Sivut kuvina
PDF
ePub
[blocks in formation]

Obviously, we may derive equation (10) by an exactly similar process in which the terms involving G do not enter. And if we wish to use an infinitesimal charge of some other shape, we may consider it as divided up into a number of cylinders, not necessarily right cylinders, such as we used above.6

Meanings of the Laws.- To find out what we can about the properties of the ether, we may now examine carefully the meanings of these five laws:

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors]

The first two of these laws contain no reference whatever to time, and deal with quantities whose existence is in no way dependent on motion or change with time. Therefore, we may infer that they probably express relations between the geometrical configurations of different parts of the ether, and show the dependence of these geometrical configurations upon the presence in the ether of the peculiar movable spots called charges, whose indestructibility and ability to be located definitely at different times (specified in equations (3) and (4), as well as the internal forces, suggest that they are due to the presence of some substances not present in the rest of the ether but freely movable through it. Since these substances can be located at any time if the vectors E and E are known at every point, the question arises whether any more information than the value of these vectors needs to be given to determine completely the configuration of the ether. A suggestion of the answer to this question is given by the fact that in applying Hamilton's Principle to problems of ordinary dynamics, the variations must be such as to give the actual configura

6 To be certain that no equations not derivable from equations (1)–(10) can be derived from (11) and (1)−(4), we need only to consider the facts that any possible variation in equation (11) can be made up of variations of the types treated above, and that the mutual energy of two independent variations of the first order is an infinitesimal of the second order.

tions exactly at the times t1 and t2, whereas, in equation (11) they must

+

be such as to give the actual vectors E and E. Hence, from analogy, we may say that these vectors are probably sufficient to specify the configuration of the ether completely.

And if this last statement is true, their time derivatives must be

+

sufficient to specify completely, not only the quantities ẞ and ẞ, but all the motions of the ether; and it seems probable that these motions at any point are specified by the values of E, E, p ẞ, and pẞ at that

+

++

point, and not by the values of the vectors H and H, which depend on the values of the other vectors at distant points. This hypothesis is further strengthened by the fact that the whole theory of the ether might be developed without any use of these vectors, replacing H wherever it occurs by

1

++

1

VxPot (E+ pẞ), and H by VxPot (EpB),

+

and therefore without any use of equations (3) and (4), except as they express the indestructibility of the charges.

Therefore we may consider equations (3) and (4) as merely equations of continuity and partial definitions of two convenient mathematical functions fully defined by equations (3) and (4) and (11) all together, and whose values at any point depend on the motions of the ether at all points, but not in any way on the motions or configurations at the point in question only. And thus, although they contain time derivatives and quantities dependent entirely on motion and existing only when there is motion, they tell us nothing about what is going to happen at some future time from what is happening now, and therefore cannot be considered as laws of motion, but only as mathematical definitions of convenient functions.

Equation (11), however, in form and substance, is essentially an equation of motion, from which no information about the geometrical configurations of the ether can be derived at any time, unless the configuration and motion at some other time, or the configurations at two other times, are specified; but without which no information about the configuration or motion at any time can be derived even if they are given at any number of other times.

[ocr errors]

Properties of the Ether. The first question that arises about the properties of the ether is, Is its structure continuous or granular?

To answer this question definitely seems impossible, but at any rate, we can say that if it is granular and if these equations are to hold, the structure must be exceedingly minute compared to the dimensions of the electrons. A further suggestion is given by the fact that in the geometrical equations, (1) and (2), the positive and negative quantities appear very similar, but seem to be more or less independent of each other; while in the equation of motion (11), and in the phenomena of vacuum tube discharges, etc., differences between the actions of the positive and negative quantities appear, that seem to show that not only are the electrons of the different signs made up differently, but that the forces are transmitted by more or less independent, and slightly different, structures in the medium. As this condition of affairs seems to be incompatible with the idea of a continuous medium we are thereby led to the conception of a medium in which there are probably two similar, but slightly different, interlacing, granular structures, whose grains and distances between them are inconceivably small, even compared to the electrons.

The question of solid or fluid character of the ether appears easier to answer; for if it were fluid, that is, if no amount of shear at any point would change the properties at that point in such a way as to affect the subsequent motions around it, a transverse wave would be impossible. And if it were quasi-elastic, with effects analogous to viscosity, that would enable it to transmit wireless telegraph waves as well as the shortest known light waves, electrostatic forces around stationary charges should be due to some effect entirely different from that which produces those of the wireless wave, so that slow continued flow of ether might occur without hindrance. But the changes of electric force near a moving electron may be much more rapid than those of the wireless wave, and yet there appears to be no viscous retardation of its motion. Furthermore, the aberration of light and experiments such as that of H. A. Wilson7 on the polarization of a dielectric cylinder rotating in a magnetic field seem to show that no flow of ether occurs in moving matter. These considerations and many others compel us to reject the fluid theory, and to say that the structures of the ether are solid. But by "solid" we must not mean possessed of ordinary solid elasticity, but merely that every particle is permanently connected to the particles near it by connections that cannot be deformed indefinitely, or even by a finite amount without affecting the subsequent motion.

7 H. A. Wilson. "Electric Effect of Rotating a Dielectric in a Magnetic Field," Roy. Soc. Proc., 73, pp. 490-492. June 22, 1904.

As we must not assume ordinary elasticity, so also we must not assume ordinary inertia of the fundamental particles. For, after all, Newton's laws of motion, that we observe for ordinary matter, appear to be only approximations to the laws that result from equation (11), the more general law of motion. And furthermore, they are by no means the only ones consistent with the relative nature of time and space, nor is there any other a priori philosophical reason for assuming that they are true, while there is good philosophical reason for assuming that Hamilton's Principle, the mathematical expression of the perfect efficiency of the fundamental machinery of nature, is at least plausible. Therefore, whatever motions of the parts of the ether it may involve, and whether or not it is easy for us, with our Newtonian mechanical training, to form a mental picture of the dynamics of these motions, the fundamental law of the dynamics of the ether, or of any mental picture of it, must be Hamilton's Principle.

A Model of the Ether. To get a mental picture of the actions of the ether, we must now make some arbitrary assumptions as to the nature of the two interlacing structures and the strains in them that

+

+

are represented by the vectors E and E. For simplicity we may think of them as nets with cubical meshes with each knot of either net in the centre of a mesh of the other, wherever the electric vectors are zero. The vector E may be a very minute displacement of one of these nets from this position, and the vector E the negative of a similar displacement of the other. If we now suppose the strings of these nets to be hollow and rigid, and the knots to be hollow boxes, so constructed that the displacements of the nets will be those of an incompressible substance, we may suppose an electric charge to be a region in which the pipes and boxes of one of the nets are filled with a liquid of high surface tension, that will expand the boxes into which it flows, and cause a divergence of the displacement of the net. An electron will then be a region of this sort, in the shape of a hollow sphere when at rest, of which every dimension, including the thickness, is very large compared with the meshes of the net. The pipes and boxes of that net that lie inside this region may be filled with a fluid whose only properties are adhesion with everything it touches and a constant hydrostatic tension, independent of its volume. For the connections between the nets we may assume anything we please.

Equations (1) and (2) are satisfied by this model, which also gives an interesting interpretation for (3) and (4). For in free ether the

+

vector H becomes a hydrokinetic flow-function for the motion of the positive net; and where there is any positive charge it is a flow-function for the motion of the net plus that of the charge. Similarly the vector H is the negative of a flow-function of the motion of the negative net and charges. And in each case, equations (5) and (6) tell us that it is the solenoidal flow-function that is required.

The equation of motion is, as we expected, one which we have some difficulty in applying. But if we split it up into equations (5) to (10), and then combine them properly, we may use in electrical problems only the vectors E and H, representing the relative displacement of the positive net from the negative, and the flow-function of the relative motion. And in gravitational problems the vectors E and H disappear entirely.

Collisions of Electrons. An interesting application of this model is to the problem of collisions of electrons, of the same or opposite signs, as in the case of a cathode particle striking an electron in the metal it hits. If they are of the same kind they will evidently become flattened as they come together. But as soon as they are within about their own length of each other, the side of either of them nearest the other will be effected not only by the displacement due to the presence of the other, but also by the displacements radiated from the other on account of its acceleration. To make the vectors balance, as required by equations (9) and (10), its acceleration must therefore be so much greater than that required by the inverse square law that they can never collide.

In the case of two electrons of different kinds, both are lengthened, and they come together faster than the inverse square law would demand. But since they may go right through each other perfectly freely, there need not be any of the destructive effects that one might expect from other theories.

8

Retarded Potentials. In calculating the values of the retarded potentials due to moving electrons it is found necessary to treat each electron as if its charge were not the same as when at rest, but changed in the ratio (1-B,) ̄1, where ẞ, is the component of ẞ in the direction towards the point at which we wish to know the potential. This has been interpreted by some writers 9 as indicating that all electromagnetic actions are due to some sort of pulsation of the electrons, and are

8 For information about retarded potentials, see Lorentz, "Theory of Electrons," Chap. 1.

9 L. de la Rive, Phil. Mag., 13, p. 279.

« EdellinenJatka »