and again. Yet, $$\alpha_{4}^{(4)}$$ must be positive since $$\alpha_{4}^{(4)}$$ is always the cosine of the imaginary angle $$\varphi$$ between the $$l$$- and the $$l'$$-axis, thus [see (58)]:

$\alpha_{4}^{(4)}=\cos\varphi=\frac{1}{\sqrt{1-\beta^{2}}}$|undefined

One can see this most easily, when one derives the general transformation from the special substitution (57) in the way discussed above. This special transformation in which the $$y$$- and $$z$$-coordinate remains unchanged, can be found, when

{{MathForm2|(70)|$$\left.\begin{array}{r} \alpha_{2}^{(2)}=\alpha_{3}^{(3)}=1,\ \alpha_{1}^{(1)}=\alpha_{4}^{(4)}=\frac{1}{\sqrt{1-\beta^{2}}},\\ \\-\alpha_{4}^{(1)}=\alpha_{1}^{(4)}=\frac{i\beta}{\sqrt{1-\beta^{2}}}\end{array}\right\} $$}}

and if one sets all other coefficients to zero. Thus their scheme looks as follows:

{{MathForm2|(71)|$$\left.\begin{array}{c|c|c|c|c|} & x' & y' & z' & l'\\ \hline x & \frac{1}{\sqrt{1-\beta^{2}}} & 0 & 0 & \frac{-i\beta}{\sqrt{1-\beta^{2}}}\\ \hline y & 0 & 1 & 0 & 0\\ \hline z & 0 & 0 & 1 & 0\\ \hline l & \frac{i\beta}{\sqrt{1-\beta^{2}}} & 0 & 0 & \frac{1}{\sqrt{1-\beta^{2}}}\end{array}\right\} $$}}

For the sub-determinants of second order $$A_{mn}^{m'n'}$$, conditions apply corresponding to formulas (63) and (65). (The upper indices are always related to rows, the lower ones to the columns of $$A$$.) As the coefficients themselves, these sub-determinants are real when the index 4 occurs not at all or twice, and purely imaginary when its appears once.

According to a known theorem, it is for an arbitrary determinant $$A$$

$A\left

By (67) and (68) it follows from that:

$A_{12}^{12}=A_{34}^{34}$

or more general if we denote as $$B_{mn}^{m'n'}$$ the sub-determinants supplementing $$A_{mn}^{m'n'}$$:

Furthermore, a known theorem concerning the representability of a determinant as the sum of the production of sub-determinants, says:

$A=\sum\limits _{m}\sum\limits _{n}A_{mn}^{m'n'}B_{mn}^{m'n'}$

By (68) and (72) we conclude from that:

We further imagine determinant $$\mathfrak{A}$$ as derived from $$A$$, by replacing the terms of the second row by the corresponding ones of the third row; it is of course zero. Thus by the theorem now employed:

$\sum\limits _{m}\sum\limits _{n}\mathfrak{A}_{mn}^{1\ 2}\mathfrak{B}_{mn}^{1\ 2}=0$

However, now it is

$\mathfrak{A}_{mn}^{1\ 2}=A_{mn}^{1\ 2},\ \mathfrak{B}_{mn}^{1\ 2}=B_{mn}^{1\ 3}$

thus by (72):

Eventually we imagine determinant $$\mathfrak{A}$$ as formed in a way, where in $$A$$ not only the terms of the second row are replaced by the ones of the third row, but also those of the first are replaced by the corresponding ones of the fourth row. Then it is:

$\mathfrak{A}_{mn}^{1\ 2}=A_{mn}^{1\ 2},\ \mathfrak{B}_{mn}^{1\ 2}=B_{mn}^{4\ 3}$

thus

Due to the equality of rows and columns in (73), (74), (75), one can permute the upper with the lower indices.

If one arranges the sub-determinants by this scheme:

$\begin{array}{

then equations (73), (74), and (75) and their (not written) generalizations have the meaning: If one squares and adds the terms in the row or the column, then one obtains the value 1; if one multiplies the corresponding terms of two different rows or columns, then one finds the value zero at the addition.

The coefficients $$\alpha_{m}^{n}$$ and the sub-determinants $$A_{m,n}^{m',n'}$$ will be used in the following in the definition of the four- and six-vectors.

In the special transformation (57), the sub-determinants [see (70)] are:

{{MathForm2|(76)|$$\left.\begin{array}{c} A_{14}^{14}=A_{23}^{23}=1,\\ \\A_{12}^{12}=A_{31}^{31}=A_{24}^{24}=A_{34}^{34}=\frac{1}{\sqrt{1-\beta^{2}}}=\cos\varphi\\ \\A_{24}^{12}=A_{31}^{34}=-A_{12}^{24}=-A_{34}^{31}=\frac{i\beta}{\sqrt{1-\beta^{2}}}=\sin\varphi\end{array}\right\} $$}}

all others are zero. The given scheme thus has the following shape:

In this way, one easily confirms the given rules.


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 entirely free from the walls of the pericarp, except at the point of attachment of the seed, forming a portion of the latter, not of the former, and has therefore no title to the name of a dissepiment, real or spurious, still given to it in systematic works, even in the Prodromus.

 28. BANKSIA, Linn. f.

Flowers hermaphrodite. Perianth regular or nearly so, straight or curved, the slender tube opening equally or along the lower side only, the limb ovoid oblong or linear, the lamini remaining long coherent, or rarely separating as the tube opens. Anthers narrow, sessile in the concave laminæ, the connective thick, usually very shortly produced beyond the cells. Hypogynous scales 4, very thin and membranous (rarely deficient?). Ovary very small and sessile; style usually longer than the perianth, rigid, curved and protruding from the slit in the perianth-tube until the end is set free by the separation of the laminæ, and then either straightened or remaining hooked or curved, rarely straight from the first and not exceeding the perianth; the stigmatic end on a level with the anthers, of a different texture but smooth, or striate and furrowed, continuous with the style or with a prominent rim at the base, the real stigma small and terminal; ovules 2, collaterally attached about the middle. Fruit a compressed capsule, opening at the broad end (or rather outer margin, for the scar of the style is lateral) in two hard often woody horizontal valves. Seeds usually 2, compressed, with a terminal membranous wing broad and rounded like the valves, the seeds separated by a plate of the same shape (the consolidated outer integuments of the inner side of the two seeds) free from the ripe seeds, simple (completely consolidated) between the nuclei, double (remaining distinct) between the wings.—Trees or shrubs. Leaves alternate or rarely verticillate or nearly so, usually narrow, entire toothed pinnatifid or pinnate, with numerous (rarely few) short teeth lobes or segments, the primary veins numerous and transverse, rarely inconspicuous or irregular and the minute reticulations numerous on the under surface, with a minute tomentum rarely wanting in the areolæ and sometimes white and covering the whole under surface, the upper surface almost always glabrous and smooth. Flowers sessile in pairs, in dense terminal cylindrical oblong or globular spikes, either terminal and sessile above the last leaves or rarely lateral or on short lateral branches ; each pair of flowers subtended by one bract and two lateral rather smaller bracteoles, both bracts and bracteoles densely woolly-villous on the sides, the tips glabrous tomentose or villous, either clavate and obtuse or truncate, or shortly acuminate, always densely imbricate in parallel spiral or rarely vertical lines. Perianth-tube very slender and entire within the bracts, ultimately splitting beyond them. In fruit the bracts and bracteoles become consolidated with the rhachis into a thick woody cone, either covered with the withered remains of the perianths amongst which the capsules are entirely concealed, or, where the flowers are wholly deciduous, the valves of the capsules protrude more or less beyond the bracts, the


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Rh by, he said "Oh, is that all? Let us go to dinner." At table he inquired what had become of Madame de Lamballe and the other female prisoners. A guest significantly drew his hand across his throat. Orleans said, "I understand," and changed the subject. One of the Englishmen, overcome with horror, had, according to Peltier, slipped away unnoticed.

Captain George Monro, probably a son of Sir Harry Monro, Bart., also saw something of the barbarities at the Abbaye, and on December 17, describing how people were stopped in the streets, watches and rings taken, and even earrings pulled off, he wrote: "I myself never move out but with pistols in my pocket, as I find them more necessary here than in Turkey." Monro remained till the condemnation of Louis XVI., and sent despatches chiefly relating to the British Jacobins in Paris, with whom he pretended to fraternise, but his position was unofficial and indeed unsafe. "He was not only suspected, but marked here," says Somers, who for a time took his place, and who deemed it prudent to disguise his political reports in mercantile language. One of Somers's letters announces that Morgan, the son of an Irish M.P., was in Paris, and had offered to assassinate George III. On the 1st February 1793 war was declared. In the following


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36 month the Convention prescribed that English residents should find six citizens to vouch for their "civisme," and if not in business nor landowners, should deposit their property in pledge, but until October 1793 they were not seriously molested.

Though he was not at the Embassy, and was ostensibly a mere visitor, Hugh Elliot may here be mentioned. Ex-Ambassador at Copenhagen, he had, with his brother Gilbert (Lord Minto), been at a military college with Mirabeau, and Pitt sent him over in October 1790 to ensure French neutrality in the event of an Anglo-Spanish war. Mirabeau, Barnave, and the other members of the diplomatic committee of the Assembly, gave him satisfactory assurances, and he returned to London at the end of October. Mirabeau had prepared, but not delivered, a long speech in favour of Spain, and threatening England with war if she attacked that country. His sudden change of tone reminds one of Rivarol's epigram, "Mirabeau is capable of anything for money, even of a good action."


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 III.


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 runs that the reconciliation was effected in a most daring way. It is said that Cortés one night left his sanctuary and suddenly appeared before the astonished Governor. His glib tongue must have served him well, for a messenger, sent to announce the prisoner's escape, found the enemies peaceably sleeping together in the same bed!

Wide estates and many slaves were bestowed on the repentant hidalgo, who now sunned himself in the Governor's favour, and soon grew rich and influential. By cultivating his lands and working his gold mines he gained in a few years a fortune of two or three thousand castellanos. "God," says Las Casas in his History of the Indies, "who alone knows at what cost of Indian lives it was obtained, will take account of it!"

Tame prosperity soon palled on Cortés. Restless still he longed for fresh adventure, as every year brought news to Cuba of the discovery of hitherto unknown shores and seas. Ponce de Leon, an unworldly-minded old knight, searching for the fabled fountain of youth, reached in his wanderings the glowing coasts of Florida. Alas, for his cherished dream! Many a silver spring he found, but none which could give back to him the long-lost vigour of his youth. Balboa from a mountain top beheld the mighty Pacific, and Cortés, as he heard on every tongue the glorious story, burned with impatience to think that others were gathering the laurels which he had hoped to wear. But Fortune the fickle is not always forgetful, and she was soon to bring to him a chance more splendid far than even his wildest dreams. 29

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 III.

AT THE BAR OF THE ASSEMBLY. Pigott—Watt—Cooper—Paul Jones—Gay—Burns—Anderson— Mrs. Freeman Shepherd—Potter—Lydia Kirkham.

was represented not merely at the capture of the Bastille, but at every stage of the Revolution, in its festivities and its tragedies, in the Convention and in the Commune, in the clubs and in the armies, in the prisons and on the scaffold. She was represented on Cloots's "Deputation of the Human Race," and at the Feast of Pikes, to which that deputation solicited admission, by Robert Pigott, who had been an opulent country gentleman. The Pigotts claimed descent from a Norman named Picot, and had for eleven generations owned an estate at Chetwynd, Shropshire. They had been strongly attached to the Stuarts, and heirlooms still preserved in the family include a ring, said to have been one of four given by Charles I. on the eve of his execution, a fragment of the Royal Oak, and a portrait on ivory of the Pretender, presented by himself to Robert Pigott's father at


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 lower indehiscent portion containing the nuclei of the seeds remaining imbedded among the bracts. The proportion of perfect capsules is usually very small in relation to the number of flowers, of which there are often from 500 to above 1000 in the same spike.


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40 Rome in 1720. This Jacobite Pigott died in 1770 and his death gave rise to a very singular lawsuit. Some hours after he had expired at Chetwynd, his son and the son of Sir William Codrington, dining at Newmarket without any idea of what had occurred, "ran their fathers' lives one against the other" for five hundred guineas. Such bets were not uncommon in those times, and were not apparently thought indecorous. The elder Pigott was seventy years of age, the elder Codrington only fifty. When Pigott learned that his father's death had occurred prior to the wager, he maintained that the bet was off; but Codrington, or rather Lord March (afterwards Duke of Queensberry), to whom he had assigned the money, insisted on payment. The case came before Lord Mansfield, who held that the impossibility of a contingency did not preclude its being the subject of a wager, if both parties were at the time unaware of that impossibility. He consequently gave judgment against Pigott. In 1774 Pigott was High Sheriff of Shropshire, as his grandfather had been before him. Two years later, sharing the belief of croakers that England was on the brink of ruin, he disposed of Chetwynd, as also, probably, of the manor of Chesterton, in Huntingdonshire. If, as is alleged, estates worth


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68 must have only light work and nourishing food. After that I asked Mary to save everything for Burkman and make his recovery her especial care. The officers made fun of me, as they were rather incredulous, and thought a bit of shamming was being practised on me, but I knew better. They never failed to comment and smile when they saw the old defender of his country coming out of the kitchen-tent, his jaws working and his mouth full, while he carried all the food his hands would hold. To tell the truth, he kept up this prescription of nourishing food long after he had quite recovered.

It became the delight of my husband and the officers to chaff me about "Old Nutriment," for such was the sobriquet they gave him. At last, even Mary began to narrate how he swept everything before him with voracious, convalescing appetite. "Why, Miss Libbie," she said to me one day, "I thought I'd try him with a can of raw tomatoes, and set them before him, asking if he was fond of them. And he just drawled out, 'Always was,' and the tomatoes were gone in no time." His laconic answer passed into a proverb with us all, when invited to partake of anything we liked.

Such a tender heart as that old soldier had! I had noticed this first in Kentucky. My horse, which I prized above all that I have ever ridden, died during my temporary absence from home. I was too greatly grieved to ask many questions about him, but one day, some time afterwards, when we were riding through a charming bit of country, Burkman approached me from the place where he usually rode behind us, and said, "I'd like to tell Mrs. Custer there's whar poor Phil lies

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.—Leaves linear or rarely lanceolate, with revolute margins or nearly flat but very white underneath, entire denticulate or pinnate with small numerous regular segments. Style remaining hooked after the perianth-limb has opened, the stigmatic end very small and not distinctly furrowed.

1. B. pulchella, ''R. Br. in Trans. Linn. Soc. x. 202, Prod. 391.'' A shrub with villous or tomentose branches. Leaves crowded, erect or incurved, linear, sometimes very narrow or almost terete, obtuse or almost acute, the margins entire and closely revolute, narrowly grooved or more broadly channelled underneath, the midrib not prominent, ¼


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 to ½ in. long. Spikes ovoid-globular, the rhachis 1 to 1½ in. long. Bracts villous. Perianth-tube densely villous, about 4 lines long, the limb acute, glabrous. Ovary villous. Style nearly 1 in. long, remaining hooked, with a very small broad stigmatic end. Fruiting cone globular, about 2 in. diameter, the capsules usually very numerous and closely packed, very flat, projecting but slightly, the margin becoming glabrous, nearly 1 in. broad when perfect.—Meissn. in Pl. Preiss. ii. 264, and in DC. Prod. xiv. 452.

2. B. Meissneri, ''Lehm.; Meissn. in Pl. Preiss. i. 582, and in DC. Prod. xiv. 452.'' A spreading shrub of 2 or 3 ft., or sometimes low and straggling, the branches slightly hoary. Leaves linear, rather crowded, very spreading or reflexed, obtuse or scarcely acute, with revolute margins, singly grooved or channelled underneath, not above ¼ in. long and thicker than in B. pulchella. Spikes ovoid and flowers smaller than in that species. Perianth-tube loosely villous, scarcely 3 lines long, the glabrous limb very small. Ovary glabrous? Style about ¾ in. long, remaining hooked with the small depressed stigmatic end of B. pulchella. Fruiting cone not seen.

3. B. nutans, ''R. Br. in Trans. Linn. Soc. x. 203, Prod. 391.'' A shrub, glabrous or nearly so except the inflorescence. Leaves crowded, very narrow-linear, almost terete, very shortly mucronate, the margins closely revolute and entire, singly grooved underneath, ½ to 1 in. long. Spikes globular or shortly oblong, erect or nodding, the rhachis from under 1 to near 2 in. long. Bracts with small glabrous tips. Perianth tube ¾ in. long, silky-villous as well as the limb. Ovary glabrous. Style remaining hooked, with a short thick stigmatic end not distinctly furrowed. Fruiting cone globular, 2 to 4 in. diameter; capsules very thick and scarcely protruding, the end in some specimens above 1 in. broad and nearly 1 in. thick, smooth and at first raised along the suture, at length depressed the thick almost turgid backs of the valves very rugose; in some specimens the capsules smaller and smoother, but perhaps not full-grown.—Meissn. in Pl. Preiss. i. 581, and in DC. Prod. xiv. 453 ; F. Muell. Fragm. iv. 108.


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 4. B. sphærocarpa, ''R. Br. in Trans. Linn. Soc. x. 208, Prod. 391.'' A shrub of 3 or 4 ft., minutely silvery or hoary-tomentose. Leaves linear, obtuse or scarcely mucronate, with closely revolute entire margins, under 1 in. long in the typical specimens, in others 2 to 3 in. long. Spikes globular or nearly so, 2 to 3 in. diameter. Perianth silky, varying from a little above 1 in. to fully 1½ in. long, the limb narrow, obtuse. Style longer than the perianth, hooked, with a small cylindrical stigmatic end. Fruiting cone globular, dense; capsules slightly prominent, glabrous, thick, with a prominent ridge at the suture, nearly 1 in. broad when perfect.—Meissn. in Pl. Preiss. i. 581, and in DC. Prod. xiv. 452; B. pinifolia, Meissn. in DC. Prod. xiv. 453.

5. B. tricuspis, ''Meissn. in Hook. Kew Journ. vii. 118, and in DC. Prod. xiv. 453''. Branches rather slender, glabrous or very slightly hoary. Leaves narrow-linear, truncate or almost notched, with a small callous point, the margins entire and closely revolute, 2 to 4 in. long. Spikes oblong-cylindrical, 5 to 6 in. long Bracts obtuse, fulvous-villous. Perianths silky-villous but all withered and revolute in our specimens. Style above 1½ in long, hooked, with a very small ovoid stigmatic end. Fruiting cone with very closely imbricate obtuse bracts, capsules very prominent, not thick, becoming glabrous, 9 to 10 lines broad.

6. B. occidentalis, ''R. Br. in Trans. Linn. Soc. x. 204, Prod. 392.'' An erect shrub of 4 or 5 ft., the branches glabrous or minutely hoary. Leaves linear, truncate notched or 3-toothed at the end, otherwise entire or with a few small teeth towards the end, the margins recurved only, showing the white under surface and prominent midrib, 2 to 4 in. long. Spikes from ovoid and 3 in. to cylindrical and twice as long. Bracts with small glabrous tips. Perianth silky-villous, about ¾ in. long, the limb narrow. Ovary villous; style about 1 in. long, hooked, the stigmatic end scarcely distinct. Fruiting cone tomentose with the closely packed bracts; capsules prominent, not very thick, rounded, tomentose-villous, becoming glabrous at the suture, about ¾ in. broad.