A History of the Theories of Aether and Electricity/Chapter 11

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A History of the Theories of Aether and Electricity
by Edmund Taylor Whittaker

Chapter XI: Conduction in Solutions and Gases, from Faraday to J. J. Thomson

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3339222A History of the Theories of Aether and Electricity — Chapter XI: Conduction in Solutions and Gases, from Faraday to J. J. ThomsonEdmund Taylor Whittaker

Chapter XI.

Conduction in Solutions and Gases, from Faraday to J. J. Thomson

The hypothesis which Grothuss and Davy had advanced^[1] to explain the decomposition of electrolytes was open to serious objection in more than one respect. Since the electric force was supposed first to dissociate the molecules of the electrolyte into ions, and afterwards to set them in motion toward the electrodes, it would seem reasonable to expect that doubling the electric force would double both the dissociation of the molecules and the velocity of the ions, and would therefore quadruple the electrolysis—an inference which is not verified by observation. Moreover it might be expected, on Grothuss' theory, that some definite magnitude of electromotive force would be requisite for the dissociation, and that no electrolysis at all would take place when the electromotive force was below this value, which again is contrary to experience.

A way of escape from these difficulties was first indicated, in 1850, by Alex. Williamson,^[2] who suggested that in compound liquids decompositions and recombinations of the molecules are continually taking place throughout the whole mass of the liquid, quite independently of the application of an external electric force. An atom of one element in the compound is thus paired now with one and now with another atom of another element, and in the intervals between these alliances the atom may be regarded as entirely free. In 1857 this idea was made by R. Clausius,^[3] of Zurich, the basis of a theory of electrolysis. According to it, the electromotive force emanating from the electrodes does not effect the dissociation of the electrolyte into ions, since a degree of dissociation sufficient for the purpose already exists in consequence of the perpetual mutability of the molecules of the electrolyte. Clausius assumed that these ions are in opposite electric conditions; the applied electric force therefore causes a general drift of all the ions of one kind towards the anode, and of all the ions of the other kind towards the cathode. These opposite motions of the two kinds of ions constitute the galvanic current in the liquid.

The merits of the Williamson-Clausius hypothesis were not fully recognized for many years; but it became the foundation of that theory of electrolysis which was generally accepted at the end of the century.

Meanwhile another aspect of electrolysis was receiving attention. It had long been known that the passage of a current through an electrolytic solution is attended not only by the appearance of the products of decomposition at the electrodes, but also by changes of relative strength in different parts of the solution itself. Thus in the electrolysis of a solution of copper sulphate, with copper electrodes, in which copper is dissolved off the anode and deposited on the cathode, it is found that the concentration of the solution diminishes near the cathode, and increases near the anode. Some experiments on the subject were made by Faraday^[4] in 1835; and in 1844 it was further investigated by Frederic Daniell and W. A. Miller,^[5] who explained it by asserting that the cation and anion have not (as had previously been supposed) the same facility of moving to their respective electrodes; but that in many cases, the cation appears to move but little, while the transport is effected chiefly by the anion.

This idea was adopted by W. Hittorf, of Münster, who, in the years 1853 to 1859, published^[6] a series of memoirs on the migration of the ions. Let the velocity of the anions in the solution be to the velocity of the cations in the ratio $v : u$ . Then it is easily seen that if $(u + v)$ molecules of the electrolyte are decomposed by the current, and yielded up as ions at the electrodes, $v$ of these molecules will have been taken from the fluid on the side of the cathode, and $u$ of them from the fluid on the side of the anode. By measuring the concentration of the liquid round the electrodes after the passage of a current, Hittorf determined the ratio $v / u$ in a large number of cases of electrolysis.^[7]

The theory of ionic movements was advanced a further stage by F. W. Kohlrausch^[8] (b. 1840, d. 1910), of Würzburg. Kohlrausch showed that although the ohmic specific conductivity $k$ of a solution diminishes indefinitely as the strength of the solution is reduced, yet the ratio $k / m$ , where $m$ denotes the number of gramme-equivalents^[9] of salt per unit volume, tends to a definite limit when the solution is indefinitely dilute. This limiting value may be denoted by $λ$ . He further showed that $λ$ may be expressed as the sum of two parts, one of which depends on the cation, but is independent of the nature of the anion; while the other depends on the anion, but not on the cation—a fact which may be explained by supposing that, in very dilute solutions, the two ions move independently under the influence of the electric force. Let $u$ and $v$ denote the velocities of the cation and anion respectively, when the potential difference per em. in the solution is unity: then the total current carried through a cube of unit volume is $mE (u + v)$ , where $E$ denotes the electric charge carried by one grammeequivalent of ion.^[10] Thus $mE (u + v)$ = total current = $k = mλ$ , or $λ = E (u + v)$ . The determination of $v / u$ by the method of Hittorf, and of $(u + v)$ by the method of Kohlrausch, made it possible to calculate the absolute velocities of drift of the ions from experimental data.

Meanwhile, important advances in voltaic theory were being effected in connexion with a different class of investigations.

Suppose that two mercury electrodes are placed in a solution of acidulated water, and that a difference of potential, insufficient to produce continuous decomposition of the water, is set up between the electrodes by an external agency. Initially a slight electric current—the polarizing current,^[11] as it is called—is observed; but after a short time it ceases; and after its cessation the state of the system is one of electrical equilibrium. It is evident that the polarizing current must in some way have set up in the cell an electromotive force equal and opposite to the external difference of potential; and it is also evident that the seat of this electromotive force must be at the electrodes, which are now said to be polarized.

An abrupt fall of electric potential at an interface between two media, such as the mercury and the solution in the present case, requires that there should be a field of electric force, of considerable intensity, within a thin stratum at the interface: and this must owe its existence to the presence of electric charges. Since there is no electric field outside the thin stratum, there must be as much vitreous as resinous electricity present; but the vitreous charges must preponderate on one side of the stratum, and the resinous charges on the other side; so that the system as a whole resembles the two coatings of a condenser with the intervening dielectric. In the case of the polarized mercury cathode in acidulated water, there must be on the electrode itself a negative charge: the surface of this electrode in the polarized state may be supposed to be either mercury, or mercury covered with a layer of hydrogen. In the solution adjacent to the electrode, there must be an excess of cations and a deficiency of anions, so as to constitute the other layer of the condenser: these cations may be either mercury cations dissolved from the electrode, or the hydrogen cations of the solution.

It was shown in 1870 by Cromwell Fleetwood Varley^[12] that a mercury cathode, thus polarized in acidulated water, shows i tendency to adopt a definite superficial form, as if the surfacetension at the interface between the mercury and the solution were in some way dependent on the electric conditions. The matter was more fully investigated in 1873 by a young French physicist, then preparing for his inaugural thesis, Gabriel Lippmann.^[13] In Lippmann's instrumental disposition, which is called a capillary electrometer, mercury electrodes are immersed in acidulated water: the anode $H 0$ , has a large surface, while the cathode $H$ has a variable surface $S$ small in comparison. When the external electromotive force is applied, it is easily seen that the fall of potential at the large electrode is only slightly affected, while the fall of potential at the small electrode is altered by polarization by an amount practically equal to the external electromotive force. Lippmann found that the constant of capillarity of the interface at the small electrode was a function of the external electromotive force, and therefore of the difference of potential between the mercury and the electrolyte.

Let $V$ denote the external electromotive force: we may, without loss of generality, assume the potential of $H 0$ , to be zero, so that the potential of $H$ is $- V$ . The state of the system may be varied by altering either $V$ or $S$ ; we assume that these alterations may be performed independently, reversibly, and isothermally, and that the state of the large electrode $H 0$ , is not altered thereby. Let $de$ denote the quantity of electricity which passes through the cell from $H 0$ , to $H$ , when the state of the system is thus varied: then if $E$ denote the available energy of the system, and $γ$ the surface-tension at $H$ , we have

$dE=\gamma dS+Vde$ ,

$γ$ being measured by the work required to increase the surface when no electricity flows through the circuit.

In order that equilibrium may be re-established between the electrode and the solution when the fall of potential at the cathode is altered, it will be necessary not only that some hydrogen cations should come out of the solution and be deposited on the electrode, yielding up their charges, but also that there should be changes in the clustering of the charged ions of hydrogen, mercury, and sulphion in the layer of the solution immediately adjacent to the electrode. Each of these circumstances necessitates a flow of electricity in the outer circuit: in the one case to neutralize the charges of the cations deposited, and in the other case to increase the surface-density of electric charge on the electrode, which forms the opposite sheet of the quasi-condenser. Let $Sf (V)$ denote the total quantity of electricity which has thus flowed in the circuit when the external electromotive force has attained the value $V$ . Then evidently

$de=d\{Sf(V)\}$ ;

so

$dE=\{\gamma +Vf(V)\}dS+VSf^{\prime }(V)dV$ .

Since this expression must be an exact differential, we have

${\frac {d\gamma }{dV}}+f(V)=0$ ;

so that $- dy / dV$ is equal to that flux of electricity per unit of new surface formed, which will maintain the surface in a constant condition ( $V$ being constant) when it is extended. Integrating the previous equation, we have

$E=S\left\{\gamma -V{\frac {d\gamma }{dV}}\right\}$ .

Lippmann found that when the external electromotive force was applied, the surface-tension increased at first, until, when the external electromotive force amounted to about one volt, the surface-tension attained a maximum value, after which it diminished. He found that $d^{2}\gamma /dV^{2}$ was sensibly independent of $V$ , so that the curve which represents the relation between $\gamma$ and $V$ is a parabola.^[14]

The theory so far is more or less independent of assumptions as to what actually takes place at the electrode: on this latter question many conflicting views have been put forward. In 1878 Josiah Willard Gibbs,^[15] of Yale (b. 1839, d. 1903), discussed the problem on the supposition that the polarizing current is simply all ordinary electrolytic conduction-current, which causes a liberation of hydrogen from the ionic form at the cathode. If this be so, the amount of electricity which passes through the cell in any displacement must be proportional to the quantity of hydrogen which is yielded up to the electrode in the displacement; so that $d\gamma /dV$ must be proportional to the amount of hydrogen deposited per unit area of the electrode.^[16]

A different view of the physical conditions at the polarized electrode was taken by Helmholtz,^[17] who assumed that the ions of hydrogen which are brought to the cathode by the polarizing current do not give up their charges there, but remain in the vicinity of the electrode, and form one face of a quasi-condenser of which the other face is the electrode itself.^[18] If $σ$ denote the surface-density of electricity on either face of this quasicondenser, we have, therefore,

$de=-d(S\sigma );\qquad {\text{so}}\qquad \sigma =d\gamma /dV$ .

This equation shows that when $dγ / dV$ is zero—i.e., when the surface-tension is a maximum—o must be zero; that is to say, there must be no difference of potential between the mercury and the electrolyte. The external electromotive force is then balanced entirely by the discontinuity of potential at the other electrode $H 0$ ; and thus a method is suggested of measuring the latter discontinuity of potential. All previous measurements of differences of potential had involved the employment of more than one interface; and it was not known how the measured difference of potential should be distributed among these interfaces; so that the suggestion of a means of measuring single differences of potential was a distinct advance, even though the hypotheses on which the method was based were somewhat insecure.

A further consequence deduced by Helmholtz from this theory leads to a second method of determining the difference of potential between mercury and an electrolyte. If a mercury surface is rapidly extending, and electricity is not rapidly transferred through the electrolyte, the electric surface-density in the double layer must rapidly decrease, since the same quantity of electricity is being distributed over an increasing area Thus it may be inferred that a rapidly extending mercury-surface in an electrolyte is at the same potential as the electrolyte.

This conception is realized in the dropping-electrode, in which a jet of mercury, falling from a reservoir into an electrolytic solution, is so adjusted that it breaks into drops when the jet touches the solution. According to Helmholtz's conclusion there is no difference of potential between the drops and the electrolyte; and therefore, the difference of potential between the electrolyte and a layer of mercury underlying it in the same vessel is equal to the difference of potential between this layer of mercury and the mercury in the upper reservoir, which difference is a measurable quantity.

It will be seen that according to the theories both of Gibbs and of Helmholtz, and indeed according to all other theories on the subject,^[19] $dγ / dV$ is zero for an electrode whose surface is rapidly increasing—e.g., a dropping electrode; that is to say, the difference of potential between an ordinary mercury electrode and the electrolyte, when the surface-tension has its maximum value, is equal to the difference of potential between a dropping-electrode and the same electrolyte. This result has been experimentally verified by various investigators, who have shown that the applied electromotive force when the surface-tension has its maximum value in the capillary electrometer, is equal to the electromotive force of a cell having as electrodes a large mercury electrode and a dropping electrode.

Another memoir which belongs to the same period of Helmholtz' career, and which has led to important developments, was concerned with a special class of voltaic cells. The most usual type of cell is that in which the positive electrode is composed of a different metal from the negative electrode, and the evolution of energy depends on the difference in the chemical affinities of these metals for the liquids in the cell. But in the class of cells now considered^[20] by Helmholtz, the two electrodes are composed of the same metal (say, copper); and the liquid (say, solution of copper sulphate) is more concentrated in the neighbourhood of one electrode than in the neighbourhood of the other. When the cell is in operation, the salt passes from the places of high concentration to the places of low concentration, so as to equalize its distribution, and this process is accompanied by the flow of a current in the outer circuit between the electrodes. Such cells had been studied experimentally by James Moser a short time previously^[21] to Helmholtz' investigation.

The activity of the cell is due to the fact that the available energy of a solution depends on its concentration, the molecules

of salt, in passing from a high to a low concentration, are therefore capable of supplying energy, just as a compressed gas is capable of supplying energy when its degree of compression is reduced. To examine the matter quantitatively, let $nf (n / V)$ denote the term in the available energy of a solution, which is due to the dissolution of n gramme-molecules of salt in a volume $V$ of pure solvent; the function $f$ will of course depend also on the temperature. Then when $dn$ gramme-molecules of solvent are evaporated from the solution, the decrease in the available energy of the system is evidently equal to the available energy of $dn$ gramme-molecules of liquid solvent, less the available energy of $dn$ gramme-molecules of the vapour of the solvent, together with $nf (n / V)$ less nf{n/(V-vdn)}, where $v$ denotes the volume of one gramme-molecule of the liquid. But this decrease in available energy must be equal to the mechanical work supplied to the external world, which is $dn . p 1 (u' - v)$ , if $p 1$ , denote the vapour-pressure of the solution at the temperature in question, and {{Wikimath|v′ denote the volume of one gramme-molecule of vapour. We have therefore

$dn.p_{1}(v^{\prime }-v)=$	-available energy of $dn$ gramme-molecules of solvent vapour
	+available energy of $dn$ gramme-molecules of liquid solvent
	+ $nf(nV)-nf\{n/(V-vdn)\}$ .

Subtracting from this the equation obtained by making $n$ zero, we have

$dn.(p_{1}-p_{0})(v^{\prime }-v)=nf(n/V)-nf\{n/(V-vdn)\}$ ,

where $p 0$ denotes the vapour-pressure of the pure solvent at the temperature in question; so that

$(p_{1}-p_{0})(v^{\prime }-v)=-(n^{2}/V^{2})f^{\prime }(n/V)v$ .

Now, it is known that when a salt is dissolved in water, the vapour-pressure is lowered in proportion to the concentration of the salt—at any rate when the concentration is small: in fact, by the law of Raoult, {{Wikimath|(p₀-p₁/p₀, is approximately equal to $nv / V$ ; so that the previous equation becomes

$p_{0}V(v^{\prime }-v)=nf^{\prime }(n/V)$ .

Neglecting $v$ in comparison with $v'$ , and making use of the equation of state of perfect gases (namely,

$p_{0}v^{\prime }=RT$ .

where $T$ denotes the absolute temperature, and $R$ denotes the constant of the equation of state), we have

$f^{\prime }(n/V)=RTV/n$ ,

and therefore

$f(n/V)=RT\log(n/V)$ .

Thus in the available energy of one gramme-molecule of a dissolved salt, the term which depends on the concentration is proportional to the logarithm of the concentration; and hence, if in a concentration-cell one gramme-molecule of the salt passes from a high concentration $c 2$ , at one electrode to a low concentration $c 1$ at the other electrode, its available energy is thereby diminished by an amount proportional to $log c 2 / c 1$ . The energy which thus disappears is given up by the system in the form of electrical work; and therefore the electromotive force of the concentration-cell must be proportional to $log c 2 / c 1$ . The theory of solutions and their vapour-pressure was not at the time sufficiently developed to enable Helmholtz to determine precisely the coefficient of $log c 2 / c 1$ in the expression.^[22]

An important advance in the theory of solutions was effected in 1887, by a young Swedish physicist, Svante Arrhenius.^[23] Interpreting the properties discovered by Kohlrausch^[24] in the light of the ideas of Williamson and Clausius regarding the spontaneous dissociation of electrolytes, Arrhenius inferred that in very dilute solutions the electrolyte is completely dissociated into ions, but that in more concentrated solutions the salt is less completely dissociated; and that as in all solutions the transport of electricity in the solution is effected solely by the movement of ions, the equivalent conductivity^[25] must be proportional to the fraction which expresses the degree of ionization, By aid of these conceptions it became possible to estimate the dissociation quantitatively, and to construct a general theory of electrolytes.

Contemporary physicists and chemists found it difficult at first to believe that a salt exists in dilute solution only in the form of ions, e.g. that the sodium and chlorine exist separately and independently in a solution of common salt. But there is a certain amount of chemical evidence in favour of Arrhenius' conception. For instance, the tests in chemical analysis are really tests for the ions; iron in the form of a ferrocyanide, and chlorine in the form of a chlorate, do not respond to the characteristic tests for iron and chlorine respectively, which are really the tests for the iron and chlorine ions.

The general acceptance of Arrhenius' views was hastened by the advocacy of Ostwald, who brought to light further evidence in their favour. For instance, all permanganates in dilute solution show the same purple colour; and Ostwald considered their absorption-spectra to be identical;^[26]this identity is easily accounted for on Arrhenius' theory, by supposing that the spectrum in question is that of the anion which corresponds to the acid radicle. The blue colour which is observed in dilute solutions of copper salts, even when the strong solution is not bine, may in the same way be ascribed to a blue copper cation. A striking instance of the same kind is afforded by ferric sulphocyanide; here the strong solution shows a deep red colour, due to the salt itself; but on dilution the colour disappears, the ions being colourless.

If it be granted that ions can have any kind of permanent existence in a salt solution, it may be shown from thermodynamical considerations that the degree of dissociation must increase as te dilution increases, and that at infinite dilution there must be complete dissociation. For the available energy of a dilute solution of volume $V$ , containing $n 1$ , gramme-molecules of one substance, $n 2$ gramme-molecules of another, and so on, is (as may be shown by an obvious extension of the reasoning already employed in connexion with concentration-cells)^[27]

$\sum _{r}n_{r}\phi _{r}(T)+RT\sum _{r}n_{r}\log(n_{r}/V)+{\text{ the available energy}}$

possessed by the solvent before the introduction of the solutes, where $φ r (T)$ depends on $T$ and on the nature of the $r th$ solute, but not on $V$ , and $R$ denotes the constant which occurs in the equation of state of perfect gases. When the system is in equilibrium, the proportions of the reacting substances will be so adjusted that the available energy has a stationary value for small virtual alterations $\partial n 1, \partial n 2, \dots\dots$ of the proportions; and therefore

$0=\sum _{r}\delta n_{r}.\phi _{r}(T)+RT\sum _{r}\delta n_{r}.\log(n_{r}/V)+RT\sum \delta n_{r}$ .

Applying this to the case of an electrolyte in which the disappearance of one molecule of salt indicated by the suffix ₁) gives rise to one cation (indicated by the suffix ₂) and one anion (indicated by the suffix ₃), we have $\partial n 1 = -\partial n 2 = -\partial n 3$ ; so the equation becomes

$0=\phi _{1}(T)-\phi _{2}(T)-\phi _{3}(T)+RT\log(n_{1}V/n_{2}n_{3})-RT$ ,

or

$n_{1}V/n_{2}n_{3}={\text{ a function of }}T{\text{ only}}$ .

Since in a neutral solution the number of anions is equal to the number of cations, this equation may be written

$n_{2}^{2}=Vn_{1}\times {\text{ a function of }}T{\text{ only}}$ ;

it shows that when $V$ is very large (so that the solution is very dilute), $n 2$ is very large compared with $n 1$ ; that is to say, the salt tends towards a state of complete dissociation.

The ideas of Arrhenius contributed to the success of Walther Nernst^[28] in perfecting Helmholtz theory of concentration-cells, and representing their mechanism in a much more definite fashion than had been done heretofore.

In an electrolytic solution let the drift-velocity of the cations under unit electric force be $u$ , and that of the anions be $v$ , so that the fraction $u /(u + v)$ of the current is transported by the cations, and the fraction $v /(u + v)$ by the anions. If the concentration of the solution be $c 1$ at one electrode, and $c 2$ , at the other, it follows from the formula previously found for the available energy that one gramme-ion of cations, in moving from one electrode to the other, is capable of yielding up an amount^[29] $RT log (c 2 / c 1)$ of energy; while one gramme-ion of anions going in the opposite direction must absorb the same amount of energy. The total quantity of work furnished when one gramme-molecule of salt is transferred from concentration $c 2$ ; to concentration $c 1$ is therefore

${\frac {u-v}{u+v}}RT\log {c_{2}}{c_{1}}$

The quantity of electric charge which passes in the circuit when one gramne-molecule of the salt is transferred is proportional to the valency $ν$ of the ions, and the work furnished is proportional to the product of this charge and the electromotive force $E$ of the cell; so that in suitable units we have

$E={\frac {RT}{\nu }}{\frac {u-v}{u+v}}\log {\frac {c_{2}}{c_{1}}}$ .

A typical concentration-cell to which this formula may be applied may be constituted in the following way:—Let a quantity of zinc amalgam, in which the concentration of zinc is $c 1$ , be in contact with a dilute solution of zinc sulphate, and let this in turn be in contact with a quantity of zinc amalgam of concentration $c 2$ . When the two masses of amalgam are connected by a conducting wire outside the cell, an electric current flows in the wire from the weak to the strong amalgam,^[30] while zinc cations pass through the solution from the strong amalgam to the weak. The electromotive force of such a cell, in which the current may be supposed to be carried solely by cations, is

${\frac {RT}{\nu }}\log {\frac {c_{2}}{c_{1}}}$

Not content with the derivation of the electromotive force from considerations of energy, Nernst proceeded to supply a definite mechanical conception of the process of conduction in electrolytes. The ions are impelled by the electric force associated with the gradient of potential in the electrolyte. But this is not the only force which acts on them; for, since their available energy decreases as the concentration decreases, there must be a force assisting every process by which the concentration is decreased. The matter may be illustrated by the analogy of a gas compressed in a cylinder fitted with a piston; the available energy of the gas decreases as its degree of compression decreases; and therefore that movement of the piston which tends to decrease the compression is assisted by a force—the "pressure" of the gas on the piston. Similarly, if a solution were contained within a cylinder fitted with a piston which is permeable to the pure solvent but not to the solute, and if the whole were immersed in pure solvent, the available energy of the system would be decreased if the piston were to move outwards so as to admit more solvent into the solution; and therefore this movement of the piston would be assisted by a force—the "osmotic pressure of the solution," as it is called.^[31]

Consider, then, the case of a single electrolyte supposed to be perfectly dissociated; its state will be supposed to be the same at all points of any plane at right angles to the axis of $x$ . Let $ν$ denote the valency of the ions, and $V$ the electric potential at any point. Since^[32] the available energy of a given quantity of a substance in very dilute solution depends on the concentration in exactly the same way as the available energy of a given quantity of a perfect gas depends on its density, it follows that the osmotic pressure $p$ for each ion is determined in terms of the concentration and temperature by the equation of state of perfect gases

$Mp=RTc$ ,

where $M$ denotes the molecular weight of the salt, and $c$ the mass of salt per unit volume.

Consider the cations contained in a parallelepiped at the place $x$ , whose cross-section is of unit area and whose length is $dx$ . The mechanical force acting on them due to the electric field is $-(vc / M) dV / dx . dx$ , and the mechanical force on them due to the osmotic pressure is $- dp / dx . dx$ . If $u$ denote the velocity of drift of the cations in a field of unit electric force, the total amount of charge which would be transferred by cations across unit area in unit time under the influence of the electric forces alone would be $-(uνe / M) dV / dx$ ; so, under the influence of both forces, it is

$-{\frac {uvc}{M}}\left({\frac {dV}{dx}}+{\frac {RT}{c\nu }}{\frac {dc}{dx}}\right)$ .

Similarly, if $v$ denote the velocity of drift of the anions in a unit electric field, the charge transferred across unit area in unit time by the anions is

${\frac {v\nu c}{M}}\left(-{\frac {dV}{dx}}+{\frac {RT}{c\nu }}{\frac {dc}{dx}}\right)$ .

We have therefore, if the total current be denoted by $i$ ,

$i-(u+v){\frac {\nu c}{M}}{\frac {dV}{dx}}-(u-v){\frac {RT}{M}}{\frac {dc}{dx}}$ ,

or

$-{\frac {dV}{dx}}dx={\frac {Mdx}{(u+v)\nu c}}i+{\frac {u-v}{u+v}}{\frac {RT}{\nu c}}{\frac {dc}{dx}}dx$ .

The first term on the right evidently represents the product of the current into the ohmic resistance of the parallelepiped $dx$ , while the second term represents the internal electromotive force of the parallelepiped. It follows that if $r$ denote the specific resistance, we must have

$u+v=M/r\nu c$

in agreement with Kohlrausch's equation;^[33] while by integrating the expression for the internal electromotive force of the parallelepiped $dx$ , we obtain for the electromotive force of a cell whose activity depends on the transference of electrolyte between the concentrations $c 1$ and $c 2$ , the value

${\frac {u-v}{u+v}}{\frac {RT}{\nu }}\int {\frac {1}{c}}{\frac {dc}{dx}}dx$ ,

or

${\frac {u-v}{u+v}}{\frac {RT}{\nu }}\log {\frac {c_{2}}{c_{1}}}$ ,

in agreement with the result already obtained.

It may be remarked that although the current arising from a concentration cell which is kept at a constant temperature is capable of performing work, yet this work is provided, not by any diminution in the total internal energy of the cell, but by the abstraction of thermal energy from neighbouring bodies. This indeed (as may be seen by reference to W. Thomson's general equation of available energy)^[34] must be the case with any system whose available energy is exactly proportional to the absolute temperature.

The advances which were effected in the last quarter of the nineteenth century in regard to the conduction of electricity through liquids, considerable though these advances were, may be regarded as the natural development of a theory which had long been before the world. It was otherwise with the kindred problem of the conduction of electricity through gases: for although many generations of philosophers had studied the remarkable effects which are presented by the passage of a current through a rarefied gas, it was not until recent times that a satisfactory theory of the phenomena was discovered.

Some of the electricians of the earlier part of the eighteenth century performed experiments in vacuous spaces; in particular, Hauksbee^[35] in 1705 observed a luminosity when glass is rubbed in rarefied air. But the first investigator of the continuous discharge through a rarefied gas seems to have been Watson,^[36] who, by means of an electrical machine, sent a current through an exhausted glass tube three feet long and three inches in diameter. "It was," he wrote, "a most delightful spectacle, when the room was darkened, to see the electricity in its passage; to be able to observe not, as in the open air, its brushes or pencils of rays an inch or two in length, but here the coruscations were of the whole length of the tube between the plates, that is to say, thirty-two inches." Its appearance he described as being on different occasions "of a bright silver hue," "resembling very much the most lively coruscations of the aurora borealis," and "forming a continued arch of lambent fame." His theoretical explanation was that the electricity "is seen, without any preternatural force, pushing itself on through the vacuum by its own elasticity, in order to maintain the equilibrium in the machine"—a conception which follows naturally from the combination of Watson's one-fluid theory with the prevalent doctrine of electrical atmospheres.^[37]

A different explanation was put forward by Nollet, who performed electrical experiments in rarefied air at about the same time as Watson,^[38] and saw in them a striking confirmation of his own hypothesis of efflux and afflux of electric matter.^[39] According to Nollet, the particles of the effluent stream collide with those of the aflluent stream which is moving in the opposite direction; and being thus violently shaken, are excited to the point of emitting light.

Almost a century elapsed before anything more was discovered regarding the discharge in vacuous spaces. But in 1838 Faraday,^[40] while passing a current from the electrical machine between two brass rods in rarefied air, noticed that the purple haze or stream of light which proceeded from the positive pole stopped short before it arrived at the negative rod. The negative rod, which was itself covered with a continuous glow, was thus separated from the purple column by a narrow dark space: to this, in honour of its discoverer, the name Faraday's dark space has generally been given by subsequent writers.

That vitreous and resinous electricity give rise to different types of discharge had long been known; and indeed, as we have seen,^[41] it was the study of these differences that led Franklin to identify the electricity of glass with the superfluity of fluid, and the electricity of amber with the deficiency of it. But phenomena of this class are in general much more complex than might be supposed from the appearance which they present at a first examination; and the value of Faraday's discovery of the negative glow and dark space lay chiefly in the simple and definite character of these features of the discharge, which indicated them as promising subjects for further research. Faraday himself felt the importance of investigations in this direction. "The results connected with the different conditions of positive and negative discharge," he wrote,^[42] "will have a far greater influence on the philosophy of electrical science than we at present imagine."

Twenty more years, however, passed before another notable advance was made. That a subject so full of promise should progress so slowly may appear strange; but one reason at any rate is to be found in the incapacity of the air-pumps then in use to rarefy gases to the degree required for effective study of the negative glow. The invention of Geissler's mercurial air-pump in 1855 did much to remove this difficulty; and it was in Geissler's exhausted tubes that Julius Plücker,^[43] of Bonn, studied the discharge three years later.

It had been shown by Sir Humphrey Davy in 1821^[44] that one form of electric discharge—namely, the arc between carbon poles—is deflected when a magnet is brought near to it. Plücker now performed a similar experiment with the vacuum discharge, and observed a similar deflexion. But the most interesting of his results were obtained by examining the behaviour of the negative glow in the magnetic field; when the negative electrode was reduced to a single point, the whole of the negative light became concentrated along the line of magnetic force passing through this point. In other words, the negative glow disposed itself as if it were constituted of flexible chains of iron filings attached at one end to the cathode.

Plücker noticed that when the cathode was of platinum, small particles were torn off it and deposited on the walls of the glass bulb. "It is most natural," he wrote, "to imagine that the magnetic light is formed by the incandescence of these platinum particles as they are torn from the negative electrode."

He likewise observed that during the discharge the walls of the tube, near the cathode, glowed with a phosphorescent light, and remarked that the position of this light was altered when the magnetic field was changed. This led to another discovery; for in 1869 Plücker's pupil, W. Hittorf,^[45] having placed a solid body between a point-cathode and the phosphorescent light, was surprised to find that a shadow was cast. He rightly inferred from this that the negative glow is formed of rays which proceed from the cathode in straight lines, and which cause the phosphorescence when they strike the walls of the tube.

Hittorf's observation was amplified in 1876 by Eugen Goldstein,^[46] who found that distinct shadows were cast, not only when he cathode was a single point, but also when it: formed an extended surface, provided the shadow-throwing object was placed close to it. This clearly showed that the cathode rays (a term now for the first time introduced) are not emitted indiscriminately in all directions, but that each portion of the cathode surface emits rays which are practically confined to a single direction; and Goldstein found this direction to be normal to the surface. In this respect his discovery established an important distinction between the manner in which cathode rays are emitted from an electrode and that in which light is emitted from an incandescent surface.

The question as to the nature of the cathode rays attracted much attention during the next two decades. In the year following Hittorf's investigation, Cromwell Varley^[47] put forward the hypothesis that the rays are composed of "attenuated particles of matter, projected from the negative pole by electricity"; and that it is in virtue of their negative charges that these particles are influenced by a magnetic field.^[48]

During some years following this, the properties of highly rarefied gases were investigated by Sir William Crookes. Influenced, doubtless, by the ideas which were developed in connexion with his discovery of the radiometer, Crookes,^[49] like Varley, proposed to regard the cathode rays as a molecular torrent: he supposed the molecules of the residual gas, coming into contact with the cathode, to acquire from it a resinous charge, and immediately to fly off normally to the surface, by reason of the mutual repulsion exerted by similarly electrified bodies. Carrying the exhaustion to a higher degree, Crookes was enabled to study a dark space which under such circumstances appears between the cathode and the cathode glow; and to show that at the highest rarefactions this dark space(which has since been generally known by his name) enlarges until the whole tube is occupied by it. Ho suggested that the thickness of the dark space may be a measure of the mean length of free path of the molecules.

"The extra velocity," he wrote, "with which the molecules rebound from the excited negative pole keeps back the more slowly moving molecules which are advancing towards that pole. The conflict occurs at the boundary of the dark space, where the luminous margin bears witness to the energy of the collisions."^[50] Thus according to Crookes the dark space is dark and the glow bright because there are collisions in the latter and not in the former. The fluorescence or phosphorescence on the walls of the tube he attributed to the impact of the particles on the glass.

Crookes spoke of the cathode rays as an "ultra-gaseous" or "fourth state" of matter. These expressions have led some later writers to ascribe to him the enunciation or prediction of a hypothesis regarding the nature of the particles projected from the cathode, which arose some years afterwards, and which we shall presently describe; but it is clear from Crookes' memoirs that he conceived the particles of the cathode rays to be ordinary gaseous molecules, carrying electric charges; and by "a new state of matter" he understood simply a state in which the free path is so long that collisions may be disregarded.

Crookes found that two adjacent pencils of cathode rays appeared to repel each other. At the time this was regarded as a direct confirmation of the hypothesis that the rays are streams of electrically charged particles; but it was shown later that the deflexion of the rays must be assigned to causes other than mutual repulsion.

How admirably the molecular-torrent theory accounts for the deviation of the cathode rays by a magnetic field was shown by the calculations of Eduard Riecke in 1881.^[51] If the axis of $z$ be taken parallel to the magnetic force $H$ , the equations of motion of a particle of mass $m$ , charge $e$ , and velocity $(u, v, w)$ are

$mdu/dt=evH,\qquad mdv/dt=-euH,\qquad mdw/dt=0$ .

The last equation shows that the component of velocity of the particle parallel to the magnetic force is constant; the other equations give

$u=A\sin(eHt/m),\qquad v=A\cos(eHt/m)$ ,

showing that the projection of the path on a plane at right angles to the magnetic force is a circle. Thus, in a magnetic field the particles of the molecular torrent describe spiral paths whose axes are the lines of magnetic force.

But the hypothesis of Varley and Crookes was before long involved in difficulties. Tait^[52] in 1880 remarked that if the particles are moving with great velocities, the periods of the luminous vibrations received from then should be affected to a measurable extent in accordance with Doppler's principle. Tait tried to obtain this effect, but without success. It may, however, be argued that if, as Crookes supposed, the particles become luminous only when they have collided with other particles, and have thereby lost part of their velocity, the phenomenon in question is not to be expected.

396 Conduction in Solutions and Gases, The alternative to the molecular-torrent theory is to suppose that the cathode radiation is a disturbance of the aether. This view was maintained by several physicists,^[53] and notably by Hertz,^[54] who rejected Varley's hypothesis when he found experimentally that the rays did not appear to produce any external electric or magnetic force, and were apparently not affected by an electrostatic field. It was, however, pointed out by FitzGerald^[55] that external space is probably screened from the effects of the rays by other electric actions which take place in the discharge tube.

It was further urged against the charged-particle theory that cathode rays are capable of passing through films of metal which are so thick as to be quite opaque to ordinary light;^[56] it seemed inconceivable that particles of matter should not be stopped by even the thinnest gold-leaf. At the time of Hertz's experiments on the subject, an attempt to obviate this difficulty was made by J.-J. Thomson,^[57] who suggested that the metallic film when bombarded by the rays might itself acquire the property of emitting charged particles, so that the rays which were observed on the further side need not have passed through the film. It was Thomson who ultimately found the true explanation; but this depended in part on another order of ideas, whose introduction and development must now be traced.

The tendency, which was now general, to abandon the electron-theory of Weber in favour of Maxwell's theory involved certain changes in the conceptions of electric charge. In the theory of Weber, electric phenomena were attributed to the agency of stationary or moving charges, which could most readily be pictured as having a discrete and atom-like existence. The conception of displacement, on the other hand, which lay at the root of the Maxwellian theory, was more in harmony with the representation of electricity as something of a continuous nature; and as Maxwell's views met with increasing acceptance, the atomistic hypothesis seemed to have entered on a period of decay. Its revival was due largely to the advocacy of Helmholtz^[58] who, in a lecture delivered to the Chemical Society of London in 1881, pointed out^[59] that it was thoroughly in accord with the ideas of Faraday,^[60] on which Maxwell's theory was founded. "If," he said, " we accept the hypothesis that the elementary substances are composed of atoms, we cannot avoid concluding that electricity also, positive as well as negative, is divided into definite elementary portions which behave like atoms of electricity."

When the conduction of electricity is considered in the light of this hypothesis, it seems almost inevitable to conclude that the process is of much the same character in gases as in electrolytes; and before long this view was actively maintained. It had indeed long been known that a compound gas might be decomposed by the electric discharge; and that in some cases the constituents are liberated at the electrodes in such a way as to suggest an analogy with electrolysis. The question had been studied in 1861 by Adolphe Perrot, who examined^[61] the gases liberated by the passage of the electric spark through steam. He found that while the product of this action was a detonating mixture of hydrogen and oxygen, there was a decided preponderance of hydrogen at one pole and of oxygen at the other

The analogy of gaseous conduction to electrolysis was applied by W. Giese,^[62] of Berlin, in 1882, in order to explain the conductivity of the hot gases of flames. "It is assumed," he wrote, "that in electrolytes, even before the application of an external electromotive force, there are present atoms or atomic groups—the ions, as they are called—which originate when the molecules dissociate; by these the passage of electricity through the liquid is effected, for they are set in motion by the electric field and carry their charges with them. We shall now extend this hypothesis by assuming that in gases also the property of conductivity is due to the presence of ions. Such ions may be supposed to exist in small numbers in all gases at the ordinary temperature and pressure; and as the temperature rises their numbers will increase."

Ideas similar to this were presented in a general theory of the discharge in rarefied gases, which was devised two years later by Arthur Schuster, of Manchester.^[63] Schuster remarked that when hot liquids are maintained at a high potential, the vapours which rise from them are found to be entirely free from electrification, from which he inferred that a molecule striking an electrified surface in its rapid motion cannot carry away any part of the charge, and that one molecule cannot communicate electricity to another in an encounter in which both molecules remain intact. Thus he was led to the conclusion that dissociation of the gaseous molecules is necessary for the passage of electricity through gases.^[64]

Schuster advocated the charged particle theory of cathode rays, and by extending and interpreting an experiment of Hittorf's was able to adduce strong evidence in its favour. He placed the positive and negative electrodes so close to each other that at very low pressures the Crookes' dark space extended from the cathode to beyond the anode. In these circumstances it was found that the discharge from the positive electrode always passed to the nearest point of the inner boundary of the Crookes' dark space—Which, of course, was in vessel; the opposite direction to the cathode. Thus, in the neighbourhood of the positive discharge, the current was flowing in two opposite directions at closely adjoining places; which could scarcely happen unless the current in one direction were carried by particles moving against the lines of force by virtue of their inertia,

Continuing his researches, Schuster^[65] showed in 1887 that a steady electric current may be obtained in air between electrodes whose difference of potential is but small, provided that an independent current is maintained in the same that is to say, a continuous discharge produces in the air such a condition that conduction occurs with the smallest electromotive forces. This effect he explained by aid of the hypothesis previously advanced; the ions produced by the main discharge become diffused throughout the vessel, and, coming under the influence of the field set up by the auxiliary electrodes, drift so as to carry a current between the latter.

A discovery related to this was made in the same year by Hertz,^[66] in the course of the celebrated researches^[67] which have been already mentioned. Happening to notice that the passage of one spark is facilitated by the passage of another spark in its neighbourhood, he followed up the observation, and found the phenomenon to be due to the agency of ultra-violet light emitted by the latter spark. It appeared in fact that the distance across which an electric spark can pass in air is greatly increased when light of very short wave-length is allowed to fall on the spark-gap. It was soon found^[68] that the effective light is that which falls on the negative electrode of the gap; and Wilhelm Hallwachs^[69] extended the discovery by showing that when a sheet of metal is negatively electrified and exposed to ultra-violet light, the adjacent air is thrown into a state which permits the charge to leak rapidly away.

Interest was now thoroughly aroused in the problem of conductivity in gases; and it was generally felt that the best hope of divining the nature of the process lay in studying the discharge at high rarefactions. "If a first step towards understanding the relations between aether and ponderable matter is to be made," said Lord Kelvin in 1893,^[70] "it seems to me that the most hopeful foundation for it is knowledge derived from experiments on electricity in high vacuum."

Within the two following years considerable progress was effected in this direction. J.J. Thomson,^[71] by a rotating-mirror method, succeeded in measuring the velocity of the cathode rays, finding it to be^[72] 1·9 x 10⁷ cm./sec.; a value so much smaller than that of the velocity of light that it was scarcely possible to conceive of the rays as vibrations of the aether. A further blow was dealt at the latter hypothesis when Jean Perrin,^[73] having received the rays in a metallic cylinder, found that the cylinder became charged with resinous electricity. When the rays were deviated by a magnet in such a way that they could no longer enter the cylinder, it no longer acquired a charge. This appeared to demonstrate that the rays transport negative electricity.

With cathode rays is closely connected another type of radiation, which was discovered in December, 1895, by W. C. Röntgen.^[74] The discovery seems to have originated in an accident: a photographic plate which, protected in the usual way, had been kept in a room in which vacuum-tube experiments were carried on, was found on development to show distinct markings. Experiments suggested by this showed that radiation, capable of affecting sensitive plates and of causing fluorescence in certain substances, is emitted by tubes in which the electric discharge is passing; and that the radiation proceeds from the place where the cathode rays strike the glass walls of the tube. The X-rays, as they were called by their discoverer, are propagated in straight lines, and can neither be refracted by any of the substances which refract light, nor deviated from their course by a magnetic field; they are moreover able to pass with little absorption through many substances which are opaque to ordinary and ultra-violet light—a property of which considerable use has been made in surgery.

The nature of the new radiation was the subject of much speculation. Its discoverer suggested that it might prove to represent the long-sought-for longitudinal vibrations of the aether; while other writers advocated the rival claims of aethereal vortices, infra-red light, and "sifted" cathode rays. The hypothesis which subsequently obtained general acceptance was first propounded by Schuster^[75] in the month following the publication of Röntgen's researches. It is, that the X-rays are transverse vibrations of the aether, of exceedingly small wavelength. A suggestion which was put forward later in the year by E. Wiechert^[76] and Sir George Stokes,^[77] to the effect that the rays are pulses generated in the aether when the glass of the discharge tube is bombarded by the cathode particles, is not really distinct from Schuster's hypothesis; for ordinary white light likewise consists of pulses, as Gouy^[78] had shown, and the essential feature which distinguishes the Röntgen pulses is that the harmonic vibrations into which they can be resolved by Fourier's analysis are of very short period.

The rapidity of the vibrations explains the failure of all attempts to refract the X-rays. For in the formula

$\mu ^{2}=1+{\frac {\sigma p^{2}}{\rho (p^{2}-n^{2})}}$

of the Maxwell-Sellmeier theory,^[79] $n$ denotes the frequency, and so is in this case extremely large; whence we have

$\mu ^{2}=1$ ,

i.e., the refractive index of all substances for the X-rays is unity. In fact, the vibrations alternate too rapidly to have an effect on the sluggish systems which are concerned in refraction.

Some years afterwards H. Haga and C. H. Wind,^[80] having measured the diffraction-patterns produced by X-rays, concluded that the wave-length of the vibrations concerned was of the order of one Ångstrom unit, that is about 1/6000 of the wavelength of the yellow light of sodium.

One of the most important properties of X-rays was discovered, shortly after the rays themselves had become known, by J. J. Thomson,^[81] who announced that when they pass through any substance, whether solid, liquid, or gaseous, they render it conducting. This he attributed, in accordance with the ionic theory of conduction, to " a kind of electrolysis, the molecule of the non-conductor being split up, or nearly split up, by the Röntgen rays."

The conductivity produced in gases by this means was at once investigated^[82] more closely. It was found that a gas which had acquired conducting power by exposure to X-rays lost this quality when forced through a plug of glass-wool; whence it was inferred that the structure in virtue of which the gas conducts is of so coarse a character that it is unable to survive the passage through the fine pores of the plug. The soon as conductivity was also found to be destroyed when an electric current was passed through the gas—a phenomenon for which a parallel may be found in electrolysis. For if the ions were removed from an electrolytic solution by the passage of a current. the solution would cease to conduct as sufficient electricity had passed to remove them all; and it may be supposed that the conducting agents which are produced in a gas by exposure to X-rays are likewise abstracted from it when they are employed to transport charges.

The same idea may be applied to explain another property of gases exposed to X-rays. The strength of the current through the gas depends both on the intensity of the radiation and also on the electromotive force; but if the former factor be constant, and the electromotive force be increased, the current does not increase indefinitely, but tends to attain a certain "saturation" value. The existence of this saturation value is evidently due to the inability of the electromotive force to do more than to remove the ions as fast as they are produced by the rays.

Meanwhile other evidence was accumulating to show that the conductivity produced in gases by X-rays is of the same nature as the conductivity of the gases from fames and from the path of a discharge, to which the theory of Giese and Schuster had already been applied. One proof of this identity was supplied by observations of the condensation of watervapour into clouds. It had been noticed long before by John Aitken^[83] that gases rising from flames cause precipitation of the aqueous vapour from a saturated gas; and R. von Helmholtz^[84] had found that gases through which an electric discharge has been passed possess the same property. It was now shown by C. T. R. Wilson,^[85] working in the Cavendish Laboratory at Cambridge, that the same is true of gases which have been exposed to X-rays. The explanation furnished by the ionic theory is that in all three cases the gas contains ions which act as centres of condensation for the vapour.

During the year which followed their discovery, the X-rays were so thoroughly examined that at the end of that period they were almost better understood than the cathode rays from which they derived their origin. But the obscurity in which this subject had been so long involved was now to be dispelled.

Lecturing at the Royal Institution on April 30th, 1897, J. J. Thomson advanced a new suggestion to reconcile the molecular-torrent hypothesis with Lenard's observations of the passage of cathode rays through material bodies. "We see from Lenard's table," he said, "that a cathode ray can travel through air at atmospheric pressure a distance of about half a centimetre before the brightness of the phosphorescence falls to about half its original value. Now the mean free path of the molecule of air at this pressure is about 10^-5 cm., and if a molecule of air were projected it would lose half its momentum in a space comparable with the mean free path. Even if we suppose that it is not the same molecule that is carried, the effect of the obliquity of the collisions would reduce the momentum to hall in a short multiple of that path.

"Thus, from Lenard's experiments on the absorption of the rays outside the tube, it follows on the hypothesis that the cathode rays are charged particles moving with high velocities that the size of the carriers must be small compared with the dimensions of ordinary atoms or molecules.^[86] The assumption of a state of matter more finely subdivided than the atom of an element is a somewhat startling one; but a hypothesis that would involve somewhat similar consequences—viz. that the so-called elements are compounds of some primordial element—has been put forward from time to time by various chemists."

Thomson's lecture drew from FitzGerald^[87] the suggestion that "we are dealing with free electrons in these cathode rays"—a remark the point of which will become more evident when wo come to consider the direction in which the Maxwellian theory was being developed at this time.

Shortly afterwards Thomson himself published an account^[88] of experiments in which the only outstanding objections to the charged-particle theory were removed. The chief of these was Hertz' failure to deflect the cathode rays by an electrostatic field. Hertz had caused the rays to travel between parallel plates of metal maintained at different potentials; but Thomson now showed that in these circumstances the rays generate ions in the rarefied gas, which settle on the plates, and annul the electric force in the intervening space. By carrying the exhaustion to a much higher degree, he removed this source of confusion, and obtained the expected deflexion of the rays.

The electrostatic and magnetic deflexions taken together suffice to determine the ratio of the mass of a cathode particle to the charge which it carries. For the equation of motion of the particle is

$m\mathbf {\ddot {r}} =e\mathbf {E} +e[\mathbf {v.H} ]$ ,

where $r$ denotes the vector from the origin to the position of the particle; $E$ and $H$ denote the electric and magnetic forces; $e$ the charge, $m$ the mass, and $v$ the velocity of the particle. By observing the circumstances in which the force $e E$ , due to the electric field, exactly balances the force $e [v.H]$ , due to the magnetic field, it is possible to determine $v$ ; and it is readily seen from the above equation that a measurement of the deflexion in the magnetic field supplies a relation between $v$ and $m / e$ ; so both $v$ and $m / e$ may be determined. Thomson found the value of $m / e$ to be independent of the nature of the rarefied gas: its amount was 10^-7 (grammes/electromagnetic units of charge), which is only about the thousandth part of the value of $m / e$ for the hydrogen atom in electrolysis. If the charge were supposed to be of the same order of magnitude as that on an electrolytic ion, it would be necessary to conclude that the particle whose mass was thus measured is much smaller than the atom, and the conjecture might be entertained that it is the primordial unit or corpuscle of which all atoms are ultimately composed.^[89]

The nature of the resinously charged corpuscles which constitute cathode rays being thus far determined, it became of interest to inquire whether corresponding bodies existed carrying charges of vitreous electricity—a question to which at any rate a provisional answer was given by W. Wien^[90] of Aachen in the same year. More than a decade previously E, Goldstein^[91] had shown that when the cathode of a discharge-tube is perforated, radiation of a certain type passes outward through the perforations into the part of the tube behind the cathode. To this radiation he had given the name canal rays. Wien now showed that the canal rays sure formed of positively charged particles, obtaining a value of $m / e$ immensely larger than Thomson had obtained for the cathode rays, and indeed of the same order of magnitude as the corresponding ratio in electrolysis.

The disparity thus revealed between the corpuscles of cathode rays and the positive ions of Goldstein's rays excited great interest; it seemed to offer a prospect of explaining the curious differences between the relations of vitreous and of resinous electricity to ponderable matter. These phenomena had been studied by many previous investigators; in particular Schuster,^[92] in the Bakerian lecture of 1890, had remarked that "if the law of impact is different between the molecules of the gas and the positive and negative ions respectively, it follows that the rate of diffusion of the two sets of ions will in general be different," and had inferred from his theory of the discharge that "the negative ions diffuse more rapidly." This inference was confirmed in 1898 by John Zeleny,^[93] who showed that of the ions produced in air by exposure to X-rays, the positive are decidedly less mobile than the negative.

The magnitude of the electric charge on the ions of gases was not known with certainty until 1898, when a plan for determining it was successfully executed by J. J. Thomson^[94] The principles on which this celebrated investigation was based are very ingenious. By measuring the current in a gas which is exposed to Röntgen rays and subjected to a known electromotive force, it is possible to determine the value of the product $nev$ , where $n$ denotes the number of ions in unit volume of the gas, $e$ the charge on an ion, and $v$ the mean velocity of the positive and negative ions under the electromotive force. As $v$ had been already determined,^[95] the experiment led to a determination of $ne$ ; so if $n$ could be found, the value of $e$ might be deduced.

The method employed by Thomson to determine $n$ was founded on the discovery, to which we have already referred, that when X-rays pass through dust-free air, saturated with aqueous vapour, the ions act as nuclei around which the water condenses, so that a cloud is produced by such a degree of saturation as would ordinarily be incapable of producing condensation. The size of the drops was calculated from measurements of the rate at which the cloud sank; and, by comparing this estimate with the measurement of the mass of water deposited, the number of drops was determined, and hence the number $n$ of ions. The value of $e$ consequently deduced was found to be independent of the nature of the gas in which the ions were produced, being approximately the same in hydrogen as in air, and being apparently in both cases the same as for the charge carried by the hydrogen ion in electrolysis.

Since the publication of Thomson's papers his general conclusions regarding the magnitudes of $e$ and $m / e$ for gaseous ions have been abundantly confirmed. It appears certain that electric charge exists in discrete units, vitreous and resinous, each of magnitude 1·5 x 10^-19 coulombs approximately. Each ion, whether in an electrolytic liquid or in a gas, carries one (or an integral number) of these charges. An electrolytic ion also contains one or more atoms of matter, and a positive gaseous ion has a mass of the same order of magnitude as that of an atom of matter. But it is possible in many ways to produce in a gas negative ions which are not attached to atoms of matter; for these the inertia is only about one-thousandth of the inertia of an atom; and there is reason for believing that even this apparent mass is in its origin purely electrical.^[96]

The closing years of the nineteenth century saw the foundation of another branch of experimental science which is closely related to the study of conduction in gases. When Röntgen announced his discovery of the X-rays, and described their power of exciting phosphorescence, a number of other workers commenced to investigate this property more completely. In particular, Henri Becquerel resolved to examine the radiations which are emitted by the phosphorescent double sulphate of uranium and potassium after exposure to the sun. The result was communicated to the French Academy on February 24th, 1896.^[97] "Let a photographic plate," he said, " be wrapped in two sheets of very thick black paper, such that the plate is not affected by exposure to the sun for a day. Outside the paper place a quantity of the phosphorescent substance, and expose the whole to the sun for several hours. When the plate is developed, it displays a silhouette of the phosphorescent substance. So the latter must emit radiations which are capable of passing through paper opaque to ordinary light, and of reducing salts of silver."

At this time Becquerel supposed the radiation to have been excited by the exposure of the phosphorescent substance to the sun; but a week later he announced^[98] that it persisted for an indefinite time after the substance had been removed from the sunlight, and after the luminosity which properly constitutes phosphorescence had died away; and he was thus led to conclude that the activity was spontaneous and permanent. It was soon found that those salts of uranium which do not phosphoresce—e.g., the uranous salts,—and the metal itself, all emit the rays; and it became evident that what Becquerel had discovered was a radically new physical property, possessed by the element uranium in all its chemical compounds.

Attempts were now made to trace this activity in other substances. In 1898 it was recognized in thorium and its compounds;^[99] and in the same year P. Curie and Madame Sklodowska Curie announced to the French Academy the separation from the mineral pitchblende of two new highly active elements, to which they gave the names of polonium^[100] and radium.^[101] A host of workers was soon engaged in studying the properties of the Becquerel rays. The discoverer himself had shown^[102] in 1896 that these rays, like the X- and cathode rays, impart conductivity to gases. It was found in 1899 by Rutherford^[103] that the rays from uranium are not all of the same kind, but that at least two distinct types are present; one of these, to which he gave the name α-rays, is readily absorbed; while another, which he named β-radiation, has a greater penetrating power. It was then shown by Giesel, Becquerel, and others, that part of the radiation is deflected by a magnetic field,^[104] and part is not.^[105] After this Monsieur and Madame Curie^[106] found that the deviable rays carry negative electric charges, and Becquerel^[107] succeeded in deviating them by an electrostatic field. The deviable or β-rays were thus clearly of the same nature as cathode rays; and when measurements of the electric and magnetic deviations gave for the ratio m/e a value of the order 10^-7, the identity of the β-particles with the cathode-ray corpuscles was fully established.

The subsequent history of the new branch of physics thus created falls outside the limits of the present work. We must now consider the progress which was achieved in the general theory of aether and electricity in the last decade of the nineteenth century.

Notes

↑ Cf. p. 78.
↑ l'hil. Mag. xxxvii (1850), p. 350; Liebig's Annalen d. Chem, u. Pharm. 1xxvii (1851) p. 87.
↑ Ann. d. Phys. сi (1857), p. 338; Phil. Mag. xv (1858), p. 94,
↑ Exper. Res. §§ 525-530.
↑ Phil. Trans., 1844, p. 1. Cf. also Pouillet, Comptes Rendus xx (1845), p. 1544.
↑ Ann. d. Phys. lxxxix (1853), p. 177; xcvii (1856), p. 1; cii (1858), p. 1: cvi (1859), pp. 337, 513.
↑ The ratio $v /(u + v)$ was termed by Hittorf the transport number of the anion.
↑ Ann. d. Phys. vi (1879), pp. 1, 145. The chief results had been communicated to the Academy of Göttingen in 1876 and 1877.
↑ A gramme-equivalent means a mass of the salt whose weight in grammes is the molecular weight divided by the valency of the ions.
↑ i.e. $E$ is 96580 coulombs.
↑ The phenomenal of voltaic polarization was discovered by Ritter in 1803. Ritter explained it by comparing the action of the polarizing current to that of a current which is used to charge a condenser. Volta in 1805 put forward the alternative explanation, that the products of decomposition set up a reverse electromotive force.
↑ Phil. Trans. clxi (1871), p. 129.
↑ Comptes Rendus lxxvi (1873), p. 1407. Phil. Mag, xlvii (1874), p. 281. Ann. de Chim. et de Phys. v (1875), p. 494, xii (1877), p. 265.
↑ Lippman, Comptes Rendus, xcv (1882), p. 686.
↑ Trans. Conn. Acad. iii (1876-1878), pp. 108, 343; Gibbs' Scientific Papers, i, p. 55.
↑ This is embodied in equation (690) of Gibbs' memoir.
↑ Berlin Monatsber., 1881, p. 945; Wiss. Abl. i, p. 925; Ann. d. Phys. xvi. (1882), p. 31. Cf. also Planck, Ann. d. Phys. xliv (1891), p. 385.
↑ The conception of double layers of electricity at the surface of separation of two bodies had been already applied by Helmholtz to explain various other phenomena—e.g., the Volta contact-difference of potential of two metals, frictional electricity, and "electric endosmose," or the transport of fluid which occurs when an electric current is passed through two conducting liquids separated by a porous barrier. Cf. Helmholtz, Berlin Monatsherichte, February 27, 1879; Ann. d. Phys. vii (1879), p. 337; Helmholtz, Wiss. Abh. 1, p. 855.
↑ E.g., that of Warburg, Ann. d. Phys. xli (1890), p. 1. In this it is assumed that the electrolytic solution near the electrodes originally contains a salt of mercury in solution. When the external electromotive force is applied, a conduction-current passes through the electrolyte, which in the body of the electrolyte is carried by the acid and hydrogen ions. Warburg supposed that at the cathode the hydrogen ions react with the salt of mercury, reducing it to metallic mercury, which is deposited on the electrode. Thus a considerable change in concentration of the salt of mercury is caused at the cathode. At the anode, the acid ions carrying the current attack the mercury of the electrode, and thus increase the local concentration of the mercuric salt; but on account of the size of the anode this increase is trivial and may be neglected.
Warburg thus supposed that the electromotive force of the polarized cell is really that of a concentration cell, depending on the different concentrations of mercuric salt at the electrodes. He found $dγ / dV$ to be equal to the amount of mercuric salt at the cathode per unit area of cathode, divided by the electro-chemical equivalent of mercury. The equation previously obtained is thus presented in a new physical interpretation.
Warburg connected the increase of the surface-tension with the fact that the surface-tension between mercury and a solution always increases when the concentration of the solution is diminished. His theory, of course, leads to no conclusion regarding the absolute potential difference between the mercury and the solution, as Helmholtz' does.
Alan electrode whose surface is rapidly increasing—e.g., a dropping electrode—Warburg supposed that the surface-density of mercuric salt tends to zero, so $dγ / dV$ is zero.
The explanation of dropping electrodes favoured by Nernst, Beilage zu den Ann. d. Phys. lviii (1896), is that the difference of potential corresponding to the equilibrium between the mercury and the electrolyte is instantaneously established; but that ions are withdrawn from the solution in order to form the double layer necessary for this, and that these ions are carried down with the drops of mercury, until the upper layer of the solution is so much impoverished that the double layer can no longer be formed. The impoverishment of the upper layer of the solution has actually been observed by Palmaer, Zeitsch. Phys. Chem. xxv (1898), p. 265; xxviii (1899), p. 257; xxxvi (1901), p. 664.
↑ Berlin Monatsber., 1877, p. 713; Phil. May (5) v (1878), p. 348; reprinted with additions in Ann. d. Phys. iii (1878), p. 201.
↑ Ann. d. Phys. iii (1878), p. 216.
↑ The formula given by Helmholtz was that the electromotive force of the cell is equal to $b (1 - n) v log(c 2 / c 1)$ , where $c 2$ and $c 1$ denote the concentrations of the solution at the electrodes, $v$ denotes the volume of one gramme of vapour in equilibrium with the water at the temperature in question, $n$ denotes the transport number for the cation (Hittorf's $1/ n$ ), and $b$ denotes $q \times$ the lowering of vapour-pressure when one gramme-equivalent of salt is dissolved in $q$ grammes of water, where $q$ denotes a large number.
↑ Zeitschrift für phys. Chem. i (1887), p. 631. Previous investigations, in which the theory was to some extent foreshadowed, were published in Bihang till Sveuska Vet. Ak. Förh. vii (1884), Nos. 13 and 14.
↑ Cf. p. 374.
↑ I.e. the ohmic specific conductivity of the solution divided by the number of gramme-equivalents of salt per unit volume.
↑ Examination of the spectra with higher dispersion does not altogether confirm this conclusion,
↑ Cf. pp. 382-383.
↑ Zeitschr. für phys. Chem. ii (1888), p. 613; iv (1889), p. 129; Berlin Sitzungsberichte, 1889, p. 83; Ann. d. Phys. xlv (1892), p. 360. Cf. also Max Planck, Ann. d. Phys. xxxix (1890), p. 161; xl (1890), p. 561.
↑ The correct law of dependence of the available energy on the temperature was by this time known.
↑ It will hardly be necessary to remark that this supposed direction of the current is purely conventional.
↑ Cf. van't Hoff, Svenska Vet.-Ak. Handlingar xxi (1886), No. 17; Zeitschrift für Phys. Chem. i (1887), p. 481.
↑ As follows from the expression obtained, supra, p. 383.
↑ Cf. p. 374.
↑ Cf. p. 241.
↑ Phil. Trans. xxiv (1705), p. 2165. Fra. Hauksbee, Physico-Mechanical Experiments, London, 1709.
↑ Phil. Trans. xlv (1748), p. 93, xlvii (1752), p. 362.
↑ Cf. ch. ii.
↑ Nollet, Recherches sur l' Electricité, 1749, troisième discours.
↑ Cf. p. 40.
↑ Phil. Trans., 1838; Exper. Res. i, § 1526.
↑ Cf. p. 44.
↑ Exper. Res., § 1623.
↑ Ann, d. Phys. ciii (1858), pp. 88, 151; civ (1868), pp. 113, 622; cv (1858), p. 67; cvii (1859), p. 77. Phil. Mag. xvi (1858), pp. 119, 408; xviii (1859), pp. 1, 7.
↑ Phil. Trans., 1821, p. 425.
↑ Ann. d. Phys. cxxxvi (1869), pp. 1, 197; translated, Annales de Chimie, xvi (1869), p. 437.
↑ Berlin Monatsberichte, 1876, p. 279.
↑ Proc. Roy. Soc. xix (1871), p. 236.
↑ Priestley in 1766 had shown that a current of electrified air tows from the points of bodies which are electrified either vitreously or resinously: cf. Priestley's History of Electricity, p. 591.
↑ Phil, Trans, clxx (1879), pp. 135, 641; Phil. Mag. vii (1879), p. 57.
↑ Phil. Mag. vii (1879), p. 57.
↑ Gött. Nach., 2 February, 1891; reprinted, Ann. d. Phys. xiii (1881), p. 191.
↑ Proc. Roy. Soc. Edinb. x (1880), p. 430.
↑ E.g. E. Wiedemann, Ann. d. Phys. x (1880), p. 202: translated, Phil. Mag. x (1880), p. 357. E. Goldstein, Ann. d. Phys. xii (1881), p. 249.
↑ Ann. d. Phys. xix (1883), p. 782.
↑ Nature, November 5, 1896; Fitz Gerald's Scientific Writings, p. 433.
↑ The penetrating power of the rays had been noticed by Hittorf, and by E. Wiedemann and Ebert, Sitzber, d. phys.-med. Soc. zu Erlangen, 11th December, 1891. It was investigated more thoroughly by Hertz, Ann. d. Phys. xlv (1892), p. 28, and by Philipp Lenard, of Bonn, Ann. d. Phys. li (1894), p. 225; lii (1894), p. 23, who conducted a series of experiments on cathode rays which had passed out of the discharge tube through a thin window of aluminium.
↑ J.J. Thomson, Recent Researches, p. 126.
↑ Cf. also G. Johnstone Stoney, Phil. Mag-, May, 1881.
↑ Journ. Chem. Soc. xxxix (1881), p. 277.
↑ Cf. p. 200.
↑ Annales de Chimie (3), lxi, p. 161.
↑ Ann. d. Phys. xvii (1882), pp. 1, 236, 519.
↑ Proc. Roy. Soc. xxxvii (1884), p. 317.
↑ In the case of an elementary gas, this would imply dissociation of the molecule into two atoms chemically alike, but oppositely charged; in electrolysis the dissociation is into two chemically unlike ions.
↑ Proc. Roy. Soc. xlii (1887), p. 371. Hittorf had discovered that very small electromotive forces are sufficient to cause a discharge across a space through which the cathode radiation is passing.
↑ Berlin Ber., 1887, p. 487; Ann, d. Phys. xxxi (1887), p. 983; Electric Waves (English ed.), p. 63.
↑ Cf. p. 357.
↑ Ry E. Wiedemann and Ebert, Ann. d. Phys. xxxiii (1898), p. 241.
↑ Ann. d. Phys. xxxiii (1888), p. 301.
↑ Proc. Roy. Soc. liv (1893), p. 389.
↑ Phil. Mag. xxxviii (1894), p. 358.
↑ The value found by the same investigator in 1897 was much larger than this.
↑ Comptes Rendus, cxxi (1895), p. 1130.
↑ Sitzungeber, der Würzburger Physikal.-Medie. Gesellschaft, 1895; reprinted, Ann. d. Phys. lxiv (1898), pp. 1, 12; translated, Nature, liii (1896), p. 274.
↑ Nature, January 23, 1896, p. 268. Fitz Gerald independently made the same suggestion in a letter to O. J. Lodge, printed in the Electrician xxxvii, p. 372.
↑ Ann. d. Phys, lix (1806), p. 321.
↑ Nature, September 3, 1896, p. 427: Proc. Camb. Phil. Soc. ix (1896), p. 215; Mem. Manchester Lit. & Phil. Soc. xli (1896-7).
↑ Journ. de Phys. v (1886), p. 354.
↑ Cf. p. 293.
↑ Proceedings of the Amsterdam Acad., March 25th, 1899 (English edition, i, p. 420), and September 27th, 1902 (English edition, v, p. 247).
↑ Nature, February 27, 1896, p. 391.
↑ J.J. Thomson and E. Rutherford, Phil. Mag. xlii (1896), p. 392.
↑ Trans. R. S. Edinb. xxx (1880), p. 337.
↑ Ann. d. Phys. xxxii (1887), p. 1.
↑ Proc. Roy. Soc., March 19, 1896; Phil. Trans., 1897, p. 265.
↑ A similar suggestion was made by E. Wiechert, Verhandl. d. physik.-öeon. Gesellsch. in Königsberg, Jan. 1897.
↑ Electrician, May 21, 1897.
↑ Phil. Mag, xliv (1897), p. 298.
↑ The value of $m / e$ for cathode rays was determined also in the same year ly W. Kaufmamn, Ann. d. Phys. lxi, p. 544.
↑ Verhundi, der physik. Gesells, zu Berlin, xvi (1897), p. 165; Ann. d. Phys. lxv (1898), p. 440.
↑ Berlin Sitzungsber., 1886, p. 691.
↑ Proc. R.S. xlvii (1890), p. 526.
↑ Phil. Mag. xlvi (1898), p. 120.
↑ Phil. Mag. xlvi (1898), p. 528.
↑ By E. Rutherfurd, Phil. Mag. xliv (1897), p. 422.
↑ Cf. p. 343.
↑ Comptes Rendus, cxxii (1895), p. 420.
↑ Ibid., cxxii (March 2nd, 1896), p. 501.
↑ By Schmidt, Ann. d. Phys., lxv (1898), p. 141; and by Madame Curie, Comptes Rendus, cxxvi (1898), p. 1101.
↑ Comptes Rendus, cxxvii (1898), p. 175.
↑ Ibid., cxxvii (1898), p. 1275.
↑ Ibid., cxxii (1896), p. 659.
↑ Phil. Mag. (5), xlvii (1899), p. 109.
↑ Giesel, Ann. d. Phys. lxix (1899), p. 834 (working with polonium); Bcequerel, Comptes Rendus, cxxix (1899), p. 996 (working with radium); Meyer and v. Schweidler, Phys. Zeitschr. i (1899), p. 113 (working with polonium and radium).
↑ Becquerel, Comptes Rendus, cxxix (1889), p. 1205); cxxx (1900), pp. 206, 372. Curie, ibid. cxxx (1900), p. 73.
↑ Comptes Rendus, cxxx (1900), p. 647.
↑ Comptes Rendus, cxxx (1900), p. 809.

[1] Cf. p. 78.

[2] 'hil. Mag. xxxvii (1850), p. 350; Liebig's Annalen d. Chem, u. Pharm. 1xxvii (1851) p. 87.

[3] Ann. d. Phys. сi (1857), p. 338; Phil. Mag. xv (1858), p. 94,

[4] Exper. Res. §§ 525-530.

[5] Phil. Trans., 1844, p. 1. Cf. also Pouillet, Comptes Rendus xx (1845), p. 1544.

[6] Ann. d. Phys. lxxxix (1853), p. 177; xcvii (1856), p. 1; cii (1858), p. 1: cvi (1859), pp. 337, 513.

[7] The ratio $v /(u + v)$ was termed by Hittorf the transport number of the anion.

[8] Ann. d. Phys. vi (1879), pp. 1, 145. The chief results had been communicated to the Academy of Göttingen in 1876 and 1877.

[9] A gramme-equivalent means a mass of the salt whose weight in grammes is the molecular weight divided by the valency of the ions.

[10] .e. $E$ is 96580 coulombs.

[11] The phenomenal of voltaic polarization was discovered by Ritter in 1803. Ritter explained it by comparing the action of the polarizing current to that of a current which is used to charge a condenser. Volta in 1805 put forward the alternative explanation, that the products of decomposition set up a reverse electromotive force.

[12] Phil. Trans. clxi (1871), p. 129.

[13] Comptes Rendus lxxvi (1873), p. 1407. Phil. Mag, xlvii (1874), p. 281. Ann. de Chim. et de Phys. v (1875), p. 494, xii (1877), p. 265.

[14] Lippman, Comptes Rendus, xcv (1882), p. 686.

[15] Trans. Conn. Acad. iii (1876-1878), pp. 108, 343; Gibbs' Scientific Papers, i, p. 55.

[16] This is embodied in equation (690) of Gibbs' memoir.

[17] Berlin Monatsber., 1881, p. 945; Wiss. Abl. i, p. 925; Ann. d. Phys. xvi. (1882), p. 31. Cf. also Planck, Ann. d. Phys. xliv (1891), p. 385.

[18] The conception of double layers of electricity at the surface of separation of two bodies had been already applied by Helmholtz to explain various other phenomena—e.g., the Volta contact-difference of potential of two metals, frictional electricity, and "electric endosmose," or the transport of fluid which occurs when an electric current is passed through two conducting liquids separated by a porous barrier. Cf. Helmholtz, Berlin Monatsherichte, February 27, 1879; Ann. d. Phys. vii (1879), p. 337; Helmholtz, Wiss. Abh. 1, p. 855.

[p380-19] E.g., that of Warburg, Ann. d. Phys. xli (1890), p. 1. In this it is assumed that the electrolytic solution near the electrodes originally contains a salt of mercury in solution. When the external electromotive force is applied, a conduction-current passes through the electrolyte, which in the body of the electrolyte is carried by the acid and hydrogen ions. Warburg supposed that at the cathode the hydrogen ions react with the salt of mercury, reducing it to metallic mercury, which is deposited on the electrode. Thus a considerable change in concentration of the salt of mercury is caused at the cathode. At the anode, the acid ions carrying the current attack the mercury of the electrode, and thus increase the local concentration of the mercuric salt; but on account of the size of the anode this increase is trivial and may be neglected.
Warburg thus supposed that the electromotive force of the polarized cell is really that of a concentration cell, depending on the different concentrations of mercuric salt at the electrodes. He found $dγ / dV$ to be equal to the amount of mercuric salt at the cathode per unit area of cathode, divided by the electro-chemical equivalent of mercury. The equation previously obtained is thus presented in a new physical interpretation.
Warburg connected the increase of the surface-tension with the fact that the surface-tension between mercury and a solution always increases when the concentration of the solution is diminished. His theory, of course, leads to no conclusion regarding the absolute potential difference between the mercury and the solution, as Helmholtz' does.
Alan electrode whose surface is rapidly increasing—e.g., a dropping electrode—Warburg supposed that the surface-density of mercuric salt tends to zero, so $dγ / dV$ is zero.
The explanation of dropping electrodes favoured by Nernst, Beilage zu den Ann. d. Phys. lviii (1896), is that the difference of potential corresponding to the equilibrium between the mercury and the electrolyte is instantaneously established; but that ions are withdrawn from the solution in order to form the double layer necessary for this, and that these ions are carried down with the drops of mercury, until the upper layer of the solution is so much impoverished that the double layer can no longer be formed. The impoverishment of the upper layer of the solution has actually been observed by Palmaer, Zeitsch. Phys. Chem. xxv (1898), p. 265; xxviii (1899), p. 257; xxxvi (1901), p. 664.

[20] Berlin Monatsber., 1877, p. 713; Phil. May (5) v (1878), p. 348; reprinted with additions in Ann. d. Phys. iii (1878), p. 201.

[21] Ann. d. Phys. iii (1878), p. 216.

[22] The formula given by Helmholtz was that the electromotive force of the cell is equal to $b (1 - n) v log(c 2 / c 1)$ , where $c 2$ and $c 1$ denote the concentrations of the solution at the electrodes, $v$ denotes the volume of one gramme of vapour in equilibrium with the water at the temperature in question, $n$ denotes the transport number for the cation (Hittorf's $1/ n$ ), and $b$ denotes $q \times$ the lowering of vapour-pressure when one gramme-equivalent of salt is dissolved in $q$ grammes of water, where $q$ denotes a large number.

[23] Zeitschrift für phys. Chem. i (1887), p. 631. Previous investigations, in which the theory was to some extent foreshadowed, were published in Bihang till Sveuska Vet. Ak. Förh. vii (1884), Nos. 13 and 14.

[24] Cf. p. 374.

[25] I.e. the ohmic specific conductivity of the solution divided by the number of gramme-equivalents of salt per unit volume.

[26] Examination of the spectra with higher dispersion does not altogether confirm this conclusion,

[27] Cf. pp. 382-383.

[28] Zeitschr. für phys. Chem. ii (1888), p. 613; iv (1889), p. 129; Berlin Sitzungsberichte, 1889, p. 83; Ann. d. Phys. xlv (1892), p. 360. Cf. also Max Planck, Ann. d. Phys. xxxix (1890), p. 161; xl (1890), p. 561.

[29] The correct law of dependence of the available energy on the temperature was by this time known.

[30] It will hardly be necessary to remark that this supposed direction of the current is purely conventional.

[31] Cf. van't Hoff, Svenska Vet.-Ak. Handlingar xxi (1886), No. 17; Zeitschrift für Phys. Chem. i (1887), p. 481.

[32] As follows from the expression obtained, supra, p. 383.

[33] Cf. p. 374.

[34] Cf. p. 241.

[35] Phil. Trans. xxiv (1705), p. 2165. Fra. Hauksbee, Physico-Mechanical Experiments, London, 1709.

[36] Phil. Trans. xlv (1748), p. 93, xlvii (1752), p. 362.

[37] Cf. ch. ii.

[38] Nollet, Recherches sur l' Electricité, 1749, troisième discours.

[39] Cf. p. 40.

[40] Phil. Trans., 1838; Exper. Res. i, § 1526.

[41] Cf. p. 44.

[42] Exper. Res., § 1623.

[43] Ann, d. Phys. ciii (1858), pp. 88, 151; civ (1868), pp. 113, 622; cv (1858), p. 67; cvii (1859), p. 77. Phil. Mag. xvi (1858), pp. 119, 408; xviii (1859), pp. 1, 7.

[44] Phil. Trans., 1821, p. 425.

[45] Ann. d. Phys. cxxxvi (1869), pp. 1, 197; translated, Annales de Chimie, xvi (1869), p. 437.

[46] Berlin Monatsberichte, 1876, p. 279.

[47] Proc. Roy. Soc. xix (1871), p. 236.

[48] Priestley in 1766 had shown that a current of electrified air tows from the points of bodies which are electrified either vitreously or resinously: cf. Priestley's History of Electricity, p. 591.

[49] Phil, Trans, clxx (1879), pp. 135, 641; Phil. Mag. vii (1879), p. 57.

[50] Phil. Mag. vii (1879), p. 57.

[51] Gött. Nach., 2 February, 1891; reprinted, Ann. d. Phys. xiii (1881), p. 191.

[52] Proc. Roy. Soc. Edinb. x (1880), p. 430.

[53] E.g. E. Wiedemann, Ann. d. Phys. x (1880), p. 202: translated, Phil. Mag. x (1880), p. 357. E. Goldstein, Ann. d. Phys. xii (1881), p. 249.

[54] Ann. d. Phys. xix (1883), p. 782.

[55] Nature, November 5, 1896; Fitz Gerald's Scientific Writings, p. 433.

[56] The penetrating power of the rays had been noticed by Hittorf, and by E. Wiedemann and Ebert, Sitzber, d. phys.-med. Soc. zu Erlangen, 11th December, 1891. It was investigated more thoroughly by Hertz, Ann. d. Phys. xlv (1892), p. 28, and by Philipp Lenard, of Bonn, Ann. d. Phys. li (1894), p. 225; lii (1894), p. 23, who conducted a series of experiments on cathode rays which had passed out of the discharge tube through a thin window of aluminium.

[57] J.J. Thomson, Recent Researches, p. 126.

[58] Cf. also G. Johnstone Stoney, Phil. Mag-, May, 1881.

[59] Journ. Chem. Soc. xxxix (1881), p. 277.

[60] Cf. p. 200.

[61] Annales de Chimie (3), lxi, p. 161.

[62] Ann. d. Phys. xvii (1882), pp. 1, 236, 519.

[63] Proc. Roy. Soc. xxxvii (1884), p. 317.

[64] In the case of an elementary gas, this would imply dissociation of the molecule into two atoms chemically alike, but oppositely charged; in electrolysis the dissociation is into two chemically unlike ions.

[65] Proc. Roy. Soc. xlii (1887), p. 371. Hittorf had discovered that very small electromotive forces are sufficient to cause a discharge across a space through which the cathode radiation is passing.

[66] Berlin Ber., 1887, p. 487; Ann, d. Phys. xxxi (1887), p. 983; Electric Waves (English ed.), p. 63.

[67] Cf. p. 357.

[68] Ry E. Wiedemann and Ebert, Ann. d. Phys. xxxiii (1898), p. 241.

[69] Ann. d. Phys. xxxiii (1888), p. 301.

[70] Proc. Roy. Soc. liv (1893), p. 389.

[71] Phil. Mag. xxxviii (1894), p. 358.

[72] The value found by the same investigator in 1897 was much larger than this.

[73] Comptes Rendus, cxxi (1895), p. 1130.

[74] Sitzungeber, der Würzburger Physikal.-Medie. Gesellschaft, 1895; reprinted, Ann. d. Phys. lxiv (1898), pp. 1, 12; translated, Nature, liii (1896), p. 274.

[75] Nature, January 23, 1896, p. 268. Fitz Gerald independently made the same suggestion in a letter to O. J. Lodge, printed in the Electrician xxxvii, p. 372.

[76] Ann. d. Phys, lix (1806), p. 321.

[77] Nature, September 3, 1896, p. 427: Proc. Camb. Phil. Soc. ix (1896), p. 215; Mem. Manchester Lit. & Phil. Soc. xli (1896-7).

[78] Journ. de Phys. v (1886), p. 354.

[79] Cf. p. 293.

[80] Proceedings of the Amsterdam Acad., March 25th, 1899 (English edition, i, p. 420), and September 27th, 1902 (English edition, v, p. 247).

[81] Nature, February 27, 1896, p. 391.

[82] J.J. Thomson and E. Rutherford, Phil. Mag. xlii (1896), p. 392.

[83] Trans. R. S. Edinb. xxx (1880), p. 337.

[84] Ann. d. Phys. xxxii (1887), p. 1.

[85] Proc. Roy. Soc., March 19, 1896; Phil. Trans., 1897, p. 265.

[86] A similar suggestion was made by E. Wiechert, Verhandl. d. physik.-öeon. Gesellsch. in Königsberg, Jan. 1897.

[87] Electrician, May 21, 1897.

[88] Phil. Mag, xliv (1897), p. 298.

[89] The value of $m / e$ for cathode rays was determined also in the same year ly W. Kaufmamn, Ann. d. Phys. lxi, p. 544.

[90] Verhundi, der physik. Gesells, zu Berlin, xvi (1897), p. 165; Ann. d. Phys. lxv (1898), p. 440.

[91] Berlin Sitzungsber., 1886, p. 691.

[92] Proc. R.S. xlvii (1890), p. 526.

[93] Phil. Mag. xlvi (1898), p. 120.

[94] Phil. Mag. xlvi (1898), p. 528.

[95] By E. Rutherfurd, Phil. Mag. xliv (1897), p. 422.

[96] Cf. p. 343.

[97] Comptes Rendus, cxxii (1895), p. 420.

[98] Ibid., cxxii (March 2nd, 1896), p. 501.

[99] By Schmidt, Ann. d. Phys., lxv (1898), p. 141; and by Madame Curie, Comptes Rendus, cxxvi (1898), p. 1101.

[100] Comptes Rendus, cxxvii (1898), p. 175.

[101] Ibid., cxxvii (1898), p. 1275.

[102] Ibid., cxxii (1896), p. 659.

[103] Phil. Mag. (5), xlvii (1899), p. 109.

[104] Giesel, Ann. d. Phys. lxix (1899), p. 834 (working with polonium); Bcequerel, Comptes Rendus, cxxix (1899), p. 996 (working with radium); Meyer and v. Schweidler, Phys. Zeitschr. i (1899), p. 113 (working with polonium and radium).

[105] Becquerel, Comptes Rendus, cxxix (1889), p. 1205); cxxx (1900), pp. 206, 372. Curie, ibid. cxxx (1900), p. 73.

[106] Comptes Rendus, cxxx (1900), p. 647.

[107] Comptes Rendus, cxxx (1900), p. 809.

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