In conclusion a few words must be said with regard to the results obtained thus far. We have found that by resolving the observed magnetic field into two, one polar and symmetri. cal about the rotation axis, the other the residual field, that this latter is apparently directed approximately equatorially. It has been amply demonstrated, I believe, that this mode of resolution is a very convenient and promising one for the study of the complex phenomena of the distribution and the secular variation. This breaking up of the total field into two has thus far been wholly arbitrary, however. To assert the actual existence of the secondary equatorial field we must endeavor to deduce it theoretically in some manner. We must show that there is some cause for such a polarization. For example, suppose we have given the primary polar field, which in fact we know to exist, though in a more complex form, can we deduce from the motions of the earth a secondary equatorial polarization as the result of self-induction? If so, what conditions of the physical constitution of the earth must we presuppose in order to get such an unsymmetrical equatorial polarization as our residual field reveals? That is, why are the secondary magnetic dip poles but 90° different in longitude?* It is evident that the solution of this question is likely to throw some light upon the secrets hidden in the bowels of the earth. It will be interesting to see whether we shall reach the same result obtained by Prof. Bigelow in a totally different manner, viz., that the earth is a magnetic shell of the thickness of about 790 miles.+

Finally, can the secular variation be explained by the reflex action of the secondary field upon the primary? This is the nature of the questions that will serve as the subject of a systematic examination in subsequent numbers.

Chicago, Aug. 15, 1895.

*The two foci obtained by constructing "the isanomalous lines of geomagnetic potential" are according to Prof W. von Bezold about 180° apart in longitude. See foot-note page 200, of previous number.

F. H. Bigelow: The Earth a Magnetic Shell, this Journal, Art. VIII, August, 1895.

AM. JOUR. SCI.-THIRD SERIES, VOL. L, No. 298.-OCTOBER, 1895.

ART. XXXVI.-Studies in the Electro-magnetic Theory.-I. The law of electro-magnetic flux ;* by M. I. PUPIN, Ph.D., Columbia College, New York.

THE law of electro-magnetic flux is a short expression for the well-known quantitative relations between electromotive force and electric flux on the one hand and magnetomotive force and magnetic flux on the other. Ohm's law is a part of it.

These relations are statements of experimental facts which we know to hold true for constant and slowly varying forces. The object of this investigation is to show the exact position which this law occupies in Maxwell's electro-magnetic theory; to point out its limitations; to show that Maxwell's electromagnetic theory of light demands a more general form of this law; and finally, to present a general form of this law of which both its ordinary form and also those forms which were assumed hypothetically in some of the recent developments of the electro-magnetic theory of light are special cases.

1. The two fundamental laws of Maxwell's Electro-dynamics. A brief statement of Maxwell's electro-magnetic theory seems desirable in this discussion. For the sake of brevity none but its most essential features will be presented, and that in such a way as to emphasize as forcibly as possible the essential differences between this theory and the older electro-magnetic theories.

The essential features of the Maxwellian theory are reducible to two, which may be called its two characteristic features. These two characteristic features can be exhibited in a very simple manner by considering the gradual change in form and in meaning which the following two well known experimental laws, relating to magneto-electric and to electro-magnetic induction, undergo as we pass from the views of old electric. theories to those of the Maxwellian theory. These laws I choose to state in the following form for reasons which will be evident presently:

First law:-A varying magnetic field induces a field of electric force in all electric conductors within its region. The electromotive force around any simple circuitt of this induced

* Read in abstract before the Am. Assoc. Adv. Science at its Springfield meeting, Aug. 30, 1895.

The expression simple circuit needs an explanation. Consider any point of the field. Pass a plane through it and in this plane draw an infinitely small area around the point under consideration. If the boundary of this elementary area be such that none of its points contain more than one branch of the boundary curve, then this boundary curve is a simple circuit around this point.

electric field is proportional to the rate of variation of the magnetic flux through any surface bounded by this circuit.

Second law:-An electric current induces a field of magnetic force

The magnetomotive force around any simple circuit of this induced magnetic field is proportional to the current passing through any surface bounded by this circuit.

For the sake of brevity the factors of proportionality are not explicitly mentioned. It is well to observe, however, that they are numerical and that by a suitable selection of electric units they could be made unity in each case.

There is evidently a considerable formal resemblance between these two experimental laws when they are stated in the form in which they have just been stated. This formal resemblance can be carried considerably further without departing from the views of pre-Maxwellian electric theories. All that is needed is a simple adjustment of the electric terminology of the preMaxwellian period. This is the step which is now in order. It leads along the shortest path from old theories to Maxwell's electro-magnetic theory.

Consider two electric conductors A and B, insulated from each other, one charged with a certain quantity of positive electricity and the other with an equal charge of negative electricity. Consider now a surface surrounding any one of these two conductors. The total electric flux through this surface is numerically equal to the total quantity of electricity enclosed by the surface. Hence it is also equal to the total integral electric current which would be obtained if the total positive charge were carried across this surface to the negatively charged conductor and thus the two conductors reduced to their neutral state. We can speak, therefore, of a total electric flux across a surface as of a fictitious integral current through this surface. We can speak of it also as of a total electric displacement in the sense that if the two conductors are neutral and a certain quantity of positive or negative electrification is transferred from one to the other and therefore displaced from one side of the bounding surface to the other we shall have a total electric flux across this surface numerically equal to the total electric displacement across the same surface. With this mental reservation, namely, that we are using certain terms in their figurative sense only, we can always employ the expressions "total electric flux," "total integral current," and "total electric displacement" as synonymous terms without departing from the views of pre-Maxwellian theories.

So far we have not restricted the fictitious total integral current or displacement to any particular paths. That shall be done now.

Let these paths be such that at any moment the total fictitious electric transference through any elementary area up to that moment is equal to the electric flux through that area at that moment. Then, as long as we remember that we are speaking of a fictitious "integral current" and a fictitious "electric displacement" through any elementary area in an electric field, we can employ these two terms as synonymous with "electric flux" through that area without departing from the views of the pre-Maxwellian period.

Again just as the electric current through any elementary area of a conductor is defined as the rate of variation of the integral current, so we can also speak of a current through the dielectric without deserting the views of old electro-magnetic theories provided that we take it as granted that this dielectric current is fictitious, since it is the rate of variation of a fictitious integral current. The expression "dielectric or displacement current" becomes, therefore, synonymous with rate of variation of the electric flux. Similarly we can substitute the expression magnetic current for the expression "rate of variation of magnetic flux."

To distinguish the real, that is the electric conduction current, from the fictitious or displacement current, the expression "conduction current" must be used when the real and not the fictitious (displacement) current is meant.

This precaution is unnecessary in the case of the magnetic current, since there is no magnetic conduction current.

The laws of magneto-electric and of electro-magnetic induction can now be stated more symmetrically, as follows:First law-A region of magnetic.

currents induces

a field of electric force in all electric conductors within that region. The electromotive force around any simple circuit in this induced electric field is proportional to the magnetic current passing through any area which is bounded

by this circuit.

Second law:-A region of electric conduction currents induces a field of magnetic force The magnetomotive force around any simple circuit in this induced magnetic field is proportional to the electric conduction current passing through any area which is bounded by this circuit.

The formal resemblance between the two laws is very striking now. It would be perfect if we either omitted the words which are in italics or filled out suitably the dotted lacunæ.

The last alternative is not admissible, because we know nothing about magnetic conduction currents, nor about magnetic conductors. The first alternative does not strike one so unfavorably. There is really no evidence against the permissibility of omitting the words in italics from the statement of the

first law. It is true that Faraday's experiments, by which he discovered that law, prove the existence of no other induced electromotive forces excepting those which Faraday detected in conductors. But these experiments neither affirm nor do they deny the presence of magneto-electric induction in dielectrics. On the other hand, Faraday's experiments on dielectric and diamagnetic substances and his speculative views' of electro-magnetic phenomena urge us with an irresistible force to the belief that electromotive forces just like magnetomotive forces are induced around every circuit, no matter whether that circuit pass through a conductor or through a dielectric (including the most perfect of all dielectrics, that is a perfect vacuum), and that just as the magnetic displacement current is real in the sense that it produces inductive effects, so the electric displacement current is not a fiction, as one who is faithful to the views of older electric theories has to assume, but it is just as real as the electric conduction current in the sense that it is an actually existing process in the dielectric, which is just as capable of inducing magnetomotive forces as the conduction current. The mechanism of this process is, of course, just as unknown to us as the mechanism of that process which is called the electric conduction current. Maxwell was the first to feel the force of this tendency of Faraday's experiments and speculations and to yield to it, making thus a radical departure from the views of old electric theories. His statement of the two laws of induction omits, therefore, the words which are in italics in the last statement of these laws. Hence the following wording of these two laws is in accordance with Maxwell's views:

First law-Every magnetic current induces a field of elec tric force. The electromotive force around any simple circuit in this induced electric field is proportional to the magnetic current passing through any area which is bounded by this circuit.

Second law :-Every electric current induces a field of magnetic force. The magnetomotive force around any simple circuit in this induced magnetic field is proportional to the electric current passing through any area which is bounded by this circuit.

In this generalized form these two laws form the foundation of Maxwell's electro-magnetic theory.* This theory may, therefore, be described broadly as that theory which generalizes the two experimental laws of magneto-electric and of electro-magnetic induction by extending the region of magneto-electric induction and of the electric current from conductors to the dielectric.

*See O. Heaviside, Phil. Mag., February, 1888; H. Hertz, Wied. Ann., xxiii, p. 84, 1884; xl, p. 577, 1890.