accuracy. The earlier experiments on nitrogen showed that a very slight trace of oxygen was sufficient to raise the kathode fall from its true value 232 to 355. In fact, the values for oxygen, nitric oxide, air, and nitrogen with a trace of oxygen, are all nearly the same, which makes it not improbable that in each case the oxygen alone acts as the carrier of the current. We have, then, finally the following values for the kathode fall— The last is enclosed in brackets in consequence of the doubts as to its accuracy. If we leave the result for this last gas out of account, it appears that the kathode fall is approximately an additive quantity. Ascribing the values 149, 116, and 184 respectively to the atoms of hydrogen, nitrogen, and oxygen, we get, by addition, 482 for water vapour and 563 for ammonia. As each of these depends on three measurements, they may be taken as agreeing with the observed values within the limits of experimental error. Hence, so far as the evidence of these experiments goes, the kathode fall is a property of the atoms rather than of the molecule. As the kathode fall is constant for all pressures and currents whilst the potential gradient along the rest of the tube is variable, we may infer that no potential difference less than the kathode fall is capable of causing a discharge through the gas. This conclusion is consistent with the experiments of Mr. Peace,* who found that the minimum difference of potential that gives a discharge in air is something over 300 volts. Assuming that the conduction is electrolytic, it seems likely, from the analogy of the electrolysis of liquids, that the kathode fall may prove to be a measure of the energy required to dissociate the gas into the ions that carry the electricity, and the present experiments were undertaken in the hope of finding some confirmation of this hypothesis. They have not, however, provided the kind of evidence that was anticipated. The results can only be reconciled with the hypothesis if further assumptions are made that would put the conduction in gases on a very different footing from the electrolytic conduction of liquids. The additive nature of the kathode fall might, for instance, be taken as an indication that the carriers of the current are provided by the disintegration of the atoms into much *Roy. Soc. Proc.,' vol. 52, p. 99. smaller particles, as has already been suggested by J. J. Thomson from entirely different evidence; but the results are too few to make further speculation on their meaning of much value. "Note on the Complete Scheme of Electrodynamic Equations. of a Moving Material Medium, and on Electrostriction." By JOSEPH LARMOR, F.R.S., Fellow of St. John's College, Cambridge. Received May 17,-Read May 26, 1898. This note forms a supplement to my third memoir on the "Dynamical Theory of the Ether," to the sections of which the references are made. 1. It is intended in the first place to express with full generality the electrodynamic equations of a material medium moving in any manner, thus completing the scheme which has been already developed subject to simplifying restrictions in the memoirs referred to. To obtain a definite and consistent theoretical basis it was necessary to contemplate the material system as made up of discrete molecules, involving in their constitutions orbital systems of electrons, and moving through the practically stagnant æther. It is unnecessary, for the mere development of the equations, to form any notion of how such translation across the æther can be intelligibly conceived: but, inasmuch as its strangeness, when viewed in the light of motion of bodies through a material medium and the disturbance of the medium thereby produced, has often led to a feeling of its impossibility, and to an attitude of agnosticism with reference to æthereal constitution, it seems desirable that a kinematic scheme such as was there explained, depending on the conception of a rotationally elastic æther, should have a place in the foundations of æther-theory. Any hesitation, resting on à priori scruples, in accepting as a working basis such a rotational scheme, seems to be no more warranted than would be a diffidence in assuming the atmosphere to be a continuous elastic medium in treating of the theory of sound. It is known that the origin of the elasticity of the atmosphere is something wholly different from the primitive notion of statical spring, being in fact the abrupt collisions of molecules: in the same way the rotational quality of the incompressible æther, which forms a sufficient picture of its effective constitution, may have its origin in something more fundamental that has not yet even been conceived. But in each case what is important for immediate practical purposes is a condensed and definite basis from which to develop the interlacing ramifications of a physical scheme: and in each case this is obtained by the use of a representation which a deeper knowledge may after 'Phil. Trans.,' A (1897). wards expand, transform, and even modify in detail. Although, however, it is possible that we may thus be able ultimately to probe deeper into the problem of æthereal constitution, just as the kinetic theory has done in the case of atmospheric constitution, yet there does not seem to be at present any indication whatever of any faculty which can bring that medium so near to us in detail as our senses bring the phenomena of matter: so that from this standpoint there is much to be said in favour of definitely regarding the scheme of a continuous rotationally elastic æther as an ultimate one. A formal scheme of the dynamical relations of free æther being postulated after the manner of Maxwell and MacCullagh, and a notion as clear as possible obtained of the æthereal constitution of a molecule and its associated revolving electrons, by aid of the rotational hypothesis, it remains to effect with complete generality the transition between a molecular theory of the ethereal or electric field which considers the molecules separately, and a continuous theory expressed by differential equations which take cognizance only of the properties of the element of volume, the latter alone being the proper domain of mechanical as distinct from molecular theory. This transformation is, as usual, accomplished by replacing summations over the distribution of molecules by continuous integrations over the space occupied by them. In cases where the integrals concerned all remain finite when the origin to which they refer is inside the matter so that the lower limit of the radius vector is null, there is no difficnlty in the transition: this is for example the case with the ordinary theory of gravitational forces. But in important branches of the electric theory of polarised media, some of the integral expressions become infinite under these circumstances; and this is an indication that it is not legitimate to replace the effect of the part of the discrete distribution of molecules which is adjacent to the point considered by that of a continuous material distribution. The result of the integration still, however, gives a valid estimate of the effect of the material system as a whole, if we bear in mind that the infinite term coming in at the inner limit really represents a finite part of the result depending solely on the local molecular configuration, a part whose actual magnitude could be determined only when that configuration is exactly assigned or known. The consideration of this indeterminate part is altogether evaded by means of a general mechanical principle which I have called the principle of mutual compensation of molecular forcives. This asserts that in such cases, when a finite portion of the effect on a molecule arises from the action of the neighbouring molecules, this part must be omitted from the account in estimating the mechanical effect on an element of volume of the medium; indeed otherwise mechanical theory would be impossible. The mutual, statically equilibrating, actions of adjacent molecules determine the structure of the medium, and any change therein involves change in its local physical constants and properties, which may or may not be important according to circumstances: but such local action contributes nothing towards polarising or straining the element of mass whose structure is thus constituted, and therefore nothing to mechanical excitation, unless at a place where there is abrupt change of density. In the memoir above mentioned this molecular principle was applied mainly to determine the mechanical stress in a polarised material medium. It necessarily also enters into the determination of the electrodynamic equations of a moving medium treated as a continuous system, and even of a magnetised medium at rest, from consideration of its molecular constitution. It is here intended only to record in precise form the general scheme that results from it, details of demonstration being for the present reserved. Everything being expressed in a continuous scheme per unit volume, let (u', v', w') denote the current of conduction, (u, v, w) the total current of Maxwell, (f, g, h) the electric displacement in the ether and (f, g, h) the electric polarisation of the molecules so that the total so-called displacement flux of Maxwell is (f+f', g+g', h+h'); let p be the volume-density of uncompensated electrons or the density of free charge, let (A, B, C) be the magnetisation, and (p, q, r) the velocity of the matter with respect to the stagnant æther. As before explained (§ 13, footnote), the convection of the material polarisation (f', g', h') produces a quasi-magnetisation (rg'-qh', ph'—rf', qƒ'—pg') which adds on to (A, B, C). Also, as before shown, the vector potential of the æthereal field, so far as it comes from the molecular electric whirls which constitute magnetisation, is given, for a point outside the magnetism, by (Imn) being the direction vector of S, and therefore is that due to sheets on the interfaces. When the point is inside the magnetism, there are still no infinities in the integral expressing F, and this transformation of it by partial integration is still legitimate. But This exception explains why the mechanical tractions on an interface, determined in § 36 as the limit of a gradual transition, are different from the forces on the Poisson equivalent interfacial distribution. the spacial differential coefficients of (F, G, H) are also involved in the forcives of the ethereal field, and with them the case is different the transformation by parts is then analytically wrong, owing to neglect of the infinite elements at the origin, while in actuality a finite portion of the whole effect arises from the influence of the neighbouring molecules. We have, therefore, by the molecular principle, to separate the infinite elements from the integrals and leave them out of account; and this is effected by employing the second form above for F, which differs from the first form only in having got rid of the local terms at the origin in its differential coefficients. Thus it is not merely convenient, but even necessary for а mechanical theory, which considers distributions instead of individual molecules, to replace magnetism by its equivalent continuous current system as here. The quasi-magnetism arising from electric convection adds to this equivalent current system the additional bodily terms together with surface sheets: thus the volume current so added has for x-component if dt dt dx dy dz) - ƒ dp - gdp df' dpf' dqf' dry dy where represents + + + or the rate of change of dt dx dy dz dt f' supposed associated with the moving matter. Combining all these parts, the current and magnetism together are completely represented as regards determination of electric effect by what we may call the total effective current (u, v1, w1) where u1 = u' + - + + dc dB df, of' (dpf', dpg', dph" together with superficial current sheets arising from the true magnetism (A, B, C) and the electric convection. equal to d(f+f') d(g+g') d(h+h') Since p is + + da dy dz |