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544 Thus in both cases the corrected expression for the longitudinal mass as derived from the energy gives the same result as that obtained from the momentum, and no other forces other than electromagnetic come into play.

On the other hand, since from what has been said above it appears that the electron will naturally retain the spherical shape as measured by the variables associated with the moving axes, it appears that some extraneous forces would be required to cause it to retain the spherical shape to an observer remaining at rest.

It is perhaps worth noting that "the principle of relativity" propounded by Bucherer in the Phil. Mag. of April 1907 is in essence identical with the statement made in the beginning of this paper. The principle referred to may be stated thus : that in the sequence of electromagnetic phenomena the giving of an additional uniform translational velocity v to the whole system of electric and magnetic bodies will not affect the phenomena observed if this velocity v is at the same time given to the observer. The transformation of space and time variables mentioned above shows a means of explaining this dependence of the electromagnetic phenomena on relative motion only; and conversely it is a comparatively simple matter to show that it is the only means. For it is required, among other things, to explain how a light-wave travelling outwards in all directions with velocity C relative to an observer A, may at the same time be travelling outwards in all directions with the same velocity relative to an observer B moving relative to A with velocity v. This can clearly not be done without some transformation of the space and time variables of the two observers.

Suppose two observers A, B, to be situated momentarily in the same spot, and let B be moving relatively to A with velocity v (measured by A in his own system of space and time). Let the direction of motion of B be A's axis of x, and let the instant of coincidence be A's time t=0.

Suppose axes of ξ η ζ to be B's system of coordinates moving with him with velocity v relative to A's axes of x y z, and coinciding with them at t=0.

Associated with a given point at a given time as marked by the values (x, y, z, t) will be unique values of (ξ, η, ζ, τ), τ being B's measure of the interval elapsed from the time of his coincidence with A, and conversely. There must therefore be a linear transformation from the variables (x, y, z, t) to (ξ, η, ζ, τ).

Consider now points on the axis of x (or ξ). Then the

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Rh transformation must be of the form

Now ξ will not in general be infinite unless x is infinite, and also when x and t are zero ξ and τ are also zero. Hence the transformation must be of the simpler form

$\begin{array}{llll} \xi=a'x+b't & & \frac{x=b''\xi-b'\tau}{\triangle}\\ & \mathrm{or} & & \triangle=(a'b-b'a)\\ \tau=ax+bt & & t=\frac{-a''\xi+a'\tau}{\triangle}\end{array}$

the coefficients a', b', a", b" being functions of the relative velocity v.

Now if a point starts from A at time t=0 and travels with B, its coordinate ξ is always zero by virtue of the relation x=vt.

Hence

$$b'=-a'v\,$$

i.e.

$$\xi=a'(x-vt)\,$$

Now consider what is involved in saying that if a point moves along the axis of x relative to A with the velocity v of light, it also moves with velocity c relative to B. If a point moves from the position x at time t to the position x+δx at time t+δt let the corresponding changes in ξ and τ be δξ, δτ.

Then

$$\delta\xi=a'\delta x+b'\delta t,\,$$

$$\delta\tau=a\delta x+b\delta t\,$$

Hence

$$\frac{\delta\xi}{\delta\tau}=\frac{a'\delta x+b'\delta t}{a\delta x+b\delta t}$$.

Hence if the point has velocity n in A's system of coordinates and $$\nu$$ in that of B

$$\nu=\frac{a'n+b'}{an+b}$$

In particular if $$n=\pm c$$, $$\nu=\pm c$$,

$$ac^{2}\pm c(b-a')-b'=0$$,

so that

$$b=a'\,$$ and $$a=\frac{b'}{c^{2}}=-\frac{va'}{c^{2}}$$.

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

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546 Thus

$$\begin{array}{l} \xi=a'(x-vt),\\ \\\tau=a'\left\{ -\frac{vx}{c^{2}}+t\right\} .\end{array}$$.

If this transformation be reversed we have

$\begin{array}{l} x=\alpha'(\xi+v\tau)\\ \\t=\alpha'\left(+\frac{v\xi}{c^{2}}+\tau\right)\ \mathrm{where}\ \alpha'=\frac{1}{a'\left(1-\frac{v^{2}}{c^{2}}\right)}\end{array}$|undefined

and α' will be the same Function of (-v) that a' is of v.

But the transformation shows that if x1 x2 be two points fixed relative to A and ξ1 ξ2 their coordinates in B at any time τ,

i. e. a line of length l as seen by A appears to be of length $$\tfrac{l}{\alpha'}$$, as seen by B moving relatively to it. But this will be the same whichever be the direction of B's motion along the axis of x, so that if $$\alpha'=f(v),\ f(v)=f(-v)$$, i.e. $$a'=\alpha'$$.

Hence

Thus the transformation is finally

$\begin{array}{l} \xi=\beta(x-vt),\\ \\\tau=\beta\left(-\frac{vx}{c^{2}}-t\right)\ \mathrm{where}\ \beta=\left(1-\frac{v^{2}}{c^{2}}\right)^{-\frac{1}{2}}\end{array}$|undefined

Now let points not on the axis of x be considered. Since the axes of x and ξ coincide at all time, y and z always vanish when η and ζ vanish.

Hence $$y=\lambda\eta$$ and $$z=\mu\zeta$$, and λ and μ will not change if the velocity of motion of B be changed from v to -v ; thus if $$\lambda=\phi(v)$$; $$\phi(v)=\phi(-v)$$.

But since by reversing the transformation

Similarly μ=1.

The general transformation between x y z t and ξ η ζ τ is therefore

$$\begin{array}{c} y=\eta,\ z=\zeta,\\ \\\xi=\beta(x-vt)+c_{1}y+d_{1}z\,\\ \\\tau=\beta\left(-\frac{vx}{c^{2}}+t\right)+c_{2}y+d_{2}z.\end{array}$$.

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 The two last equations reducing to those above for y=0 and z=0.

But our hypothesis requires that the equation

shall be a result of the equation

This is so if, and only if, c1, c2, d1, d2 are all zero.

Thus we have arrived exactly at the transformation as given above, and, as Einstein has shown (loc. cit.), in order that the electromagnetic equations may be invariant under this transformation the electric and magnetic vectors in the two systems must be correlated in the manner done in this paper.

Bucherer in the paper referred to does not take into account this necessary modification of coordinates, and therefore when in the latter part of it he evaluates the electromagnetic mass of the electron on the assumption that it is spherical he is in reality considering the Abraham electron, and so obtains Abraham's expression for its mass.

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 duty which he discharged successfully with Abraham Cowley, but in 1648 he was suspected by the Parliamentary authorities, and thought it wiser to cross the Channel. He helped in the removal of the young duke of York to Holland, and for some time he served Queen Henrietta Maria in Paris, being entrusted by her with despatches for Holland. In 1650 he was sent to Poland in company with Lord Crofts to obtain money for Charles II. They succeeded in raising £10,000. After two years spent at the exiled court in Holland, Denham returned to London and being quite without resources, he was for some time the guest of the earl of Pembroke at Wilton. In 1655 an order was given that Denham should restrict himself to some place of residence to be selected by himself at a distance of not less than 20 m. from London; subsequently he obtained from the Protector a licence to live at Bury St Edmunds, and in 1658 a passport to travel abroad with the earl of Pembroke. At the Restoration Denham’s services were rewarded by the office of surveyor-general of works. His qualifications as an architect were probably slight, but it is safe to regard as grossly exaggerated the accusations of incompetence and peculation made by Samuel Butler in his brutal “Panegyric upon Sir John Denham’s Recovery from his Madness.” He eventually secured the services of Christopher Wren as deputy-surveyor. In 1660 he was also made a knight of the Bath.

In 1665 he married for the second time. His wife, Margaret, daughter of Sir William Brooke, was, according to the comte de Gramont, a beautiful girl of eighteen. She soon became known as the mistress of the duke of York, and the scandal, according to common report, shattered the poet’s reason. While Denham was recovering, his wife died, poisoned, it was said, by a cup of chocolate. Some suspected the duchess of York of the crime, but the Comte de Gramont says that the general opinion was that Denham himself was guilty. No sign of poison, however, was found in the examination after Lady Denham’s death. Denham survived her for two years, dying at his house near Whitehall in March 1669. He was buried on the 23rd in Westminster Abbey. In the last years of his life he wrote the bitter political satires on the shameful conduct of the Dutch War entitled “Directions to a Painter,” and “Fresh Directions,” continuing Edmund Waller’s “Instructions to a Painter.” The printer of these poems, with which were printed one by Andrew Marvell, was sentenced to stand in the pillory. In 1667 Denham wrote his beautiful elegy on Abraham Cowley.

 DÉNIA, a seaport of eastern Spain, in the province of Alicante; on the Mediterranean Sea, at the head of a railway from Carcagente. Pop. (1900) 12,431. Dénia occupies the seaward slopes of a hill surmounted by a ruined castle, and divided by a narrow valley on the south from the limestone ridge of Mongó (2500 ft.), which commands a magnificent view of the Balearic Islands and the Valencian coast. The older houses of Dénia are characterized by their flat Moorish roofs (azoteas) and view-turrets (miradores), while fragments of the Moorish ramparts are also visible near the harbour; owing, however, to the rapid extension of local commerce, many of the older quarters were modernized at the beginning of the 20th century. Nails, and woollen, linen and esparto grass fabrics are manufactured here; and there is a brisk export trade in grapes, raisins and onions, mostly consigned to Great Britain or the United States. Baltic timber and British coal are largely imported. The harbour bay, which is well lighted and sheltered by a breakwater, contains only a small space of deep water, shut in by deposits of sand on three sides. In 1904 it accommodated 402 vessels of 175,000 tons; about half of which were small fishing craft, and coasters carrying agricultural produce to Spanish and African ports.

Dénia was colonized by Greek merchants from Emporiae (Ampurias in Catalonia), or Massilia (Marseilles), at a very early date; but its Greek name of Hemeroskopeion was soon superseded by the Roman Dianium. In the 1st century, Sertorius made it the naval headquarters of his resistance to Rome; and, as its name implies, it was already famous for its temple of Diana, built in imitation of that at Ephesus. The site of this temple can be traced at the foot of the castle hill. Dénia was captured by the Moors in 713, and from 1031 to 1253 belonged successively to the Moorish kingdoms of Murcia and Valencia. According to an ancient but questionable tradition, its population rose at this period to 50,000, and its commerce proportionately increased. After the city was retaken by the Christians in 1253, its prosperity dwindled away, and only began to revive in the 19th century. During the War of the Spanish Succession (1701–14), Dénia was thrice besieged; and in 1813 the citadel was held for five months by the French against the allied British and Spanish forces, until the garrison was reduced to 100 men, and compelled to surrender, on honourable terms.

 DENIKER, JOSEPH (1852– ) French naturalist and anthropologist, was born of French parents at Astrakhan, Russia, on the 6th of March 1852. After receiving his education at the university and technical institute of St Petersburg, he adopted engineering as a profession, and in this capacity travelled extensively in the petroleum districts of the Caucasus, in Central Europe, Italy and Dalmatia. Settling at Paris in 1876, he studied at the Sorbonne, where he took his degree in natural science. In 1888 he was appointed chief librarian of the Natural History Museum, Paris. Among his many valuable ethnological works mention may be made of Recherches anatomiques et embryologiques sur les singes anthropoides (1886); Étude sur les Kalmouks (1883); Les Ghiliaks (1883); and Races et peuples de la terre (1900). He became one of the chief editors of the Dictionnaire de géographie universelle, and published many papers in the anthropological and zoological journals of France.

 DENILIQUIN, a municipal town of Townsend county, New South Wales, Australia, 534 m. direct S.W. of Sydney, and 195 m. by rail N. of Melbourne. Pop. (1901) 2644. The business of the town is chiefly connected with the interests of the sheep and cattle farmers of the Riverina district, a plain country, in the main pastoral, but suited in some parts for cultivation. Deniliquin has a well-known public school.

 DENIM (an abbreviation of serge de Nîmes), the name originally given to a kind of serge. It is now applied to a stout twilled cloth made in various colours, usually of cotton, and used for overalls, &c.

 DENINA, CARLO GIOVANNI MARIA (1731–1813), Italian historian, was born at Revello, Piedmont, in 1731, and was educated at Saluzzo and Turin. In 1753 he was appointed to the chair of humanity at Pignerol, but he was soon compelled by the influence of the Jesuits to retire from it. In 1756 he graduated as doctor in theology, and began authorship with a theological treatise. Promoted to the professorship of humanity and rhetoric in the college of Turin, he published (1769–1772) his Delle revoluzioni d’Italia, the work on which his reputation is mainly founded. Collegiate honours accompanied the issue of its successive volumes, which, however, at the same time multiplied his foes and stimulated their hatred. In 1782, at Frederick the Great’s invitation, he went to Berlin, where he remained for many years, in the course of which he published his Vie et règne de Frédéric II (Berlin, 1788) and La Prusse littéraire sous Frédéric II (3 vols., Berlin, 1790–1791). His Delle revoluzioni della Germania was published at Florence in 1804, in which year he went to Paris as the imperial librarian, on the invitation of Napoleon. At Paris he published in 1805 his Tableau de la Haute Italie, et des Alpes qui l’entourent. He died there on the 5th of December 1813.

 DENIS, SAINT, first bishop of Paris, patron saint of France. According to Gregory of Tours (Hist. Franc. i. 30), he was sent into Gaul at the time of the emperor Decius. He suffered martyrdom at the village of Catulliacus, the modern St Denis. His tomb was situated by the side of the Roman road,

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[Received May 1st, 1909.]

1. The absence, as far as experiment can detect, of any phenomenon arising from the Earth's motion relative to the electromagnetic æther has been fully accounted for by Lorentz and Einstein, provided the hypothesis of electromagnetism as the ultimate basis of matter be accepted, so that the only available means of estimating the distance between two points is the measuring of the time of propagation of effects between the bodies, such propagation taking place in accordance with the equations of the electron theory. It has been proved not only within the limits of experimental accuracy, but exactly, that any actual effect is completely obscured by the fact that the observer necessarily shares in the motion of the earth, and has therefore different measures of time and space from those which he would have if he did not do so. The foundation of this theory of relativity is the set of relations subsisting between the space and time measures of two observers having a uniform relative velocity. If this is v and the axis of x is taken in the direction of v, these relations are

where

$$\beta=\left(1-v^{2}/c^{2}\right)^{-\frac{1}{2}}.$$

The analytical result obtained is that, if e, h, u are vectors and ρ a scalar satisfying the equations

$$\begin{array}{rl} \frac{1}{c}\left(\frac{\partial e}{\partial t}+\rho u\right)= & \mathsf{curl}\ h,\\ \\-\frac{1}{c}\frac{\partial h}{\partial t}= & \mathsf{curl}\ e,\\ \\\mathsf{div}\ e= & \rho,\\ \\\mathsf{div}\ h= & 0,\end{array}$$

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78 so that e, h are the electric and magnetic intensities, ρ is the density of electricity, and u the velocity of convection in a sequence of electromagnetic phenomena in the æther, the observer being at rest relative to the æther, then the quantities E, H, U, Ρ given by the equations

of which the last is deduced without further assumption from (1), satisfy the same equations in (X, Y, Z, T).

Thus E, H, U, Ρ are quantities which can consistently represent a sequence of phenomena in which the second observer is supposed to be at rest in the æther.

The relations given above are reciprocal, and it is proved that the charges in corresponding elements of volume at corresponding times are the same in magnitude. Thus it is impossible for either observer to deduce from any experimental observation that he is at rest in the true æther and that the other is moving or vice versa.

This being so, the relations connecting the values, as estimated by the two observers of all physical quantities, must be such as to leave the constitutive equations for material media invariant also. This aspect of the question has been discussed by Minkowski, Frank, and Mirimanoff. The two former agree in stating that the Lorentz equations must be modified if the theorem of relativity is to be maintained in its entirety; but the last-named shews that this may be avoided, and that the relativity is complete and exact provided a change is made in the constitutive equations as ordinarily given. The modification required is of the second order of small quantities only in the equation

which has been verified to the first order by H. A. Wilson, and is of the first order in u/c in the equation $$B=\mu H$$. Thus the suggested equations are not contrary to experiment. The first part of the present paper is a verification of Mirimanoff's results by a process of averaging.

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1909.] The second part deals with a new transformation or, rather, group of transformations, which shews that a uniform translation through space is not the only type of motion which it will be impossible to detect so long as our measures of time and space are of electrodynamic origin. It has been pointed out by Minkowski that in a space of four dimensions in which the coordinates are $$\left(x,y,z,ct\sqrt{1-1}\right)$$, the geometrical transformation employed by Einstein, is simply a finite rotational displacement of the whole space about y = 0, z = 0. The equation $$\nabla^{2}V=0$$, i.e.,

is known to be invariant for such a transformation.

But this equation is invariant for a larger group of transformations than that of rotations, viz., for the group of conformal transformations in the four dimensional space, which, as is known, is built up out of inversions with respect to the hyperspheres of the space. The question arises whether the theorem of relativity also holds for the types of motion of an electromagnetic system derived from one another by such a transformation. The following investigation shews that this is so, and develops a scheme of correlation between the physical quantities in the system and its transformation. The constitutive equations take exactly the same form as in the first part of the paper.

The motions that arise in these transformations are naturally a good deal more complicated than in that of Einstein. In that case, a fixed configuration transforms into one every point of which has the same velocity of translation. But, in the present case, under the simplest operation of the group a fixed system becomes one in which the whole is expanding or contracting radially about a point in a certain way, which, though analytically simple, is difficult to describe geometrically. But the important property of it is, that any sphere which is expanding with velocity equal to that of light transforms into a sphere expanding (or contracting) with equal velocity.

It may be remarked here that it appears to be impossible for a uniform velocity of rotation of an electromagnetic system to be obscured in the way in which the types of motion above mentioned are. For, without considering the electromagnetic equations at all, if a disturbance propagated with velocity c equally in all directions is to transform into the like, it follows that the space (x, y, z, ict) of the one system must be conformal

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80 with the space (X, Y, Z, icT) of the other: inasmuch as

must be a consequence of

We may, however, prove that no conformal transformation in this space exists which will transform every point at rest in the space (x, y, z) into a point moving with angular velocity about a fixed line in the (XYZ) space. Thus, as in material dynamics, the equations of electrodynamics will not be preserved in the same form for a set of axes rotating uniformly relatively to the space in which they are satisfied by actual phenomena.

2. The following relations arising immediately from the Lorentz-Einstein transformation are used in what follows.

The velocities of a moving point in the two systems are connected thus:

{{MathForm2|(2)|$$\left.\begin{array}{l} W_{X}=\left(w_{x}-v\right)/\left(1-\frac{vw_{x}}{c^{2}}\right)\\ \\W_{N}=w_{n}/\left(1-\frac{vw_{x}}{c^{2}}\right)\end{array}\right\} ,$$}}

the suffixes n and N denoting components in any direction perpendicular to v.

The relative coordinates of two moving points (x, y, z), (x', y', z') :

{{MathForm2|(3)|$$\left.\begin{array}{l} X'-X=(x'-x)/\beta\left(1-\frac{vw'_{x}}{c^{2}}\right)\\ \\N'-N=(n'-n)+vw'_{n}(x'-x)/c^{2}\left(1-\frac{vw'_{x}}{c^{2}}\right)\end{array}\right\} ,$$}}

The relative velocity of two moving points:

{{MathForm2|(4)|$$\left.\begin{array}{l} W'_{X}-W_{X}=(w'_{x}-w_{x})/\beta^{2}\left(1-\frac{vw_{x}}{c^{2}}\right)\left(1-\frac{vw'_{x}}{c^{2}}\right)\\ \\W'_{N}-W_{N}=(w'_{n}-w_{n})/\beta\left(1-\frac{vw_{x}}{c^{2}}\right)-vw'_{n}\left(w'_{x}-w_{x}\right)/\beta c^{2}\left(1-\frac{vw_{x}}{c^{2}}\right)\left(1-\frac{vw'_{x}}{c^{2}}\right)\end{array}\right\} .$$}}

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1909.] If a denote a volume which is moving with velocity $$\left(w_{x},w_{y},w_{z}\right)$$, the corresponding volume A is given by

In applying the transformation to the consideration of the number of free electrons per unit volume, it must be remembered that they are moving with different velocities, and that this velocity affects the relative volumes which they occupy in the two systems of coordinates.

But the magnitude of the charge of each electron and the number of them are unchanged. If, therefore, we confine our attention to a group of electrons all having the same component of velocity $$w'_{x}$$ in the one system and $$W'_{x}$$ in the other, the volumes occupied, δa, δA, are connected by the relation (5).

Thus this group contributes to the density of the free electricity as measured in the two systems different amounts, $$\delta\rho=\sum e/\delta a$$, $$\delta\mathsf{P}=\sum e/\delta A$$. Using (5), we have

Let $$w_{x}$$ be the mass velocity of the moving medium as distinguished from $$w'_{x}$$ the velocity of the individual electron.

Then

$$\begin{array}{ll} \delta\mathsf{P} & =\frac{\beta\sum e}{\delta a}\left\{ \left(1-\frac{vw_{x}}{c^{2}}\right)-\frac{v}{c^{2}}\left(w'_{x}-w_{x}\right)\right\} .\\ \\ & =\beta\left(1-\frac{vw_{x}}{c^{2}}\right)\delta\rho-\frac{\beta v}{c^{2}}\frac{\sum e\left(w'_{x}-w_{x}\right)}{\delta a}\\ \\ & =\beta\left(1-\frac{vw_{x}}{c^{2}}\right)\delta\rho-\frac{\beta v}{c^{2}}\delta j_{x},\end{array}$$

where $$\delta j_{x}$$ is the contribution of the same group of electrons to the conduction current through the medium.

This last relation is true for each particular value of $$w'_{x}$$, so that on summing for all electrons, we have

an equation identical with Minkowski's and Mirimanoff's.

Similarly the contributions of the same group of electrons to the

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82 current density in the two systems are,

these being vector equations.

Using equations (3) and (5) these give

$$\begin{array}{ll} \delta J_{x} & =\frac{\sum e\left(w'_{x}-w_{x}\right)}{\delta a\beta\left(1-\frac{vw_{x}}{c^{2}}\right)}=\frac{\delta j_{x}}{\beta\left(1-\frac{vw_{x}}{c^{2}}\right)}=\beta\left(1+\frac{vW_{x}}{c^{2}}\right)\delta j_{x},\\ \\\delta J_{y} & =\frac{\sum e}{\delta a}\left\{ \left(w'_{y}-w_{y}\right)-\frac{vw_{y}\left(w'_{x}-w_{x}\right)}{c^{2}\left(1-\frac{vw_{x}}{c^{2}}\right)}\right\} \\ \\ & =\delta j_{y}-\frac{\beta v}{c^{2}}W_{y}\delta j_{x},\\ \\\delta J_{z} & =\delta j_{z}-\frac{\beta v}{c^{2}}W_{z}\delta j_{x}.\end{array}$$

Summing for all electrons, we obtain Minkowski's equation,

{{MathForm2|(7)|$$\left.\begin{array}{l} J_{x}=\beta j_{x}\left(1+\frac{vW_{x}}{c^{2}}\right)\\ \\J_{y}=j_{y}-\frac{\beta vW_{y}}{c^{2}}j_{x}\\ \\J_{z}=j_{z}-\frac{\beta vW_{z}}{c^{2}}j_{x}\end{array}\right\} .$$}}

So far no distinction has been made between various types of motion of the electrons; but now they must be differentiated.

Consider first the so-called polarization electrons which move with the body, but have an electric moment. Taking δa, δA to be the volumes occupied by such a polarized element, its contributions to the polarization in the two systems are related as follows:

$$\delta P_{x}=\frac{\sum eX}{\delta A},$$

where X is measured relative to the centre of the element.

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1909.] Using (4) and (5) this becomes

{{MathForm2|(a)|$$\left.\begin{array}{cll} & \delta P_{x} & =\frac{\sum ex}{\delta a}=\delta p_{x},\dots\\ \\\mathsf{Similarly,} & \delta P_{x} & =\frac{\sum eY}{\delta A}\\ \\ & & =\frac{\sum e}{\delta a}\left\{ \beta\left(1-\frac{vw_{x}}{c^{2}}\right)y+\frac{\beta vw_{y}x}{c^{2}}\right\} \\ \\ & & =\beta\left(1-\frac{vw_{x}}{c^{2}}\right)\delta p_{y}+\frac{\beta vw_{y}}{c^{2}}\delta p_{x},\dots\\ \\\mathsf{and} & \delta P_{z} & =\beta\left(1-\frac{vw_{x}}{c^{2}}\right)\delta p_{z}+\frac{\beta vw_{z}}{c^{2}}\delta p_{x},\dots\end{array}\right\} .$$}}

Since the polarization electrons have no velocity relative to the body, they contribute nothing to the magnetization in either system.

Consider next the magnetization electrons, that is, systems of electrons spinning round centres fixed in the body and having a magnetic moment; possibly also an electric moment in a moving system.

If these have an electric moment, consider the contribution to the polarization

$$\delta P_{X}=\frac{\sum eX}{\delta A}=\frac{\sum e}{\delta a}\beta\left(1-\frac{vw'_{x}}{c^{2}}\right)X=\frac{\sum ex}{\delta a},$$

where we consider in the first place only those electrons having a definite value of $$w'_{x}$$ and afterwards effect a summation.

Similarly $$\delta P_{Y}$$, i.e., $$\sum eY/\delta A$$, becomes, after reduction,

$$\beta\left(1-\frac{vw_{x}}{c^{2}}\right)\delta p_{y}+\frac{\beta vw_{y}}{c^{2}}\delta p_{x}+\frac{\beta v}{c}\delta m_{z},$$

and likewise

$$\delta P_{z}=\beta\left(1-\frac{vw_{x}}{c^{2}}\right)\delta p_{z}+\frac{\beta vw_{z}}{c^{2}}\delta p_{x}-\frac{\beta v}{c}\delta m_{y}.$$

In the same manner for the same group of electrons

$$\delta M_{X}=\frac{1}{2c}\sum e\left\{ Y\left(W'_{Z}-W_{Z}\right)-Z\left(W'_{Y}-W_{Y}\right)\right\} /\delta A,$$

which, on reduction, becomes

$$\delta m_{x}-\left(\beta vW_{Y}\delta m_{y}+\beta vW_{Z}\delta m_{z}\right)/c^{2},$$

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84 and, similarly,

$$\delta M_{N}=\beta\left(1+\frac{vw_{x}}{c^{2}}\right)\delta m_{n}.$$

From these, by addition, the formulae of transformation are seen to be exactly those given by Mirimanoff, viz.,

{{MathForm2|(8)|$$\left.\begin{array}{l} P_{X}=p_{x}\\ \\P_{Y}=\beta\left(1-\frac{vw_{x}}{c^{2}}\right)p_{y}+\frac{\beta vw_{y}}{c^{2}}p_{x}+\frac{\beta v}{c^{2}}m_{z}\\ \\P_{Z}=\beta\left(1-\frac{vw_{x}}{c^{2}}\right)p_{z}+\frac{\beta vw_{z}}{c^{2}}p_{x}-\frac{\beta v}{c^{2}}m_{y}\end{array}\right\} ,$$}}

{{MathForm2|(9)|$$\left.\begin{array}{l} M_{X}=m_{x}-\beta v\left(W_{Y}\delta m_{y}+W_{Z}\delta m_{z}\right)/c^{2}\\ \\M_{N}=\beta\left(1+\frac{vw_{x}}{c^{2}}\right)m_{n}\end{array}\right\} .$$}}

It is thus verified, by direct method, that the equations of the field as given by Lorentz for moving ponderable bodies are left unchanged by a transformation in which the pairs of vectors $$(e,b),\ (d,h-[pw]/c)$$ are correlated in the same manner as were (e, h) for the free aether.

The constitutive equations now become

{{MathForm2|(10)|$$\left.\begin{array}{rl} D_{X}= & \epsilon E_{X}\\ \\\left(1-\frac{v^{2}}{c^{2}}\right)D_{N}= & \left(\epsilon-\frac{v^{2}}{c^{2}}\right)E_{N}+\frac{1}{c}[v,\ \epsilon B-H]_{N}\end{array}\right\} ,$$}}

{{MathForm2|(12)|$$\left.\begin{array}{l} J_{X}=\sigma E_{X}/\beta\\ J_{N}=\beta\sigma\left(E_{n}+[v,B]_{n}/c\right)\end{array}\right\} .$$}}

The first and last of these appear to be the exact forms of the better known approximate equations

$$\begin{array}{l} D=\epsilon E+(\epsilon-1)[w,B]/c,\\ J=\sigma(E+[w,B]),\end{array}$$

and reduce to these, if $$\mu=1$$ and if $$v^{2}/c^{2}$$ is neglected.

The middle equation has as yet no experimental corroboration.

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1909.]

M. Planck in a recent paper has shewn, from thermodynamic considerations, that the pressure of radiation in equilibrium in an enclosure must be independent of the motion of that enclosure, and obtains relations between the energy per unit volume, and the temperature of the radiation when considered at rest and in motion in turn. Inasmuch as energy and pressure are independent of thermodynamics, it seems appropriate to shew here that Planck's results for those quantities are an immediate result of the foregoing transformation.

Taking the expression for the force per unit area on a moving surface,

$$p'=\frac{1}{8\pi}\left\{ 2e'e_{\nu}+2h'h_{\nu}-n(ee'+hh')\right\},$$

this being a vector equation, we adapt it to the case where the surface is moving in the direction of the axis of x with velocity v.

The equations of the transformation are

$$\begin{array}{lll} e'_{x}=E_{X}, & e'_{y}=E_{Y}/\beta, & e'_{z}=E_{Z}/\beta,\\ h'_{x}=H_{X}, & h'_{y}=H_{Y}/\beta, & h'_{z}=H_{Z}/\beta,\end{array}$$

Let the element of area upon which p' acts be ds, the normal having direction cosines (l, m, n), and let the transformed area be dS, with normal (L, M, N).

Then

$$lds=LdS,\ \beta mds=MdS,\ \beta nds=NdS.$$

Making these substitutions in p' and reducing, we obtain

$$p'_{x}ds=P_{X}dS-\frac{v}{4\pi c}[EH]_{N}dS,$$

where $$P_{X}$$ is the component pressure per unit area on dS at rest as given by the electric and magnetic intensities E, H.

Likewise

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86 Now pass to the consideration of the equilibrium radiation within a cavity in a body. The pressure is normal to the surface and the vector $$\frac{\sigma}{4\pi}[EH]$$ vanishes at all points.

Thus

$$P_{X}=LP,\ P_{Y}=MP,\ P_{Z}=NP.$$

Hence

$$\begin{array}{lrl} p'_{x}ds= & LPdS & =Plds,\\ \\p'_{y}ds= & \frac{1}{\beta}MPdS & =Pmds,\\ \\p'_{z}ds= & \frac{1}{\beta}NPdS & =Pnds,\end{array}$$

Thus the pressure in the moving cavity is in the direction (l, m, n), and is equal to P.

Let the expression for the energy of the field be now similarly treated.

$$\frac{1}{8\pi}\left(e^{2}+h^{2}\right)=\frac{1}{8\pi}\left\{ E{}_{X}^{2}+H_{X}^{2}+\beta^{2}\left(1+\frac{v^{2}}{c^{2}}\right)\left(E_{Y}^{2}+H_{Y}^{2}+E_{Z}^{2}+H_{z}^{2}\right)+\frac{4\beta^{2}v}{c}\left(E_{Y}H_{z}-E_{z}H_{Y}\right)\right\} .$$

In the cavity at rest when the radiation is in equilibrium, the mean values of $$\left(E_{X}^{2}+H_{X}^{2}\right),\ \left(E_{Y}^{2}+H_{Y}^{2}\right),\ \left(E_{Z}^{2}+H_{Z}^{2}\right)$$ are each equal to $$\frac{1}{3}\left(E^{2}+H^{2}\right)$$, and that of $$\left(E_{Y}H_{Z}-E_{Z}H_{Y}\right)$$ is zero, the average being taken over any interval of time very small, but large compared with the periods of the constituent radiation.

Hence, denoting mean values by a stroke,

$$\frac{1}{8\pi}\overline{\left(e^{2}+h^{2}\right)}=\frac{1}{8\pi}\overline{\left(E^{2}+H^{2}\right)}\left\{ \frac{1}{3}+\frac{2\left(c^{2}+v^{2}\right)}{3\left(c^{2}-v^{2}\right)}\right\},$$

or

$$\overline{\epsilon}=\frac{3c^{2}+v^{2}}{3\left(c^{2}-v^{2}\right)}\overline{E}.$$

In identical manner, Planck's equation

$$\overline{g}=\frac{4v}{3\left(c^{2}-v^{2}\right)}\overline{E}$$

is obtained.

Finally,

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1909.]

4. The group of conformal transformations in four dimensions referred to in the introduction can be built up of transformations by inversion in the hyperspheres of the space.

To establish the new theorem of relativity it is sufficient to consider a single member of the group, i.e., a single inversion. Let the centre of inversion be taken as origin.

The analysis is best carried out in spherical polar coordinates, so that the geometrical correlation of the two systems is

the angular coordinates remaining unaltered.

Geometrical relations arising immediately are first given.

If u, U are the velocities of a moving point and the corresponding point in the other system,

In particular, if v is the velocity in either system of a point corresponding to a stationary point in the other,

and is a radial velocity.

As in the former transformation,

and, using this, the above equations become

If we write $$k^{2}\Lambda=\left(R^{2}-c^{2}T^{2}\right)$$, and consider a small volume at rest in

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88 the (r, t) system, we find that to an observer in the (R, T) system its dimensions radially are diminished in the ratio $$-1:\beta\Lambda$$, while transversely they are diminished in the ratio $$1:\Lambda$$.

Thus a minute sphere, in the one system at rest, is observed in the other to be an oblate spheroid in the other, the axis of revolution being in the direction of the radius R, and the ratio of the axes being $$1:\beta$$.

Passing to the field equations of the electron theory, and putting them in terms of polar coordinates, it is found by direct transformation that the following transformation leaves them invariant, λ being written for $$\left(r^{2}-c^{2}t^{2}\right)/k^{2}$$:

$$\begin{array}{rl} E_{R} & =\lambda^{2}e_{r},\\ E_{\theta} & =\lambda^{2}\beta\left(-e_{\theta}+vh_{\phi}/c\right),\\ E_{\phi} & =\lambda^{2}\beta\left(e_{\phi}-vh_{\theta}/c\right),\\ H_{R} & =-\lambda^{2}h_{r},\\ H_{\theta} & =\lambda^{2}\beta\left(h_{\theta}+ve_{\phi}/c\right),\\ H_{\phi} & =\lambda^{2}\beta\left(h_{\phi}-ve_{\theta}/c\right),\\ \mathsf{P} & =-\lambda^{3}\beta\left(1-u_{r}v/c^{2}\right)\rho,\end{array}$$

together with the transformations in velocity and coordinates given above.

It is a consequence of this result that, if e, h, u, ρ are the electric and magnetic intensities, the velocity of convection, and the density of electricity in a given set of phenomena, then another set of phenomena consistent with the fundamental equations is expressed by the quantities E, H, U, P, so obtained if expressed in terms of the variables R and T. The relations between the two sets of quantities are completely reciprocal.

Further, there is exact correspondence of charges. For, if the ratio of a small element of volume $$r^{2}\sin\theta\delta r\ \delta\theta\ \delta\phi$$ moving with velocity u, to the corresponding element $$R^{2}\sin\theta\delta R\ \delta\theta\ \delta\phi$$ moving with velocity U bo calculated, it is found to be

$$-\lambda^{3}\left(1-vu_{r}/c^{2}\right),$$

that is $$P/\rho.$$.

Hence it is impossible for an observer to discriminate between the sequence of electromagnetic phenomena as he knows them and the sequence obtained therefrom by this transformation. So long as his measures of time and space are dependent on electrically constituted

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1909.] apparatus, the change in the space coordinates will be obscured by the change in the measure of time which will be automatically made.

It may be worth while to give the following forms of the transformation:

or, in vector notation,

5. The theorem of relativity being established, it is possible to deduce from any known set of phenomena another which geometrically is the transformation of it. The values of physical quantities in the new phenomena will be obtained from those in the old by the equations obtained in the last section. The results so obtained are quite analogous to those in § 2, and will be given below. Two quantities that are invariant will be mentioned first.

The following equation is immediately established:

$$\left[E_{R}^{2}+E_{\theta}^{2}+E_{\phi}^{2}-H_{R}^{2}-H_{\theta}^{2}-H_{\phi}^{2}\right]=\lambda^{4}\left[e_{r}^{2}+e_{\theta}^{2}+e_{\phi}^{2}-h_{r}^{2}-h_{\theta}^{2}-h_{\phi}^{2}\right].$$

Thus, if l, L denote the differences of the magnetic energies in the two systems per unit volume,

$$L=\lambda^{4}l.$$

Now

$$\frac{\partial(R,T)}{\partial(r,t)}=\frac{1}{\lambda^{2}}$$ and $$\frac{R}{r}=\frac{1}{\lambda^{2}}.$$

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Rh shot by the police or others. The true reason is, that after the coming of the whites the women cease to bear children. In their wild state the blacks are subject to a certain set of laws and code of morality (though a strange one) binding on them through the superstitious veneration in which they hold the old men of the tribe. When the white man arrives among them, his moral ascendancy makes them lose all respect for the elders, and consequently all morality is at an end. I am afraid nothing can be devised to replace their old system. It would be almost as useless for whites to try and make animals moral as the Queensland aborigines.

With regard to "Humanity's" trooper's thrilling story of Native Police warfare. I simply do not believe it. There is nothing a half-civilised blackfellow so delights in as the making up of stories of this description, and I have often wondered at the cleverness they show in regulating the amount of sensation according to the supposed credulity of the listener. I can assure you I have had blacks give me the most circumstantial accounts of murders of whites by blacks, with all the minute details, and I have found afterwards there has not been a shadow of foundation for the stories to rest upon. I have had considerable experience of Native Police affairs, and I can safely say that as a rule the officers find the "dispersing" business a most painful and disagreeable duty, and one never to have recourse to unless under great provocation. It seems to me a great shame to persistently run down a force which (as every reasonable, thinking pioneer must admit) has been to a great extent the means of saving life and property in all outside districts. We have heard of instances of great cruelty on the part of one or two officers, but then one black sheep does not make the whole flock black, and as a rule I am sure that the officers are as much averse to anything in the shape of cruelty as "Humanity" himself. With regard to the boys, those who come in or are brought in from the bush—"stolen," as "Humanity" puts it—lead not only useful but happy lives, and are brought to as high a state of civilisation as they are capable of attaining. As a rule they are kindly treated, well fed and clothed, and can always obtain a holiday. But the holidays are not generally of long duration, for, although fond of change, they soon tire of wild life and begin to long for the comforts to which they have grown accustomed. This, however, is the case only with those who have been brought in young.

And now with regard to the grown-up blacks. As to the success or otherwise of the missionary station spoken of by "R. S." I cannot say anything, but I do know that missionary boys are notoriously the most unmitigated scoundrels, combining polished hypocrisy with their own natural vices. With regard to making reserves for blacks, whatever success may have attended this plan amongst the half-civilised coast blacks, I am quite sure that it never would answer amongst the wild blacks. In the first place force would have to be used to keep them within the prescribed boundaries, which would lead to all the old evils complained of, and confinement to one place is so totally opposed to their nomadic instincts that any restraint of that kind would be intolerably irksome to them. But the great objection to any such scheme is this: I have already pointed out that the main source of evil amongst the blacks is the upsetting of their social laws, and I think all who have seen much of outside blacks will agree with me that under any such white supervision as proposed the relation between the sexes would be more unsatisfactory than ever. And in connection with this matter I allow the whites have been much to blame, and I would be the last to palliate their conduct. But practically this is an evil which no Government measures can reach, and which must exist as long as human nature is what it is. At the same time, as I have already pointed out, this is not the fruitful source of bad blood and ill-feeling between the races which it is represented to be.—Yours, &c., —Queenslander, June 12, 1880.

,—I have read all that you have published in favour of a more humane treatment of our blacks and of a reform in the Native Police system with that view, and also all that has been said against your proposals; and I wish to assure you that I entirely concur with your views, and hope they may be carried out. I have always thought it monstrous that the actions of the Native Police were not allowed to be properly witnessed and regularly reported, and consider that the reverse should be the rule. The force should protect as well as punish blacks. There ought to be degrees of punishment too, and far more trouble should be taken to find out the real

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90 Hence

and therefore for a closed system, if we integrate through the whole volumes and corresponding times,

Thus, if the electrodynamic equations are based on the principle of least action, the action being the time integral of the difference of the magnetic and electric energies of the system, we may say that the action is invariant under the transformation.

The other invariant is expressed in the equation

Thus, if the electric and magnetic vectors are at right angles at any point in the one system, they are also at right angles at the corresponding point and instant in the other.

Consider now the conditions which hold at a reflecting surface which is at rest in the (r, t) system, viz.,

where (l, m, n) are the direction cosines of the normal to the surface referred to the directions of $$(r,\theta,\phi)$$ respectively.

The mechanical force per unit charge moving with the surface in the transformed system is

Hence

But if we consider corresponding areas ds, dS and their projections, we obtain

δr being an element of length at rest.

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1909.] Hence

$$\frac{m\ ds}{M\ dS}=\frac{n\ ds}{N\ dS}=-\lambda^{2}\beta,$$

so that

$$\frac{l}{L}=\frac{-m/\beta}{M}=\frac{-n/\beta}{N}=\frac{1}{\kappa}$$ (say);

and therefore

$$\frac{F_{R}}{L}=\frac{F_{\theta}}{M}=\frac{F_{\phi}}{N}.$$

Again,

$$LH_{r}+MH_{\theta}+NH_{\phi}=-\lambda^{2}\kappa\left(lh_{r}+mh_{\theta}+nh_{\phi}\right),$$

since

$$me_{\phi}-ne_{\phi}=0.$$

Hence

$$LH_{r}+MH_{\theta}+NH_{\phi}=0.$$

Thus the conditions for a moving reflecting surface are satisfied at the transformed surface.

Thus a perfect reflector transforms into a perfect reflector.

6. There is no difficulty in adapting the analysis of § 2 to the new transformation if the following form of it is noticed.

The differential elements in the transformation are connected by the equation

where, as before,

$$\lambda=\left(r^{2}-c^{2}t^{2}\right)/k^{2},\ v=2c^{2}rt\left(r^{2}+c^{2}t^{2}\right),\ \beta=\left(1-v^{2}/c^{2}\right)^{-\frac{1}{2}}.$$

Thus within a small element of volume the space time coordinates are changed by a transformation of the same form as the Lorentz-Einstein, save for the magnification factor $$\lambda^{-1}$$, and a difference of sign, which itself disappears in a sequence of two such transformations, e.g., in an infinitesimal transformation of the group.

The same is true of the fundamental equations connecting the magnetic and electric intensities in the two systems.

If, therefore, we confine our attention to a portion of matter contained within a volume which is small, but large enough to allow of the process of averaging commonly employed in molecular physics, we shall obtain results similar to those of the last section. It will be enough here to give the equations without the analysis, referring to the corresponding equations of previous sections.

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92 The velocity of a moving point

{{MathForm2|(2')|$$\left.\begin{array}{l} W_{R}=-\frac{w_{r}-v}{1-\frac{vw_{r}}{c^{2}}}\\ \\W_{N}=\frac{w_{n}}{\beta\left(1-\frac{vw_{r}}{c^{2}}\right)}\end{array}\right\} ,$$}}

where $$W_{N}$$ is the velocity in any direction perpendicular to R.

The relative velocity of two moving points

{{MathForm2|(4')|$$\left.\begin{array}{l} W'_{R}-W_{R}=-\frac{w'_{r}-w_{r}}{\beta^{2}\left(1-\frac{vw_{r}}{c^{2}}\right)\left(1-\frac{vw'_{r}}{c^{2}}\right)}\\ \\W'_{N}-W_{N}=\frac{w'_{n}-w_{n}}{\beta\left(1-\frac{vw_{r}}{c^{2}}\right)}-\frac{w'_{n}v\left(w'_{r}-w_{r}\right)}{\beta c^{2}\left(1-\frac{vw_{r}}{c^{2}}\right)\left(1-\frac{vw'_{r}}{c^{2}}\right)}\end{array}\right\} .$$}}

The distance between two points (r, t), (r', t') at the time T corresponding to (r, t),

{{MathForm2|(3')|$$\left.\begin{array}{l} \lambda\delta R=-\frac{\delta r}{\beta\left(1-vw'_{r}/c^{2}\right)}\\ \\\lambda\delta N=\delta n+\frac{vw'_{n}\delta r}{c^{2}\left(1-vw'_{r}/c^{2}\right)}\end{array}\right\}.$$}}

The element of volume

The density of free electricity

The current density

{{MathForm2|(7')|$$\left.\begin{array}{rl} \lambda^{-3}J_{R}= & \beta\left(1+vW_{R}/c^{2}\right)j_{r}\\ -\lambda^{-3}J_{N}= & j_{n}-\beta vW_{N}j_{r}/c^{2}\end{array}\right\} .$$}}

The polarization

{{MathForm2|(8')|$$\left.\begin{array}{rl} \lambda^{-2}P_{R}= & p_{r}\\ \lambda^{-2}P_{\theta}= & -\beta\left(1-vw_{r}/c^{2}\right)p_{\theta}-\beta vw_{\theta}p_{r}/c^{2}-\beta vm_{\phi}/c\\ \lambda^{-2}P_{\phi}= & -\beta\left(1-vw_{r}/c^{2}\right)p_{\phi}-\beta vw_{\phi}p_{r}/c^{2}+\beta vm_{\theta}/c\end{array}\right\} .$$}}

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1909.] The magnetization

{{MathForm2|(9')|$$\left.\begin{array}{rl} -\lambda^{-2}M_{R}= & m_{r}-\beta v\left(W_{\theta}m_{\theta}+W_{\phi}m_{\phi}\right)/c^{2}\\ \lambda^{-2}M_{N}= & \beta\left(1-vW_{R}/c^{2}\right)m_{n}\end{array}\right\} .$$}}

7. The question now arises as to whether in a ponderable body the transformations above given between (E, H) and (e, h) stand as they are, or whether they should be really between (E, B) and (e, b). Examination shews that relativity is fully maintained if the latter is adopted in accordance with Mirimanoff's paper on the Lorentz-Einstein transformation. We know, changing (H, h) into (B, b) that the transformation obtained leaves invariant the equations

$$\begin{array}{rl} -\frac{1}{c}\frac{\partial b}{\partial t}= & \mathsf{curl}\ e\\ \\0= & \mathsf{div}\ b.\end{array}$$

The transformation being now

{{MathForm2|(10')|$$\left.\begin{array}{ll} E_{R}=\lambda^{2}e_{r}, & B_{R}=-\lambda^{2}b_{r}\\ E_{\theta}=\lambda^{2}\beta\left(-e_{\theta}+vb_{\phi}/c\right), & B_{\theta}=\lambda^{2}\beta\left(b_{\theta}+ve_{\phi}/c\right)\\ E_{\phi}=\lambda^{2}\beta\left(-e_{\phi}-vb_{\theta}/c\right), & B_{\phi}=\lambda^{2}\beta\left(b_{\phi}-ve_{\theta}/c\right)\end{array}\right\} ,$$}}

with the use of (8'), (9'), the following equations are deduced:

{{MathForm2|(11')|$$\left.\begin{array}{ll} D_{R}=\lambda^{2}d_{r}, & Q_{R}=-\lambda^{2}q_{r}\\ D_{\theta}=\lambda^{2}\beta\left(-d_{\theta}+vq_{\phi}/c\right), & Q_{\theta}=\lambda^{2}\beta\left(q_{\theta}+vd_{\phi}/c\right)\\ D_{\phi}=\lambda^{2}\beta\left(d_{\phi}-vq_{\theta}/c\right), & Q_{\phi}=\lambda^{2}\beta\left(q_{\phi}-vd_{\theta}/c\right)\end{array}\right\} ,$$}}

where the new quantities are defined by

$$\begin{array}{ll} D=E+P, & d=e+p,\\ B=M+H, & b=m+h,\\ \\Q=H-\frac{1}{c}[PU], & q=h-\frac{1}{c}[pu].\end{array}$$

These together with (6'), (7') will, by comparison with the transformation in its original form, leave unchanged the equations

$$\begin{array}{rl} \frac{1}{c}\left(\frac{\partial d}{\partial t}+\rho u\right)= & \mathsf{curl\ }q,\\ \\\rho= & \mathsf{div}\ d.\end{array}$$

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94 These equations expanded become the second and third of Lorentz' equations for ponderable bodies.

If, now, at a point (xyzt) the velocity u vanishes, and we assume that consequently $$d=\epsilon e,\ b=\mu h$$, we at once obtain from (7'), (10'), (11') equations identical with (10), (11), (12), § 2.

Thus, if we assume those equations together with Lorentz' fundamental equations, to be the correct scheme of the electrodynamics of moving bodies, we arrive at the conclusion that that scheme is invariant, not only under the extension of the transformation of Lorentz and Einstein given by Mirimanoff, but also under the transformation obtained by inversion in the four dimensional space here developed, and therefore also under a transformation obtained by combining any number of such operations.

8. In a paper in the Annalen der Physik, starting from the equation

$$\frac{1}{c}\left(\frac{\partial D}{\partial t}+S\right)=\mathsf{curl}\ H,$$

Einstein and Laub deduce that the conditions to be satisfied at a surface of discontinuity in the material media are that the following quantities must be continuous in crossing the surface:

$$D_{n},\ \left\{ H+\frac{1}{c}[DW]\right\} _{\overline{n}},\ B_{n},\ \left\{ E-\frac{1}{c}[BW]\right\} _{\overline{n}},$$

n denoting the direction of the normal, and $$\overline{n}$$ of a tangent line to the surface. S, as in Minkowski's paper, denotes the sum of the conduction and convection currents.

If we apply the extended transformation to these conditions, we find that they are not conserved. But if we take the Lorentz equation which is obtained from that above by the substitution of the vector for

$$Q=H-\frac{1}{c}[PW]$$

the conditions are that

$$D_{n},\ \left\{ Q+\frac{1}{c}[DW]\right\} _{\overline{n}},\ B_{n},\ \left\{ E-\frac{1}{c}[BW]\right\} _{\overline{n}},$$

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1909.] shall be continuous, provided there is no true electricity on the surface of separation.

There is no difficulty in proving that these conditions are conserved under Mirimanoff's extended transformation.

It will be proved now that they are also conserved under the transformation by inversion, and therefore by the most general transformation of the whole group which leaves the fundamental equations invariant.

From the equations (18'), we obtain if we write

{{MathForm2|(14')|$$\left.\begin{array}{l} H'_{R}=-\frac{\lambda^{2}}{1-\frac{vw_{r}}{c^{2}}}\left\{ h'_{r}-\frac{v}{c^{2}}\left(w_{r}h'_{r}+w_{\theta}h'_{\theta}+w_{\phi}h'_{\phi}\right)\right\} \\ \\H'_{\theta}=\frac{\lambda^{2}}{\beta\left(1-\frac{vw_{r}}{c^{2}}\right)}h'_{\theta}\\ \\H'_{\phi}=\frac{\lambda^{2}}{\beta\left(1-\frac{vw_{r}}{c^{2}}\right)}h'_{\phi}\end{array}\right\} .$$}}

If we consider an element of length $$(\delta s,\delta S)$$ of which the components are $$(\delta r,\delta n), (\delta R,\delta N)$$ in the two systems respectively, $$\delta n, \delta N$$ being the components perpendicular to r, R respectively, we have equations analogous to (4),

$$\begin{array}{l} \delta R=-\frac{\lambda^{-1}\delta r}{\beta\left(1-\frac{vw_{r}}{c^{2}}\right)},\\ \\\delta N=\lambda^{-1}\left\{ \delta n+\frac{vw_{n}\delta r}{c^{2}\left(1-\frac{vw_{r}}{c^{2}}\right)}\right\} .\end{array}$$

Hence

$$\begin{array}{ll} H'_{R}\delta R+H'_{N}\delta N & =\frac{\lambda}{\beta\left(1-\frac{vw_{r}}{c^{2}}\right)}\left\{ \frac{h'_{r}\delta_{r}}{1-\frac{vw_{r}}{c^{2}}}+h'_{n}\delta_{n}-\frac{v\left(w_{r}h'_{r}+w_{n}h'_{n}\right)}{c^{2}\left(1-\frac{vw_{r}}{c^{2}}\right)}+\frac{vw_{n}h_{n}}{c^{2}\left(1-\frac{vw_{r}}{c^{2}}\right)}\right\} \\ \\ & =\frac{\lambda}{\beta\left(1-\frac{vw_{r}}{c^{2}}\right)}\left\{ h'_{r}\delta_{r}+h'_{r}\delta_{n}\right\} ,\end{array}$$

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96 or

$$H'_{s}\delta S=\frac{\lambda}{\beta\left(1-\frac{vw_{r}}{c^{2}}\right)}h'_{s}\delta s.$$

Thus, since $$\delta S/\delta s$$ does not depend on the physical properties of the medium, and is consequently continuous in crossing the surface of discontinuity, H' will be continuous in those directions in which h' is continuous, and discontinuous in those directions in which h' is discontinuous.

Exactly the same is to be proved of $$E'=E-\frac{1}{c}[BW]$$ by means of (10'). Taking the component of D perpendicular to the same element, we have

$$\begin{array}{ll} D_{R}\delta N-D_{N}\delta R= & \lambda d_{r}\left\{ \delta_{n}+\frac{vw_{n}\delta_{r}}{c^{2}\left(1-\frac{vw_{r}}{c^{2}}\right)}\right\} \\ \\ & -\frac{\lambda\delta r}{1-\frac{vw_{r}}{c^{2}}}\left\{ d_{n}\left(1-\frac{vw_{r}}{c^{2}}\right)+\frac{v}{c^{2}}d_{r}w_{R}-\frac{v}{c}h'_{n_{1}}\right\} \end{array}$$

where $$h'_{n_{1}}$$ denotes the component of $$h'$$ in the direction perpendicular to r and n.

Thus, on simplifying,

$$D_{R}\delta N-D_{N}\delta R=\lambda\left(d_{r}\delta_{n}-d_{n}\delta_{r}\right)-\frac{\lambda v\delta r\ h'_{n_{1}}}{c\left(1-\frac{vw_{r}}{c^{2}}\right)}.$$

Thus D will be continuous in any direction in which d is continuous, provided the component of $$h'$$ perpendicular to that direction is continuous.

Hence, with what has just been proved as to the continuity of h', it follows that the condition of the continuity of the normal component of d is conserved ; and in the same manner it may be shewn of the normal component of b.

9. Let the physical quantities considered in § 3 be now treated in the light of the inversion in four-dimensional space.

If the elements ds, dS are corresponding elements of area of which dS is at rest in the (R, T) system, and (l, m, n), (L, M, N) are the direction cosines of the respective normals, by considering the projections of the

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1909.] areas, we have

$$\frac{l\ ds}{L\ dS}=\frac{r^{2}}{R^{2}}=\frac{1}{\Lambda^{2}},$$

where

in which $$\delta r,\delta R$$ are corresponding elements of a radius, $$\delta R$$ being at rest in the (R, T) system.

Hence

$$\frac{\delta r}{\delta R}=-\frac{1}{\Lambda\beta},$$

and

$$\frac{m\ ds}{M\ dS}=\frac{n\ ds}{N\ dS}=-\frac{1}{\Lambda^{2}\beta},$$

Using these and the final form of the transformation of the electric vectors in § 3, the following equations are obtained exactly as in the last section:

$$\begin{array}{l} p_{r}ds=\Lambda^{2}\left\{ P_{r}-\frac{v}{4\pi c}[EH]_{\nu}\right\} dS,\\ \\p_{\theta}ds=-\Lambda^{2}P_{\theta}dS/\beta,\\ \\p_{\phi}ds=-\Lambda^{2}P_{\phi}dS/\beta,\end{array}$$

p and P being the mechanical forces per unit area according as the element is observed in the (r, t) or (R, T) system.

If the element dS belongs to the wall of a region in which there is radiation in equilibrium $$[EH]_{\nu}$$ is zero as before if the time-average be taken, and P is normal to dS.

Thus

$$\begin{array}{l} p_{r}ds=\Lambda^{2}PL\ dS=\Lambda^{4}Pl\ ds,\\ \\p_{\theta}ds=-\frac{\Lambda^{2}}{\beta}PM\ dS=\Lambda^{4}Pm\ ds,\\ \\p_{\phi}ds=-\frac{\Lambda^{2}}{\beta}PN\ dS=\Lambda^{4}Pn\ ds.\end{array}$$

Thus p is normal to ds and is equal to $$\Lambda^{4}P.$$.

For the energy the transformation gives

$$\begin{array}{ll} \epsilon & =\frac{1}{8\pi}\left\{ e^{2}+h^{2}\right\} \\ \\ & =\frac{\Lambda^{4}}{8\pi}\left\{ E_{R}^{2}+H_{R}^{2}+\beta^{2}\left(1+\frac{v^{2}}{c^{2}}\right)\left(E_{\theta}^{2}+E_{\phi}^{2}+H_{\theta}^{2}+H_{\phi}^{2}\right)-\frac{4v\beta^{2}}{c}\left(E_{\theta}H_{\phi}-E_{\phi}H_{\theta}\right)\right\} ,\end{array}$$

title: Page:CunninghamExtension.djvu/21


text:
Page:CunninghamExtension.djvu/22

98 so that for the radiation in equilibrium

Thus

This is the same equation as that obtained in § 3. Exactly in the same way for the momentum is found the result

title: Page:CunninghamExtension.djvu/22


text:
Author:Alexander John Philip

Works

 * A Dickens Dictionary (1909)
 * Dickens's Honeymoon and Where He Spent it. (1912)
 * The Business of Bookbinding (1912)

title: Author:Alexander John Philip


text:
Page:Feilberg.djvu/41

Rh criminal in each case, whether black or white. All this would require a mixed force such as you advocate. As to the truth of those accounts which you have given of particular outrages on blacks, and which have gone unpunished, I see no reason to doubt them, many acts of equal inhumanity of the kind having to my own knowledge gone unpunished, and I have no doubt the same thing is going on now, and will do so till the people of this country are convinced of their responsibility in the matter; and I only hope you will go on exposing the rottenness of our present system of dealing with the blacks till you have made the facts widely known, and then something like justice will be done—if our boasted civilisation is not a sham to all who cannot buy its fruits. As on all previous occasions when anyone has attempted to draw public attention to the evils complained of, your efforts in favor of humanity on this occasion have so far called forth little but hostile critisism, scorn, and ridicule; or at least friendly advice to leave the question to those who understand it. Emboldened by the uniform success of such tactics previously, writers have again come forward proclaiming themselves the champions of the pioneer settlers, ready to defend them against misrepresentation and slander; denouncing your ideas as wildly absurd, remarkable chiefly for their impracticability, and so on, however seemingly benevolent; at the same time expecting implicit belief in their own statements on account of their (the writers') wide experience, practical common sense, and intimate knowledge of the nature and character of the blacks, &c. We are told that it is only people who are unacquainted with the hard realities of frontier life who ask for these reforms—visionary sentimentalists, dwellers in towns, and so forth; and the charge has generally been a telling one, on account of persons not allowing their names to appear when writing against the abuses in question; and so people take critics such as "Never-Never" at their word, and decide that these matters are properly understood only by such persons as he is described to be (by himself), and that they must be settled in the natural way—"the survival of the fittest"—which, interpreted by "Never-Never," means that every scoundrel who has a mind to should be allowed by society (as indeed he is now in most places) to rob and murder the blacks with as much freedom from punishment as if they were pigeons. Such conduct may be according to wild beast nature, even if men degraded themselves, as some do, to the level of wild beasts, for then they cease to be men, to all intents and purposes. And so, with all due respect to "Never-Never's," profound, wisdom, we may be allowed to call his conception of human nature a beggarly one, if he thinks it would be natural for men to let the weak and defenceless fall victims to the worst passions of mankind; this sort of thing is called (rightly so) inhuman, unnatural, and brutal. Besides, if allowed, where is it to stop? Are the Chinese to be the next we are to conclude want shooting, "being useless?" And what about the useless members of society of our own color; are they all to perish whilst society looks on to see "Never Never's" notion of a natural law work out? The natural result would be the mutual destruction of all—not the survival of the fittest. Let us look at the question fairly; use our reason and experience; grant that we have a plain duty to perform towards every black as well as white person—to protect as well as to punish him—and I fully expect we should find the task an easy one, and easier year by year. And I maintain that the blacks are, in the greater part of the country, wonderfully easy to manage, and very useful. As to the Native Police force, I am of opinion that with some slight alteration as to its composition, and a radical one as to its methods, it would be the most suitable for the greater part of the country; it is a cheap and competent force, and only wants to be properly worked. I think those who have bad the best opportunities of judging the question should, in justice not only to the blacks but to themselves and "outsiders" generally, give their experiences and express their views. It is usually represented that outsiders get accustomed to seeing blacks brutally ill-treated and become indifferent to it, and that the desire for reform comes from people living in civilised parts. I think this is a mistake, and can only say that, after having worked on stations with blacks for about 20 years in old districts and on the very outside stations, I never saw the necessity for anything at all like the extermination policy; on the other hand, I know that within two years of the first settlement of some Western rivers the blacks have ceased to trouble the whites and have become very useful in many ways; and I am pretty sure that very few people could be found in the West who approve or ever pratrice such a plan as "Never Never" advocates. There are a few, I know, and I hold that they should be punished most severely when possible; at all events never encouraged in the way they are. As to our being committed to a course of harsh and cruel treatment of blacks, having unjustly taken their country, I think it nonsense. I don't admit it was their country, and want proof of it;

title: Page:Feilberg.djvu/41


text: File:ComstockInertia.djvu

title: File:ComstockInertia.djvu


text:
Author talk:Horace Barnet Samuel

Probate
Name: 	Horace Barnett Samuel, 9 Templars-avenue Golders Green and of Broad-street House Old Broad-street Death Date: 	9 Mar 1950 Death Place: 	The Hospital of St. John and St, Elizabeth, London, England Probate Date: 	13 Jun 1950 Probate Registry: 	London, England
 * Calendar of the Grants ...

relict Esther Sauel

1939 Register
Name: 	Horace B Samuel Gender: 	Male Marital status: 	Married Birth Date: 	28 May 1883 Residence Year: 	1939 Address: 	King Arthurs Castle Hotel Residence Place: 	Camelford, Cornwall, England Occupation: 	Barrister-at-Law (Not Practising) Financial Consultant & Journalist Schedule Number: 	9 Sub Schedule Number: 	20 Enumeration District: 	WBAR Registration district: 	289-1

Estelle Samuel 14 May (18)93

Census
Name: 	Horace Barnett Samuel Age in 1911: 	27 Relation to Head: 	Son Gender: 	Male Birth Place: 	Edgbaston, Birmingham Civil Parish: 	Kensington Street address: 	56 Pembridge Villas, Kensington W Marital status: 	Single Occupation: 	BARRISTER AT LAW Registration district: 	Kensington Registration District Number: 	2 Sub-registration district: 	Kensington Central ED, institution, or vessel: 	3 Piece: 	142 Household Members: Name 	Age Selim Samuel 	58 Amy Samuel 	48 Horace Barnett Samuel 	27 Enid Nellie Samuel 	22 Margaret Mary Samuel 	21 Jesse Cook 	47 Annie Greedy 	43 Jane Mears 	37 Adelaide Earl 	18 Florence Hutchinson 	37
 * 1911 England

title: Author talk:Horace Barnet Samuel


text: Index:ComstockInertia.djvu

title: Index:ComstockInertia.djvu


text:
User talk:76.23.104.88

Welcome

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title: User talk:76.23.104.88


text: 1911 Encyclopædia Britannica/Dénia

title: 1911 Encyclopædia Britannica/Dénia


text: 1911 Encyclopædia Britannica/Deniker, Joseph


text:
User talk:92.23.85.200

Welcome

Hello, and welcome to Wikisource! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers:

 * Help pages
 * Style Guide
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You are welcome to continue editing articles without logging in, but you may wish to  [ create an account] . Doing so is free, requires no personal information, and provides several benefits. For one thing, if you continue to edit without a username, your IP address is used to publicly identify you.

In any case, I hope you enjoy donating your time to grow the Wikisource library that is free for everyone to use! In discussions, please sign your comments using four tildes ( ~ ); this will automatically produce your IP address (or username if you're logged in) and the date. If you need help, ask me on my talk page, or ask your question here (click  [ edit] ) and place  before your question. Again, welcome! — billinghurst  sDrewth  11:17, 22 January 2011 (UTC)

<div> formatting
Hi again. The formatting <div class="indented-page> is now redundant, and in fact where it is used it does not allow the page number to show on the left hand side of the page. Feel free to remove it from the DNB biographies if you so wish. Thanks. — billinghurst  sDrewth  11:17, 22 January 2011 (UTC)

Rev. John Field (puritan).
does not appear to have his own article in DNB00 or 1901 supplement, yet he is mentioned in both of his sons’ articles and. They were born in London and the only possibility, John Field (1525?-1587) (DNB00) resided around York. Need to research some more locate his bio… Thanks for your help on this project…JamAKiska (talk) 22:06, 1 February 2011 (UTC)

Found it… made the Supplement while his son’s made the original.JamAKiska (talk) 02:27, 2 February 2011 (UTC)

Lord Fitzwalter link changed. (double redirect).
JamAKiska (talk) 22:24, 1 February 2011 (UTC)

title: User talk:92.23.85.200


text: 1911 Encyclopædia Britannica/Deniliquin

title: 1911 Encyclopædia Britannica/Deniliquin


text: 1911 Encyclopædia Britannica/Denim

title: 1911 Encyclopædia Britannica/Denim


text: 1911 Encyclopædia Britannica/Denina, Carlo Giovanni Maria


text:
Page:ComstockInertia.djvu/1

1-10
1. the inertia of matter has or has not a complete electromagnetic explanation is a question that it will perhaps take many years to answer with any degree of certainty. The experiments of Kaufmann seem to prove that in the case of a single electron the mass is entirely of this origin; and it is impossible therefore to avoid the conclusion that at least a fraction of ordinary material inertia is also electromagnetic. Doubtless there is a psychological cause for our reluctance to accept the electromagnetic explanation as complete, constant familiarity with ponderable bodies having blinded us to the possibility of anything being more fundamental ; but certain it is, that if we free ourselves from prejudice as much as possible and adopt the well-tried policy of choosing the simplest theory which adequately represents the phenomena, —the theory that is, which involves the least number of variables,— we must decide in favour of the complete electromagnetic explanation, which involves only the aether and its properties.

2. The complexity of the Zeeman effect and the relations

title: Page:ComstockInertia.djvu/1


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Page:ComstockInertia.djvu/2

 between the wave-lengths of the spectral lines, make it seem probable that if matter is to be considered as an electrical system, it must be much more complex than a system composed entirely of electrons separated by distances great in comparison to their size. It becomes therefore of interest to see whether any relations can be found between the mass of an electric system in general, and any of its other properties. It will be found that a general relation does exist, which is not only of considerable interest in itself, but also suggests other relations.

3. The straightforward calculation of the mass of an electric system possessing any distribution of charge and any internal velocities below that of light presents considerable difficulty; for such calculation involves the use of the scalar and vector potential, and these are not effective instantaneously at all parts of the system. Any expression for the mass of the system calculated in this way will therefore involve terms which vary in an extremely complicated way with the internal velocities when these are not very small. The same is true with respect to the velocity of the system as a whole. In the following discussion the problem is attacked in an entirely different way, which is not open to this objection.

As the constraints of the system are intimately involved, it will be well first to consider them.

4. The position of internal constraints in general electrical theory is a very fundamental one. By "constraints" are meant rigid connexions of any kind. These act merely as reactions to the electrical forces, and do not contribute to the virtual work. If the electrical laws are to hold universally, i. e., for minute distances as well as for greater ones, it is obvious that no electrical system can exist as such unless there are such constraints to balance the electrical forces. Even a single electron would dissipate itself through the mutual repulsion of its elements, were it not for some form of internal constraint. Besides holding the system together, as it were, these constraints also act in another important way. They may become, in common with all geometrical constraints, paths of energy flow. We are accustomed to think of the Poynting vector as representing completely the energy flow in a purely electrical system, but of course this is not in general true.

Take as a simple example the case of a large plane aircondenser moving in a direction perpendicular to the plane of its plates. If the condenser is charged there is obviously a transference of energy at a rate equal to the internal energy multiplied by the velocity of movement. The Poynting

title: Page:ComstockInertia.djvu/2


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Page:ComstockInertia.djvu/3

 vector is, however, zero. The energy transfer is not through space in the ordinary sense, but is along the constraint which holds the condenser-plates apart. The plate in the rear picks up, as we may say, the energy of the field, and after it has been transmitted to the forward plate by means of the constraint it is there set down again. On the other hand, when the condenser is moving parallel to the plane of the plates, there is no energy How along the constraint and the Poynting vector adequately represents the transfer of energy. So also in the case of a single moving electron, the rate of transfer of energy is not given by the integration of the Poynting vector through all space, but differs from this by an amount corresponding to the energyflow along the constraints in the body of the electron. This does not mean that there is any energy associated with the constraints, for of course rigid constraints can neither absorb nor give out energy; there is no storing up, but merely a transfer.

5. It is not difficult to find an expression for this rate of transfer. If the constraint is a simple linear one the transfer of energy along its direction is evidently

$-lTv'\,$,

where ($$l$$) is the length of the constraint, ($$v'$$) the velocity with which it is moving along its length, and ($$T$$) the tension along it. The amount of energy ($$Tv'$$) per sec. is put at the forward end, and is instantly available at the rear end at a distance ($$l$$). If the velocity ($$v$$) makes an angle ($$\theta$$) with the constraint, $$v'=v\cos \theta$$, and the transfer in the direction of $$v$$ is

$-lTv\cos^{2}\theta\,$.

Another type of transfer enters when there is shearing stress in the constraint, a transfer that is which is in a direction perpendicular to that of velocity. It must be remembered that, the constraints are described in no way except geometrically.

If we consider therefore the general case where the stress in the body of the constraint is represented mathematically by the nine stresses commonly used in the theory of elasticity, namely, $$X_{x}, Y_{x}, Z_{x}, X_{y}, Y_{y}, Z_{y}, X_{z}, Y_{z}, Z_{z}$$, then there is a rate of transfer of energy in the x-direction through unit volume given by

$f'_{x}=-(v_{x}+X_{x}v_{y}Y_{x}+v_{z}Z_{x})\,$.

title: Page:ComstockInertia.djvu/3


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Page:ComstockInertia.djvu/4

 This can be readily shown by a consideration of the figure. When the velocity of the element is along (x) there is an amount of work $$(v_{x} \cdot X_{x}) dy \cdot dz$$ done per second on the element by the tension ($$X_{x}$$) applied at the surface ($$O'A'$$), and this energy is instantly available at the surface ($$OA$$), where it is given out. The distance over which the energy is transmitted being $$dx$$ (the thickness of the element), the rate of energy-flow is

$-v_{x}X_{x} dy dz dx=-v_{x}X_{x} d\tau\,$,

where ($$d\tau$$) is the element of volume.

In like manner the velocity ($$v_{y}$$) and the shearing stress ($$Y_{x} \cdot dy \cdot dz$$) cause energy to be taken up at the surface ($$O'A'$$) and given out at the surface ($$OA$$), and we have the rate of flow along the x-axis

$-v_{y}Y_{x} d\tau\,$;

and finally the velocity ($$v_{z}$$) and the shearing stress ($$Z_{x} \cdot dy \cdot dz$$) give

$-v_{z}Z_{x} d\tau\,$.

Hence adding we have, if we call ($$f'_{x}$$) the density of flow along $$x$$,

$f'_{x}d\tau =-(v_{x}X_{x}+v_{y}Y_{x}+v_{z}Z_{x})d\tau\,$.

Obtaining the corresponding equations in similar way we have finally for the three components of the density of energyflow along the constraints in any system

title: Page:ComstockInertia.djvu/4


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Page:ComstockInertia.djvu/5

 For the total density of energy-flow ($$f_{x}, f_{y}, f_{z}$$) we must of course add to the above the components of the Poynting vector. Writing as usual $$X, Y, Z$$ and $$\alpha, \beta, \gamma$$ for the electric and magnetic force intensities and calling ($$V$$) the velocity of light, we have

These equations give the density of the total energy-flow through any purely electrical system, in which the ordinary electrical laws hold universally.

6. Consider an isolated electrical system moving as a whole through space with the constant velocity ($$v_{1}$$). A constant velocity will be possible if the system retains on the average the same internal structure. The total average rate of transfer of energy corresponding to the movement of such a system is evidently ($$v_{1} \cdot W$$), where $$W$$ is the total contained energy. Another expression for the same thing is to be obtained by integrating throughout the system the components along ($$v_{1})$$ of $$(f_{x}, f_{y}, f_{z}$$) given in equations (2). In order that the velocity ($$v_{1}$$) may appear explicitly, however, it is necessary that the velocity ($$v$$), which was used in equations (2), be written as the sum of ($$v_{1}$$) and another velocity ($$v_{2}$$). Then ($$v_{2}$$) is the velocity with respect to axes moving with the system.

If $$l, m, n$$ are the direction cosines of the constant velocity ($$v_{1}$$), we have for the total energy-flow ($$F$$) in the direction of ($$v_{1}$$),

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 Since the proof of equations (1) is equally valid for relative motion, the integrals involving ($$v_{2}$$) in the above correspond to flow of energy with respect to axes moving with the system. There is also an implicitly involved internal term in each of the Poynting vector integrals. Since the system is isolated, the sum of such " internal " terms must on the average vanish. There remains therefore to represent the actual average rate of transfer of energy through space only the explicit Poynting terms and the terms involving ($$v_{1}$$).

The electromagnetic momentum corresponding to any electrical system is given by the components

which, except for the factor $$V^2$$, are the same as the integrals of the components of the Poynting vector throughout the system. Hence equation (3) may be written

Also if the electrical system here dealt with is to represent a material body, we may assume that the resultant momentum ($$M$$) is in the direction of the velocity, and hence

$M=lM_{x}+mM_{y}+nM_{z}\,$.

This may be considered as due to the fact that the lack of symmetry necessarily involved in the intimate structure of any electromagnetic system has become a symmetrical average in particles large enough to be dealt with. This symmetrical point may of course have been reached in the case of single atoms. We may now write (3A) in the form

7. To reduce this expression further requires some relation to be established between the stresses and the electric and magnetic force intensities. This process is closely analogous to the derivation of the Maxwell stress in the free aether

title: Page:ComstockInertia.djvu/6


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Page:ComstockInertia.djvu/7

 except that we here have to deal with, besides the forces in the constraint, only the electromagnetic force on electricity embedded in the constraint, and we have nothing to do with hypothetical stresses in the free aether.

If ($$\rho$$) represents the electric density and ($$\Im_{x}$$) the $$x$$-component of the total electromagnetic force on unit charge embedded in the constraint, we have

{{MathForm2|(6)|$$\left.\begin{array}{l} \frac{\partial X_{x}}{\partial x}+\frac{\partial X_{y}}{\partial y}+\frac{\partial X_{z}}{\partial z}=-\rho\Im_{x}\\ \\\quad=-\rho X-\frac{\rho}{V}(v_{y}\gamma-v_{z}\beta)\\ \\\quad=-\rho X-(k_{y}\gamma-k_{z}\beta)\end{array}\right\} $$,}}

where ($$k$$) is the density of convection current caused by the movement with velocity ($$v$$) of the electricity of density ($$\rho$$).

Making use temporarily of the vector terminology for the sake of brevity and calling the electric force ($$E$$), the magnetic ($$H$$), and the sign [ ] denoting the vector product, we have

$-\rho\Im_{x}=-\rho E_{x}-[kH]$.

Since div $$E = 4\pi\rho$$ and $$curl\ H-\tfrac{1}{V}\tfrac{\partial E}{\partial t}=4\pi k$$,

Now it is an easily verifiable identity that

and hence, remembering that div $$H = 0$$, equation (7) becomes

title: Page:ComstockInertia.djvu/7


text:
Page:ComstockInertia.djvu/8

 Since

$[E,\mathrm{\ curl}\ E]_{x}=-\left[E,\ \frac{1}{V}\frac{\partial H}{\partial t},\ H\right]_{x},$,

the last two terms in the bracket of (9) become together

$-\frac{1}{V^{2}}\frac{\partial}{\partial t}[EH]_{x}$,|undefined

which is minus the time rate of the density of momentum at the point. The time rate, however, refers to a point fixed in space, and to change to a point moving with the system we make use of the usual expression and write

where $$m_{x}$$ is the $$x$$-component of the density of momentum and ($$l, m, n$$) are, as formerly, the direction cosines of the constant velocity ($$v_{1}$$). The operator $$\frac{\partial'}{\partial t}$$ now refers to the rate of change at a point moving with the system.

Substituting (10) for the two last terms in (9) and noticing that

$\frac{\partial m_{x}}{\partial t}$ may be written $\frac{\partial}{\partial x}\frac{\partial'}{\partial t}\int_{R}^{P}m_{x}dx$,|undefined

where the integration is to be taken from $$R$$ (meaning merely from a point outside the system where $$m_{x}$$ is zero) up to the point $$P$$ in question, account being taken of any discontinuity at the bounding surface, we have in place of (9)

This equation gives us what we were seeking, namely, the values of $$X_{x}, X_{y}$$, and $$X_{z}$$ in terms of the electric and magnetic forces and the density of the momentum.

title: Page:ComstockInertia.djvu/8


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Page:ComstockInertia.djvu/9

 Thus we may take

Similar values are of course to be found for the other six components of stress in the constraint $$Y_{x}, Y_{y}, Y_{z}, Z_{x}, Z_{y}, Z_{z}$$.

8. The values for these nine stress-components are now to be substituted in equation (5). In doing this it is to be noticed that the last term in each of equations (12) will, after substitution, furnish a term of the type

and this, being a time derivative, gives an average value of zero when the time is allowed to increase indefinitely, since all quantities in the system remain finite. Also

$l^2 + m^2 + n^2 = 1\,$.

Making the substitution in (5) and simplifying, we have as the value for ($$v_{1}W$$)

Now the first integral represents the total included energy, the two parts of the second integral represent the squares of the components of the electric and magnetic forces in the direction of the motion of the system, and the last integral represents the momentum in the direction of motion, which in this case is the whole momentum $$M$$, since we have assumed that $$M$$ and $$v_{1}$$ are in the same direction.

Calling

$\frac{1}{8\pi}\int\left\{ (lE_{x}+mE_{y}+nE_{z})^{2}+(lH_{x}+mH_{y}+nH_{z})^{2}\right\} d\tau=W_{L}$

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Page:ComstockInertia.djvu/10

 the longitudinal energy of the system, and

$W-W_{L}=W_{T}\,$

the transverse energy of the system, we may rewrite equation (14) as

and hence

This gives the total momentum of any isolated, purely electrical system, which has on the average the same internal structure, in terms of its transverse energy, i. e., the energy represented by the components of the electric and magnetic forces which are perpendicular to the velocity of the system. The mass of the system is then

If, as we have assumed in deriving this expression, the system possesses the same momentum for uniform translation in any direction, this formula for the mass can contain terms of even powers only in the ratio of the velocity of the system to the velocity of light. If we neglect terms of the second and higher orders $$W_{T}$$ has the same value as for $$v_{1}=0$$, which from symmetry of the system must be two-thirds the total energy $$W$$. Therefore

if second order terms he neglected. This formula would apply with extreme accuracy for the electromagnetic mass of ponderable bodies, for no such bodies have in nature a velocity large enough to make $$\left(\tfrac{v_{1}}{V}\right)^{2}$$ appreciable.

It should be noticed that in equation (17) second order terms may enter in either ($$W_{T}$$) or its derivative with respect to $$v_{1}^2$$. In fact such terms do enter for two reasons. In the first place, the setting of the body in motion requires work and hence adds new energy, through a second order term; and secondly there is an effect due to the change which

title: Page:ComstockInertia.djvu/10


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Page:ComstockInertia.djvu/11

 motion causes in the velocity of propagation through the system of electrical disturbances. This is seen in the simple case of a moving electron where the crowding of the lines towards the equator with increase of velocity is only partly due to the added energy. It is evident, therefore, that for velocities so great that the second order terms cannot be neglected, the mass depends on complicated terms which vary with the internal structure and motions of the system, and it does not appear as if a general expression for the mass of a system for such high velocities could be found.

The second order terms may in the future make themselves experimentally manifest through an increase of mass of rapidly moving a-particles.

9. Expression (18) may readily be verified for simple symmetrical systems. For a single charged conducting sphere of radius (a) the mass for slow velocities is well-known to be

$\frac{2}{3}\frac{1}{V^{2}}\frac{e^{2}}{a}=\frac{2}{3}\frac{e}{V^{2}}\frac{e}{a}=\frac{4}{3V^{2}}\frac{1}{2}$ (e. Potential) $=\frac{4}{3V^{2}}W$.|undefined

An interesting verification of equation (16) for the special case of a general, rigid, electrostatic system in translatory motion has been furnished me privately by Mr. G. F. C. Searle. He obtains for such a system (Phil. Mag. Jan. 1907, p. 129) the expression

where $$M_{x}$$ is the momentum of the entire system along the direction ($$x$$) of motion, ($$T$$) is the total magnetic energy due to this motion, and ($$c_{1}$$) is the common translatory velocity possessed by all parts of the system.

Now it is well known that where the Faraday tubes move through space uniformly, as in the present case, the magnetic force ($$H$$) is given in terms of the electric force ($$E$$) by the expression

$H=\frac{v_{1}}{V}E\ \sin\theta$,|undefined

($$\theta$$) representing the angle between ($$E$$) and the velocity of motion ($$v_{1}$$), and ($$H$$) being in a direction perpendicular both to ($$E$$) and ($$v_{1}$$). In the present notation

$H=\frac{v_{1}}{V}\sqrt{E_{y}^{2}+E_{z}^{2}}$,|undefined

and hence we have

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Page:ComstockInertia.djvu/12

 Combining (19) and (20) we can obtain

and remembering that

$T=\frac{1}{8\pi}(H_{y}^{2}+H_{z}^{2})$

we have finally

which, since ($$v_{1}$$) is along ($$x$$), is identical with (16). Thus (16) is verified for the case where the moving system possesses no internal motion.

Perhaps the simplest symmetrical system containing magnetic as well as electric energy is that formed by a great, number of charged spheres moving in straight lines out from a common centre, with velocities small enough so that the fourth and higher powers may be neglected. They are to he at distances from each other great in comparison with their size, and at equal distances from the common centre. If the system be now given the slow velocity ($$v_{1}$$) as a whole the total momentum accompanying this motion may be determined. Because of limited space the calculation will not be here given, but if it be carried out along established lines it will be found that the mass is $$\tfrac{4}{3}\tfrac{1}{V^{2}}$$ times the sum of the electric and magnetic energies, thus verifying equation (18).

10. We conclude that, if ordinary material mass has an electromagnetic basis, such mass for slow velocities is proportional to the total electromagnetic energy-content of the body, and the laws of conservation of mass and energy become closely related if not identical. In any case the expression given represents the electromagnetic part of the total mass whatever that may be.

The Atomic Weights.
11. If the conclusion of the last article is correct a diminution in mass should follow a loss of energy in material transformations. Calculation shows, however, that in the

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 case of the powerful reaction between hydrogen and oxygen forming water, the change of mass would only he of the order $$10^{-13}$$ gram. In the case of radioactivity, however, the energy change is very much greater and an appreciable effect is to be expected. Thus if a radium atom gives off an $$\alpha$$-particle of mass ($$m$$) with velocity ($$\mu$$), then there should be a diminution in the sum of the masses of the \alpha -particle and the remaining atom equal to

$\frac{4}{3}\frac{1}{V^{2}}\left(\frac{1}{2}m\mu^{2}\right)$,|undefined

since $$\tfrac{1}{2}m\mu^{2}$$ represents the energy lost, and this, calling $$m = 4$$ (using gram-atomic weight) and $$\mu = 2.5 \cdot 10^{9}$$, gives

$\Delta(\mathrm{Mass})=-1.7\cdot10^{-2}\mathrm{gram}$;

an amount large enough to cause discrepancies in calculating the atomic weights of radioactive substances from the number of $$\alpha$$-particles lost. Since $$\Delta$$(Mass) is proportional to the square of the velocity of the $$\alpha$$-particle, its value would be greatly increased by a slight error in the determination of ($$\mu$$) and the effect could easily be much larger.

12. A consideration of some interest is the following. If we adopt the disintegration theory, we are obliged to think of the various atoms as combinations or groups, more or less modified, of the lighter atoms. If there were perfect conservation of mass this would introduce a certain uniformity in the relations between the atomic weights, a uniformity which apparently does not exist. On the other hand, if we take into consideration the inevitable change of mass when the electromagnetic energy of the system is modified, the atomic weights will involve a correction term depending upon the change in this energy, and hence they will no longer bear simple, exact relations to each other. In a highly important paper (Zeitschrift fur Anorg. Chemie, xiv. p. 66, 1897) Rydberg has shown that the atomic weights of the first twenty -seven elements of the periodic system approximate to whole numbers very much more closely than chance could bring about. He has also shown that the atomic weights of these elements are best considered as the sum of two parts ($$N+D$$) where $$N$$ is an integer and $$D$$ is a fraction, in general positive and smaller than unity. If $$M$$ is the number of the element in the system (called by Rydberg the "Ordnungszahl"), then $$N$$ is equal to $$2M$$ for the elements of even valence and $$2M+1$$ for the elements of odd valence. Below is given a table showing the various quantities. I have used,

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 however, the International Atomic Weight values for 1907 instead of those Rydberg used.

The orderly arrangement of the series is striking. It will be noticed that in three cases only are the $$D$$'s greater than unity and only in two cases are they negative.

Rydberg points out that although the heavier elements do not conform well to this scheme, i. c., do not in general give the small fractional values of ($$D$$) noticed above, yet this is in reality no valid objection, for the numerical values of the weights of heavier elements depend much more on the value of the arbitrary unit chosen than do those of the lighter weight elements, and hence they can have little influence one way or the other in estimating the validity of the curious relations he sets forth.

The whole question is of course whether these differences represent real physical deviations from something or whether they are merely mathematical remainders. Rydberg certainly believes them to represent physical realities, and considering the before-mentioned overwhelming improbability that the approximation of the atomic weights to whole numbers is due to chance, we can hardly doubt that he is right.

13. Now it is to be noticed that these deviations find a ready explanation when the conclusions of the present paper are combined with the theory, so much favoured recently, that one element breaks down into two or more others with an accompanying expulsion of energy. The deviations are then to be explained as resulting from loss of mass accompanying the dissipation of energy. On the other hand, if no such loss of mass takes place, the existence of these deviations in the

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 table of atomic weights becomes a well-nigh insuperable difficulty in the path of the evolutionary theory of the elements.

If we follow the present suggestion, we must search for the components of an element, not by comparing atomic weights, but by comparing the corresponding values of $$N$$, for the atomic weights deviate because of the lost mass accompanying the dissipation of internal energy.

Very recently Sir W. Ramsay has announced several striking discoveries which seem to add much weight to the disintegration theory, and, indirectly, to the views here set forth. Ho found helium, neon, or argon appearing as a product of radium emanation according to the exterior conditions imposed, and he found lithium appearing when a copper-sulphate solution was left in the presence of the emanation. Prof. Ramsay states. I believe, that every source of error was eliminated and that the results were obtained many times.

14. It should be noticed that this theory of loss of mass and its consequences does not require that the whole material mass should be of electromagnetic nature. It only requires that the energy lost in the transformations, explosive or otherwise, should be at the expense of internal electromagnetic energy, i. e., that the forces which expel the $$\alpha$$-particles should be electric or magnetic.

Respecting gravitation.
15. The experiments of many investigators have shown that up to a high degree of accuracy the ratio of mass to weight for different substances is the same. Now if the mass is proportional to the internal energy as here suggested, instead of being proportional to the number of electric nuclei as might be supposed, the conclusion is apparently forced upon us that gravitational attraction is between quantities of confined energy, and not between quantities of "matter" in any other sense.

On this basis, the weight of a calorie at the earth's surface would be of order $$10^{-11}$$ dyne. This is apparently too small to explain the temperature gradient in the earth although the calculation, depending as it does on the mechanical force on confined energy due to a temperature gradient, would certainly depend to a large degree on the medium.

If we assume this gravitational effect, it is interesting to ask whether free energy would also show an attraction for itself. If so, the energy radiated from a gravitational centre like the sun would leave some of itself behind along its path

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 as it moved through space, and it might be possible to account in this way for some of the energy which is ordinarily thought of as totally dissipated.

Another conclusion which is suggested by the foregoing is that, assuming the loss of mass accompanying dissipation of energy, the sun's mass must have decreased steadily through millions of years. If too, our conclusion respecting the gravitating quality of confined energy be correct, the gravitation constant of the sun has also decreased and the distances of the planets must have increased accordingly. This last increase of planetary distance can be calculated by making the angular momentum of the planet about the sun a constant, and allowing the mass of the planet, together with the gravities of both sun and planet, to grow less with time.

So little is known as to the former radiating power of the sun that no even approximate calculation can be made, but it is not difficult to show that the order of magnitude is such as might make the increase in the planetary distances not altogether negligible during great lapses of time.

A Proof from a different Point of View.
16. The proof of expression (17) which has been given has the advantage of entering intimately into the structure of the general system and showing the part that non-electrical forces in the form of constraints must play if the fundamental laws of electrical action are to hold for every infinitesimal element of the finite volume occupied by any electrical system. Although this is assumed in every mathematical derivation of the mass of an electron, and in fact in all problems of a similar nature, many will doubtless object to this assumption on the ground that probably the ordinary electrical laws do not apply when the distance between "elements of charge," so called, is comparable with the diameter of an electron.

Although it is difficult to see how a coherent mathematical theory of electricity can at present be formed without this assumption, yet it was thought best to add a more general proof of (17). The following is therefore given as avoiding the explicit use of constraints.

17. The statement of the law of the conservation of energy for an element of volume in any electrical system possessing electrical charges in motion, is the well-known expression

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 Here ($$w$$) is the density of the total electromagnetic energy, $$S_{x}, S_{y}, S_{z}$$ are the components of the Poynting vector, $$\Im_{x},\Im_{y},\Im_{z}$$ are the components of the total electromagnetic force on unit charge, $$(\rho$$) is the density of electrification at the given point, and $$v_{x}, v_{y}, v_{z}$$ represent the velocity through space of this electrification. Thus

$W=\frac{1}{8\pi}\{(X^{2}+Y^{2}+Z^{2})+(\alpha^{2}+\beta^{2}+\gamma^{2})\}$,

where $$X, Y, Z$$, and $$\alpha, \beta, \gamma$$, are the electric and magnetic force intensities respectively, and

$\Im_{x}=X+(v_{y}\gamma-v_{z}\beta)$,

$\Im_{y}=Y+(v_{z}\alpha-v_{x}\gamma)$,

$\Im_{z}=Z+(v_{x}\beta-v_{y}\alpha)$.

Equation (23) states merely that the rate of increase of energy in an elementary volume is equal to the activity of any foreign (i. e., non-electrical) forces which may act therein minus the outward flow of energy.

Now suppose we consider an electromagnetic system bounded by a rigid surface ($$AB$$), which moves uniformly through space with the velocity ($$v_{1}$$) along the axis of ($$x$$); and further suppose that the volume inside this closed surface is divided into two parts by the plane partition ($$CD$$) which is perpendicular to the x-axis and which, although fixed in the moving system, coincides at a given instant with the plane ($$C'D'$$) fixed in space. If this system be considered as isolated, then no disturbance passes through the bounding surface ($$AB$$).

In equation (23) the time derivative of the energy-density

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