370 



NATURE 



[February i8, 1892 



Mossotti's theory, which is the starting-point of all 

 mathematical theories of polarization, the conducting 

 particles are spherical, and therefore, if e be the ratio of 

 the volume of the spheres to the total volume of the 

 medium, the value of K is (i + 2f)/(i - e). It is here 

 assumed that the specific inductive capacity of the insu- 

 lating dielectric is unity. Poincard, however, sees no 

 reason for making this particular assumption, and takes 

 it as X, a quantity which, if Maxwell's theory be the true 

 one, must be exceedingly small. This involves, as already 

 stated, K = X(/)(€), where (^(e) is a quantity which be- 

 comes very great when e = i. Thus, according to 

 Maxwell's theory, the conducting particles are separated 

 by infinitely thin insulating partitions, so that they prac- 

 tically fill the whole space. Of course, the physical fact 

 may be very different from that here supposed : the theory 

 only furnishes a picture, not perhaps altogether clear and 

 intelligible, of the structure of the medium and its func- 

 tions. 



It may be said that the infinitely small inductive capa- 

 city, X, of the medium, itself requires physical explana- 

 tion. This is quite true ; but so also does the specific 

 inductive capacity equal to unity assumed for vacuum or 

 air in the ordinary theories. In fact, such dielectric 

 theories as have been put forward, involving merely 

 polarization of the medium, only give an explanation 

 of the difference between the electric behaviour of one 

 medium and another, and furnish none whatever of the 

 Teal ratiojtale of the propagation of electric action. 



That the value of (/> may be finite, it is necessary that 

 the values of the volume density, p, and the surface density, 

 <t\ may be infinitely small, since 



J \r 



dS. 



Here p is the volume density due to the surface distribu- 

 tions on the opposite faces of the partitions between the 

 conducting particles, and this, it is easy to see, will be 

 infinitely small. Also, a-' is the sum of the actual density 

 (surface density of charge) on the surface of the con- 

 •ductors, and the density, which is the surface manifesta- 

 tion of the polarization of the conducting particles, or 

 <r' = a- — {If -{• vig + nh). This also can be conceived 

 as exceedingly small, so that ^ may have a finite value. 



Further reasons for preferring the theory of Maxwell 

 are discussed in chapter vi., which is entitled "The Unity 

 of Electric Force." This chapter consists of an exposi- 

 tion of Hertz's modification of Maxwell's electromagnetic 

 theory — a modification, it is to be remarked, practically 

 given also, but in vector form, by Mr Oliver Heaviside, 

 in various papers in the Philosophical Magazine. When 

 made, it exhibits a striking parallelism between the equa- 

 tions of electric and magnetic force, and leads to some 

 remarkable theorems. Using Maxwell's equations, and 

 deviating slightly from Poincar^'s mode of presenting the 

 equations, we have, if k now denote electric conductivity 

 of the medium, and P, Q, R components of electric force — 



C,,+ K d\^^.(dy _d_^\ 

 \ 47r dt ) 4ir \dy dzj 



■with two similar equations for Q and R. But also we 

 have — 



M cla ^_ I /a'R _ ^\ 



4ir dt 47r \ (/j/ dz J 



with two similar equations for ^ and y. These last may, 



by the introduction of a non-existent quantity, g, be 



written — 



\ /\TT dt J i\ir\dy dz J 



&c., &c. 



The quantity g, Heaviside points out, is the proper 

 magnetic analogue to k, and may therefore be called 

 the magnetic conductivity. Its reciprocal would be the 



NO. I 164. VOL. 45] 



true magnetic resistivity of the medium. Of course, in 

 an insulating medium k is also zero. 



According to Maxwell's theory, P, O, R, a, /3, y fulfil 

 the equations — 



'^ + '!^ + ''"^ = o, 



dx dy dz 



<i± j^d^ j^ dy 

 . dx dy dz 



= o ; 



the first, since dujdx -\- dvjdy -f- dwjdz = o, and the 

 second because the magnetic force in the medium, being 

 supposed purely inductive, nmst fulfil the solenoidal con- 

 dition, except at the (vortex) origin of the disturbance. 



There is therefore, in Maxwell's theory, a perfect reci- 

 procity of relation between the electric and magnetic 

 quantities. Hence we might infer, from the magnetic 

 phenomena following from electric currents or flow of 

 electricity, an analogous set of electric phenomena follow- 

 ing from the flow of magnetism. Now we know that if 

 a magnet varies in strength it produces an electromotive 

 force of components P, O, R, at every point of the sur- 

 rounding space. This we may suppose due to a current 

 of magnetism flowing from one end of the magnet to the 

 other, and thus producing the variation in the magnet's 

 strength. The directions of the components of electric 

 force at any point are in fact coincident with those of the 

 components of vector potential produced by the magnet at 

 that point, and are equal to the time-rates of variation of 

 these components. But why should this not be regarded 

 as an electrostatic field in the ordinary sense of the term ? 

 F"or example, a current of electricity, flowing round a 

 closed circuit, produces a magnetic field equivalent to that 

 which would be produced by a magnetic shell of proper 

 strength, and having its edge coincident with the circuit. 

 Of this current a closed solenoid varying continuously in 

 magnetic strength (for example, a closed solenoid in which 

 the magnetizing current is varying m strength) is the 

 magnetic analogue, and ought in the same way to be 

 equivalent to an electric shelly in the sense of producing 

 an identical electric field. Such a shell ought to be sub- 

 ject in an electric field to dynamical action ; and further, 

 two such varying solenoids ought to exert the same mutual 

 dynamical action, as would the two equivalent electric 

 shells if placed in the same configuration. The second 

 of these conclusions asserts that the dynamical action on 

 such a shell depends only on the electric field in which 

 it is placed, and that its action on the other varying 

 solenoid is due to its producing exactly the same electric 

 field as the equivalent electric shell would produce. This 

 is what Poincare gives as Hertz's principle of the unity of 

 electric force. 



Of course, it is to be noticed that the second conclusion 

 does not follow from the first. We cannot reason that 

 because the mutual action of an electric shell and a vary- 

 ing solenoid is the same as that of two electric shells, 

 therefore the mutual action of two solenoids is the same 

 as that of two electric shells. 



If, however, we assume that the dynamical action on 

 a closed varying solenoid depends only on the electric 

 field in which it is placed, we can say that the mutual 

 action of two varying solenoids is the same as that of their 

 equivalent electric shells. 



M. Poincare calculates the work done in effecting a 

 relative displacement of two such varying solenoids, and 

 finds that it is equal to the change in the electrostatic 

 energy of the system, as the change in the electrokinetic 

 energy is all accounted for otherwise. Now, the electro- 

 static energy of a system is given as we have seen by the 

 equation — 



where 



"=/{sM2)'^K-^'x^'^''}^^' 



(K - A)/47r . dYldt + d<^ldx. 



