February iS, 1892] 



NATURE 



369 



■y' is flowing. Now, if in the insulating dielectric two 

 like quantities of electricity, .?, e\ would produce a re- 

 pulsive force of magnitude ee'lXr'-, the electrostatic 

 potential ^, due to the free electricity of density p, is 

 given by the equation- 



\<t> = p'. 



(/tS'/t 



Thus, from the foregoing values of F, G, H, three 

 equations are obtained, viz., 



V'^'F = - 47r« + (I - k)\dmJxdl, 



with two similar equations for v-G and v-H. These 

 give 



uYldx + dGldy + dnidz = - k\d<^ldt, 

 which is zero if ^ or X be zero. 



If, now, the magnetic inductive capacity be taken for 

 the moment as unity, and the magnetic force (a, ^3, y) be 

 so defined that T is y times the magnetic induction 

 through the circuit in which the current whose numerical 

 magnitude is y flows, we deduce easily the equations— 



a = d\{ldy - d(^ldz, &c. ; 

 so that there follow 



dy _d$ __ 

 dy dz 

 &c.. 



dxdt 

 &C. 



These coincide with the equations of currents given by 

 Maxwell when the last terms are omitted. We must 

 therefore either put X = o, or (since d(^\dt is not in general 

 zero) = o to reduce to Maxwell's theory. 



Poincare next proves that unless the k in Helmholtz's 

 theory be not less than zero, the sum of the electrostatic 

 and electrokinetic energies may diminish indefinitely 

 from an infinitely small value, so that there would be 

 unstable equilibrium. This affords another reason for 

 rejecting the theory of Weber, in which k = - \. 



In chapter v. Poincard passes from the theory of Helm- 

 holtz to that of Maxwell. He first considers magnetic 

 and dielectric polarization according to a modified and 

 corrected version of the theory of Poisson. He supposes 

 the dielectric space filled with conducting particles sepa- 

 rated by other material, the dielectric proper, which com- 

 pletely insulates these bodies from one another. These 

 conducting bodies are supposed to be electrically polar- 

 ized, so that electric displacement (/, g, h) is set up in 

 the medium. A parallel theory of magnetic polarization 

 is considered, and the electric displacement is simply the 

 electric analogue of the intensity of magnetization of the 

 medium — that is, the magnetic moment at each point per 

 unit of volume. If « be the ratio of the volume occupied 

 by the conducting particles to the whole volume of the 

 dielectric space in which they are embedded, the specific 

 inductive capacity, K, of the medium is found to beX(^(e), 

 where (/)(e) is a function analogous to (i + 2e)/(i - e) (the 

 corresponding function for the case in which the conduct- 

 ing particles are spheres) in that it becomes infinite when 

 f = I. 



According to Poincard, X is the specific inductive 

 capacity of the dielectric medium proper, or insulator 

 between these conducting bodies, and is very small. In 

 order, therefore, that K may be finite, it is necessary that 

 f may be very nearly equal to unity. 



The electrostatic potential, 0, at any point is that due 

 to the free electricity present on conductors, and to the 

 electricity developed throughout the medium by its 

 polarization. The electrification, in fact, consists of two 

 parts— a volume density on the dielectric depending on 

 the electric displacement, and of amount. - {d/l dx -\- 

 <^.^ldy -f dhjdz), and a resultant surface density, <t' = (t — 

 {Jf-\-mg-\-7ih), where o- is the surface density of the 

 electricity present in the form of charges on conductors, 

 and If -\- mg -\- 7ih (in which /, m, n are the direction 

 cosines of the normal to the surface directed inwards 

 NO. I 164, VOL. 45] 



towards the dielectric) is the surface distribution due to 

 the polarization. By speaking of this as the electrifica- 

 tion, what is meant is that this electrification existing in 

 the dielectric medium proper would give the observed 

 potential, <^, at each point. Thus 



^-A\?r^\''Vv 



The value of the electrostatic energy, U, according to 

 this dielectric theory, is given by 



If there are applied electromotive forces, X, Y, Z, their 

 values are given by 



with similar equations for Y and Z, Hence, if X, Y, Z=o 

 —that is, if there is nothing but electrostatic action— the 

 electrostatic energy is given by 



If X, Y, Z are not zero, the electrostatic energy becomes 



27r/K. j 2(/2)rt'a! (which is Maxwell's expression), provided 



X = o. We have to inquire what reasons can be adduced 

 for puttmg X = o in this theory. 



M. Poincard shows that the velocity of propagation of 

 a wave of [ongitudinal displacement in Helmholtz's theory 

 is ^. K/(K - X)ytX. On the other hand, the velocity of 

 propagation of a wave of transverse displacement is 

 V i//it(K — X). Thus there will be no longitudinal wave 

 if one of the conditions, X- = o, X = o, K = X, hold. The 

 last condition would make the velocity of propagation of 

 waves of transverse displacement infinite, and must be 

 rejected for every medium, even so-called vacuum, in 

 which light is propagated. Poincare adopts the second 

 hypothesis. 



Further, if to X a positive value sensibly greater than 

 zero be assigned, a wave of transverse displacement will 

 be given a velocity sensibly greater than that of light. 

 Thus, to pass to Maxwell's theory, it is necessary to 

 make X insensible, as this gives for the velocity of waves 

 of transverse displacement his value i/VmK, which is 

 known by experiment to be the velocity of light. 



The adoption of this value of X leads to all the electro- 

 magnetic equations of Maxwell, and to the conditions 

 J = o, duldx -\- dv'dy + divldz = o, the last of which 

 expresses that electricity considered as the analogue of a 

 fluid is incompressible — that is, that all currents flowing 

 are closed currents. 



It may be pointed out here that this conclusion agrees 

 with that arrived at by Mr. R. T. Glazebrook in his com- 

 parison of Maxwell's electromagnetic equationswith those 

 of Helmholtz and Lorentz, and in his further paper on the 

 general equationsof the electromagnetic field (Proc. Camb. 

 Phil. Soc, vol. v.. Part ii., 1884). Mr. Glazebrook's result 

 is that *, the electrostatic potential of Helmholtz's paper 

 must be zero everywhere in order to pass from Helm- 

 holtz's theory to that of Maxwell. But * is what Poincare 

 has denoted by X<^, so that the conditions are the same. 



It has been attempted to make the transition to 

 Maxwell's theory by putting k = o. This would not 

 suffice alone, as Poincard points out. For, while giving 

 at once J = o, it would fail to give Maxwell's velocity for 

 transverse waves unless further X = o, which by itself 

 would suffice to effect the transition irrespective of the 

 value of k. 



Poincard thus supposes, in opposition to the older 

 dielectric theories, that even vacuum, or the ether of 

 the inter-planetary spaces, consists of polarizable con- 

 ducting matter embedded in an insulating medium of 

 infinitely small inductive capacity, X. According t» 



