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 The value of $$\mathbf{\Psi}$$ in terms of $$h$$ thus becomes

Equation (11) now becomes

so that instead of the cylindrical coordinates $$x$$ and $$\rho(=\sqrt{y^{2}+z^{2}})$$ we, can take $$h$$ and $$\phi$$ where

From (18) we have in terms of $$h$$ and $$\phi$$

$\frac{dh}{dx}=\frac{(h^{2}-l^{2})\cos\phi}{h^{2}-l^{2}\cos^{2}\phi},\ \frac{dh}{d\rho}=\frac{h\sqrt{h^{2}-l^{2}}\sin\phi\sqrt{\alpha}}{h^{2}-l^{2}\cos^{2}\phi}$.|undefined

Hence

I now pass on to calculate the total energy possessed by the ellipsoid when in motion along its axis of figure. In making the calculation I shall suppose that $$a^2>ab^2$$, i.e., that $$l^2$$ is positive. The case in which $$a^2<ab^2$$ can be deduced by the appropriate mathematical transformation.

I have shown {§ 22} that the total energy, viz. the volume integral of $$\frac{\mathrm{K}\mathbf{E}^{2}+\mu\mathbf{H}^{2}}{8\pi}$$, due to the motion of a charge on any surface, is

$\mathrm{W}=\frac{1}{2}q\mathbf{\Psi}_{0}+2\mathrm{T}$,

where $$\Psi_{0}$$ is the value of the convection-potential at the surface of the body, and T is the magnetic part of the energy, viz., the volume integral of $$\mu H^{2}/8\pi$$.

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 Now in fig. 4 let the ellipsoid PQ be determined by $$h$$, and the ellipsoid RS by $$h+dh$$.

Let the angular coordinate of P and S be $$\phi$$, and let that of Q and R be $$\phi+d\phi$$. Then the area PQRS

$\begin{align} & =\frac{d(x,\rho)}{d(h,\phi)}dh\ d\phi=\left(\frac{dx}{dh}\frac{d\rho}{d\phi}-\frac{dx}{d\phi}\frac{d\rho}{dh}\right)dh\ d\phi\\ & =\frac{h^{2}-l^{2}\cos^{2}\phi}{\sqrt{\alpha}\sqrt{h^{2}-l^{2}}}dh\ d\phi \end{align}$.|undefined

Now if the area PQRS revolve about the axis O$$x$$ the volume of the ring traced out is

$2\pi\rho\frac{d(x,\rho)}{d(h,\phi)}dh\ d\phi=\frac{2\pi(h^{2}-l^{2}\cos^{2}\phi)\sin\phi}{\alpha}dh\ d\phi$.

Thus for the magnetic part of the energy we have

$\begin{align} \mathrm{\mathrm{T}} & =\int\int\frac{\mu\mathrm{H}^{2}}{8\pi}2\pi\rho\frac{d(x,\rho)}{d(h,\phi)}dh\ d\phi\\ & =\frac{\mu q^{2}u^{2}}{4}\int\int\frac{h^{2}\sin^{3}\phi\ dh\ d\phi}{(h^{2}-l^{2})(h^{2}-l^{2}\cos^{2}\phi)} \end{align}$|undefined

Since $$\lambda$$ goes from 0 to $$\infty$$, $$h$$ goes from $$a$$ to $$\infty$$. The limits of $$\phi$$ are 0 and $$\pi $$.

Now

$\begin{align} \int_{0}^{\pi}\frac{\sin^{2}\phi\ d\phi}{h^{2}-l^{2}\cos^{2}\phi} & =-\frac{1}{l^{2}}\left[\cos\phi-\frac{h^{2}-l^{2}}{2hl}\log\frac{h+l\ \cos\phi}{h-l\ \cos\phi}\right]_{0}^{\pi}\\ & =\frac{1}{l^{2}}\left\{ 2-\frac{h^{2}-l^{2}}{hl}\log\frac{h+l}{h-l}\right\} \end{align}$.|undefined

Hence

$\begin{align} \mathrm{T} & =\frac{\mu q^{2}u^{2}}{4l^{2}}\int_{a}^{\infty}\left(\frac{2h^{2}}{h^{2}-l^{2}}-\frac{h}{l}\log\frac{h+l}{h-l}\right)dh\\ & =\frac{\mu q^{2}u^{2}}{4l^{2}}\left[h-\frac{h^{2}-l^{2}}{2l}\log\frac{h+l}{h-l}\right] \end{align}$|undefined

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

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 When $$h$$ is large the quantity in [ ]

$=h-l\left(\frac{h^{2}}{l^{2}}+1\right)\left(\frac{l}{h}+\frac{l^{2}}{3h^{3}}\dots\right)$|undefined

vanishing when $$h=\infty$$.

Thus, making use of $$\mu\mathrm{K}v^{2}=1$$ we have for the magnetic energy

$\mathrm{T} =\frac{q^{2}u^{2}}{4l\mathrm{K}v^{2}}\left\{ \frac{a^{2}-l^{2}}{2l^{2}}\log\frac{a+l}{a-l}-\frac{a}{l}\right\}$.|undefined

Now by (17) we have at the surface of the ellipsoid

$\mathbf{\Psi}_{0}= \frac{q\alpha}{\mathrm{K}}\int_{a}^{\infty}\frac{dh}{h^{2}-l^{2}}=\frac{q\alpha}{2\mathrm{K}l}\log\frac{a+l}{a-l}$.|undefined

Hence the total electromagnetic energy of the ellipsoid is

Here we must remember that $$l^{2}=a^{2}-\alpha b^{2}$$.

(A) Energy of Heaviside Ellipsoid. If we put $$a/l$$= S and make S large we have

This corresponds to the Heaviside ellipsoid, for when $$S=\infty$$, $$a^{2}=ab^{2}$$. The energy of the same ellipsoid at rest is

$\frac{q^{2}\sqrt{\alpha}}{2\mathrm{K}a}\cdot\frac{v}{u}\sin^{-1}\frac{u}{v}$.|undefined

(B) Energy of a Sphere. Putting $$b = a$$ we have $$l = au/v$$, and thus

If $$u$$ is small compared with $$v$$ we have

$\mathrm{W}=\frac{q^{2}}{2\mathrm{K}a}\left(1+\frac{2}{3}\frac{u^{2}}{v^{2}}+\cdot\cdot\cdot\right)$.|undefined

It will be found that as far as $$u^2 /v^2$$ the magnetic energy is

$\frac{q^{2}u^{2}}{3\mathrm{K}av^{2}}=\frac{\mu q^{2}u^{2}}{3a}$|undefined

as has been found by Mr. Heaviside. It follows from this

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 that as far as terms in $$u^2 / v^2$$ the electric part of the energy is unaltered by the motion.

(C) Energy of a very slender Ellipsoid. When the ellipsoid is so slender that $$b^2 /a^2$$ may be neglected in comparison with unity we have

When $$u / v$$ is small, this becomes

$\mathrm{W}=\frac{q^{2}}{2\mathrm{K}a}\left\{ \left(1+\frac{u^{2}}{v^{2}}\right)\log\frac{2a}{b}+\frac{1}{2}\frac{u^{2}}{v^{2}}\right\}$.|undefined

(D) Energy of a Disk.

When $$a^2 < \alpha b^2$$ the ellipsoid is more oblate than Heaviside's, and $$l^2$$ becomes negative. In this case let us write

$r^{2}=b^{2}-\frac{a^{2}}{\alpha}$,|undefined

so that $$r$$ is the radius of the disk which is the "image" of the ellipsoid $$a, b$$. Then writing $$\sqrt{-1}=i$$ we have from (23)

$\mathrm{W}=\frac{q^{2}}{4\mathrm{K}ir\sqrt{\alpha}}\left(1-\frac{u^{2}a^{2}}{v^{2}r^{2}\alpha}\right)\log\frac{1+i\sqrt{a}r/a}{1-i\sqrt{\alpha}r/a}+\frac{q^{2}u^{2}a}{2Kv^{2}r^{2}\alpha}$.|undefined

But

$\frac{1}{i}\log\frac{1+xi}{1-xi}=2\left(x-\frac{x^{3}}{3}+\frac{x^{5}}{5}\dots\right)=2\tan^{-1}x$,|undefined

so that (23) becomes

When $$a = 0$$ we find for the energy of a disk of radius $$r$$ moving along its axis

In all these cases it will be found that when $$u = v$$ the energy becomes infinite, so that it would seem to be impossible to make a charged body move at a greater speed than that of light.

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text: Index:LarmorDrift.djvu

title: Index:LarmorDrift.djvu


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 IN a recent paper by Prof. D. B. Brace (Phil. Mag. April 1904, p. 318) the author removes by very refined experimenting all trace of doubt from Lord Rayleigh's conclusion that motion of transparent solids through the æther does not induce any double refraction, even to the second order of the ratio of the velocity of the translation to that of radiation; but he infers from this the non-existence of the second-order deformation of the solid due to its translation, suggested by FitzGerald and by H. A. Lorentz to account for Michelson's earlier demonstrated absence of effect on optical interferences over long paths in free aæther. As he remarks, it had previously been suggested by Lord Rayleigh that such an inference might possibly follow from this result. The object of this note is to explain that the inference in question is the opposite to that which I still hold to be the natural result of the theory of the motion of molecular aggregates through æther, as hitherto developed.

The argument of Prof. Brace proceeds on the basis that the whole effect of the convection through the æther is to introduce new forces between the molecules, causing the shrinkage aforesaid along the direction of convection; and it can be readily granted that if this were all, double refraction must result. But both the line of argument suggested as probable by Lorentz, and the molecular analysis offered by me some years later, proceed by comparing a system shrunk in the FitzGerald-Lorentz manner and convected through the æther, with the same system unshrunk and at rest, and finding a complete correspondence between them as regards the states and activities of the individual molecules. As the argument is somewhat complex and has been misunderstood, a brief re-statement of the result may prove useful.

We are to compare the field of physical activity of a system of molecules at rest, with the field of the identically same configuration of molecules in uniform translatory motion through æther. If small quantities of the order of the square of the ratio of the velocity of convection to that of radiation (v/c) are neglected, the Maxwellian physical equations for the

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 second system, referred of course to axes of co-ordinates moving along with it, can be reduced to the form belonging to the same system at rest, by the transformation first developed by Lorentz: namely, each point in space is to have its own origin from which time is measured, its "local time" in Lorentz's phraseology, and then the values of the electric and magnetic vectors

(f, g, h) and (a, b, c)

at all points in the æther between the molecules in the system at rest, are the same as those of the vectors

$\left(f,\ g-\frac{v}{4\pi c^{2}}a,\ h+\frac{v}{4\pi c^{2}}b\right)$ and (a, b + 4πvh, c-4πvg)|undefined

at the corresponding points in the convected system at the same local times. This correspondence can, in fact, be shown to locate the electrons at corresponding points in the two systems, and to make them equal; if, then, they are held in rigid connexion, or more generally if their states of orbital motion in the molecules are conserved, the effect of translatory motion of the system with velocity v is to transform the æthereal field around them and between them as here specified. The fields of æthereal activity are not identical, but where one vanishes at any point so does the other at the same point. This conclusion was reached by Lorentz, who pointed out that it carried with it a null result for all recognizable optical tests of convection in the system, up to the first order, with the one exception of the Doppler effect which is involved in the "local" time measurements, and which is only a partial exception because it refers to radiation coming from outside the system.

Does, however, the system of electrons need to be constrained in order to prevent change of configuration when being convected? The force acting on an individual electron e is thereby changed from

$4\pi c^{2}e\left(f,\ g-\frac{v}{4\pi c^{2}}c,\ h+\frac{v}{4\pi c^{2}}b\ \right)$ to 4πc²e(f, g, h).|undefined

If there is a magnetic field (a, b, c) there will thus be alteration: if there is no sensible average magnetic field, even among the molecules, we may perhaps fairly assume, with Lorentz, that no constraint is needed in order to prevent change in molecular configuration in the system due to convection. Anyhow, the absence of recognizable optical result to the first order is certain, as the physical constants of the system in bulk must be unaltered to that order.

But the brilliant experimenting of Michelson and Morley

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 had already led to the recognition of absence of optical result up to the second order of the ratio of the velocities. Thus the question was suggested whether the above correspondence between the resting and convected systems can be effectively extended up to the second order. It is, in fact, found that the Maxwellian circuital equations of æthereal activity, in the ambient æther, referred to axes moving along with the uniform velocity of convection v. can be reduced to the same form as for axes at rest, up to and including (v/c)², but not (v/c)³, by adopting a local time ε-1/2(t - vx/c²) as before, but with a new unit ε-1/2, and also a reduced unit of length parallel to x equal to ε-1/2, where here and in what follows ε represents 1+v²/c², the units of length along y and z remaining unaltered. It is found that for two æther-fields, one referred to fixed axes and the other to moving axes, standing in this mutual correlation, the electrons, or poles, in approaching which the æthereal electric vector becomes infinite as er-1, are situated at corresponding points and are of equal values: the relation, exact to the second order, is now that

(f, g, h) and (a, b, c)

in the field belonging to the fixed system of poles correspond to

$\mathsf{E}^{\frac{1}{2}}\left(\mathsf{E}^{-\frac{1}{2}}f,\ g-\frac{v}{4\pi c^{2}}c,\ h+\frac{v}{4\pi c^{2}}b\right)$|undefined

and

$\mathsf{E}^{\frac{1}{2}}\left(\mathsf{E}^{-\frac{1}{2}}a,\ b+4\pi vh,\ c-4\pi vg\right)$|undefined

for the field belonging to the convected system; where ε is 1 + v²/c², as above, the factor ε1/2 being needed to make corresponding poles equal in value instead of merely proportional.

If each pole or electron is connected with a molecule possessing extraneous mass, and it may be having an extraneous field of gravitational and other force of its own, and thereby interacting with other molecules, we shall want to know the forces exerted on that molecule by the surrounding æther, in order to form its own equations of motion, which must be combined with those of the æther-field around it in order to constitute a complete system. But if such other forces are molecularly insignificant, or better, if the electron is a mere passive pole — nucleus of beknottedness in some way — in the æther, conditioned and controlled entirely by the æther around it, just as a vortex ring is conditioned by the fluid in which it subsists and is also carried along thereby, then, as in the familiar hydrodynamics of vortices, the motion of the æther determines the motion of the entirely passive

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 electrons, and the idea of force acting between them and the æther is dispensed with.

If, then, matter is for physical purposes a purely æthereal system, if it is constituted of simple polar singularities or electrons, positive and negative, in the Maxwellian æther, the nuclei of which may be either practically points or else small regions of æther with internal connexions of pure constraint, the propositions above stated for the first order are extended to the second order of v/c, with the single addition of the FitzGerald-Lorentz shrinkage in the scale of space, and an equal one in the scale of time, which, being isotropic, is unrecognizable.

On such a theory as this the criticism presents itself, and was in fact at once made, that one hypothesis is needed to annul optical effects to the first order; that when these were found to be actually null to the second order another hypothesis had to be added; and that another hypothesis would be required for the third order, while in fact there was no reason to believe that they were not exactly null to all order. Such a train of remarks indicates that the nature of the hypotheses has been overlooked. And if indeed it could be proved that the optical effect is null up to the third order, that circumstance would not demolish the theory, but would rather point to some finer adjustments than it provides for: needless to say the attempt would indefinitely transcend existing experimental possibilities.

As, then, the theory contains no further power of immediate adaptation, what are the hypotheses on which it rests, and how far are they gratuitous hypotheses introduced for this purpose alone? Up to the first order the electron hypothesis, that electricity is atomic, suffices by itself, as Lorentz was the first to show. Yet, even if the nature of the particles of the cathode discharge had never been made out, and the Zeeman effect had never been discovered, the facts known to Ampere and Faraday were sufficient to demonstrate that no other conception of electricity than the atomic one is logically self-consistent.

Up to the second order the hypothesis that matter is constituted electrically — of electrons — is required in addition. For this there is no independent evidence except perhaps the general simplicity of the correlations of physical law. The circumstance that positive electrons have not yet been isolated naturally counts considerably on the other side; yet the theory puts no limit to the size and inertia and complexity of an electron, it only prescribes that it must be a collocation of æther poles connected together by some sort of pure constraint, but with no extraneous activities.

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 Any rival theory must on the threshold give an account of the Michelson null optical result, of Trouton's null electric result for convection of a charged condenser, and of Rayleigh's absence of double refraction now rendered thoroughly secure by Brace.

As electrons are already held to be a reality on various grounds, theoretical and experimental, it would appear therefore that there is much to be said for a benevolent attitude to the proposition that all the interactions of matter, so far as the laws of physics and chemistry extend, are to be described as phenomena occurring in and through the aether, and thus differentiated from the more recondite world of vital growth and change which they make manifest to our senses. This principle does not yet, so far as one can see, stand in the way of any other branch of physical science, while it accounts for the very remarkable absence of influence of the earth's motion through space on the most sensitive phenomena, and is almost led up to thereby.

It is pertinent to the present subject to refer to Mr. Sutherland's recent remarks (Phil. Mag. April, p. 406) on the magnetic effect of electric convection, in relation to the mysterious action of a dielectric varnish that has been announced by Crémieu and Pender. The discrepancy in the conservation of energy, there described, applied to the domain of electric polarization, is too startling to have been overlooked by the current theory; and accordingly closer consideration gets rid of the difficulty. When an electron e is transferred in an electric field from a place where the potential is V1 to a place where it is V2, the fores acting on it, being e multiplied by the gradient of V, does work equal to e(V1-V2). When, however, the electron is embedded in a piece of dielectric matter which is so transferred, the force acting on the electron itself is diminished by the presence of the surrounding polarized matter, and so the work done on the electron is less than before: but now the electric polarization induced by the electron in this surrounding matter is also acted on by the electric field, and if we add the work done on it during the movement, we shall get the same total work as before for the system that is moved, and there will be no discrepancy to be otherwise explained.

Cambridge, April 7, 1904.

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text: Index:MillerKelvin.djvu

title: Index:MillerKelvin.djvu


text: More Celtic Fairy Tales/Dream of Owen O'Mulready


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Rh LXXXVI. Extract from a Letter dated Cleveland, Ohio, August 5th, 1904, to Lord Kelvin from Profs. and.

N 1887 Michelson and Morley made an experiment on the relative motion of the earth and the luminiferous ether. They could detect no relative motion equal to one sixth of the earth's velocity in its orbit.

To explain this result, FitzGerald and Lorentz suggested that the stone slab on which the apparatus was constructed might have its dimensions changed by its drift through the ether.

There was a remote chance of detecting such an effect by repeating the experiment of 1887, but with different materials. If the FitzGerald-Lorentz effect exists, it may affect all materials to the same amount, independently of the nature of the material. But it is also possible that the effect is one which depends on the physical properties of the material, so that pine may be affected more than sandstone. In this case, if sandstone gives no displacement in an experiment like that of 1887, an apparatus supported by pine, which would be compressed more than sandstone, would give an effect of the sign opposite to that suggested by the original simple theory.

Such an experiment has now been made. We first made a structure of pine, which we floated on mercury as in 1887. While this structure was new, we obtained good observations with it. But after the wood had been affected by steam-heat for one season, it was not possible to maintain it in adjustment for five minutes. We therefore made a structure in which the distance of the mirrors depends on pine rods, but all other circumstances depend on steel. Two very stiff steel girders cross symmetrically, and are floated on mercury. Two holders, each carrying four mirrors, are fixed at the ends of two arms of this cross. Two other mirror-holders are suspended freely. Pine rods reach from the fixed holders to the free holders, and springs maintain stable contact. These pine rods are carried in brass tubes which constitute a sort of truss; the distance of the two sets of mirrors depends solely on these pine rods.

Observations were made by noting the place of the central black fringe on a kind of eyepiece micrometer. They were made at sixteen equidistant azimuths, and commonly at a rate of one revolution in little more than a minute, readings being made to the tenth of a wave-length. Two times were

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754 selected in which the motion of the earth was in the plane of the apparatus. In the first part of July, the direction of motion at these two times apparently differed by 115°; the morning and evening sets of observations were combined by taking into account this difference. In all, more than 250 turns were observed.

The velocity of the earth in July may be assumed to be 33·5 kilometres a second. The length of path in our apparatus was 32·2 metres, and the effect to be expected (on the original theory) is 1·4 wave-lengths.

We have established that, if there is any effect, it is not more than 0·015 wave-length.

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text: 1911 Encyclopædia Britannica/Denain

title: 1911 Encyclopædia Britannica/Denain


text: 1911 Encyclopædia Britannica/Denbigh, William Feilding, 1st Earl of


text: Index:MillerTheory.djvu

title: Index:MillerTheory.djvu


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[Plate IX.]

N 1887 Michelson and one of the present writers made an experiment "On the Relative Motion of the Earth and the Luminiferous Æther". We found that, if there were any effect, it was not sensibly larger than one-fortieth of the amount expected. To explain this result, FitzGerald and Lorentz suggested that the motion of translation of a solid through the æther produces a contraction in the direction of the drift, with extension transversely, the amount of which is proportional to the square of the ratio of velocities of translation and of light.

Such a contraction can be imagined in two ways. It may be thought to be independent of the physical properties of the solid and governed only by geometric conditions; so that sandstone and pine, if of the same form, should be affected in the same ratio. On the other hand, the contraction may depend upon the physical properties of the solid; so that pine-timber would doubtless suffer a greater compression than sandstone. If the compression annul the expected effect in one apparatus, it may in another apparatus give place to an effect other than zero, perhaps with the contrary sign.

We have now completed an experiment in which two different pine-structures have been used, and in which the optical parts have been so enlarged as to produce an effect 2.3 times as great as the apparatus of 1887. The object was to determine whether there is any difference between the behaviour of sandstone and of pine.

When Michelson and Morley got a null result in 1887, it was thought sufficient to give the theory for merely the maximum and the minimum expected in the four principal azimuths, without mention of the phenomena at intermediate azimuths. The theory also neglected powers higher than the second of the ratio of the velocities. Recently, Dr. Hicks has published a profound and elaborate discussion of the theory, obtained by methods which are not approximate.

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 He develops expressions for angles of reflexion, for wave-length after reflexion, and for the conditions which determine the network of parallelograms formed by the two systems of wave-fronts. The diagonals of these parallelograms are alternately lines of maximum and minimum disturbance in the æther, so that they define the interference phenomena. These expressions are not only rigorous, but also general, applying to any adjustment whatever of the optical parts of the apparatus, and form a welcome contribution to the thorough understanding of the theory of the Michelson and Morley experiment.

In one passage he says that a term added by him "may entirely modify the nature of the changes produced as the direction of the drift alters"; and some appear to think that the inference from the earlier experiment is involved in doubt by this discussion. It is therefore well to examine again the theory.

Let D, Pl. IX. fig. 1, be a plane-parallel glass plate, silvered so thinly that equal quantities of light are transmitted and reflected. S being a source of light, part of the light passes through D, moves on to the plane mirror II, where it is reflected back to D; here, part is transmitted and lost, part is reflected to the observer at T. The other part of the entering light is reflected at the first surface of D, reflected again by the mirror I, is in part reflected by D, and lost, in part transmitted through D, and proceeds towards T. If distances and angles are suitable, the reunited rays between D and T will produce interference phenomena. If distances are equal, we may obtain interference phenomena in white light. In one of the usual adjustments of distances and angles, parallel fringes are seen when the eye or the telescope is made to give distinct vision of one of the mirrors I or II. The fringes apparently coincide with these surfaces. A central fringe is black; on either side are coloured fringes, less and less distinct till they fade away into uniform illumination. If the path of either ray is shortened, the fringes move rapidly to one side. If we engrave a scale on I or II, we can, after any alteration of one of the paths, restore with great accuracy and ease the former relations by bringing the central dark fringe to its original place on this scale. If the motion of the earth through the æther were the cause of this change of path, we could measure the amount of change by measuring the displacement.

Suppose, fig. 2, that the apparatus moves in the direction of the arrow through the stationary æther. While the ray of light moves from D to I and returns to D, the mirror D

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 moves to the new position D2. The angle of reflexion from D is no longer equal to the angle of incidence. The ray moving from D towards II finds the latter in the position II2, returns to D2, and is reflected from D2 nearly in the same direction with the ray from I. In four azimuths of the apparatus the coincidence is exact; for all others, the ray I and the ray II are inclined at a small angle which, at its maximum, is numerically equal to $$v^2 /\mathrm{V}^2$$, v and V being the velocities of the apparatus and of light. Since the angle, the total aberration, cannot be observed, being annulled by the motion of the observing telescope at T, we can hope to detect merely this aberration of the second degree, namely, the small angle between the emergent rays I and II.

With the adjustments just supposed, there are four methods of measuring interference phenomena which in turn measure the angle sought. We may use a micrometer in the telescope, or a scale engraved on I or on II; we may use mechanical compensation to return a displaced fringe to its marked position, or we may use optical compensation.

In another adjustment the fringes are made infinitely broad. We are then limited to the last pair of methods. This pair, especially the last method, is capable of very great precision. When Michelson and Morley set up the first apparatus in which they utilized this method, the mean error of a setting, in which the observer did not himself see the reading, was less than the two-hundredth part of a wave-length. Since the theory of the apparatus in this special case is simpler, the discussion will assume this adjustment.

Accordingly, let the angles I B D, II B D, fig. 3 (PI. IX.), be equal to each other and to 45°. Let the three planes intersect in a common point B. For brevity, imagine that the mirrors themselves are produced so as to intersect in this point. Assume that the system is moving through the æther in a direction making an angle of 67½° with the direction of the light entering the telescope, as indicated in fig. 2. The velocities of the apparatus and of light being denoted by v, V, assume that $$v/\mathrm{V}\ =\ 1/5$$.

A certain wave-front enters the apparatus, making with II an angle which is to be specified. If some given ray enters the apparatus so as to pass axially through the telescope, rays making an angle of 5 minutes on either side of it will pass through our actual apparatus. Almost any ray, wisely selected, may be used to determine what we desire to know about the whole pencil. For instance, we might select the ray which, after reflexion from II, shall

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 return to the point in the mirror D at which it first passed through this semitransparent mirror; or the ray which, after reflexion from I, returns to D with the same exactness. The simplest treatment is possible when we select that system of wave-fronts which make with the mirror II an angle $$\sin^{-1}\frac{v}{\mathrm{V}}\cos\alpha$$, where is the azimuth of the apparatus measured from the position in which its motion through the æther is parallel to the axis of the observing telescope. The azimuth of the motion assumed in figs. 2 and 3 is 67° 30'.

We will now examine the condition of the wave-fronts in the apparatus, fig. 3, at two specified instants, using two diagrams to avoid confusing the numerous lines. In fig. 3 are shown nine wave-fronts making the specified angle with II. The wave-fronts of the transmitted fraction are denoted by accented letters. Seven have not yet reached the mirror II; b' intersects II in the common point B; a' has been reflected from II, and its upper part begins to he reflected from D. At the same instant are shown the wave-fronts of the other system by unaccented letters. All have been in part reflected from D; c begins to be reflected from I; b intersects I in the common point; a is quite cleared from the mirror I. In fig. 4 we follow the same nine wave-fronts. Of the transmitted waves, a' has quite cleared the system of mirrors; b' is just clearing D; five are but partly reflected from D; h' passes through the common point B, and is beginning to be reflected by D; i' has not yet reached II. Of the other system of waves, a and b have cleared the system of mirrors; five are passing through the semitransparent mirror D; h passes through the common point B, having just finished its reflexion from I; i is just beginning this reflexion. For all azimuths except four the general conditions are those of the diagram, but the amount and direction of the various inclinations alter with the azimuth.

The wave-front h is established in its whole length when it passes through the common point. The wave-front corresponding to it in the other system is, this instant, infinitely short, and a' is the first to be established in its entirety. But the position of a fictitious wave of this system, h", is determined by two conditions—first, that it be parallel to a', and, secondly, that it coincide with the infinitely short wave-front h' at the common point B. Except at four certain azimuths, these two wave-fronts, in the same phase, and intersecting in a common point B, will be inclined to each other at a small angle. To measure or to detect this inclination is to measure or to detect the secondary aberration

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 which interests us. If we could measure the perpendicular distance between these wave-fronts at a sufficient distance from B, we should know the angle between them. But h" is only a fictitious line. What we cannot measure between h and h" we can measure between a and a, provided we can determine the point of intersection between a and a, and provided this be found in a convenient position. We have therefore to determine the point of intersection of a and a', knowing that of h and h".

The observing telescope is shown at T, fig. 5. Its axis is parallel to B II. We will show that the phase-difference of a and a' is constant at all points on any line parallel to the line B II, or to the axis of the telescope.

If we write, ', not for wave-lengths, but for the perpendicular distance between consecutive wave-fronts of the same phase, and , ' for the total aberration of the wave-fronts of the two systems, we have to show that $$\frac{\lambda'}{\cos\delta'}-\frac{\lambda}{\cos\delta}$$ is identically zero for eight specified equidistant azimuths, and is not greater than $$0.3\frac{v^{4}}{\mathrm{V}^{4}}$$ for other azimuths. Each of these quantities is determined by a complicated expression; and the equality specified can be most readily determined by trigonometric computation.

To prove the proposition, therefore, we will take that azimuth where, according to Dr. Hicks, the shifting of the intersection is a maximum, and we will assume the extreme case where the velocity of the apparatus is half that of light.

In fig. 6, the mirrors D, I, and II are accordingly supposed to move in the direction of the arrow. Let be the period of the waves of light incident on D; according to the previous specification, the angle between these wave-fronts and the plane of II is $$\sin^{-1}\frac{v}{\mathrm{V}}\cos\alpha$$; that is, they are parallel to II. Lay off on c d, the line of motion of a certain point of the mirror D, the positions of this point at the times 0,, 2, &c. Positions of D and of I and II at certain times are also noted in the same way: all numerical subscripts denote times. The source moves with the apparatus, and therefore, with the assumed ratio of velocities, the apparent wave-length of the light incident at D is half the wave-length in the case of rest, and is half the distance described by a wave-front in the unit of time. Let the initial position be one in which a wave-front passes through the given point in the mirror and through the point O in the line of motion. At the time $$t=\tau$$ the mirror is at 1, and the wave-front in

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 question cuts the line of motion in 2, and intersects the mirror in e. The wave-front reflected from D at $$t=0$$ will have reached the point f1, and the tangent ef1 establishes the reflected wave-front. At the times 2, 3, &c., this particular disturbance will be found at f2, f3, f4, &c.

When D is at a position 2, a new disturbance will have been established at g, which, at the time $$t=5\tau$$, will be found at g5. In the same way, h5, i5, k5 will have arranged themselves in the line f5 k5. At the time $$t=10\tau$$ the six wave-fronts will have been reflected from I, and will be placed along the line o10 f10. The angle f10 O I o is equal to the aberration of the wave-fronts after reflexion from D. As, at this azimuth, the angles of incidence and of reflexion at I are equal, this angle is also the aberration of the emergent rays.

Part of the wave-front f, indicated by f, will be transmitted through the mirror D. It will overtake the mirror II at the time $$t=8\tau$$, when II will have position marked II8, fig. 7. Returning after reflexion, it will take the position noted for the times $$t=9\tau$$, $$t=10\tau$$; and meeting the mirror D at $$t=10\tfrac{2}{3}\tau$$, it will be reflected as shown f11, at an angle whose tangent is given by the formula below. The following wave-front g' will be reflected one period later, at II9, and it is shown in several positions. The wave-front f, belonging to the other system, having passed through the mirror, and having reached the line, Sd, at $$t=10\tau$$, is shown at f11.

In fig. 8 is shown the position of the wave-fronts below the mirror D for the time $$t=15\tau$$. f15, and f'15 have moved along the paths indicated, while the other wave-fronts have moved in a corresponding manner, their position at the time $$t=15\tau$$ being as shown in the figure. The wave-fronts of the unaccented system are placed on the line op; the aberration of the system is equal to the angle. The wave-fronts of the accented system are placed on the line qr; the corresponding aberration is the angle '; the line T15q being the position of the axis of the observing telescope at the time $$t=15\tau$$. Produce the planes of the wave-fronts, draw line l, l', parallel to qT15, the axis of the telescope, each terminated by the planes of consecutive wave-fronts. Their lengths are $$l=\frac{\lambda}{\cos\delta},\ l'=\frac{\lambda'}{\cos\delta'}$$. It is to be proved that $$l=l'$$.

Putting for the angle of incidence and ' for the angle of reflexion, we have for both reflexions at D, =45°. For this azimuth there is no change of angles at I and II. The

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 angles of reflexion are given by the equation

$\tan\frac{\phi'}{2}\ =\ \tan\frac{\phi}{2}\cdot\frac{\mathrm{V}-u}{\mathrm{V}+u}$

where u is the velocity of the surface perpendicular to its own plane; negative, if moving away from the approaching light. With the assumed adjustments, we therefore have for the reflexion of the unaccented system at D:

For the reflexion of the accented system we have:

By equation (3) of Dr. Hicks's paper, putting L1 for the perpendicular distance between wave-fronts of the light incident on D from the moving source,

$\frac{\lambda}{\mathrm L_{1}}\ =\ \frac{4-0.5}{4+0.5-2}=1.4$; and $\frac{\lambda'}{\mathrm L_{1}}=\frac{4-1}{4+1-4}\cdot\frac{4-0.5}{4+0.5+2}=\frac{21}{13}$.|undefined

Therefore

$\frac{\lambda}{\cos\delta}\ =\ \mathrm L_{1}1.4\div0.8=1.75\ \mathrm L_{1},$

and

$\frac{\lambda'}{\cos\delta'}\ =\ \mathrm L_{1}\frac{21}{13}\div\frac{12}{13}\ =\ 1.75\ \mathrm L$;

accordingly,

$\frac{\lambda'}{\cos\delta'}\ =\ \frac{\lambda}{\cos\delta}\ =\ l'\ =\ l$.

Therefore, if the intersection of k15 and i15 is on the line xy parallel to qT15, the intersection of i15, and h15 is also on the same line; that is, the phase-difference of the two sets of waves is constant along any line parallel to the axis of the observing telescope. The same thing may easily be proved for any one of eight equidistant points of the circumference commencing from the point where the motion of the apparatus is parallel to the axis of the telescope.

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 The equality is exact for the central ray. For rays inclined as much as five minutes of arc on either side of the central ray,

$\frac{\cos\delta'}{\lambda'}-\frac{\cos\delta}{\lambda}\ \leqq\ 0.2\frac{v^{3}}{\mathrm{V}^{3}}.$|undefined

At azimuths other than those specified, the quantity $$\frac{\cos\delta'}{\lambda'}-\frac{\cos\delta}{\lambda}$$ is not greater than $$0.3\frac{v^4}{\mathrm V^4}$$ for central rays, and not greater than $$0.5\frac{v^3}{\mathrm V^3}$$ for extreme rays. We may set side by side the magnitudes of this disturbing effect for central rays at several azimuths according to rigorous computation and according to Dr. Hicks's approximate formula.

It will be seen that the effect detected by Dr. Hicks proves, by rigorous computation, to be entirely negligible for the central rays. Its extreme value for marginal rays is not greater than $$0.5\frac{v^{3}}{\mathrm{V}^{3}}$$, which is entirely too small to influence the observations. This result is very satisfactory. It is proved for the specified adjustment of the angles, but it is easy to see that the rotation of mirror I about a perpendicular line in its surface, by a quantity like ten seconds of arc, will not change all relations of residual aberrations by important amounts. It is therefore established, at least for the adjustment specified, that the wave-fronts a and a' of fig. 4. intersect in the line B II, if the wave-fronts h and h" do, rigorously for eight principal positions, very approximately for all other positions. If, then, we can measure the linear distance between a and a' at some convenient position T, we may determine the angle between the wave-fronts a and a', which is the same as the angle between h and h", the

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 Dinbych means a fort. There is a goblin well at the castle. Historically, David (Dafydd), brother of the last Llewelyn, was here (aet. Edward I.) perhaps on a foray; also Henry Lacy, who built the castle (aet. Edward I.), given to the Mortimers and to Leicester (under Edward III. and Elizabeth, respectively).

 DENBIGHSHIRE (Dinbych), a county of N. Wales, bounded N. by the Irish Sea, N.E. by Flint and Cheshire, S.E. by Flint and Shropshire, S. by Montgomery and Merioneth, and W. by Carnarvon. Area, 662 sq. m. On the N. coast, within the Denbighshire borders and between Old Colwyn and Llandulas, is a wedge of land included in Carnarvonshire, owing to a change in the course of the Conwy stream. (Thus, also, Llandudno is partly in the Bangor, and partly in the St Asaph, diocese.) The surface of Denbighshire is irregular, and physically diversified. In the N.W. are the bleak Hiraethog (“longing”) hills, sloping W. to the Conwy and E. to the Clwyd. In the N. are Colwyn and Abergele bays, on the S. the Yspytty (Lat. Hospitium) and Llangwm range, between Denbigh and Merioneth. From this watershed flow the Elwy, Aled, Clywedog, Merddwr and Alwen, tributaries of the Clwyd, Conwy and Dee (Dyfrdwy). Some of the valleys contrast agreeably with the bleak hills, e.g. those of the Clwyd and Elwy. The portion lying between Ruabon (Rhiwabon) hills and the Dee is agricultural and rich in minerals; the Berwyn to Offa’s Dyke (Wâl Offa) is wild and barren, except the Tanat valley, Llansilin and Ceiriog. One feeder of the Tanat forms the Pistyll Rhaiadr (waterspout fall), another rises in Llyncaws (cheese pool) under Moel Sych (dry bare-hill), the highest point in the county. Aled and Alwen are both lakes and streams.

Geology.—The geology of the county is full of interest, as it develops all the principal strata that intervenes between the Ordovician and the Triassic series. In the Ordovician district, which extends from the southern boundary to the Ceiriog, the Llandeilo formation of the eastern slopes of the Berwyn and the Bala beds of shelly sandstone are traversed east and west by bands of intrusive felspathic porphyry and ashes. The same formation occurs just within the county border at Cerrig-y-Druidion, Langum, Bettys-y-coed and in the Fairy Glen. Northwards from the Ceiriog to the limestone fringe at Llandrillo the Wenlock shale of the Silurian covers the entire mass of the Hiraethog and Clwydian hills, but verging on its western slopes into the Denbighshire grit, which may be traced southward in a continuous line from the mouth of the Conway as far as Llanddewi Ystrad Enni in Radnorshire, near Pentre-Voelas and Conway they are abundantly fossiliferous. On its eastern slope a narrow broken band of the Old Red, or what may be a conglomeratic basement bed of the Carboniferous Limestone series, crops up along the Vale of Clwyd and in Eglwyseg. Resting upon this the Carboniferous Limestone extends from Llanymynach, its extreme southern point, to the Cyrnybrain fault, and there forks into two divisions that terminate respectively in the Great Orme’s Head and in Talargoch, and are separated from each other by the denuded shales of the Moel Famma range. In the Vale of Clwyd the limestone underlies the New Red Sandstone, and in the eastern division it is itself overlaid by the Millstone Grit of Ruabon and Minera, and by a long reach of the Coal Measures which near Wrexham are 4 m. in breadth. Eastward of these a broad strip of the red marly beds succeeds, formerly considered to be Permian but now regarded as belonging to the Coal Measures, and yet again between this and the Dee the ground is occupied—as in the Vale of Clwyd—by the New Red rocks. As in the other northern counties of Wales, the whole of the lower ground is covered more or less thickly with glacial drift. On the western side of the Vale of Clwyd, at Cefn and Plâs Heaton, the caves, which are a common feature in such limestone districts, have yielded the remains of the rhinoceros, mammoth, hippopotamus and other extinct mammals.

Coal is mined from the Coal Measures, and from the limestone below, lead with silver and zinc ores have been obtained. Valuable fireclays and terra-cotta marls are also taken from the Coal Measures about Wrexham.

The uplands being uncongenial for corn, ponies, sheep and black cattle are reared, for fattening in the Midlands of England and sale in London. Oats and turnips, rather than wheat, barley and potatoes, occupy the tilled land. The county is fairly wooded. There are several important farmers’ clubs (the Denbighshire and Flintshire, the vale of Conway, the Cerrig y druidion, &c.). The London & North-Western railway (Holyhead line), with the Conway and Clwyd valleys branches, together with the lines connecting Denbigh with Ruabon (Rhiwabon), via Ruthin and Corwen, Wrexham with Connah’s Quay (Great Central) and Rhosllanerchrhugog with Glyn Ceiriog (for the Great Western and Great Central railways) have opened up the county. Down the valley of Llangollen also runs the Holyhead road from London, well built and passing through fine scenery. At Nantglyn paving flags are raised, at Rhiwfelen (near Llangollen) slabs and slates, and good slates are also obtained at Glyn Ceiriog. There is plenty of limestone, with china stone at Brymbo. Cefn Rhiwabon yields sandstone (for hones) and millstone grit. Chirk, Ruabon and Brymbo have coal mines. The great Minera is the principal lead mine. There is much brick and pottery clay. The Ceiriog valley has a dynamite factory. Llangollen and Llansantffraid (St Bridgit’s) have woollen manufactures.

The area of the ancient county is 423,499 acres, with a population in 1901 of 129,942. The area of the administrative county is 426,084 acres. The chief towns are: Wrexham, a mining centre and N. Wales military centre, with a fine church; Denbigh; Ruthin, where assizes are held (here are a grammar school, a warden and a 13th-century castle rebuilt); Llangollen and Llanrwst; and Holt, with an old ruined castle. The Denbigh district of parliamentary boroughs is formed of: Denbigh (pop. 6483), Holt (1059), Ruthin (2643), and Wrexham (14,966). The county has two parliamentary divisions. The urban districts are: Abergele and Pensarn (2083), Colwyn Bay and Colwyn (8689), Llangollen (3303), and Llanrwst (2645). Denbighshire is in the N. Wales circuit, assizes being held at Ruthin. Denbigh and Wrexham boroughs have separate commissions of the peace, but no separate quarter-session courts. The ancient county, which is in the diocese of St Asaph, contains seventy-five ecclesiastical parishes and districts and part of a parish.

The county was formed, by an act of Henry VIII., out of the lordships of Denbigh, Ruthin (Rhuthyn), Rhos and Rhyfoniog, which are roughly the Perfeddwlad (midland) between Conway and Clwyd, and the lordships of Bromfield, Yale (Iâl, open land) and Chirkland, the old possessions of Gruffydd ap Madoc, arglwydd (lord) of Dinas Brân. Cefn (Elwy Valley) limestone caves hold the prehistoric hippopotamus, elephant, rhinoceros, lion, hyena, bear, reindeer, &c.; Plâs Heaton cave, the glutton; Pont Newydd, felstone tools and a polished stone axe (like that of Rhosdigre); Carnedd Tyddyn Bleiddian, “platycnemic (skeleton) men of Denbighshire” (like those of Perthi Chwareu). Clawdd Coch has traces of the Romans; so also Penygaer and Penbarras. Roman roads ran from Deva (Chester) to Segontium (Carnarvon) and from Deva to Mons Heriri (Tomen y mur). To their period belong the inscribed Gwytherin and Pentrefoelas (near Bettws-y-coed) stones. The Valle Crucis “Eliseg’s pillar” tells of Brochmael and the Cairlegion (Chester) struggle against Æthelfrith’s invading Northumbrians, 613, while Offa’s dike goes back to the Mercian advance. Near and parallel to Offa’s is the shorter and mysterious Watt’s dike. Chirk is the only Denbighshire castle comparatively untouched by time and still occupied. Ruthin has cloisters; Wrexham, the Brynffynnon “nunnery”; and at both are collegiate churches. Llanrwst, Gresford and Derwen boast rood lofts and screens; Whitchurch and Llanrwst, portrait brasses and monuments; Derwen, a churchyard cross; Gresford and Llanrhaiadr (Dyffryn Clwyd), stained glass. Near Abergele, known for its sea baths, is the ogof (or cave), traditionally the refuge of Richard II. and the scene of his capture by Bolingbroke in 1399.

 DENDERA, a village in Upper Egypt, situated in the angle of the great westward bend of the Nile opposite Kena. Here was the ancient city of Tentyra, capital of the Tentyrite nome, the sixth of Upper Egypt, and the principal seat of the worship of Hathor [Aphrodite] the cow-goddess of love and joy. The old Egyptian name of Tentyra was written ’In·t (Ant), but the pronunciation of it is unknown: in later days it was ’In·t-t-ntr·t, “ant of the goddess,” pronounced Ni-tentôri, whence . The temple of Hathor was built in the 1st century, being begun under the later Ptolemies (Ptol. XIII.) and finished by Augustus, but much of the decoration is later. A great

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 angle of aberration of the second degree, which it is the object to detect.

We have shown that the wave-lengths of the two rays, when resolved in the direction in which alone they are used, are equal. One other point as to wave-lengths must be considered. We use wave-lengths to measure a length of less than 0.0002 mm., to determine the angle h B h". Is the scale of variable value? The light from a source moving with the apparatus has its wave-length modified by the motion. Dr. Hicks gives the formula for this effect in equation (4), page 17. If with this and the equation (2) we compute the wave-length resolved in the axis, at the azimuths where the effect is a maximum, and for the velocity ratio 100, the two minima are 0.9899995 L and 0.9899505 L, while the two maxima are both 0.9999500 L, where L is the wave-length in the case of no motion. For the ratio 10,000 these quantities differ from unity by about a hundredth part as much, and the inequality is negligible, even if we had to multiply this unit by a large number. But we have to do with only a fraction of the unit.

We next inquire as to the amount and the laws of aberration produced by reflexions from the mirrors of the apparatus. These can be developed in a series of powers of the velocity ratio, and of sines and cosines of the azimuth and of its multiples. But numerical estimates seem desirable, and the formulæ are such that these can more easily be obtained from trigonometrical computation. For the actual velocity ratio the computation is not easy, because trigonometrical tables of fifteen decimal places are not available. Imagine, then, three different apparatus, each of the dimensions proper to the special value of the velocity ratio for which it is specially designed. One apparatus, for the ratio 10, may have the length B II, fig. 3, equal to 10² L; another, for the ratio 100, may have the length 100² L; and the third, for the ratio 1000, may have the length 1000² L. What we can readily learn for the ratio 10, with seven place logarithms, will apply to the ratio 100, except for the circumstance that angles are not so small that sines and arcs are identical in value. What we compute for the ratio 100 with ten place logarithms tells us everything we desire to know for the ratio 1000 and for the actual ratio.

We have computed the aberrations of the two rays I and II, for certain azimuths with the velocity-ratios 10 and 1000, and for 18 azimuths of the apparatus with the velocity-ratio 100. From these aberrations we subtract that part of the aberration which is annulled by the motion of the telescope,

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 and then decompose these residual aberrations into terms depending on the squares and on the cubes of the ratio of velocities. To a thousandth part of the residual aberrations, their difference is represented by the equation

$\delta'-\delta\ =\ \frac{v^2}{\mathrm{V}^2}\cos 2\alpha+\frac{v^3}{\mathrm{V}^3}(\sqrt{0.5}\sin{2\alpha}+\tfrac{1}{6}\sin{4\alpha}+\cos\alpha)$.

For the velocity-ratio 10,000 the agreement would be much closer. Fig. 9 (Pl. IX.) A shows the laws of the variation in the residual aberrations of the two rays I and II, coming from the mirrors I and II. The unit for A and C is $$\sin^{-1}{\frac{v^2}{\mathrm{V}^2}}$$, and for B and D, $$\sin^{-1}{\frac{v^3}{\mathrm{V}^3}}$$. The curves A are nearly represented by $$\frac{v^2}{\mathrm{V}^2}(\sin{2\alpha}\pm\tfrac{1}{2}\cos{2\alpha})$$. Subtracting the ordinates given by this expression from the actual ordinates, we get the residuals shown (after multiplication by the reciprocal of the velocity-ratio) at B. C shows the difference of the curves I. and II. of A, and thus gives directly the angle of divergence of the emergent wave-fronts which is the object of our study. D gives the difference between this curve and the sine curve $$\delta'-\delta\ =\ \frac{v^2}{\mathrm{V}^2}\cos{2\alpha}$$. The latter curve shows that the difference of the aberrations of I and II is at an undisturbed maximum at 90° and at 270°; at 0°, it is less than the undisturbed maximum by the quantity $$\frac{v^3}{\mathrm{V}^3}\cos\alpha$$, or $$\frac{v^3}{\mathrm{V}^3}$$; at 180° it is greater by the same quantity.

It may be thought that the adjustment of the angles between the mirrors which has been assumed will limit too narrowly the use of the apparatus. We may simply say that experience with mirrors as nearly plane as those used by us has shown us that the method of observation supposed would suffice for angles of aberration at least twenty-five times that expected if the velocity-ratio is 10,000.

Since the experiment gives a null result, it is not worth the space to prove that what is true of this adjustment is true with sufficient approximation for an adjustment which differs from the assumed adjustment only by the rotation of mirror I by an angle of ten seconds around a perpendicular axis passing through its surface. Instead, we may compare the results here obtained with those of Dr. Hicks.

In the first place, he declares that the position of the fringes is displaced by aberration. This point is eliminated from our discussion by the fact that the fringes are infinitely

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 wide. We simply remark that, if we understand rightly his statement, this aberration is annulled by the motion of the telescope. Also, his discussion contains a term expressing the fact that the waves of one system gain upon those of the other while passing towards the observer. We have shown that, in the conditions assumed (and realized), this effect is nil for central rays in the eight principal azimuths, and is small in all others. At its maximum, for central rays, it is $$0.3\frac{v^4}{\mathrm{V}^4}$$. With our present large apparatus, whose length is $$54\times10^6\lambda$$, the gain of one wave-front over the other in the whole length is much less than $$10^{-6}\lambda$$.

In the theory of 1887, powers of the velocity-ratio higher than the second were expressly regarded as negligible. Dr. Hicks virtually supplies one such term. He writes, displacement of fringes $$=\ \frac{\frac{1}{2}\xi^2\mathrm{L}\cos{2\alpha}}{\sin(\mathrm{B-A})-\frac{1}{2}\xi^{2}\cos{2\alpha}}$$, where is the velocity-ratio, L is the length of path in the apparatus, from D to I, fig. 5, and B—A is the difference between the angles DB I and DB II. Without the small term in the denominator, this gives precisely the same value as the expression in the paper of 1887, as a simple numerical computation shows. The effect of the small term is the following:—the value of the denominator is decreased or increased by $$\tfrac{1}{2}\frac{v^2}{\mathrm{V}^2}$$ at alternate quadrants, and the value of the fraction is therefore increased or decreased at alternate quadrants. But, according to the present solution, the expression should have a mean value at 90° and 270°, and have, further, a maximum at 180° and a minimum at 0°. At three quadrants we agree, but at the fourth we differ by twice the term in question. The difference is easily explained and is negligible, especially in view of the null result of experiment.

It should be noted that, when there is aberration of the wave-front, there are four closely related magnitudes. One is the distance travelled by the wave-front in the period; a second is the perpendicular distance between consecutive wave-fronts, called in Dr. Hicks's paper; a third is the distance between wave-fronts, resolved parallel to some line dictated by the geometric conditions of the case; and the fourth is the distance between wave-fronts in the line of sight, which is the true wave-length. The perpendicular distance between wave-fronts is used rightly, as we conceive, in establishing the conditions of the network of intersecting wave-fronts in Dr. Hicks's admirable paper. But in one

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 paragraph, which is entirely distinct from the rest of the discussion, he uses an expression which is not sufficiently approximate; e. g., if the expression be taken to mean the wave-length as stated above, and accordingly used to compute the number of waves in a given length in the line of vision, it differs from the truth by $$\frac{v^2}{\mathrm{V}^2}$$, precisely doubling the result found otherwise.

We assert, then, that the theory of 1887 is correct to terms of the order retained, which were sufficient; that Dr. Hicks's theory agrees with it precisely as to numerical amount and sign of the effect, and that a third examination of the theory gives results differing from those of the two others only by negligible terms of the third order.

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 rectangular enclosure of crude bricks, measuring about 900 ✕ 850 ft., contains the sacred buildings: it was entered by two stone gateways, in the north and the east sides, built by Domitian. Another smaller enclosure lies to the east with a gateway also of the Roman period.

The plan of the temple may be supposed to have included a colonnaded court in front of the present façade, and pylon towers at the entrance; but these were never built, probably for lack of funds. The building, which is of sandstone, measures about 300 ft. from front to back, and consists of two oblong rectangles; the foremost, placed transversely to the other, is the great hypostyle hall or pronaos, the broadest and loftiest part of the temple, measuring 135 ft. in width, and comprising about one-third of the whole structure; the façade has six columns with heads of Hathor, and the ceiling is supported by eighteen great columns. The second rectangle contains a small hypostyle hall with six columns, and the sanctuary, with their subsidiary chambers. The sanctuary is surrounded by a corridor into which the chambers open: on the west side is an apartment forming a court and kiosk for the celebration of the feast of the New Year, the principal festival of Dendera. On the roof of the temple, reached by two staircases, are a pavilion and several chambers dedicated to the worship of Osiris. Inside and out, the whole of the temple is covered with scenes and inscriptions in crowded characters, of ceremonial and religious import; the decoration is even carried into a remarkable series of hidden passages and chambers or crypts made in the solid walls for the reception of its most valuable treasures. The architectural style is dignified and pleasing in design and proportions. The interior of the building has been completely cleared: from the outside, however, its imposing effect is quite lost, owing to the mounds of rubbish amongst which it is sunk. North-east of the entrance is a “Birth House” for the cult of the child Harsemteu, and behind the temple a small temple of Isis, dating from the reign of Augustus. The original foundation of the temple must date back to a remote time: the work of some of the early builders is in fact referred to in the inscriptions on the present structure. Petrie’s excavation of the cemetery behind the temple enclosures revealed burials dating from the fourth dynasty onwards, the most important being mastables of the period from the sixth to the eleventh dynasties; many of these exhibited a peculiar degradation of the contemporary style of sculpture.

The zodiacs of the temple of Dendera gave rise to a considerable literature before their late origin was established by Champollion in 1822: one of them, from a chamber on the roof, was removed in 1820 to the Bibliothèque Nationale in Paris. Figures of the celebrated Cleopatra VI. occur amongst the sculptures on the exterior of the temple, but they are purely conventional, without a trace of portraiture. Horus of Edfu, the enemy of the crocodiles and hippopotami of Set, appears sometimes as the consort of Hathor of Dendera. The skill displayed by the Tentyrites in capturing the crocodile is referred to by Strabo and other Greek writers. Juvenal, in his seventeenth satire, takes as his text a religious riot between the Tentyrites and the neighbouring Ombites, in the course of which an unlucky Ombite was torn to pieces and devoured by the opposite party. The Ombos in question is not the distant Ombos south of Edfu, where the crocodile was worshipped; Petrie has shown that opposite Coptos, only about 15 m. from Tentyra, there was another Ombos, venerating the hippopotamus sacred to Set.

 DENDROCOMETES (so named by F. Stein), a genus of suctorian Infusoria, characterized by the repeatedly branched attached body; each of the lobes of the body gives off a few retractile tentacles. It is parasitic on the gills of the so-called freshwater shrimp Gammarus pulex.

 DENE-HOLES, the name given to certain caves or excavations in England, which have been popularly supposed to be due to the Danes or some other of the early northern invaders of the country. The common spelling “Dane hole” is adduced as evidence of this, and individual names, such as Vortigern’s Caves at Margate, and Canute’s Gold Mine near Bexley, naturally follow the same theory. The word, however, is probably derived from the Anglo-Saxon den, a hole or valley. There are many underground excavations in the south of the country, also found to some extent in the midlands and the north, but true dene-holes are found chiefly in those parts of Kent and Essex along the lower banks of the Thames. With one exception there are no recorded specimens farther east than those of the Grays Thurrock district, situated in Hangman’s Wood, on the north, and one near Rochester on the south side of the river.

The general outline of the formation of these caves is invariably the same. The entrance is a vertical shaft some 3 ft. in diameter falling, on an average, to a depth of 60 ft. The depth is regulated, obviously, by the depth of the chalk from the surface, but, although chalk could have been obtained close at hand within a few feet, or even inches, from the surface, a depth of from 45 to 80 ft., or more, is a characteristic feature. It is believed that dene-holes were also excavated in sand, but as these would be of a perishable nature there are no available data of any value. The shaft, when the chalk is reached, widens out into a domed chamber with a roof of chalk some 3 ft. thick. The walls frequently contract somewhat as they near the floor. As a rule there is only one chamber, from 16 to 18 ft. in height, beneath each shaft. From this excessive height it has been inferred that the caves were not primarily intended for habitations or even hiding-places. In some cases the chamber is extended, the roof being supported by pillars of chalk left standing. A rare specimen of a twin-chamber was discovered at Gravesend. In this case the one entrance served for both caves, although a separate aperture connected them on the floor level. Where galleries are found connecting the chambers, forming a bewildering labyrinth, a careful scrutiny of the walls usually reveals evidence that they are the work of a people of a much later period than that of the chambers, or, as they become in these cases, the halls of the galleries.

Isolated specimens have been discovered in various parts of Kent and Essex, but the most important groups have been found at Grays Thurrock, in the districts of Woolwich, Abbey Wood and Bexley, and at Gravesend. Those at Bexley and Grays Thurrock are the most valuable still existing.

It is generally found that the tool work on the roof or ceiling is rougher than that on the walls, where an upright position could be maintained. Casts taken of some of the pick-holes near the roof show that, in all probability, they were made by bone or horn picks. And numerous bone picks have been discovered in Essex and Kent. These pick-holes are amongst the most valuable data for the study of dene-holes, and have assisted in fixing the date of their formation to pre-Roman times. Very few relics of antiquarian value have been discovered in any of the known dene-holes which have assisted in fixing the date or determining the uses of these prehistoric excavations. Pliny mentions pits sunk to a depth of a hundred feet, “where they branched out like the veins of mines.” This has been used in support of the theory that dene-holes were wells sunk for the extraction of chalk; but no known dene-hole branches out in this way. Chrétien de Troyes has a passage on underground caves in Britain which may have reference to dene-holes, and tradition of the 14th century treated the dene-holes of Grays as the fabled gold mines of Cunobeline (or Cymbeline) of the 1st century.

Vortigern’s Caves at Margate are possibly dene-holes which have been adapted by later peoples to other purposes; and excellent examples of various pick-holes may be seen on different parts of the walls.

Local tradition in some cases traces the use of these caves to the smugglers, and, when it is remembered that illicit traffic was common not only on the coast but in the Thames as far up the river as Barking Creek, the theory is at least tenable that these ready-made hiding-places, difficult of approach and dangerous to descend, were so utilized.

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text: 1911 Encyclopædia Britannica/Dendera

title: 1911 Encyclopædia Britannica/Dendera


text: Index:MillerContraction.djvu

title: Index:MillerContraction.djvu


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A result was obtained in 1887, in an experiment to detect, if possible, a difference of velocity of light in different directions owing to the motion of the apparatus towards or away from waves of light in the stationary ether. FitzGerald and Lorentz then suggested that the dimensions of the apparatus might be modified by its motion through the ether. If this modification depend on the resilience or other physical properties of the materials, it may perhaps be detected by experiment.

We have constructed two apparatus with which to examine this question. In the first, we replaced the sandstone used in 1887 by a structure of white pine. A strong cross was built up of planks, fourteen inches wide and two inches thick, and fourteen feet long. One was laid east and west, then one across it north and south, and so on. They were slightly notched where they crossed. On their intersection was secured a cast-iron bedplate for certain optical parts of the apparatus. At the ends, after filling the spaces between the planks, were bolted iron supports for our mirrors. The whole was placed on a round float, which in turn rested in a basin of mercury.

Our sixteen mirrors were each four inches in diameter. The mirrors rested each on the points of three adjusting screws, against which they were held by springs. On the bedplate at the intersection of the arms of the cross were placed a plane half-silvered mirror and a compensating plate; these had been, as is usual, cut from the same plane-parallel disk.

Figure 1 is a diagram, not to scale, of the optical arrangements. Light from a source S reaches the mirror D. Part is transmitted, reaching the mirror II. It is successively reflected to 2, 3, 4, 5, 6, 7, and 8. From 8 it returns by the same path to D, where part is reflected

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322 to the observer at T. Another part of the incident ray is reflected along the other arm of the cross, is similarly passed to and fro, returned, and at last transmitted to the observer. In the apparatus actually used, mirror 5 lay above 3, rather than to one side of it; Figure 2 shows this arrangement. The whole path of the light along these mirrors was enclosed and covered, to lessen the effect of air currents and other local disturbances. An acetylene flame was carried as a source of light. A telescope magnifying thirty-five diameters gave distinct vision of mirror 8, at whose surface the interference fringes are apparently localized.

[[File:MorleyMiller1905.png|thumb|500px|center|

The mirrors, being silvered and polished, were put in place, and the lengths of the two paths were measured with a split rod and then made nearly equal. Establishing interferences in sodium light, we found the central part of a series of some seven hundred interferences which are brighter than the adjoining three hundred. With no long search, we could see interferences in white light, although we had provided no screw for moving a mirror with its surface always parallel to a given surface. This we had avoided, in order to have everything about the two arms as symmetrical as possible.

We now computed the direction and velocity of the motion of the centre of the apparatus by compounding the annual motion in the orbit of the earth with the motion of the solar system towards a certain point in the heavens. During part of August, the whole of September, and nearly all of October, this motion never coincides with the plane of our apparatus. For other dates, there are two hours in each day when the motion is in the desired plane, except for two days when the two hours

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Rh coalesce into one. At the beginning of June the two hours are about 11h. 20m. A.M., mean solar time, and 9h. 50m. P.M. At the time of our last set of observations, July 5th to July 9th, the hours were 11h. 40m. A.M. and 8h. 20m. P.M., local mean time.

After many trials, with filar micrometer, and with scale on mirror 8, we found it advisable to accumulate a great number of observations made as rapidly as might be. What we had to do, in presence of all the local disturbances of density of the air which sometimes made observation impossible and always made it difficult, was as if we were trying to measure the solar atmospheric tide. If we could vary the period of this tide at will by controlling the revolutions of the earth, we should doubtless get a result sooner by accelerating the latter and making a great number of observations in a given time, rather than by retarding the period in order to measure with very great precision the hourly height of a barometer. We therefore proceeded as follows. One observer walked around with the moving apparatus, his eye at the telescope, while he maintained the rotation by an occasional gentle pull on a cord so fixed as not to bring any strain to bear on the cross arms of the apparatus. The room was darkened. The other observer also went around with the apparatus; as an index showed the azimuth of the apparatus to be that indicated by one of sixteen equidistant marks, he called out the number or some other signal. The first observer replied with the reading for the given azimuth, which the second observer recorded. The next azimuth was called at the proper instant, the reading given, and so on. Half the time, perhaps, the observations were interrupted before they became numerous enough to be useful, being stopped by excessive displacement of fringes owing to temperature changes and the like. But patience is a possession without which no one is likely to begin observations of this kind. Runs of twenty and thirty turns, involving 320 or 480 readings, were not uncommon. A run of thirty turns meant that the observer, who could sometimes make a turn of sixteen readings in sixty-five or seventy-five seconds, walked half a mile while making the severe effort involved in keeping his eye at the moving eyepiece without the least interruption for half an hour. The work is, of course, somewhat exhausting.

Observation with this apparatus could not begin till the month of August, and we had to stop without having accomplished as much as was desirable. During the busy season of the school year, observation is impossible. We had therefore expected to resume our work in June. But we then found that our pine apparatus had so much suffered from the dryness of the building that we could not maintain the adjustment of our

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324 fringes. We could not, in the time, build another apparatus of timber which had not been dried all winter, nor was it thought well to construct another apparatus closely resembling the first. While planning a new apparatus, we made a couple of experiments to show, what was well enough known, that difference of magnetic attraction on the iron parts of our apparatus could not disturb our observations. We suspended two massive pieces of iron at the ends of one arm, so that one should be in the lines of magnetic force of the earth's field, and the other transverse to them, these relations being reversed on reversing the position of the apparatus. But observations with this load of iron gave the same result as before. Next we placed an analytical balance on one arm, with which to weigh a bur of iron at the extremity of that arm. It was so placed that at one azimuth the bar was nearly in the lines of force, and at another was transverse to them. If there were a difference of half a milligram in twelve hundred grams, it would have been detected, but no such difference existed. We found by trial how much a weight of a hundred grams displaced our fringes, and so learned, as was known before, that the influence of the earth's magnetism could not be a disturbing factor.

The Rumford Committee of the American Academy of Arts and Sciences having made a grant in aid of this experiment, we carried out our original plan of making a steel structure so rigid as to permit easy and satisfactory observation. In this new apparatus all the optical parts are carried by a steel frame built of plate and angle-iron, somewhat like a bridge girder. A cubical steel box, fourteen inches on each edge, constitutes the centre of the structure, which is in the form of a cross. To each of the four sides of this cube are firmly attached arms, each about six feet and a half in length. Each arm is made of steel plates, three eighths of an inch thick, eighteen inches wide at one end, and six inches wide at the other, standing on edge, and kept fourteen inches apart by suitable plates, angle-irons, and other braces; thus are formed hollow beams of great rigidity, especially in a vertical direction. This framework is shown in Plate 1, from which it is seen that the structure is in effect two rigid beams, each fourteen feet long, crossing at right angles, and symmetrical as regards strength and rigidity.

On two ends of the cross, S and T, Figure 1, are two upright cast-iron frames, fastened by bolts, each of which carries four mirrors, marked 2, 4, 6, and 8. Against the corners of each of these frames rest four pine rods, three quarters of an inch in diameter and fourteen feet long. Each rod is supported throughout its length by a brass tube an inch in

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Rh diameter; each pair of tubes is joined together in a vertical truss, as shown in Plate 1. Against the farther end of these rods there rest the frames which hold the two sets of mirrors, I and II, Figure 1. Each of the latter frames is freely suspended by two thin steel ribbons and is held firmly against the pine rods, and through these against one of the two fixed mirror holders; the pressure is applied by means of adjustable spiral springs. Thus the distance between tho opposite systems of mirrors depends upon the pine rods only. This construction permits the convenient substitution of distance rods of other material, so that experiments might be easily made to test the theory that the dimensions of different materials are differently affected by motion of translation through the ether. The diagonal mirrors are carried by adjustable supports bolted to the steel frame near its centre.

The observing telescope of an inch and a half aperture with a magnifying power of thirty-five is attached to a support bolted to the steel frame. The acetylene lamp and the four-inch condensing lens stand on a wooden shelf as far as convenient from the mirrors, which are protected by asbestos screens with air spaces. The whole path of light through the apparatus is enclosed by a wooden cover made of pine seven eighths of an inch thick, having doors and glass windows where these are required. The observer's eyes are protected from extraneous light by a dark cloth.

The entire apparatus, weighing about nineteen hundred pounds, rests upon a circular wooden platform about five feet in diameter. An annular projection on the under side of the platform is immersed in mercury of such depth as to float the platform and the apparatus. The mercury is contained in an annular cast-iron trough of such dimensions as to leave a clearance of about half an inch between the iron and the wooden float. A small pin at the centre of the iron trough enters a socket in the wooden float, so as to keep the float from touching the sides of the trough.

Plate 1 shows the steel framework and float, together with the trusses which are to support the distance pieces. The mirror frames and the telescope are in position, but lamp and lens are not in position. Plate 2 shows the apparatus as it appeared at the time of the observations.

With this apparatus, fringes adjusted on a certain Monday remained in adjustment throughout the whole of the week during which our observations continued. Observations were made in precisely the same manner as with the previous apparatus.

We obtained 260 complete observations, consisting each of readings at

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326 sixteen azimuths around a circumference. At the date of the observations, the annual motion of the earth, together with the motion of the solar system, may be taken as 33.5 kilometers a second. It is assumed that the solar system is moving towards a point whose right ascension is 277.5°, and whose north declination is 35°, with a velocity of eleven miles a second. The velocity of light being 300,000 kilometers a second, the ratio of the squares of the velocities is $$0.72\times10^{8}$$. The length of path of a ray in our apparatus was 3224 centimeters, in which distance there are contained $$5.5\times10^{7}$$ wave lengths of sodium light. The expected effect being doubled by rotation through 90, the displacement of fringes expected on the simple kinematic theory will be $$11\times10^{7}\div0.72\times10^{8}$$. This is 1.5 wave length.

As was indicated, there were two times in the day when observation was advisable. The direction of the motion with reference to a fixed line on the floor of the room being computed for the two hours, we were able to superimpose those observations which coincided with the line of drift for the two hours of observation. Doing this, and subtracting a constant so as to make the algebraic sum of the observations equal to zero, we get a certain result. Then adding the first term to the ninth, and so on, since the effect repeats itself in a circumference, we get our final result, as follows: —

Result of observations at various azimuths.

Azimuth mark 1 denotes that the telescope of the apparatus was directed N. 29° E.; 3, N. 16° W.; 5, N. 61° W., &c.

These numbers may be confidently pronounced to be due to errors of observation. We computed from them several curves of the theoretical form, having their origin at sixteen equidistant points in the half circumference; this was done by the method of least squares. The most probable of these curves had an amplitude of 0.0073 wave lengths, and its zero was half-way between the azimuths marked 4 and 5. The average of the given observations is 0.0076 wave lengths; after subtracting the ordinates of the computed curve, the mean residual was 0.0066 wave lengths. The sum of the squares of the residuals before was $$565\times10^{-4}$$; afterwards, it was $$329\times10^{-4}$$.

We may therefore declare that the experiment shows that if there is any effect of the nature expected, it is not more than the hundredth part

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Rh

 * }

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Rh of the computed value. If pine is affected at all, as has been suggested, it is affected to the same amount as is sandstone. If the ether near the apparatus did not move with it, the difference in velocity was less than 3.5 kilometers a second, unless the effect on the materials annulled the effect sought

Some have thought that the former experiment only proved that the ether in a certain basement room was carried along with it. We desire to place the apparatus on a hill, covered only with a transparent covering, to see if any effect could be there detected. As the Rumford Committee have allowed us thus to utilize an unexpended balance, we hope to make the experiment in this form, should it be possible to make observations in trying conditions.

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 There are three purposes for which dene-holes may have been originally excavated: (a) as hiding-places or dwellings, (b) draw-wells for the extraction of chalk for agricultural uses, and (c) storehouses for grain. For several reasons it is unlikely that they were used as habitations, although they may have been used occasionally as hiding-places. Other evidence has shown that it is equally improbable that they were used for the extraction of chalk. The chief reasons against this theory are that chalk could have been obtained outcropping close by, and that every trace of loose chalk has been removed from the vicinity of the holes, while known examples of chalk draw-wells do not descend to so great a depth. The discovery of a shallow dene-hole, about 14 ft. below the surface, at Stone negatives this theory still further. The last of the three possible uses for which these prehistoric excavations were designed is usually accepted as the most probable. Silos, or underground storehouses, are well known in the south of Europe and Morocco. It is supposed that the grain was stored in the ear and carefully protected from damp by straw. A curious smoothness of the roof of one of the chambers of the Gravesend twin-chamber dene-hole has been put forward as additional evidence in support of this theory. One other theory has been advanced, viz. that the excavations were made in order to get flints for implements, but this is quite impossible, as a careful examination of a few examples will show.

 DENGUE (pronounced deng-ga), an infectious fever occurring in warm climates. The symptoms are a sudden attack of fever, accompanied by rheumatic pains in the joints and muscles with severe headache and erythema. After a few days a crisis is reached and an interval of two or three days is followed by a slighter return of fever and pain and an eruption resembling measles, the most marked characteristic of the disease. The disease is rarely fatal, death occurring only in cases of extreme weakness caused by old age, infancy or other illness. Little is known of the aetiology of “dengue.” The virus is probably similar to that of other exanthematous fevers and communicated by an intermediary culex. The disease is nearly always epidemic, though at intervals it appears to be pandemic and in certain districts almost endemic. The area over which the disease ranges may be stated generally to be between 32° 47′ N. and 23° 23′ S. Throughout this area “dengue” is constantly epidemic. The earliest epidemic of which anything is known occurred in 1779–1780 in Egypt and the East Indies. The chief epidemics have been those of 1824–1826 in India, and in the West Indies and the southern states of North America, of 1870–1875, extending practically over the whole of the tropical portions of the East and reaching as far as China. In 1888 and 1889 a great outbreak spread along the shores of the Aegean and over nearly the whole of Asia Minor. Perhaps “dengue” is most nearly endemic in equatorial East Africa and in the West Indies. The word has usually been identified with the Spanish dengue, meaning stiff or prim behaviour, and adopted in the West Indies as a name suitable to the curious cramped movements of a sufferer from the disease, similar to the name “dandy-fever” which was given to it by the negroes. According to the New English Dictionary (quoting Dr Christie in The Glasgow Medical Journal, September 1881), both “dengue” and “dandy” are corruptions of the Swahili word dinga or denga, meaning a sudden attack of cramp, the Swahili name for the disease being ka-dinga pepo.

 DENHAM, DIXON (1786–1828), English traveller in West Central Africa, was born in London on the 1st of January 1786. He was educated at Merchant Taylors’ School, and was articled to a solicitor, but joined the army in 1811. First in the 23rd Royal Welsh Fusiliers, and afterwards in the 54th foot, he served in the campaigns in Portugal, Spain, France and Belgium, and received the Waterloo medal. In 1821 he volunteered to join Dr Oudney and (q.v.), who had been sent by the British government via Tripoli to the central Sudan. He joined the expedition at Murzuk in Fezzan. Finding the promised escort not forthcoming, Denham, whose energy was boundless, started for England to complain of the “duplicity” of the pasha of Tripoli. The pasha, alarmed, sent messengers after him with promises to meet his demands. Denham, who had reached Marseilles, consented to return, the escort was forthcoming, and Murzuk was regained in November 1822. Thence the expedition made its way across the Sahara to Bornu, reached in February 1823. Here Denham, against the wish of Oudney and Clapperton, accompanied a slave-raiding expedition into the Mandara highlands south of Bornu. The raiders were defeated, and Denham barely escaped with his life. When Oudney and Clapperton set out, December 1823, for the Hausa states, Denham remained behind. He explored the western, south and south-eastern shores of Lake Chad, and the lower courses of the rivers Waube, Logone and Shari. In August 1824, Clapperton having returned and Oudney being dead, Bornu was left on the return journey to Tripoli and England. In December 1826 Denham, promoted lieutenant-colonel, sailed for Sierra Leone as superintendent of liberated Africans. In 1828 he was appointed governor of Sierra Leone, but after administering the colony for five weeks died of fever at Freetown on the 8th of May 1828.

 DENHAM, SIR JOHN (1615–1669), English poet, only son of Sir John Denham (1559–1639), lord chief baron of the exchequer in Ireland, was born in Dublin in 1615. In 1617 his father became baron of the exchequer in England, and removed to London with his family. In Michaelmas term 1631 the future poet was entered as a gentleman commoner at Trinity College, Oxford. He removed in 1634 to Lincoln’s Inn, where he was, says John Aubrey, a good student, but not suspected of being a wit. The reputation he had gained at Oxford of being the “dreamingest young fellow” gave way to a scandalous reputation for gambling. In 1634 he married Ann Cotton, and seems to have lived with his father at Egham, Surrey. In 1636 he wrote his paraphrase of the second book of the Aeneid (published in 1656 as The Destruction of Troy, with an excellent verse essay on the art of translation). About the same time he wrote a prose tract against gambling, The Anatomy of Play (printed 1651), designed to assure his father of his repentance, but as soon as he came into his fortune he squandered it at play. It was a surprise to everyone when in 1642 he suddenly, as Edmund Waller said, “broke out like the Irish rebellion, three score thousand strong, when no one was aware, nor in the least expected it,” by publishing The Sophy, a tragedy in five acts, the subject of which was drawn from Sir Thomas Herbert’s travels. At the beginning of the Civil War Denham was high sheriff for Surrey, and was appointed governor of Farnham Castle. He showed no military ability, and speedily surrendered the castle to the parliament. He was sent as a prisoner to London, but was soon permitted to join the king at Oxford.

In 1642 appeared Cooper’s Hill, a poem describing the Thames scenery round his home at Egham. The first edition was anonymous: subsequent editions show numerous alterations, and the poem did not assume its final form until 1655. This famous piece, which was Pope’s model for his Windsor Forest, was not new in theme or manner, but the praise which it received was well merited by its ease and grace. Moreover Denham expressed his commonplaces with great dignity and skill. He followed the taste of the time in his frequent use of antithesis and metaphor, but these devices seem to arise out of the matter, and are not of the nature of mere external ornament. At Oxford he wrote many squibs against the roundheads. One of the few serious pieces belonging to this period is the short poem “On the Earl of Strafford’s Trial and Death.”

From this time Denham was much in Charles I.’s confidence. He was entrusted with the charge of forwarding letters to and from the king when he was in the custody of the parliament, a

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text: 1911 Encyclopædia Britannica/Dene-holes

title: 1911 Encyclopædia Britannica/Dene-holes


text:
Author:Horace Barnet Samuel

Works

 * (tr.) The Red and the Black (1916)
 * (tr.) The Genealogy of Morals (1921)

title: Author:Horace Barnet Samuel


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title: Index:CunninghamPrinciple.djvu


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 N his discussion of the electromagnetic mass of a moving electron (Theorie der Elektrizität, ii. p. 205), Abraham raises an objection to the Lorentz conception of an electron as having, at rest, a spherical shape, but in motion the shape of an oblate spheroid the ratio of whose axes is $$\sqrt{1-v^{2}/c^{2}}$$, v being the velocity relative to the tether, and c the velocity

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Rh of light. The present paper reconsiders Abraham's discussion and comes to the conclusion that the objection is not valid. The discussion was suggested by the fact that it has been proved that Maxwell's equations represent equally well the sequence of electromagnetic phenomena relative to a set of axes moving relative to the aether, as relative to a set of axes fixed in the aether. More explicitly this is stated as follows : —

If there are two sets of rectangular axes (A, A') coinciding at a certain instant, of which A' is moving relative to A with velocity v in the direction of the axis of x, which is conceived as at rest, and if x, y, z, t be space and time variables associated with A, and x', y' z', t' similar variables associated with A', then the equations

transform identically into the equations

the accented and unaccented magnitudes being connected by the relations

$\begin{array}{ll} x'=\beta(x-vt], \\y'=y, \\z'=z, & \beta=\left(1-v^{2}/c^{2}\right)^{-\frac{1}{2}} \\t'=\beta\left(t-\frac{vx}{c^{2}}\right),\end{array}$|undefined

$$\begin{array}{l} E'=\beta\left(\frac{E_{x}}{\beta},\ E_{y}-vH_{z},\ E_{z}+vH_{y}\right),\\ \\H'=\beta\left(\frac{H_{x}}{\beta},\ H_{y}-vE_{z},\ H_{z}+vE_{y}\right)\end{array}$$

Further, if $$\rho=\tfrac{1}{4\pi}div\ E$$, and $$\rho'=\tfrac{1}{4\pi}div\ E'$$, the volume integrals taken through corresponding regions $$\tau,\ \tau',\ \int_{\tau}\rho d\tau$$ and $$\int_{\tau'}\rho'd\tau'\!$$ are identically equal, giving an exact correspondence as regards distribution of electric charge.

Thus the above transformation renders the electromagnetic equations of a system independent of a uniform translation of the whole system through the aether.

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540 According to this transformation, as a more geometrical correspondence, the length of a line in the direction of the axis of x moving with the axes A', as measured in the coordinates x', y', z', t', is greater than its length measured in the coordinates x, y, z, t in the ratio $$1:\left(1-v^{2}/c^{2}\right){}^{\frac{1}{2}}$$, so that Lorentz's hypothesis of the reduction in the dimensions of a body when it moves relatively to an observer is reduced by this geometrical correspondence to the assumption that in the variables associated with axes moving with it its shape is unaltered — an assumption suggested by the fact that the electromagnetic equations referred to those variables are independent of the motion through the aether, and by the attempt to form a purely electromagnetic theory of matter. Thus if the single electron at rest has a spherical configuration, and there are no other than electromagnetic forces, we should expect it in motion to have a spherical configuration when measured by the variables x' y' z' which means that as measured by the variables x, y, z it will have the spheroidal shape as suggested by Lorentz.

The electron as conceived by Abraham, on the other hand, is spherical always as regards the variables x, y, z, and a prolate spheroid as regards x' y' z' the ratio of the axes being $$1:\left(1-v^{2}/c^{2}\right){}^{\frac{1}{2}}$$.

In either case the electromagnetic mass is defined as the ratio of the external mechanical force on the electron to the acceleration of the centre of the electron, and Abraham develops two expressions for the longitudinal mass, i. e. for the ratio of the force in the direction of motion to the acceleration in the same direction, viz.: $$\tfrac{dG}{dv}$$ and $$\tfrac{1}{v}\frac{dW}{dv}$$ for the case of the so-called quasistationary motion, G being the electromagnetic momentum and W the electromagnetic energy. For the latter case these two expressions are proved identical, but for the Lorentz electron they are not equal, and Abraham deduces that W cannot be the whole energy of the electron. But the fact is that in this case the mass as above defined is not equal to $$\tfrac{1}{v}\frac{dW}{dv}$$, this expression being obtained on the assumption that the electron is "rigid" (Theorie der Elek. ii. p. 155). If the change in the shape of the electron with the change in velocity is taken into account, it will be found that the mass as obtained from the change in momentum is identical with the mass as obtained from the change in the energy, as it clearly must be, since a quantity defined in a perfectly definite manner cannot from consistent equations be shown to have two different values.

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Rh The assumption on which the theory is built is that the forces from sources exterior to the electron balance those due to the electron itself: this is the assumption that there is no inertia other than electromagnetic, and we deduce the equation

where W is the electromagnetic energy, and A is the work done by the forces due to the electron itself.

If v0 is the velocity of the centre of the electron, v=(v0 + v1) the velocity of the charge at any point of it, F the mechanical force per unit charge, we have

(vF) being the vector product of v and F.

If ξ η ζ are the coordinates relative to the centre of the electron of the element of charge whose velocity is v, and x y z of the same element when the electron is at rest, ξ=βx, η=y, ζ=z; so that the velocity v1 of the charge relative to the centre is

in the direction of the axis of x.

Thus, if Fx is the component of F in that direction,

K being the total mechanical force on the electron;

where the suffix 0 in the last integral refers to the corresponding quantities when the electron is at rest, so that the region of integration is spherical.

For quasistationary motion W is a function of v0 only, and therefore

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542 Hence

$$\frac{dW}{dv_{0}}f=+v_{0}K+\frac{v_{0}f}{c^{2}\beta}\int\rho_{0}x\left(F_{x}\right)_{0}d\tau_{0}$$

and therefore

$$f\left\{ \frac{1}{v_{0}}\frac{dW}{dv_{0}}-\frac{1}{c^{2}\beta}\int\rho_{0}x\left(F_{x}\right)_{0}d\tau_{0}\right\} =K$$,

so that the longitudinal mass is equal to

$$\frac{1}{v_{0}}\frac{dW}{dv_{0}}-\frac{1}{c^{2}\beta}\int\rho_{0}x\left(F_{x}\right)_{0}d\tau_{0}$$

It is the second term in this expression that is neglected by Abraham, and which he has to account for by assuming the energy of the electron to be made up of W together with a term not electromagnetic in origin.

We proceed to evaluate this expression in the two cases (i.) of a sphere with uniform volume density; (ii.) of a sphere with uniform surface charge.

(i.) Volume Distribution.

$\begin{array}{l} G=\frac{4}{5}\frac{e^{2}}{ac^{2}}\frac{v_{0}}{\beta};\\ \\W=\frac{3}{5}\frac{e^{2}}{a\beta}\left(1+\frac{v_{0}^{2}}{3c^{2}}\right).\end{array}$|undefined

Hence

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Rh $$\therefore$$

(ii.) Surface Distribution.

In this case the integral $$\int\rho_{0}x\left(F_{x}\right)_{0}d\tau_{0}$$ becomes $$\int\sigma_{0}x\left(F_{x}\right)_{0}dS_{0}$$, over the surface of the sphere of radius a, and $$F_{x}=2\pi\sigma_{0}\cos\theta$$.

Thus

and

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