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  [q. v.], to adhere to his society (5 Dec. 1775). But Copley joined the Academy.

At the request of the Earl of Bute Greenwood made a journey, in July 1771, into Holland and France purchasing paintings; he afterwards visited the continent, buying up the collections of Count van Schulembourg and the Baron Steinberg. In 1776 he was occupying Ford's Rooms in the Haymarket as an art auctioneer. In this business he continued to the end of his life, removing in 1783 to Leicester Square, where he built a commodious room adjoining his dwelling-house, and communicating with Whitcomb Street.

He died while on a visit at Margate, 16 Sept. 1792, and was buried there. His wife, who survived him a few years, was buried at Chiswick, close to the tomb of Hogarth.

A small half-length portrait of Greenwood in mezzotint, by W. Pether, bearing an artist's pallet and brushes and an auctioneer's mallet, was afterwards published. A three-quarter length, by Lemuel Abbot, and a miniature by Henry Edridge, are in possession of his grandson, Dr. John D. Greenwood, ex-principal of Nelson College, New Zealand. The portrait of himself as a young man, in coloured crayon, mentioned by Van Eynden and Van der Willigen, is now in the possession of the writer of this article.

Greenwood was not, as has been said, father of Thomas Greenwood, the scene-painter at Drury Lane Theatre, who died 17 Oct. 1797. His eldest son, Charnock-Gladwin, died an officer in the army at Grenada, West Indies; the second, John, succeeded him in business; James returned to Boston; and the youngest, Captain Samuel Adam Greenwood, senior-assistant at the residency of Baroda, died at Cambray in 1810.

GREER, SAMUEL MCURDY(1810–1880), Irish politician, eldest son of the Rev. Thomas Greer, presbyterian minister of Dunboe, and Elizabeth Caldwell, daughter of Captain Adam Caldwell, R.N., was born at Springvale, co. Derry, in 1810, educated at the Belfast Academy and Glasgow University, and was called to the Irish bar in 1833. His life was devoted to constitutional agitation for such reforms in Irish land tenure as were necessary to make the union tolerable as a permanent arrangement. It was about 1848 that Greer first began to take an active part in political life, and although never a very prominent figure in public, his influence and popularity in his native county were very great. He was one of the originators of the tenant league, formed in 1850 by himself, Sir John Gray, proprietor of the 'Freeman's Journal,' Dr. M'Knight, editor of the 'Londonderry Standard,' Frederick Lucas, and John Francis Maguire. They demanded for the Irish tenant what have since been known as the three F's —fixity of tenure, fair rents, and free sale. Greer was one of the few Ulstermen of any weight or position— [q. v.] was another—who adopted these principles. He contested the representation of co. Derry four times, and that of the city of Londonderry twice, being successful only once, in 1857. Although almost continuously defeated he was in reality more than any other man the creator of the liberal party in Ulster. He practically retired in 1870, before the movement in favour of home rule had attained its later importance. Most of the reforms for which he struggled—tenant right, vote by ballot, &c.—had already been conceded. He probably would not have approved the policy afterwards developed by Mr. Parnell's party, and dissented from their cardinal principle of standing entirely aloof from both English parties. There was, therefore, nothing to prevent him from accepting the recordership of Londonderry in 1870. He held this office until 1878, when he was appointed county court judge of Cavan and Leitrim. He died in 1880.

GREETING, THOMAS (fl. 1675), musician, published in 1675 'The Pleasant Companion, or new Lessons and Instructions for the Flagelet.' Pepys engaged him to teach his wife an 'art that would be easy and pleasant for her' (1 March 1666–7); in the following year Greeting sent the Duke of Buckingham's musicians to Pepys's house to play dance music.

GREG, PERCY (1836–1889), author, son of [q. v.], was born at Bury in 1836, and died in London on 24 Dec. 1889. His career during the greater part of his life was that of a journalist, and in his later years that of a novelist and historian. He contributed largely to the 'Manchester Guardian,' 'Standard,' and 'Saturday Review,' and obtained much distinction as a political writer. But, although endowed with great ability, he lacked the equity that characterised his father, and always tended to violent extremes; in youth a secularist, in middle life


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 WILHELM MEISTER'S APPRENTICESHIP

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


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 a spiritualist, in his later years a champion of feudalism and absolutism, and in particular an embittered adversary of the American Union. The violence of his political sympathies has entirely spoiled his attempted 'History of the United States to the Reconstruction of the Union,' 1887, which can only be regarded as a gigantic party pamphlet. His ultimate convictions, political and religious, found expression in two volumes of essays, 'The Devil's Advocate,' 1878, and 'Without God; Negative Science and Natural Ethics,' 1883; and in a series of novels displaying considerable imagination and invention: 'Across the Zodiac,' 1880; 'Errant,' 1880; 'Ivy cousin and bride,' 1881; 'Sanguelac,' 1883; and 'The Verge of Night,' 1885. Of his sincerity there could be no question, and his polemical virulence did not exclude a tender vein of lyrical poetry, pleasingly manifested in his early poems, published under the pseudonym of Lionel H. Holdreth, and in his 'Interleaves' (1875).

GREG, ROBERT HYDE (1795–1875), economist and antiquary, born in King Street, Manchester, on 24 Sept. 1795, was son of Samuel Greg, a millowner near Wilmslow, Cheshire, and brother of [q. v.] and [q. v.] His mother was Hannah, daughter and coheiress of Adam Lightbody of Liverpool, and a descendant of, the nonconformist [q. v.] He was educated at Edinburgh University, and before joining his father in business as a cotton manufacturer, travelled in Spain, Italy, and the East. In 1817 he entered the Literary and Philosophical Society of Manchester, and afterwards contributed to its 'Memoirs' some interesting papers on topics chiefly suggested by his observations abroad. Their titles are:
 * 1) 'Remarks on the Site of Troy, and on the Trojan Plain,' 1823.
 * 2) 'Observations on the Round Towers of Ireland,' 1823.
 * 3) 'On the Sepulchral Monuments of Sardis and Mycenæ,' 1833.
 * 4) 'Cyclopean, elasgic, and Etruscan Remains; or Remarks on the Mural Architecture of Remote Ages,' 1838.

He took a leading part in public work in Manchester, aiding in the foundation of the Royal Institution, the Mechanics' Institution, and in the affairs of the Chamber of Commerce, of which for a time he was president. He was an ardent liberal politician, and rendered valuable assistance in money and advocacy in the agitations for parliamentary reform and the repeal of the corn laws. In 1837 he wrote a pamphlet on the 'Factory Question and the Ten Hours Bill.' He was elected M.P. for Manchester in September 1839, during his absence from England. He took the seat against his will and he retired in July 1841. In the meantime he published a speech on the corn laws, which he had delivered in the House of Commons in April 1840, and a letter to Henry Labouchere, afterwards Lord Taunton, 'On the Pressure of the Corn Laws and Sliding Scale, more especially upon the Manufacturing Interests and Productive Classes,' 1841, 2nd ed. 1842.

He was much interested in horticulture, and in practical and experimental farming, which he carried on at his estates at Norcliffe, Cheshire, and Coles Park, Hertfordshire. In this connection he wrote three pamphlets: 'Scottish Farming in the Lothians,' 1842; 'Scottish Farming in England,' 1842; and 'Improvements in Agriculture,' 1844.

He married, 14 June 1824, Mary, eldest daughter of Robert Philips of the Park, Manchester; by her he had four sons and two daughters. Greg died at Norcliffe Hall on 21 Feb. 1875, and was buried at the Unitarian chapel, Dean Row, Wilmslow, Cheshire, being followed to the grave by nearly five hundred of his tenants and employés, and by many others.

GREG, SAMUEL (1804–1876), philanthropist, was fourth son of Samuel Greg, a mill-owner at Quarry Bank, near Wilmslow, Cheshire, by his wife Hannah, and therefore a brother of [q. v.] and [q. v.] He was born in King Street, Manchester, 6 Sept. 1804, and educated at Unitarian schools at Nottingham and Bristol. After leaving Bristol he spent two years at home learning mill-work, and in the autumn of 1823 went to Edinburgh for a winter course of university lectures. In 1831, with his youngest brother, William Rathbone Greg, he studied and practised mesmerism with great enthusiasm, and to such practice he attributed his subsequent ill-health. He took the Lower House Mill, near the village of Bollington, in 1832, and having fitted it up with the requisite machinery, commenced working with hands imported from the neighbouring districts of Wilmslow, Styall, and other places. For about fifteen years the mill and the workpeople were his all-absorbing objects of consideration and pursuit. Some account of his proceedings is found in two letters which in 1835 he addressed to Leonard Horner,


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 inspector of factories, and which were printed for private circulation. He first established a Sunday school, next a gymnasium, then drawing and singing classes, baths and libraries, and finally he instituted the order of the silver cross in 1836 as a reward for good conduct in young women. In 1847 he was employed in making experiments on new machinery for stretching cloth. This idea was unpopular in the mill, and the workpeople, instead of coming to him to talk the matter over, surprised him by turning out. Other troubles followed, and it was not long before he was obliged to retire altogether from business, a comparatively poor man. In 1854 he wrote and published 'Scenes from the Life of Jesus,' a work of which a second edition was printed in 1869. His 'Letters on Religious Belief' appeared in 1856, but came to a conclusion after the seventh letter. He entertained Kossuth on 22 March 1857, at his residence, the Mount, Bollington, and in the same year commenced giving Sunday evening lectures to working people in Macclesfield, a practice which he continued for the remainder of his life. During 1867 he gave scientific lectures to a class of boys. In 1863 he formed the acquaintance of Dean Stanley, with whom he afterwards continued a pleasant intercourse. After a long illness he died at Bollington, near Macclesfield, 14 May 1876. In June 1838 he married Mary Needham of Lenton, near Nottingham, by whom he had a family. She was the writer in 1855 of 'Little Walter, a Mother's first Lessons in Religion for the younger classes.'

GREG, WILLIAM RATHBONE (1809–1881), essayist, born at Manchester in 1809, was son of Samuel Greg, merchant, and brother of [q. v.] and [q. v.] His father became owner of a mill near Wilmslow in Cheshire, where William Rathbone's childhood was passed. After receiving his education under Dr. Lant Carpenter at Bristol, and afterwards at the university of Edinburgh, Greg became in 1828 manager of one of his father's mills in Bury, and in 1832 commenced business on his own account. In 1835 he married Lucy, daughter of [q. v.], a physician of Manchester. In 1842 he won a prize offered by the Anti-Corn Law League for the best essay on 'Agriculture and the Corn Laws.' In the same year he was induced by concern for his wife's health to settle in the neighbourhood of Ambleside. The removal unfavourably affected his business, and after a long struggle to avert failure he ultimately relinquished it in 1850. His literary and speculative pursuits had also probably interfered with his success in trade, for in 1851 he came before the world with his 'Creed of Christendom,' the outcome of long study and thought. Mr. Morley has recorded the effect in its day of this contribution to 'dissolvent literature;' it must be said that no work hostile to received opinions was ever so little of a polemic against them, or more distinguished by candour and urbanity. Greg now took distinct rank as an author, writing in 1852 no fewer than twelve articles for the four leading quarterlies, mostly on political or economical subjects. His essay on Sir Robert Peel in the 'Westminster Review,' vol. lviii., was the finest tribute called forth by the statesman's death. His 'Sketches in Greece and Turkey' appeared in 1853. In 1856 Sir George Cornewall Lewis bestowed on him a commissionership at the board of customs, which restored him to independence. From 1864 to 1877 he was comptroller of the stationery office. He had in the interim lost his first wife, and married the daughter of of the 'Economist' [q. v.] The only other marked incidents of his life during this period were the successive publications of his works: 'Political Problems for our Age and Country,' 1870; 'Enigmas of Life,' 1872; 'Rocks Ahead, or the Warnings of Cassandra,' 1874; 'Mistaken Aims and Attainable Ideals of the Working Classes,' 1876. He continued to be an extensive contributor to the periodical press, and his essays were collected three times, as 'Essays on Political and Social Science' (1853), 'Literary and Social Judgments' (2nd edit. 1869, 4th edit. 1877), and 'Miscellaneous Essays' (1882 and 1884). He died at Wimbledon 15 Nov. 1881. His son Percy is separately noticed.

In Greg ardent philanthropy and disinterested love of truth were curiously allied to an almost epicurean fastidiousness, which made him unduly distrustful of the popular element in politics. He would have wished to see public affairs controlled by an enlightened oligarchy, and did not perceive that such an oligarchy was incompatible with the principles which he had himself admitted. Little practical aid towards legislation, therefore, is to be obtained from his writings. It was Greg's especial function to discourage unreasonable expectations from political or even social reforms, to impress his readers with the infinite complexity of modern


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 "Wilhelm still held that lovely hand"


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1922 Encyclopædia Britannica/Eichhorn, Hermann von

EICHHORN, HERMANN VON (1848-1918), German field-marshal, was born at Breslau Feb. 13 1848. He took part, as a young officer, in the campaigns of 1866 and 1870-1. In 1897 he was appointed chief of the staff of the VI. Army Corps at Breslau, in 1901 divisional and in 1907 corps commander. In 1905 he was promoted to the rank of general of the infantry and in 1913 to that of Generaloberst, while in the same year he was appointed inspector-general of the VII. Army Inspection at Saarbrücken. At the outbreak of the World War he was incapacitated in consequence of an accident, but was able to play a part in the battle of Soissons in Jan. 1915. In that month he was appointed to the command of the X. Army, which was engaged in the great battle of the Masurian Lakes in the following February. In Aug. he took Kovno and afterwards the fortresses of Grodno and Olita, and continued his victorious advance into Russia. From 1916-8 Eichhorn was in command of the army group known by his name in Courland. In Dec. 1917 he was raised to the rank of field-marshal and was sent to the Ukraine as chief-in-command of the German troops on the eastern front. He was assassinated at Kiev July 30 1918.


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 With His Life by

GEORGE HENRY LEWES

In Fourteen Volumes

Volume I.

PUBLISHED BY

LONDON AND BOSTON


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, in one of his letters written in response to a communication from a sentimental young countess, who wrote him anonymously regarding his "Werther," gives a contrasting picture of himself in two phases. The one is a carnival Goethe, in a laced coat and other consistent finery, illuminated by the unmeaning magnificence of sconces and chandelier, amidst all kinds of people, kept at the card-table by a pair of beautiful eyes, and in varying dissipation driven from party to concert and from concert to ball, and with all the fascination of frivolity paying court to a pretty Blondine: a sentimental Goethe, with affectedly gloomy deep feelings. The other is a wholesome German Goethe, in a gray beaver coat and boots, with brown silk cravat, eagerly detecting the breath of spring in the cool February air, and waiting for his dear wide world to open out once more. This Goethe, ever living, striving, and working in himself, seeks to express, according to his powers, sometimes the innocent feelings of youth in little poems, the strong spices of life in various dramas, the forms of his friends and his neighbourhood and his beloved household possessions with chalk on gray paper, never asking if any of his work is destined to last, for the reason that the very act of working makes him keep rising higher and higher, and he will leap at no ideal, but fight and play, leaving his feelings to develop of themselves.

On the one hand, he found pleasure in the gay coquetries of fashionable life; on the other, he declared that his greatest happiness was to live with the best men of his time. Goethe, when he wrote that, was in his twenty-sixth year. He had drunk deeply of the cup of life and had recognised the danger vii


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Translation:On the Theory of Relativity I: Four-dimensional Vector Algebra

by.

In this and some subsequent studies I want to illustrate, how remarkably simplified the electrodynamic concepts and calculations become, when we allow ourselves to be led by the deep space-time understanding of. For the friend who suddenly passed away, the following would hardly provide any new insight; yet as elucidation of 's ideas it may be welcomed by some.

The content of the relativity principle can be formulated according to : In the physical equations only space-time vectors are allowed to occur, i.e. quantities that have vector-character in the four-times manifold of space and time, the an "world". Therefore when passing from one to a new reference system, its components can be transformed according to the scheme of the coordinate transformation for that manifold (" transformation"). Instead of absolute space of the older theory, the absolute world takes its place, i.e. the connection of space and time by the speed of light $$c$$, whose invariability now constitutes the absolute substrate of electrodynamics.

In this first part I confine myself to the algebraic relations of the space-time vectors. A second part "vector analysis" should illustrate the differential properties of the four-dimensional vector-fields. In particular I attached great importance to the geometric description (as far as possible) of the definition and the transformation of the vectors. The required concepts and calculations, that give also a full replacement for the matrix calculus used by, are (when admitting imaginary coordinates) direct generalizations of the known three-dimensional vector-procedure.

§ 1. Four- and Six-vectors.
It's well known that in space of three dimensions we have to distinguish between two kinds of vectors, vectors of first kind or polar vectors and vectors of second kind or axial vectors, also denoted as rotors or plan-quantities. A vector of first kind is a straight line provided with a direction-sense, its components are the perpendicular projections upon the coordinate-axes. A vector of second kind is a plane surface provided with a rotation sense, its components are the perpendicular projections of the surface upon the coordinate-planes. The components of the first ones are thus related to one axis for each component, those of the second ones are related to two axes for each component, thus the first has to be written with one and the latter with two indices. In mechanics, force and velocity belong to the first, torque and angular velocity to the second. E. g. the components of an angular velocity were to be written as $$\omega_{yz},\ \omega_{zx},\ \omega_{xy}$$; only after passing to 's "supplement", i.e. the perpendicular upon the rotating plane, the ordinary notation $$\omega_{x},\ \omega_{y},\ \omega_{z}$$ is justified. Also the magnetic field strength $$\mathfrak{H}$$ belongs to the same class of vectors, as it was emphasized by. The ordinary notation $$\mathfrak{H}_{x}\dots$$ that is also maintained in the following, should consequently be replaced by $$\mathfrak{H}_{yz}\dots$$.

In space of four dimensions the vectors of first kind have four components, "four-vectors", those of second kind have six components, "six-vectors". There also exist vectors of third kind, that again become a four-component vector.

A four-vector is e.g. a position vector from the origin to a space-time point (x y z l) with components

$x,y,z,l=ict,\ i=\sqrt{-1}$

l as well as x, y, z is a length; the expression shall remind of a "way of light". The magnitude of that vector, i.e. the length of the corresponding four-dimensional distance, is:

$\sqrt{x^{2}+y^{2}+z^{2}+l^{2}}=\sqrt{x^{2}+y^{2}+z^{2}-c^{2}t^{2}}$|undefined

Two vectors $$\left(x_{1}y_{1}z_{1}l_{1}\right)$$ and $$\left(x_{2}y_{2}z_{2}l_{2}\right)$$ shall be called perpendicular to each other, if

$x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}+l_{1}l_{2}=0$

The coordinate vector (x y z l) is the type of all four-vectors. A quadruple of quantities only then deserves the name four-vector, when it is transformed by a coordinate change in the same manner as the coordinate vector, i.e. covariant with it. Here, the length of any four-vector is invariant; it is its unique invariant. Especially concerning the coordinate vector, the invariance of its length expresses the principle of the constant propagation of light. By the way, due to the imaginary character of its fourth components, the length of its four-vector can be zero, without that the vector and its components vanish.

A vector of first kind is also the "four-density P", that summarizes the concept of velocity $$\mathfrak{v}$$ and density $$\varrho$$ of a moving charge, with the components:

its length becomes

The vector-character of P is given as follows: We consider the worldline of a charge element de, i.e. the successive space-time locations of it in the xyzl-manifold. The worldline element between to adjacent points 1 and 2

dx, dy, dz, dl

is surely a four-vector covariant with the coordinate-vector. Afterwards we use the important principle of the reference-frame independence of charge, by which the electric mass is fundamentally distinguished before the material mass. This principle is of course in accordance with 's equations and is mentioned by as a consequence of it. By that:

$de=\varrho dS=\varrho'dS'$

dS = dx dy dz means the three-dimensional space-element, as well as $$dS'$$ is the same in a new "primed" reference system, that emerges from the original by a mere rotation around 0. In Fig. 1, $$dS$$ and $$dS'$$ appear as two arbitrary bisections through the world-line-canal, that corresponds to the limits of de. If we put each of such bisection-pairs through the points 1 and 2, then one cuts out two equal four-dimensional space elements dΣ:

$d\Sigma=dS\ dl=dS'dl'$

where $$dl, dl'$$ are the heights of the two space elements dΣ, and at the same time the components of the worldline-element 12 into the axes $$l, l'$$, which as the fourth axes of our unprimed and primed reference system must stand perpendicularly upon $$dS$$ and $$dS'$$. By division of the two preceding equations it is given, that

$\frac{\varrho}{dl}=\frac{\varrho'}{dl'}=\frac{de}{d\Sigma}$

is a reference-frame-independent scalar quantity. If we thus multiply the four-vector (dx, dy, dz, dl) by $$i\varrho/dl$$, then it keeps its vector-character; exactly the four-vector P from equation (1) occurs by that.

When passing to the space-time vectors of second kind, we consider in space of four dimensions a plane-section of specific magnitude and location, where we are only concerned with the area and the orientation, but not with the form or the absolute location; the location can be changed by a parallel displacement, the form can be arbitrarily changed by a transformation of same content. Yet this geometric image is still too special; it has only five independent specification-parts, where its magnitude is given by one, and its location by four numbers. But we have to consider, that to a plane in space of four dimensions, also the perpendicular plane (die ensemble of all those lines that are perpendicular to all lines of the first plane) are uniquely specified. In this normal plane we now imagine a second plane-section of specific magnitude. The quintessence of two such plane-sections perpendicular to each other, is the geometric image of a vector of second kind. It depends on six specification-sections, by adding to the five mentioned parts, the magnitude of the plane-section (that is perpendicular to it) as the sixth specification-section. We therefore speak about a six-vector. As its pure right-angled components in a xyzl coordinate space, we denote the sum of the perpendicular projections of the two plane-sections upon one of the six coordinate planes ''xy, yz, .. zl''; the components of a six-vector shall therefore be written by two indices each, that must be mutually different, and their order is of importance.

To the special space-time vector of second kind, the simple plane-section, we can kinematically relate a simple rotation; to the general six-vector, the plane-pair, we relate a general rotation or a helix. Namely we have to note, that for the simplest rotation operation that let unchanged all points of a point in four dimensions, we have to speak not about a rotation axis, but about a rotation plane; that rotation plane it then given by the plane of our plane section, the rotation quantity is given by the magnitude of it. As in three dimensions, a helix is composed of a certain rotation and a translation to the axis of this translation, i.e. a rotation around the infinitely distant axis perpendicular to it, likewise the general rotation in four dimensions emerges from the composition of two simple rotations around two mutually perpendicular planes, in agreement with the constitution of our general six-vector from two perpendicular plane-sections. By this remark the expression "helix" for the general rotation may be justified.

A vector of second kind is e.g. the six-vector f of the electromagnetic field with the components

{{MathForm1|(2)|$$\left\{ \begin{array}{lrlrlr} f_{yz}= & \mathfrak{H}_{x}, & f_{zx}= & \mathfrak{H}_{y}, & f_{xy}= & \mathfrak{H}_{z},\\ f_{xl}= & -i\mathfrak{E}_{x}, & f_{yl}= & -i\mathfrak{E}_{y}, & f_{zl}= & -i\mathfrak{E}_{z},\end{array}\right.$$}}

The meaning of the six-vector and the relation of its components to two coordinate axis chosen in a certain order, corresponds to the general definition:

That the electric and magnetic forces $$\mathfrak{E}$$ and $$\mathfrak{H}$$ are combined by to the higher unity of the six-vector f,  is characterizing for the relativity theory, in which (depending on the reference system) the same field can appear as purely electric or as electromagnetic. E.g a uniform moving charge shows the co-moving observer only the "electric side" of the six-vector, but to the non-co-moving observer also its "magnetic side".

That the electromagnetic field is indeed to be understood as a six-vector, i.e. that when changing the reference frame its components are transformed accordingly to the type of the four-dimensional plane-section or plane-pair, follows from the fact, that due to 's equations the field is connected with the four-density P by the divergence condition (see later).

To better become acquainted with the six-vector, we consider the special case of a single plane-section $$\varphi$$ of area 1 and give it the form of a parallelogram that emerges from the origin. The two sides of it may be the four-vectors u, v. Then the projection of that plane-section upon the xy-plane is specified by the projections $$u_{x}u_{y}v_{x}v_{y}$$ of the four-vectors, namely

Between the thus formed six components of this specific six-vector, of which only five are independent from each other by the counting on p. 753, the identity exists:

as it can directly be proven by multiplication of the relevant determinants. Furthermore the area $$\left|\varphi\right|$$ of the surface section must be defined as the square root of the square sum of its six components. In our case we thus have:

On the other hand we consider the surface-section $$\varphi^{*}$$ (perpendicular to $$\varphi$$) of area 1, that we can also denote as the "supplement" of that, and claim that its components are

or in general, when we indicate the indices that are nor illustrated by brackets

where in respect to the order of the indices ik(ik) it must be defined, that it must emerge from the order x y z l by an even number of permutations.

For verification we represent $$\varphi^{*}$$ again by two four-vectors u*v*, which individually are perpendicular to uv. That means (see the above definition of perpendicularity) the existence of the following equations:

$$\begin{array}{ll} u_{x}^{*}u_{x}+u_{y}^{*}u_{y}+u_{z}^{*}u_{z}+u_{l}^{*}u_{l} & =0,\\ u_{x}^{*}v_{x}+u_{y}^{*}v_{y}+u_{z}^{*}v_{z}+u_{l}^{*}v_{l} & =0,\\ v_{x}^{*}u_{x}+v_{y}^{*}u_{y}+v_{z}^{*}u_{z}+v_{l}^{*}u_{l} & =0,\\ v_{x}^{*}v_{x}+v_{y}^{*}v_{y}+v_{z}^{*}v_{z}+v_{l}^{*}v_{l} & =0.\end{array}$$

By multiplying them by $$v_{l},\ u_{l},\ v_{l},\ u_{l}$$ respectively, and subtracting the second from the first and the fourth from the third, we obtain:

$$\begin{array}{ll} u_{x}^{*}\varphi_{xl}+u_{y}^{*}\varphi_{yl}+u_{z}^{*}\varphi_{zl} & =0,\\ v_{x}^{*}\varphi_{xl}+v_{y}^{*}\varphi_{yl}+v_{z}^{*}\varphi_{zl} & =0.\end{array}$$

Furthermore, if we multiply these two equations by $$v_{z}^{*}u_{z}^{*}$$ or $$v_{y}^{*}u_{y}^{*}$$ respectively, and subtract them from each other, then it follows

$\varphi_{xz}^{*}\varphi_{xl}+\varphi_{yz}^{*}\varphi_{yl}=0$ or $\varphi_{xy}^{*}\varphi_{xl}+\varphi_{zy}^{*}\varphi_{zl}=0$

Due to $$\varphi_{xz}^{*}=-\varphi_{zx}^{*}$$ we can write the proportion:

$\varphi_{yz}^{*}:\varphi_{xy}^{*}:\varphi_{zx}^{*}=\varphi_{xl}:\varphi_{yl}:\varphi_{zl}$

Accordingly it is given:

$\varphi_{yz}:\varphi_{xy}:\varphi_{zx}=\varphi_{xl}^{*}:\varphi_{yl}^{*}:\varphi_{zl}^{*}$

and thus, when λ means the factor of proportionality:

$\varphi_{ik}^{*}=\lambda\varphi_{(ik)}$

However, as $$\varphi^{*}$$ as well as $$\varphi$$ should have the same area 1, then $$\lambda^{2}=1$$ in accordance with the claim before (3d).

The identity (3a) can be written, by using the expression of § 3A, also in this way:

{{MathForm1|(3a)|$$\left\{ \begin{array}{ll} \left(\varphi\varphi^{*}\right) & =\varphi_{yz}\varphi_{yz}^{*}+\varphi_{xy}\varphi_{xy}^{*}+\dots+\varphi_{zl}\varphi_{zl}^{*}\\ & =2\left(\varphi_{yz}\varphi_{xl}+\varphi_{zx}\varphi_{yl}+\varphi_{xy}\varphi_{zl}\right)=0\end{array}\right.$$}}

The general six-vector f now emerges from the two special ones $$\varphi$$ and $$\varphi^{*}$$ in the form

$$\varrho$$ and $$\varrho^{*}$$ are the areas of the two mutually perpendicular surface-sections of orientation $$\varphi$$ and $$\varphi^{*}$$, of which f is composed. To f we form the "dual vector f*", its "supplement" by permutation of the surfaces $$\varrho$$ and $$\varrho^{*}$$. Namely, f* may be defined in consequence of (4):

From that it follows e.g. for the yz-components of this dual vector

in the same way

and in general

so that f and f* are, with respect to each other, in the same relation, as the special surface-section $$\varphi$$ with $$\varphi^{*}$$ perpendicular to it.

Also the quantities $$\left|f\right|^{2}$$ and (ff*) can be expressed by $$\varrho$$ and $$\varrho^{*}$$ in accordance with (3b, e).

{{MathForm1|(5a)|$$\left\{ \begin{array}{c} \left|f\right|^{2}=f_{yz}^{2}+f_{zx}^{2}+\dots f_{zl}^{2}\\ =\varrho^{2}\left|\varphi\right|^{2}+\varrho^{*2}\left|\varphi^{*}\right|^{2}+2\varrho\varrho*(\varphi\varphi*)=\varrho^{2}+\varrho^{*2}\end{array}\right.$$}}

{{MathForm1|(5b)|$$\left\{ \begin{array}{c} \left(ff^{*}\right)=f_{yz}f_{yz}^{*}+f_{xy}f_{xy}^{*}+\cdots+f_{zl}f_{zl}^{*}\\ =\varrho^{2}\left(\varphi\varphi^{*}\right)^{2}+\varrho^{*2}\left(\varphi^{*}\varphi\right)+\varrho\varrho*\left(\left|\varphi\right|^{2}+\left|\varphi^{*}\right|^{2}\right)=2\varrho\varrho*\end{array}\right.$$}}

Both have to be denoted as invariants of the six-vector, as they have, like the surfaces $$\varrho$$ and $$\varrho^{*}$$, a geometric meaning independent of the location of the coordinate system. We can, for example, denote $$\left|f\right|$$ as (geometric) surface sum, and (ff*) as (double) surface product. In case f means the electromagnetic force, we have by (2) and (5a, b)

The expression L means "Lagrangian function of unit volume".

However, besides the components related to the coordinate planes, we can also speak of its components related to the coordinate spaces and coordinate axes.

By the component related to coordinate space, we understand a surface section that lies in this space, which we obtain when we project the two surface sections $$\varrho$$ and $$\varrho^{*}$$ upon this space, and form the geometric sum of their two projections. E.g. the component of f in relation to the space of x y z, is the three-dimensional vector of second kind:

$\left(f_{yz},\ f_{zx},\ f_{xy}\right)=\mathfrak{H}$

By the component related to a coordinate axis, we understand a surface-section obtained by splitting off (from the surface sections $$\varrho$$ and $$\varrho^{*}$$) its components in relation to the space perpendicular with this axis, and the two remaining surface-sections are combined geometrically. If we intersect the resulting surface with the space perpendicular to the axis, then a three-dimensional vector of first kind in this space arises. E.g. the component of f in relation to the axis of l, is the three-dimensional vector of first kind:

that also can be interpreted as the special four-vector of the vanishing l-component. In general $$f_{j}$$ shall be the four-vector that is perpendicular to the j-axis:

The component of the supplement f* related to the j-axis has the corresponding meaning:

and in particular

Besides vectors of first and second kind, also vectors of third kind exist in four-dimensional space, which are again four-vectors given not by a distance, but by a three-dimensional space-section, and which were written with three indices as projections of the space-section upon the four coordinate spaces, e.g.

$\mathfrak{A}_{yzl},\ \mathfrak{A}_{zlx},\ \mathfrak{A}_{lxy},\ \mathfrak{A}_{xyz}$

Geometrically, to such a vector of third kind we can attribute a vector of first kind, its "supplement", namely the line perpendicular to the space section whose length is equal to the size of the space-section. The components of the latter

$\mathfrak{A}_{x},\ \mathfrak{A}_{y},\ \mathfrak{A}_{z},\ \mathfrak{A}_{l}$

are one after the other equal to the components of the vector of third kind. The replaceability of the vector of third kind by its supplement is reaching even further, like the relation of the supplement with the vector of second kind in the case of three dimensions, in so far as in this case the supplement (due to the even dimension number) correctly reproduces the behavior of the vector of third kind, not only regarding coordinate transformations of determinant +1, but also regarding such of determinant -1 (mirroring, inversion). As the summarizing denotation for the vector of first and third kind, the name "four-vector" is recommended.

Since we can geometrically compose any two space sections by vectorial addition of their supplements, we don't obtain a more general vector of third kind through a space-pair as through a single space-section. Indeed the latter has already four mutually independent specification-sections - its size and the three data that are necessary for the definition of its location.

An important example (compare § 3B and §4) for a vector of third kind, we will find in the electrodynamic force (Pf).

§ 2. Component formation into arbitrary directions and planes, and its connection with the Lorentz transformation.
It's characteristic for the vector-concept and its independence of the coordinate system, that we can speak of its components into arbitrary directions (planes as regards the six-vector). By the component of a four-vector $$P$$ into the $$x$$-axis we have to think of the perpendicular projection upon this axis, i.e.

If $$x'$$ is a "space-like" axis, then $$\cos(x',x)$$, $$\cos(x',y)$$, $$\cos(x',z)$$ become real, $$\cos(x',l)$$ becomes purely imaginary, so that (see (1) and 7)) $$P_{x'}$$ becomes real. On the other hand, $$P_{l'}$$ is purely imaginary when $$l'$$ is a "time-like" axis.

By the component of a six-vector $$f$$ into an arbitrary plane, which we think as given by two mutually perpendicular directions $$x'y'$$ contained in it, we understand the sum of the perpendicular projections of the surface sections (that forms the vector f) upon this plane; it can be calculated from the two-row determinant of the direction cosine:

{{MathForm1|(8)|$$\left\{ \begin{array}{lll} f_{x'y'} & =f_{xy}\left|{\cos(x'x)\cos(x'y)\atop \cos(y'x)\cos(y'y)}\right| & +f_{yz}\left|{\cos(x'y)\cos(x'z)\atop \cos(y'y)\cos(y'z)}\right|\\ \\ & +f_{zx}\left|{\cos(x'z)\cos(x'x)\atop \cos(y'z)\cos(y'x)}\right| & +f_{xl}\left|{\cos(x'x)\cos(x'l)\atop \cos(y'z)\cos(y'l)}\right|\\ \\ & +f_{yl}\left|{\cos(x'y)\cos(x'l)\atop \cos(y'y)\cos(y'l)}\right| & +f_{zl}\left|{\cos(x'z)\cos(x'l)\atop \cos(y'z)\cos(y'l)}\right|\end{array}\right.$$}}

For verification we consider at first the special six-vector $$\varphi$$ (pure surface section of magnitude 1) and illustrate those as in §1 in the form of a parallelogram by two four-vectors u,v. Then the surface-section (projected upon the x'y'-plane, i.e. the sought component $$\varphi_{x'y'}$$, is specified by the projections of the four-vectors $$u_{x'},u_{y'},v_{x'},v_{y'}$$ upon the $$x'$$- and $$y'$$-axis as in equation (3) by:

$\varphi_{x'y'}=u_{x'}v_{y'}-v_{x'}u_{y'}$

If we substitute into $$u_{x'},\dots$$ its value from (7) and write, according to (3), $$u_{x}v_{y}-u_{y}v_{x}$$ and $$\varphi_{xy}\dots$$ for the sub-determinants, then exactly the equation (8) for the special case $$f=\varphi$$ is given. The same equations also arise for the surface-section $$\varphi^{*}$$ (perpendicular to $$\varphi$$) and therefore also for the sum $$\varrho\varphi+\varrho^{*}\varphi^{*}$$, that means the general six-vector by equation (4). The component rule (8) for the six-vector is also a necessary consequence of that for the four-vector (7).

Regarding the reality-relations, it is similar for the components of the four-vector: $$f_{x'y'}$$ consists of only real, $$f_{x'l'}$$ of only imaginary terms, when the axes $$x', y'$$ are space-like, and $$l'$$ is time-like.

Furthermore we consider the quadruple of four-vectors $$f_h$$ derived from the six-vector f, and ask for the expression of the components $$f_{x'}$$ of f into the new $$x'$$-axis (which thus means a surface-section parallel to x', or a four-vector perpendicular to this axis) from the components $$f_{x},f_{y},f_{z},f_{l}$$ into the original axes. Indeed, by stating $$f_{x},f_{y},f_{z},f_{l}$$, also $$f_{x'}$$ must be given. We claim, that the following connection exists:

Equation (9) is formally in agreement with (7), but also essentially differs from it, because on the right-hand side of (9) there are four directed four-vectors, on the side of (7) there are four undirected components. Equation (9) is of course to be proved due to the vector-property of f, equation (8). We denote the vector-sum that occurs in (9) by $$\Sigma$$:

$\Sigma=f_{x}\cos(x'x)+f_{y}\cos(x'y)+f_{z}\cos(x'z)+f_{l}\cos(x'l)$

and form the components into the axis x y z l by equation (6a):

$$\begin{array}{lllll} \Sigma_{x} & = & f_{yx}\cos(x'y) & +f_{zx}\cos(x'z) & +f_{lx}\cos(x'l)\\ \Sigma_{y} & =f_{xy}\cos(x'x) & & +f_{zy}\cos(x'z) & +f_{ly}\cos(x'l)\\ \Sigma_{z} & =f_{xz}\cos(x'x)+ & f_{yz}\cos(x'y) & & +f_{lz}\cos(x'l)\\ \Sigma_{t} & =f_{xl}\cos(x'x)+ & f_{yl}\cos(x'y) & +f_{zl}\cos(x'z)\end{array}$$

If we multiply one after the other by $$\cos(j'x)$$, $$\cos(j'y)$$, $$\cos(j'z)$$, $$\cos(j'l)$$ (by $$j'$$ we understand an arbitrary axis at first), and sum up, then by (7) the components of $$\Sigma$$ into this axis emerge, and namely it is given:

 * for $$j'=x'$$ the sum 0,
 * for $$j'=y'$$ the expression (8),
 * for $$j'=z'$$ the same expression by permutation of $$y'$$ with a new direction $$z'$$ (perpendicular to $$x'$$) etc.

The right-hand side of (9) indeed represents the quadruple of that magnitude, denoted by us as $$f_{x'}$$ by equation (6), namely the quadruple

$\left(0\ f_{x'y'}\ f_{x'z'}\ f_{x'l'}\right)$

For the sake of completeness, also the behavior of a vector of third kind in its projection upon an arbitrary space $$x'y'z'$$ may be mentioned. Its component into space $$x'y'z'$$ would be calculated by a four-parametric expression, in which the three-row sub-determinants of the direction cosine occur as coefficients, yet for which we can put (by the conditions of orthogonality that exist between them) the direction cosine of the common perpendicular $$l'$$. The four-vector of third kind is also projected the same way as the four-vector of first kind forming its supplement.

Now, we prove to ourselves, that the projection- and component-formulas developed here, are identical with the transformation formulas of relativity theory. For this sake we consider, as usual, the special case, that the new reference system of $$x'y'z'l'$$ corresponds in the two middle axes with the original one of x y z l, so that one from the other can obtained by a rotation in the xl-plane (around the yz-plane) (rotation angle $$\varphi$$);  namely the $$x'$$-axis may be space-like, thus the $$l'$$-axis is time-like. Then

{{MathForm1|(10)|$$\left\{ \begin{array}{lrl} \cos(x'x)= & \cos(l'l) & =\cos\varphi\\ \cos(x'l)= & -\cos(l'x) & =\sin\varphi\\ \cos(y'y)= & \cos(z'z) & =1\\ \cos(y'x)= & \dots & =0\end{array}\right.$$}}

If we apply the component formula (7) to the coordinate vector (x y z l), then we obtain

{{MathForm1|(10a)|$$\left\{ \begin{array}{lrl} x'= & x\cos\varphi+l\sin\varphi, & y'=y\\ l'= & -x\sin\varphi+l\cos\varphi, & z'=z\end{array}\right.$$}}

Now $$\cos\varphi>1$$, since by the remark on p. 760 $$\varphi=x_{1}/\sqrt{x_{1}^{2}+l_{1}^{2}}$$ and $$l_{1}^{2}<0$$, thus $$\varphi$$, $$\sin\varphi$$ and $$\operatorname{tg}\varphi$$ are purely imaginary, and $$\left|\operatorname{tg}\varphi\right|<1$$. Thus we have, when $$\beta$$ denotes a real fraction:

Consequently the equations (10a) become identical with the transformation and show, that $$v=\beta c$$ means the velocity, by which (in space of xyz) the new system of axis is uniformly displaced in the x-direction against the original one.

At the same time equation (8) shows, as to how the six-vector of the field is transformed (or "projected"). We have by (8) and (10):

$$\begin{array}{ll} f_{x'y'}=f_{xy}\cos\varphi-f_{yl}\sin\varphi, & f_{z'x'}=f_{zx}\cos\varphi+f_{zl}\sin\varphi,\\ f_{y'z'}=f_{yz}, & f_{x'l'}=f_{xl},\\ f_{y'l'}=f_{yl}\cos\varphi+f_{xy}\sin\varphi, & f_{z'l'}=f_{zl}\cos\varphi-f_{zx}\sin\varphi,\end{array}$$

or because of (10b) and (2):

{{MathForm1|(10c)|$$\left\{ \begin{array}{lll} \mathfrak{H}_{x'}=\mathfrak{H}_{x}, & \mathfrak{H}_{y'}=\frac{\mathfrak{H}_{y}+\beta\mathfrak{E}_{z}}{\sqrt{1-\beta^{2}}}, & \mathfrak{H}_{z'}=\frac{\mathfrak{H}_{z}+\beta\mathfrak{E}_{y}}{\sqrt{1-\beta^{2}}},\\ \\\mathfrak{E}_{x'}=\mathfrak{E}_{x}, & \mathfrak{E}_{y'}=\frac{\mathfrak{E}_{y}+\beta\mathfrak{H}_{z}}{\sqrt{1-\beta^{2}}}, & \mathfrak{E}_{z'}=\frac{\mathfrak{E}_{z}+\beta\mathfrak{H}_{y}}{\sqrt{1-\beta^{2}}}.\end{array}\right.$$}}

By that also these well-known transformation formulas of the field appear to be the immediate consequence of our geometrical interpretation of the six-vector and its component formation derived from the projection of four-dimensional lines. About 's equation, from which these formulas were derived by and, we didn't speak at all (except in so far, as we have denoted f as six-vector by referring to those equations). Generally, according to and  the relativity principle is to be seen as a superior principle above electrodynamics.

A. Scalar products.
Two four-vectors P and $$\Phi$$ will be scalar-multiplied by the rule

$(P\Phi)=P_{x}\Phi_{x}+P_{y}\Phi_{y}+P_{z}\Phi_{z}+P_{l}\Phi_{l}$

E.g., if P means the four-density, $$\Phi$$ the four-potential (combination of scalar and vector potential, see later), then $$(P\Phi)$$ becomes the "electro-kinetic potential" of. The square of the length of a four-vector is equal to the scalar product by itself, e.g. $$\left|P\right|^{2}=(PP)$$ see (1a).

In case of arbitrary motion of P and $$\Psi$$, the formula taken from ordinary vector calculus

$(P\Phi)=\left

defines the direction cosine of two arbitrary directions; it was used in this sense already at the beginning of § 2, footnote. The definition of perpendicularity from the beginning of § 1 is a special case of this.

Also two six-vectors f and F can be scalar-multiplied by the rule

$(fF)=f_{xy}F_{xy}+f_{yz}F_{yz}+\dots f_{zl}F_{zl}$

Into this rule belong the two invariants of the six-vector f, see (5a, b)

$\left

B. Vector products.
Two four-vectors u, v, will be vector-multiplied by a rule, which is analogous to the ordinary three-dimensional vector product. Like in that place a vector of second case emerges from two vectors of second kind, here a six-vector results from two four-vectors, namely a special six-vector $$\varphi$$ (simple surface section) with the invariant $$\left(\varphi\varphi^{*}\right)=0$$. The multiplication rule already used in § 1 reads

$\varphi_{xy}=u_{x}v_{y}-u_{y}v_{x},\dots$

$$\varphi$$ means the parallelogram formed by u and v in terms of size and location.

We can also vector-unify a four-vector and a six-vector, namely to a four-vector of third kind. If the six-vector is in particular a simple surface section, then the vector of third kinds means the parallelepiped directly formed by the four-vector and the supplement of that surface-section. In the general case it will be constructed as the geometric sum of two such parallelepipeds, yet it can always be replaced (see the end of §1) by a four-vector of first kind. This state of facts is the same as in ordinary vector calculus, where a vector of first kind emerges by vector-multiplication of a vector of first and second kind. The formal scheme of that multiplication is taken by us from this three-dimensional case.

$$\mathfrak{A}=\left(\mathfrak{A}_{x}\mathfrak{A}_{y}\mathfrak{A}_{z}\right)$$ be the vector of first kind and $$\mathfrak{B}=\left(\mathfrak{B}_{yz},\mathfrak{B}_{zx},\mathfrak{B}_{xy}\right)$$ the vector of second kind. From the latter we form three special vectors of first kind, namely

$$\begin{array}{cc} \mathfrak{B}_{x}=\left(\mathfrak{B}_{xx},\mathfrak{B}_{xy},\mathfrak{B}_{xz}\right),\\ \mathfrak{B}_{y}=\left(\mathfrak{B}_{yx},\mathfrak{B}_{yy},\mathfrak{B}_{yz}\right), & \mathfrak{B}_{ik}=-\mathfrak{B}_{ki},\ \mathfrak{B}_{ii}=0,\\ \mathfrak{B}_{z}=\left(\mathfrak{B}_{zx},\mathfrak{B}_{zy},\mathfrak{B}_{zz}\right),\end{array}$$

in general $$\mathfrak{B}_{j}$$, from which all are perpendicular to the j-axis because of $$\mathfrak{B}_{jj}=0$$. Like in §1 equation (6a), they can also be called the components of the surface section $$\mathfrak{B}$$ into the j-axis, and originally they denote the surface sections parallel to this axis. As j-component of the vector product of $$\mathfrak{A}$$ and $$\mathfrak{B}$$, we now have to define the scalar product of the two vectors of first kind $$\mathfrak{A}$$ and $$\mathfrak{B}_j$$, namely

$\left(\mathfrak{A}\mathfrak{B}_{j}\right)=\mathfrak{A}_{x}\mathfrak{B}_{jx}+\mathfrak{A}_{y}\mathfrak{B}_{jy}+\mathfrak{A}_{z}\mathfrak{B}_{jz}$

This is in agreement with the ordinary rule of the vector product and gives e.g. for $$j=x$$:

$\left(\mathfrak{A}\mathfrak{B}_{x}\right)=\mathfrak{A}_{y}\mathfrak{B}_{xy}-\mathfrak{A}_{z}\mathfrak{B}_{zx}$

By that we define in the four-dimensional case the product (PF) from an arbitrary four-vector P and a six-vector f by its four components for j = x, y, z, l, by means of formula:

Any of this components it thus given by scalar-multiplication of the four-vector P with the four-vector $$f_j$$ (perpendicular to the j-axis, defined in (6a), and derived from f). In the next paragraph we consider the electrodynamic force $$\mathfrak{F}$$ as an important example of this. In this place, at first the already mentioned geometric meaning of this vector of third kind may be derived, in the special case, where $$f=\varphi$$ represents a simple surface-section.

We replace $$\varphi$$ as a parallelogram by two four-vectors u, v and go over to the dual surface section $$\varphi^{*}$$ consisting of two four-vectors u*, v* parallel to u, v. Then the components of $$(P\varphi)$$ prove to be equal to the four three-row sub-determinants $$D_{x},D_{y},D_{z},D_{l}$$ of the scheme

When neglecting the first colon it is namely given:

$$\begin{array}{ll} D_{x}= & P_{y}\left(u_{z}^{*}v_{l}^{*}-u_{l}^{*}v_{z}^{*}\right)+P_{z}\left(u_{l}^{*}v_{y}^{*}-u_{y}^{*}v_{l}^{*}\right)\\ & +P_{l}\left(u_{y}^{*}v_{z}^{*}-u_{z}^{*}v_{y}^{*}\right)=P_{y}\varphi_{zl}^{*}+P_{z}\varphi_{ly}^{*}+P_{l}\varphi_{yz}^{*}\end{array}$$

For that we can write by (5a):

$D_{x}=P_{y}\varphi_{xy}+P_{z}\varphi_{xz}+P_{l}\varphi_{xl}$

which is in agreement with our definition (11) of $$P\varphi_{xl}$$.

$$D_x$$ means by the known formulas of space geometry, a parallelepiped situated in the yzl-space formed by the vectors $$\left(P_{y},P_{z},P_{l}\right)$$, $$\left(u_{y}^{*},\ u_{z}^{*},\ u_{l}^{*}\right)$$, $$\left(v_{y}^{*},\ v_{z}^{*},\ v_{l}^{*}\right)$$, i.e. by the projections of the four-vectors P, u*, v* upon the yzl-space; $$D_x$$ is at the same time also the projection of the parallelepiped (formed by these four-vectors, situated in four-dimensional space) upon the coordinate space of yzl. By the definition of the end of §1, $$D_{x},\ D_{y},\ D_{z},\ D_{l}$$ indeed represent the components of a vector of third kind. By denoting this components with $$D_{x},\dots$$ instead of the more extensive $$D_{yzl}\dots$$, we already executed the passage from the space section to its supplement (the line perpendicular to the space section). That the quantity $$\left(D_{x},\ D_{y},\ D_{z},\ D_{l}\right)$$ considered as a vector of first kind, is perpendicular to the parallelepiped (formed by the vectors P, u*, v*), can directly be seen. We only have to supplement the above 3 × 4-row determinant scheme by addition of the four components of P or u* or v* to a vanishing four-row determinant, and obtain:

If we are dealing with the product of P with a general six-vector $$f=\varrho\varphi+\varrho^{*}\varphi^{*}$$, then we shall consider in the same way the two parallelepipeds D and D*, which are formed by the vectors P, u*, v* and P, u, v, respectively. Then $$(Pf)=\varrho D+\varrho^{*}D^{*}$$ represents the geometric sum of these two space sections, which we can imagine as to be constructed by the supplements (perpendicular to that space section), as the resultant of the two lines $$\varrho D$$ and $$\varrho^{*}D^{*}$$.

C. Tensor products.
A vector of second kind gives, completely multiplied by itself, a tensor. We explain the way of constitution at first in the three-dimensional case.

If $$\mathfrak{B}$$ is a three-dimensional vector of second kind with components $$\mathfrak{B}_{yz},\ \mathfrak{B}_{zx},\ \mathfrak{B}_{xy}$$, and $$\mathfrak{B}_{j},\ \mathfrak{B}_{h}$$ are whatever two of the three special vectors derived from it (see p. 765), then it is in general for j, h = x, y, z

$T_{jh}=\left(\mathfrak{B}_{j}\mathfrak{B}_{h}\right)=\mathfrak{B}_{jx}\mathfrak{B}_{hx}+\mathfrak{B}_{jy}\mathfrak{B}_{hy}+\mathfrak{B}_{jz}\mathfrak{B}_{hz}$

From that it directly follows

$T_{jh}=T_{hj},\ T_{jj}\ne0$

The thus defined three-dimensional tensor quantity has six components, and differs essentially from our four-dimensional six-vector, for which $$f_{ik}=-f_{ki},\ f_{ii}=0$$ is given. The six components of T can be most conveniently, geometrically interpreted as coefficients of the equation of a surface of second degree.

In the four-dimensional case we accordingly start with a six-vector f, and define a tensor (ff) by all possible scalar products of the special four-vectors $$f_{j},f_{h}$$ (derived by equation (6a) from f) for h, j = x, y, z, l. From that, at first 16 tensor components follow

$\left(f_{j}f_{h}\right)=f_{jx}f_{hx}+f_{jy}f_{hy}+f_{jz}f_{hz}+f_{jl}f_{hl}$

which, however, are reduced to 10 because of $$\left(f_{j}f_{h}\right)=\left(f_{h}f_{j}\right)$$. The calculation gives, if $$f=(\mathfrak{H},-i\mathfrak{E})$$ especially means the field vector, for j = h:

$$\begin{array}{ll} \left(f_{x}f_{x}\right)= & \mathfrak{H}_{y}^{2}+\mathfrak{H}_{z}^{2}-\mathfrak{E}_{x}^{2}\\ \left(f_{y}f_{y}\right)= & \mathfrak{H}_{z}^{2}+\mathfrak{H}_{x}^{2}-\mathfrak{E}_{y}^{2}\\ \left(f_{z}f_{z}\right)= & \mathfrak{H}_{x}^{2}+\mathfrak{H}_{y}^{2}-\mathfrak{E}_{z}^{2}\\ \left(f_{l}f_{l}\right)=- & \mathfrak{E}_{x}^{2}-\mathfrak{E}_{y}^{2}-\mathfrak{E}_{z}^{2}\end{array}$$

and for $$j\ne h$$

$$\begin{array}{llr} \left(f_{x}f_{y}\right)= & \left(f_{y}f_{x}\right)= & -\left(\mathfrak{H}_{x}\mathfrak{H}_{y}+\mathfrak{E}_{x}\mathfrak{E}_{y}\right)\\ \left(f_{y}f_{z}\right)= & \left(f_{z}f_{y}\right)= & -\left(\mathfrak{H}_{y}\mathfrak{H}_{z}+\mathfrak{E}_{y}\mathfrak{E}_{u}\right)\\ \left(f_{z}f_{x}\right)= & \left(f_{x}f_{z}\right)= & -\left(\mathfrak{H}_{z}\mathfrak{H}_{x}+\mathfrak{E}_{z}\mathfrak{E}_{x}\right)\\ \left(f_{x}f_{l}\right)= & \left(f_{l}f_{x}\right)= & i\left(\mathfrak{E}_{y}\mathfrak{H}_{z}-\mathfrak{E}_{z}\mathfrak{H}_{y}\right)\\ \left(f_{y}f_{l}\right)= & \left(f_{l}f_{y}\right)= & i\left(\mathfrak{E}_{z}\mathfrak{H}_{x}-\mathfrak{E}_{x}\mathfrak{H}_{z}\right)\\ \left(f_{z}f_{l}\right)= & \left(f_{l}f_{z}\right)= & i\left(\mathfrak{E}_{x}\mathfrak{H}_{y}-\mathfrak{E}_{y}\mathfrak{H}_{x}\right)\end{array}$$

Besides we also consider the tensor (f*f*) which is formed by the dual six-vector $$f^{*}=(-i\mathfrak{E,H})$$, whose components emerge form the preceding by permutation of $$\mathfrak{H}$$ by $$-i\mathfrak{E}$$ and $$\mathfrak{E}$$ by $$+i\mathfrak{H}$$, e.g.

$$\begin{array}{ll} \left(f_{x}^{*}f_{x}^{*}\right)= & -\mathfrak{E}_{y}^{2}-\mathfrak{E}_{z}^{2}+\mathfrak{H}_{x}^{2}\\ \left(f_{x}^{*}f_{y}^{*}\right)= & \left(f_{x}^{*}f_{y}^{*}\right)=+\left(\mathfrak{E}_{x}\mathfrak{E}_{y}+\mathfrak{E}_{x}\mathfrak{E}_{y}\right)\\ \left(f_{x}^{*}f_{l}^{*}\right)= & \left(f_{l}^{*}f_{x}^{*}\right)=i\left(\mathfrak{H}_{y}\mathfrak{E}_{z}-\mathfrak{H}_{z}\mathfrak{E}_{y}\right)\end{array}$$

and eventually go over to the tensor (composed by both):

with the components

{{MathForm1|(12a)|$$\left\{ \begin{array}{ll} T_{xx} & =\frac{1}{2}\left(-\mathfrak{H}_{x}^{2}+\mathfrak{H}_{y}^{2}+\mathfrak{H}_{z}^{2}-\mathfrak{E}_{x}^{2}+\mathfrak{E}_{y}^{2}+\mathfrak{E}_{z}^{2}\right)\\ \\T_{yy} & =\frac{1}{2}\left(-\mathfrak{H}_{y}^{2}+\mathfrak{H}_{z}^{2}+\mathfrak{H}_{x}^{2}-\mathfrak{E}_{y}^{2}+\mathfrak{E}_{z}^{2}+\mathfrak{E}_{x}^{2}\right)\\ \\T_{zz} & =\frac{1}{2}\left(-\mathfrak{H}_{z}^{2}+\mathfrak{H}_{x}^{2}+\mathfrak{H}_{y}^{2}-\mathfrak{E}_{z}^{2}+\mathfrak{E}_{x}^{2}+\mathfrak{E}_{y}^{2}\right)\\ \\T_{ll} & =\frac{1}{2}\left(-\mathfrak{H}_{x}^{2}-\mathfrak{H}_{y}^{2}-\mathfrak{H}_{z}^{2}-\mathfrak{E}_{x}^{2}-\mathfrak{E}_{y}^{2}-\mathfrak{E}_{z}^{2}\right)\end{array}\right.$$}}

{{MathForm1|(12b)|$$\left\{ \begin{array}{ll} T_{xy}=T_{yx} & =-\mathfrak{H}_{x}\mathfrak{H}_{y}+\mathfrak{E}_{x}\mathfrak{E}_{y}\\ T_{yz}=T_{zx} & =-\mathfrak{H}_{y}\mathfrak{H}_{z}+\mathfrak{E}_{y}\mathfrak{E}_{z}\\ T_{zx}=T_{xz} & =-\mathfrak{H}_{z}\mathfrak{H}_{x}+\mathfrak{E}_{z}\mathfrak{E}_{x}\end{array}\right.$$}}

{{MathForm1|(12c)|$$\left\{ \begin{array}{ll} T_{xl}=T_{lx} & =i\left(\mathfrak{E}_{y}\mathfrak{H}_{z}-\mathfrak{E}_{z}\mathfrak{H}_{y}\right)\\ T_{yl}=T_{ly} & =i\left(\mathfrak{E}_{z}\mathfrak{H}_{x}-\mathfrak{E}_{x}\mathfrak{H}_{z}\right)\\ T_{zl}=T_{lz} & =i\left(\mathfrak{E}_{x}\mathfrak{H}_{y}-\mathfrak{E}_{y}\mathfrak{H}_{x}\right)\end{array}\right.$$}}

As it can be seen, the components $$T_{ik}$$ are identical with the above components $$\left(f_{i}f_{k}\right)$$; the components $$\left(T_{ii}\right)$$ that are different from $$\left(f_{i}f_{i}\right)$$, satisfy the relation $$T_{xx}+T_{yy}+T_{zz}+T_{ll}=0$$. The three components (12c) are, after multiplication by -ic, equal to the components of 's energy flux, the first three components (12a) and the three components (12b) represent 's stresses, while the fourth component (12a) is the negatively taken energy density.

Also now, it is most convenient to interpret the 16 tensor components as coefficients in the equation of a three-dimensional "space of second degree" located in four-dimensional space.

The product formations discussed here, do all correspond to 's matrix scheme. Without doubt this offers (in mathematical view) the advantage of greatest generality and simplicity. On the contrary, in the preceding it was tried to emphasize the special geometric meaning of the single products by ordinary vector calculus.

All these products are, by their geometric meaning, independent of the choice of reference frame, namely in the following way: The scalar products (considered in A) are pure invariants of the Lorentz transformation. The vector products (considered in B) were transformed covariantly as a six- or a four-vector; their components in the new reference frame are thus only connected to those of the old reference frame by the formulas of §2.

For the vector product (Pf) we can calculate this directly from equation (9). Namely, if we scalar-multiply the four-vectors located at the left and right sides with the four-vector P, then

$\left(Pf_{x'}\right)=\left(Pf_{x}\right)\cos(x'x)+\left(Pf_{y}\right)\cos(x'y)+\left(Pf_{z}\right)\cos(x'z)+\left(Pf_{l}\right)\cos(x'l)$

i.e. that formula which, by §2 equation (7), expresses the $$x'$$-component of a four-vector by the old components. (Pf) is thus a true four-vector and $$\left(Pf_{x'}\right)$$ its projection upon the $$x'$$-direction.

At the same time also the new tensor components are calculated from the old tensor components by the simple scheme of the tensor transformation, whose coefficients are quadratically formed from the direction cosine of the new against the old coordinate axes.

§ 4. The electrodynamic force $$\mathfrak{F}$$.
The covariant formation (Pf) from the four-density P and the field-vector f (discussed in the previous paragraph under (11)) requires a special interest. Its formal calculation gives:

$$\begin{array}{ll} \mathfrak{F}_{x}=\left(Pf_{x}\right) & =\varrho\left(\frac{\mathfrak{v}_{x}}{c}f_{xx}+\frac{\mathfrak{v}_{y}}{c}f_{xy}+\frac{\mathfrak{v}_{z}}{c}f_{xz}+if_{xl}\right)\\ \\ & =\varrho\left(\frac{\mathfrak{v}_{y}}{c}\mathfrak{H}_{z}-\frac{\mathfrak{v}_{z}}{c}\mathfrak{H}_{y}+\mathfrak{E}_{x}\right)\end{array}$$

and similarly for the y,z-components; in contrast for the l-component:

$$\begin{array}{ll} \mathfrak{F}_{l}=\left(Pf_{l}\right) & =\varrho\left(\frac{\mathfrak{v}_{x}}{c}f_{lx}+\frac{\mathfrak{v}_{y}}{c}f_{ly}+\frac{\mathfrak{v}_{z}}{c}f_{lz}+if_{ll}\right)\\ \\ & =\frac{i\varrho}{c}\left(\mathfrak{v}_{x}\mathfrak{E}_{x}+\mathfrak{v}_{y}\mathfrak{E}_{y}+\mathfrak{v}_{z}\mathfrak{E}_{z}\right)\end{array}$$

In the expression of the ordinary three-dimensional vector calculation we thus find for j = x, y, z or l:

$$\mathfrak{F}_{j}$$ thus represents the j-component of the electrodynamic force (acting on the unit volume), and $$\mathfrak{F}_{l}$$ the electrical work (multiplied with i/c) on the moving charge per unit volume and time. The latter shall be denoted energetic, the first dynamic components.

If we decompose $$f=\varrho\varphi+\varrho^{*}\varphi^{*}$$ into two simple surface sections, then $$F=\varrho D+\varrho^{*}D^{*}$$, as discussed in the previous paragraph, is decomposed into two simple space sections, whose suplements are perpendicular to the four-density P. Since the latter (see the beginning of §1) has the direction of the worldline element of the charge (its components were proportional to dx, dy, dz, dl), then also the force $$\mathfrak{F}$$ is perpendicular to the worldline of its point of contact. The corresponding requirement is to be extended to an arbitrary force, as long as we exclude dissipative processes like the generation of an heat etc.

The fourth energetic component of $$\mathfrak{F}$$ has in no way only a formal meaning, as it might appear at first. Rather (when changing frames) it is located directly at the real dynamic components. If it is (e.g. like in equation (10a)) a reference frame $$x'y'z'l'$$ emerging from the original one by a rotation in the xl-plane, then

The energetic supplementary terms $$\mathfrak{F}_{l}\sin\varphi$$ (or $$\mathfrak{-F}_{l'}\sin\varphi$$) are small and of order $$1/c^2$$, since $$\mathfrak{F}_{l}$$ (or $$\mathfrak{F}_{l'}$$) as well as $$\sin\varphi$$ contain the factor $$1/c$$. Nevertheless this supplementary terms are indispensable for carrying out the principle of relativity, as it will be shown later by the example of the simple law.

If the dynamic components $$\mathfrak{F}_{x}$$ and $$\mathfrak{F}'_{x}$$ for any force $$\mathfrak{F}$$ in two relatively moving systems are known, e.g. by sufficiently exact measurements, then we can (by comparison according to the preceding formulas) specify the energetic components $$\mathfrak{F}_{l}$$ and $$\mathfrak{F}_{l'}$$. Conversely, we can also say: A force (e.g. also gravitation) is only then physically known, i.e. in the sense of the relativity principle specifiable for an arbitrary reference frame, when also its energetic component $$\mathfrak{F}_{l}$$ is known.

If we especially choose the primed system as "co-moving", so that the point of contact of force $$\mathfrak{F}$$ is at rest in this reference frame, then $$dx'=dy'=dz'=0$$ and also $$\mathfrak{F}_{l'}=0$$, since the force on the worldline of its point of contact is perpendicular. Only in this case the energetic component vanishes out of the formulas for the determination of $$\mathfrak{F}_{x}\dots\mathfrak{F}_{l}$$, thus having the simple form (x-direction = direction of relative motion):

These are the typical transformation formulas for the passage from the "co-moving" to the "resting system", valid for any vector of first (or third) kind, especially for the electrodynamic force related to unit volume. We assume, that any "specific" force, i.e. calculated per unit volume, also behaves as a vector of first kind.

It is different for the entire force $$\mathfrak{K}$$ or $$\mathfrak{K}'$$ in the stationary or co-moving system, of which the first, due to its relation to an arbitrary coordinate system, has actually no place in relativity theory. By the specific force is can be thus defined:

if dS = dy dy dz is the three-dimensional volume element in the stationary, $$ds'=dy'dy'dz'$$ is that in the co-moving system. The latter differs from the first, depending on the mutual location of both reference frames. By addition of dS, $$dS'$$, the vector character of the right-hand side will thus be disturbed. The entire force doesn't simply behave as a vector of first kind.

From Fig. 1 it follows

If, what we don't presuppose, the primed system with its $$l'$$-axis is orientated into the world-line of the contact point of $$dS/dS'=\sqrt{1-\beta^{2}}$$, then the $$x'y'z'$$-space and its volume element $$dS'$$ is perpendicular to this worldline, and $$dS'$$ is the perpendicular projection of dS in this space. The projection angle is equal to the angle of the two considered perpendiculars $$l'$$ and l, thus equal to $$\varphi$$. The circumstance, that $$\varphi$$ is imaginary, has the consequence that the projection $$dS'$$ is greater than the projected element dS, opposing to our image in Fig. 1, that operates with a real $$\varphi$$. The ratio $$dS/dS'=\sqrt{1-\beta^{2}}$$ is the known contraction.

From (15), (15a) and (14a) it directly follows:

$$\begin{array}{l} \mathfrak{K}_{x}=\int\mathfrak{F}_{x}dS=\cos\varphi\int\mathfrak{F}_{x'}dS=\int\mathfrak{F}_{x'}dS'=\mathfrak{K}_{x'}\\ \\\mathfrak{K}_{y}=\int\mathfrak{F}_{y}dS=\int\mathfrak{F}_{y'}dS=\frac{1}{\cos\varphi}\int\mathfrak{F}_{y'}dS'\frac{1}{\cos\varphi}\mathfrak{K}_{y'}\ \mathrm{etc}.\end{array}$$

and summarizing:

These formulas (14b), when compared with (14a), prove the preceding explanation, that $$\mathfrak{K}$$ doesn't behave like a vector. Namely, if we denote, in formal analogy to our vector notation, by $$\left|\mathfrak{K}\right|^{2}$$ and $$\left|\mathfrak{K}'\right|^{2}$$ the square sum of the unprimed and primed components (where $$\mathfrak{K}'_{l}$$ is zero according to definition equation (15)), then because of (14b), $$\left|\mathfrak{K}\right|$$ is not becoming equal to $$\left|\mathfrak{K}'\right|$$, but

$\left

while from (14a), as it must be the case for a four-vector, it follows of course $$\left|\mathfrak{F}\right|=\left|\mathfrak{F}'\right|$$.

We also especially consider the case of a normal pressure acting in the primed system, that acts upon a surface $$\sigma'$$ at rest in the primed system, which gives rise to pressure $$p'$$ that coincides with the perpendicular $$n'$$ and which is measured per unit area. We can interpret this case, either as a distribution of infinitely small total-forces $$\mathfrak{K}$$ or infinitely great specific forces $$\mathfrak{F}$$, according to one or the other of the following two schemes

$\mathfrak{K}_{n'}=p'd\sigma'$ or $p'=\mathfrak{F}_{n'}dn'$

The first standpoint was taken by. The latter should be preferable because of the vivid vectorial behavior of $$\mathfrak{F}$$. From this standpoint we think of $$\mathfrak{F}$$ as being distributed over a layer of thickness $$dn'$$ which is adjacent to surface $$\sigma'$$, similarly as we interpret the surface charge as the limit of a spatial distribution.

If we erect upon this three-dimensional layer $$\sigma'dn'$$ (located in $$x'y'z'$$-space) a four-dimensional cylinder parallel to the $$l'$$-axis, then this encloses the entirety of all worldlines of the layer points. The force $$\mathfrak{F}'$$, that should be perpendicular to $$\sigma'$$ in $$x'y'z'$$-space, also coincides with the perpendicular of that four-dimensional cylinder, since it is also perpendicular to the worldline at its contact point. If we lay any inclined intersection as xyz-space through the cylinder, then it intersects (from the cylinder) the image $$\sigma dn$$ of the layer that appears to an observer in the xyzl-system. In Fig. 2, the (by p. 753, Note 2 completely specified) normal plane (laid through the considered point of $$\sigma$$) to the tangential plane of $$\sigma$$, is chosen as drawing plane. This not only contains the normal n of $$\sigma$$ in xyz-space, but also the cylinder-normal $$n'$$ as well as the l-axis, since all three directions are perpendicular to $$\sigma$$. In the $$n'$$-direction falls the direction of vector $$\mathfrak{F}$$; by direction and size we have $$\mathfrak{F}=\mathfrak{F}_{n'}$$. For the reference frame xyzl it is decomposed in two components, which both can be seen in the figure: the energetic component $$\mathfrak{F}_{l}$$ and the dynamic $$\mathfrak{F}_{n}$$.

On the other hand, the component of $$\mathfrak{F}$$ perpendicular to the drawing plane in the tangential plane of $$\sigma$$ doesn't exist, i.e. also in the unprimed system (after separation of its energetic component) the force $$\mathfrak{F}$$ and its corresponding pressure p normal to its contact-area $$\sigma$$. Regarding the magnitude of p, it directly follows from the figure: If $$\alpha$$ is the angle between n and $$n'$$ (where $$\alpha$$ will lie between 0 and $$\varphi$$ and will assume these most extreme values, when $$n'$$ lies especially perpendicular or parallel to the direction of relative motion of the primed and unprimed system). Then it is obviously given

$\mathfrak{F}_{n}=\mathfrak{F}_{n'}\cos\alpha$ and simultaneously $dn'=dn\cos\alpha$

thus

$\mathfrak{F}_{n}dn=\mathfrak{F}_{n'}dn'$ or $p=p'$

Because the unprimed observer defines its pressure p by the dynamic component $$\mathfrak{F}_{n}$$ of $$\mathfrak{F}$$, and the layer-thickness dn (seen by him) by $$\mathfrak{F}_{n}dn$$ in the same way as the primed by $$\mathfrak{F}_{n'}dn'$$. The equality of these two pressures, fundamental for 's thermodynamics of moving systems, thus follows here again as purely geometric statement concerning the behavior of four-dimensional vectors and its projection.

When operating with the total-force $$\mathfrak{K}$$, such a direct geometric reasoning can hardly be carried out, because $$\mathfrak{K}$$ doesn't behave like a vector.

We again come back to the electrodynamic meaning of $$\mathfrak{F}=(Pf)$$. While this was, at the beginning of this paragraph, explained by the formal agreement with Lorentz's force-definition, equation (13), it should be directly derived from the relativity principle now.

We consider a unit volume at rest in the primed system, in which the electric charge is distributed by the density $$\varrho'$$. Since the electric field-strength is defined as the moving force upon the charge 1, then the force $$\varrho'\mathfrak{E}'$$ is acting on our unit volume. By this the three dynamic components of the specific force $$\mathfrak{F}$$ are specified; the energetic component is to be put equal to zero, since the contact-point of $$\mathfrak{F}$$ is at rest in the primed system, and its worldline is perpendicular to $$\mathfrak{F}$$. The four components of $$\mathfrak{F}$$ thus read:

$\mathfrak{F}_{x'}=\varrho'\mathfrak{E}_{x'},\ \mathfrak{F}_{y'}=\varrho'\mathfrak{E}_{y'},\ \mathfrak{F}_{z'}=\varrho'\mathfrak{E}_{z'},\ \mathfrak{F}_{l'}=0$

On the other hand, the vector (Pf) for the $$x'y'z'l'$$-axis gives by equation (11), since $$P_{x'}=P_{y'}=P_{z'}=0$$, $$P_{l'}=i\varrho'$$ (see equation (1)) and $$f_{x'l'}=-i\mathfrak{E}_{x'}$$ etc. (see equation (2));

$$\begin{array}{ll} \left(Pf_{x'}\right)=P_{l'}f_{x'l'}=\varrho'\mathfrak{E}_{x'}, & \left(Pf_{y'}\right)=P_{l'}f_{y'l'}=\varrho'\mathfrak{E}_{y'},\\ \left(Pf_{z'}\right)=P_{l'}f_{z'l'}=\varrho'\mathfrak{E}_{z'}, & \left(Pf_{l'}\right)=P_{l'}f_{l'l'}=0.\end{array}$$

This vector thus agrees for the primed system in all four components with the specific force $$\mathfrak{F}$$. Under the single assumption, that the force behaves like a four-vector, $$\mathfrak{F}$$ must be given for any reference system by the four-vector (Pf). In relativity theory, 's force-definition directly follows from the definition of the electric field-strength by the single assumption, that the specific force behaves as a four-vector perpendicular to the worldline of the contact-point. In the original absolute-theory of, in contrast, this force-definition had to be introduced as a new fact of experience besides the field-equations. It seems to me a particularly nice achievement of relativity theory, that it makes this fact of experience unnecessary as such, by deriving it from much more general principles.

Following this, a remark of a more critical nature is appended. The relativity principle should be valid only for stationary or quasi-stationary motions of both reference systems against each other. A derivation of 's force-concept from relativity theory, consequently guarantees it only for the realm of these motions. While in relativity theory this concept was postulated as generally valid, we have (when we consider the relativity principle as its true source) to let it open at all, whether in the expression of $$\mathfrak{F}$$ for a real accelerated motion there might occur supplementary terms that might depend on the acceleration of first and higher order. As it seems, all realizable motions fall under the class of quasi-stationary motions, thus such supplementary terms would hardly have a practical interest. Yet in a principle view it seems useful to me, to consider its possibility.

(Received March 17, 1910.)


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1922 Encyclopædia Britannica/Einem, Karl von

EINEM, KARL VON (1853-      ), Prussian general, was born at Hertzberg in the Harz Jan. 1 1853. He entered the Prussian army in 1870 and rose to the rank of major-general in 1900. In the same year he was entrusted with the organization of the German section of the international military expedition to Peking. In 1903 he was raised to the rank of lieutenant-general and appointed Minister of War, an office which he held till 1909. He had meanwhile been promoted to be a general of the cavalry, and in 1909 he was placed in command of the VII. Army Corps, which under Kluck he led in the advance through Belgium in 1914. In Sept. 1914 he was appointed to the command of the III. Army (the army of the Aisne), which he successfully handled throughout the heavy fighting in Champagne in Feb. 1915. He continued his defence of his section of the German position with this army throughout 1917 and the early months of 1918.


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 Mother Twitchett had but one eye, And a long tail which she let fly;

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348 thumb, dividing the life-line, and running toward the little finger. She bound it up in silence, with a significant, reflective look. He asked, once or twice, "Aurelia, how could you hurt your friend?"

"Hush!" replied she, laying her finger on her mouth: "Hush!"


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122

4em

One leg is a leg of mutton; two legs, a man; three legs, a stool; four legs, a dog.

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true love lives far from me, Perrie, Merrie, Dixie, Dominie. Many a rich present he sends to me, Petrum, Partrum, Paradise, Temporie, Perrie, Merrie, Dixie, Dominie.

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1922 Encyclopædia Britannica/Einstein, Albert

EINSTEIN, ALBERT (1879–  ), German-Swiss physicist, was born of Jewish parents at Ulm in the kingdom of Württemberg on May 14 1879. His boyhood was spent at Munich where his father, who owned electro-technical works, settled in the early ’eighties. The family migrated to Italy in 1894, whilst Albert Einstein went to the Cantonschule at Aarau in Switzerland, where he passed the abiturienten examination, the indispensable preliminary to any professional career in Central Europe, two years later. He attended lectures while supporting himself by teaching mathematics and physics at the polytechnic school at Zürich until 1900 and finally, after a year as tutor at Schaffhausen, was appointed examiner of patents at the patent office at Berne, where, having become a Swiss citizen, he remained until 1909. It was during this period that he took his Ph.D. degree at the university of Zürich and published his first papers on physical subjects. These were so highly thought of that in 1909 he was appointed extraordinary professor of theoretical physics at the university of Zürich. In 1911 he accepted the chair of physics in Prague, only to be induced to return to his own polytechnic school at Zürich as full professor in the following year. In 1914 his preëminence had become so evident that a special position was created for him in Berlin, where he was elected a member of the Royal Academy of Sciences and given a sufficient stipend to enable him to devote all his time to research without any restrictions or duties whatsoever. He was elected a foreign member of the Royal Society in 1921, having also been made previously a member of the Amsterdam and Copenhagen Academies, while

the universities of Geneva, Manchester, Rostock and Princeton conferred honorary degrees on him.


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126

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text: Author:Frederick Alexander Lindemann

title: Author:Frederick Alexander Lindemann


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48

THREE blind mice, see how they run! They all ran after the farmer's wife, Who cut off their tails with a carving-knife; Did you ever see such a thing in your life? As three blind mice!

1, 2, 3, 4, 5! I caught a hare alive; 6, 7, 8, 9, 10! I let her go again.

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text:
1922 Encyclopædia Britannica/Eisner, Kurt

EISNER, KURT (1867-1919), Bavarian Socialist politician and author, was born in Berlin on May 14 1867. He became a journalist, and at an early stage of his career had the first of his many experiences of imprisonment for the subversive tendency of his writings. He was successively on the editorial staff of the Vorwaerts in Berlin 1898-1905 and of other socialist newspapers at Nürnberg and Munich. On the outbreak of the World War he at first seemed to be going to side with the Government, but, after having obtained some private knowledge of the way in which German public opinion had been duped, he turned against his own party, the Social Democrats, and attacked them for supporting the war. In Jan. 1918 he was prosecuted at Munich on a charge of treason for inciting munition workers to strike. He was released from prison on the ground that he was a candidate for the Reichstag, and recovered his liberty in time to arrange the mass meeting on the Theresienwiese at Munich on Nov. 7 1918, which the same day led to the overthrow of the Bavarian monarchy, the flight of the King, and the institution of a Bavarian revolutionary Government under the presidency of Eisner. A red-haired Jew, he possessed a magnetic and artistic temperament, and had various special methods of arousing and restraining the revolutionary masses, including orchestral and vocal concerts of high excellence in the formerly royal theatres and the opera house of Munich. His policy followed extreme lines in the sense of furthering the Workmen's and Soldiers' Councils system, while at the same time he manifested a Bavarian particularism of his own in his efforts to maintain his conceptions of republican government in conjunction with the Councils in Bavaria as against the centralizing tendencies of the Berlin policy. It was with difficulty that he was induced to agree to the arrangements for reëstablishing the Federal system of the German Reich and for the election of a National Constituent Assembly. Meanwhile a Bavarian Assembly had been elected, and the Bavarian reactionaries feared that, when it assembled, Eisner's influence might continue to predominate or might even be fortified. He was, further, obnoxious to them on account of his revelations as to the origin of the war, and at an international Socialist conference at Berne he had urged the German delegates to make a clean breast of Germany's war guilt. He was on his way to open this Assembly, when he was shot dead in the street by a young Count Arco on Feb. 21 1919. This crime was speedily followed by the Bolshevist chaos into which Munich was for a brief period plunged in April.

 * (G. S.)


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 themselves to the task of examining the fossils of this tertiary formation, and making a complete list of the species found in it. Bowerbank's anatomical studies, which were pursued with considerable attention, prepared his mind by a stern discipline for the study of the sponges, to which he subsequently devoted himself for many years. At the same time he occupied his leisure by examining the moss agates, and the minute structure of shells and corals.

In 1840 he published a volume on the 'Fossil Fruits of the London Clay,' which remains a standard work; indeed, the only one in which these very interesting remains are thoroughly described and accurately figured. In 1842 Bowerbank was elected a fellow of the Royal Society. In 1847, after the reading of a paper by Professor Prestwich at the rooms of the Geological Society, Bowerbank invited the leading geologists to meet him in the tearoom. He then proposed the establishment of a society for the publication of undescribed British fossils. He was supported in this by Buckland, De la Beche, Fitton, and others, and thus was founded the Palæontographical Society. From 1844 to 1864 Bowerbank was in the habit of receiving at his residence, once a week, professed geologists and young amateurs who showed a real fondness for this science, which was still struggling against the prejudices which dogmatic teaching had fostered. Every young and earnest geologist found in him a sincere friend and always a willing instructor. Bowerbank's classification of the spongidæ, his observations on their spiculate elements, and his papers on the vital powers of the sponges, remain splendid examples of unwearying industry and careful observation. On his retirement from the active labours of life, his fervent desire was to finish his great work on the sponges, and unremittingly he gave all the energies of his well-trained mind to this object, until the failure of brain-power compelled intervals of entire repose. Happily he reached the last plate of his great work. When half of it was drawn his powers began to fail him, and he became sadly depressed. The finishing tasks were postponed from day to day, then resumed for a few hours, to be again deferred, until 8 March 1877, when death closed for ever the labours of a well-spent life.

Bowerbank was always a most indefatigable collector, and in 1864 his collection had arrived at a state which truly merited the name of magnificent. It was purchased by the British Museum, and forms a well-known and most important division of the natural history section of this national establishment. The catalogue of scientific papers published by the Royal Society credits Bowerbank with forty-five papers. These appeared in the 'Journal of the Microscopic Society,' 'The Annals and Magazine of Natural History,' the 'Journal of the Geological Society,' the 'Reports of the British Association,' and the publications of the Zoological and Linnean Societies. 'The Pterodactyles of the Chalk,' published in the 'Proceedings of the Zoological Society,' was one of Bowerbank's most important memoirs. He paid great attention to the question of silicification, and some admirable papers on this interesting subject are scattered through the journals named. His 'Contributions to a General History of the Spongidæ,' which is in the 'Proceedings of the Zoological Society,' deserves especial attention. Bowerbank's first published paper was 'Observations on the Circulation of the Blood in Insects,' which appeared in 1833. His last was a 'Report on a Collection of Sponges found at Ceylon by E. W. H. Holdsworth,' printed in 1873.

 BOWERS, GEORGE HULL, D.D. (1794–1872), dean of Manchester, born in Staffordshire in 1794, was the son of Mr. Francis Bowers. He was sent to the Pembroke grammar school, and thence proceeded to Clare College, Cambridge. After a successful university career he was appointed perpetual curate of Elstow, Bedfordshire. He graduated B.A. in 1819, proceeding B.D. in 1829, and D.D. in 1849. He was select preacher of his university in 1830. In 1832 he became rector of St. Paul's, Covent Garden. On the death of Dean Herbert in 1847 he was nominated by Lord John Russell to the deanery of Manchester, an office which he held until 26 Sept. 1871. He was not a frequent preacher in Manchester, but his pulpit discourses were at once simple and scholarly, and his delivery effective.

His chief writings are:
 * 1) 'Sermons preached before the University of Cambridge.'
 * 2) 'A Letter to the Archbishop of Canterbury on a Proposed School for Sons of Clergymen,' London, 1842.
 * 3) 'A Scheme for the Foundation of Schools for the Sons of Clergymen and others,' London, 1842; this led to the establishment of Marlborough School, of which, conjointly with the Rev. C. E. Plater, he was founder. Similarly Rossall and Haileybury owed their origin to Bowers's suggestion, and the latter gained much on its establishment from Bowers's personal help and  experience.


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 Oh, there's none so rare, As can compare With King Cole and his fiddlers three.

I'd as much money as I could spend, I never would cry, "Old chairs to mend! Old chairs to mend! Old chairs to mend!" I never would cry, "Old chairs to mend!" If I'd as much money as I could tell, I never would cry, "Old clothes to sell! Old clothes to sell! Old clothes to sell!" I never would cry, "Old clothes to sell!"

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 We have a fault, Florian is kind enough to inform us:—

I suspect that this criticism is more just than his rendering of English surnames and his spelling of Scottish words.


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Rh much about his works, I went to a bookseller at Alais to ask if he had them.

"The works of Florian!" he exclaimed. "We have his statue in the place."

"Yes; but that is the work of the sculptor Gaudez, not of Florian himself."

"Les œuvres de Florian—mais—" The man looked puzzled. "He lived a very long time ago. What did he write?"

"I fancy, fables."

"Ah, monsieur! you mistake. That was La Fontaine."

"There is an 'F' in each," said I, "as there is a river in Macedon, and there is also a river in Monmouth, and there is salmon in both." Of course, the allusion was lost on him.

"I think his works have never been reprinted," said the bookseller. "I will tell my child to ask the schoolmaster about him."

Now I happen to possess at home an edition of Florian, printed in the year III. of the Republic, 1797, and on my return I read some of his works—as much as was possible. Among them is an "English novel," very complimentary to our nation at the opening, but full of the most amusing blunders. The characters are Sir Edouerd Selmours, Mistriss Hartlay, a M. Pikle, and a Mekelfort. Florian gives a translation into French verse of "Auld Robin Gray," but in an evil moment appended the original Scottish text, which is rendered thus— and so on.


text:
Author:James Duff Brown

Works

 * Biographical Dictionary of Musicians (1886)
 * A handbook of library appliances (1892)
 * Guide to the Formation of a Music Library (1893)
 * British Musical Biography with (1897)
 * Adjustable classification for libraries (1898)
 * Manual of Library Classification and Shelf Arrangement (1898)
 * Characteristic songs and dances of all nations with (1901)
 * Manual of Library Economy (1903)
 * Classified list of current periodicals (1904)
 * Manual of Practical Bibliography (1906)
 * Subject Classification (1906)
 * The Small Library: a guide to the collection and care of books (1907)
 * Guide to librarianship (1909)
 * Library Classification and Cataloguing (1912)
 * A British library itinerary
 * (with Henry Richard Tedder)

title: Author:James Duff Brown


text: Index:Observations on an autograph of Shakespeare, and the orthography of his name.djvu


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Page:Observations on an autograph of Shakespeare, and the orthography of his name.djvu/3

 OBSERVATIONS

ON AN

AND THE

ORTHOGRAPHY OF HIS NAME.

COMMUNICATED

TO THE SOCIETY OF ANTIQUARIES

BY

IN A LETTER TO

LONDON:

THOMAS RODD, GREAT NEWPORT STREET.

MDCCCXXXVIII.

Price One Shilling.

STEVENS AND PARDON, PRINTERS, BELL YARD, TEMPLE BAR.


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 RECENTLY PUBLISHED BY T. RODD.

NEW PARTICULARS REGARDING THE WRITINGS of SHAKESPEARE. By J. P. COLLIER, Esq. crown 8vo. 3s. 6d.

ON THE SONNETS of SHAKESPEARE; identifying the Person to whom they are addressed, and elucidating several points in the Poet's History. By J. Boaden, Esq. 8vo. 2s. 6d.

TRADITIONARY ANECDOTES of SHAKESPEARE, collected in the year 1693, now first published. 8vo. One shilling.

SHAKSPERE'S WILL, now first published according to the original in the Prerogative Court. 8vo. One shilling.

ON SALE BY T. RODD.

SHAKESPEARE'S COMEDIES, HISTORIES, and TRAGEDIES, second impression, 1632, russia, good copy, 8I. 8s.

SHAKESPEARE'S COMEDIES, HISTORIES, and TRAGEDIES, third impression, 1664, russia, very fine copy, 24l.

Shakespeare's Comedies, Histories, and Tragedies. Fourth edition, 1685, good copy, 3l. 10s.

Shakespeare's Works, by Rowe, plates, the first edition with illustrations, 7 vols. 8vo. 1709, 18s.

Shakespeare's Works, by Rowe, large paper, 7 vols. 2l. 12s. 6d.

Shakespeare's Twenty Plays, printed from the original edition, in 4to. 4 vols. 8vo. 1l.

Shakespeare, Œuvres complettes, par Letourneur, revue et cor- rigee par Guizot, 13 tom, large paper, Paris, 1821, 41.

Shakespeare's Schauspiele, von Eschenburg, 12 vols. 6vo, Zurich, 1775, 16s.


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Page:Observations on an autograph of Shakespeare, and the orthography of his name.djvu/7

 OBSERVATIONS

ON AN

AUTOGRAPH OF SHAKSPERE,

&c &c,


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Page:Robert's Rules of Order - 1915.djvu/67

 The motion to adjourn can be repeated if there has been any intervening business, though it is simply progress in debate. The assembly may decline to adjourn in order to hear one speech or to take one vote, and therefore it must have the privilege of renewing the motion to adjourn when there has been any progress in business or debate. But this high privilege is liable to abuse to the annoyance of the assembly, if the chair does not prevent it by refusing to entertain the motion when evidently made for obstructive purposes, as when the assembly has just voted it down, and nothing has occurred since to show the possibility of the assembly's wishing to adjourn. [See Dilatory Motions, 40.] The motion to adjourn, like every other motion, cannot be made except by a member who has the floor. When made by one who has not risen and addressed the chair and been recognized, it can be entertained only by general consent. It cannot be made when the assembly is engaged in voting, or verifying the vote, but is in order after the vote has been taken by ballot before it has been announced. In such case the ballot vote should be announced as soon as business is resumed. Where much time will be consumed in counting ballots the assembly may adjourn, having previously appointed a time for the next meeting, or, still better, may take a recess as explained in the next section. No appeal, or


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Page:Observations on an autograph of Shakespeare, and the orthography of his name.djvu/9

 OBSERVATIONS

ON AN

AND THE

ORTHOGRAPHY OF HIS NAME.

COMMUNICATED

TO THE SOCIETY OF ANTIQUARIES

BY

IN A LETTER TO

Shakesper.jpg

LONDON:

THOMAS RODD, GREAT NEWPORT STREET.

MDCCCXXXVIII.


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Page:Robert's Rules of Order - 1915.djvu/68

 question of order, or inquiry, should be entertained after the motion to adjourn has been made, unless it is of such a nature that its decision is necessary before an adjournment, or unless the assembly refuses to adjourn, when it would be in order.

Before putting the motion to adjourn, the chair, in most organizations, should be sure that no important matters have been overlooked. If there are announcements to be made they should be attended to before taking the vote, or at least, before announcing it. If there is something requiring action before adjournment, the fact should be stated and the mover requested to withdraw his motion to adjourn. The fact that the motion to adjourn is undebatable does not prevent the assembly's being informed of business requiring attention before adjournment. Members should not leave their seats until the chair has declared the assembly adjourned. An adjournment sine die — that is, without day — closes the session and if there is no provision for convening the assembly again, of course the adjournment dissolves the assembly. But, if any provision has been made whereby another meeting may be held, its effect is simply to close the session. In an assembly, as a convention, which meets regularly only once during its life, but whose bylaws provide for calling special meetings, an adjournment sine die means only the ending


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are you going, my pretty maid?" "I'm going a-milking, sir," she said. "May I go with you, my pretty maid?" "You're kindly welcome, sir," she said.

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 of the regular session of the convention, which, however, may be reconvened as provided in the by-laws. If called to meet again the assembly meets as a body already organized.

When the motion to adjourn is qualified in any way, or when its effect is to dissolve the assembly without any provision being made for holding another meeting of the assembly, it loses its privilege and is a main motion, debatable and amendable and subject to having applied to it any of the subsidiary motions.

In committees where no provision has been made for future meetings, an adjournment is always at the call of the chair unless otherwise specified. When a special committee, or the committee of the whole, has completed the business referred to it, instead of adjourning, it rises and reports, which is equivalent to adjournment without day.

The Effect upon Unfinished Business of an adjournment, unless the assembly has adopted rules to the contrary, is as follows: (a) When the adjournment does not close the session [63], the business interrupted by it is the first in order after the reading of the minutes at the next meeting, and is treated the same as if there had been no adjournment, an adjourned meeting being legally the continuation of the meeting of which it is an adjournment. (b) When the adjournment closes a


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 Books on Sale by T. Rodd. Becket's Shakespeare's Himself again ; or, the Language of the Poet Asserted, 2 vols. 8vo. 5s. 6d. Boaden's Inquiry into the Authenticity of the various Portraits of Shakspeare, ivithfive portraits, 8vo. 5*. The same on large paper,. proof»impressions of the portraits, 12s. Chalmers' Apology for the Believers in the Shakspeare Papers ; with Supplemental Apology and Appendix, 2 vols. 8vo. 18s. Chalmer's Supplemental Apology, 8vo. 3s. 6d. Chalmers' Account of the Incident from which the Title and part of the Story of Shakspeare's Tempest were derived, 8vo. forty copies printed, 11. 5s. Collier's New Facts regarding the Life of Shakespeare, Svo. scarce, 5s. 6d. Davies's Dramatick Miscellanies, 3 vols. 8vo. 8s. Dennis's Original Letters (on Shakespeare, &c.) 8vo. 4s. 6d. Drake's Shakespeare and his Times, 2 vols. 4to. 21. Drake's Memorials of Shakespeare, 8vo. 7s. 6d. Edwards's Canons of Criticism, Svo. 1765, 4s. Farmer's Essay on the Learning of Shakespeare, portrait, 8vo. 4s. Felton's Hints towards a New Edition of Shakespeare, 4to. 7s. 6d. Griffith's Morality of Shakespeare's Drama Illustrated, 8vo. 3s. 6d. Hardinge's Another Essence of Malone, two parts, 8vo. 4s. Ha zlitt's Characters of Shakespeare's Plays, 8vo. 8s. Heath's Revisal of Shakespeare's Text, 8vo. 4s. Ireland's Miscellaneous Papers under the Hand and Seal of Shakespeare, folio, l. Ireland's Authentic Account of the Shakesperian Manuscripts, 8vo. Is. 6d. Jackson's Restorations and Illustrations of Seven Hundred Passages in Shakespeare, 8vo. 4s. 6d.


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 as often as quarterly, the unfinished business should be taken up, just where it was interrupted, at the next succeeding session previous to new business; provided that, in a body elected, either wholly or in part, for a definite time (as a board of directors one-third of whom are elected annually), unfinished business falls to the ground with the expiration of the term for which the board, or any part of it, was elected.
 * in an assembly having regular sessions

(c) When the adjournment closes a session in an assembly which does not meet as often as quarterly, or when the assembly is an elective body, and this session ends the term of a portion of the members, the adjournment puts an end to all business unfinished at the close of the session. The business may be introduced at the next session, the same as if it had never been before the assembly. 18. Take a Recess.† This motion is practically a combination of the two preceding, to


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Author:George Bryce

Works

 * Manitoba
 * A Short History of the Canadian People
 * Laura Secord: A Study in Patriotism (1907)
 * (in part)
 * (in part)

title: Author:George Bryce