:62 



NA TURE 



[February 19, 1903 



waves are surging along its surface. This provides a 

 knowledge of the ratio in which the distance of the open 

 end from the nearest node falls short of half a wave ; 

 and, the other successive nodes being practically equi- 

 distant, it thus affords a knowledge of the free periods in 

 terms of the length of the wire. Finite curvature of the 

 wire does not sensibly affect things ; this was elucidated 

 very clearly by Pocklington in 1897/ and his analytical 

 device for replacing the electrification and electric flow 

 on the wire by a series of changing electric doublets situ- 

 ated along it, the fields of disturbance of which are simply 

 expressible, is here largely employed. Consider, in fact, 

 a system of doublets of moment o- (in the magnetic 

 sense) per unit length distributed along the length j- of 

 the wire ; they are equivalent to a current of intensity 

 dcr/di, a charge of line-density -dvldx, and two point- 

 charges -o^and +<r 2 at the ends; as the true current 

 vanishes at the ends, o- must be constant there, and so 

 will vanish too. This kind of theory leads very directly, j 

 in § 67, to the character of the forced oscillation on the 

 wire that is established by an impressed magnetic field in 

 the surrounding region, which is symmetrical and there- 

 fore ranged in circles around the wire as axis. Each 

 infinitesimal ring of impressed alternating magnetic force 

 is propagated out directly into wider rings until it meets 

 the point of the wire under consideration, but in addition 

 it travels to the open end of the wire and thence down 

 its ength, the signs being such that the two parts cancel 

 at the end ; the amplitudes of these two interfering 

 systems of rays of magnetic force are not attenuated 

 with increase of the distance traversed, because each 

 point of the wire is equidistant from all elements of 

 the source. It is their interference that constitutes the 

 standing waves on the wire. We have here rings of 

 magnetic disturbance radiated from the outside sources, 

 converging on the wire through its open end, and 

 travelling down it ; it would appear that the author's 

 restriction to symmetry may largely be dispensed with. 

 The conditions are now reversed, and a system of 

 standing oscillations on the wire pouring out radiation 

 into space is contemplated ; that occurs only through 

 open ends, the oscillatory surging on the perfectly 

 conducting wire elsewhere being capable of adjusting 

 itself locally, like waves on a musical cord, without having 

 to constrain any radiation. If we know the dis- 

 tribution of the radiation from the open end of a straight 

 wire, over the infinite sphere, we can, by reversal of the 

 motions and treating the infinite spherical surface as a 

 region of sources of disturbance, deduce by the previous 

 analysis the positions of the nodes on the wire. In 

 applying this method, the author considers (§ 78), for 

 reasons not obvious, that an open end radiates uniformly 

 over the hemisphere in front of it. 2 In the dis- 

 cussion of the Hertzian wire resonator which follows, 

 the two contiguous ends are taken to constitute a 

 Hertzian oscillating doublet, and this determines the re- 



1 Proc. Camb. Phil. Soc. In this powerful paper, the radiation from a 

 complete circular wire comes in evidence, in a second-order approximation, 

 through a very slight damping of the free oscillations. On the view above 

 described, there should be no such effect ; yet, on the othpr hand, the 

 electricities can be separated to the two sides of the ring bv an electric 

 field, and should surge back in vibratory manner when released. 



- It appears that tire assumption of a considerably different law would 

 not much affect the result. 



NO. I738, VOL. 67] 



quired distribution of radiation at infinite distance ; the 

 reversed radiation is supposed to affect the two ends in- 

 dependently. One feels more confidence here than in 

 the previous case of a single end ; and the results are, in 

 fact, in very close agreement with experimental measure- 

 ments by Sarasin and de la Rive. The modification 

 arising from arming the ends with small balls or plates is 

 also gone into. 



The author's verification of the known form of the 

 wave-fronts near the open end of a wire, namely confocal 

 paraboloids with focus at that end, also comes from the 

 reversed motion as above. It appears that this result 

 holds whatever be the distribution of the radiation over 

 the infinite sphere, the magnetic force around the end 

 being of the form A tanAtl. The author adverts to the 

 transverse wave-fronts travelling along the wire towards 

 the end and finally bending round near the end into 

 paraboloids as it is approached ; the wave-front may be 

 considered as detained on the wire because the magnetic 

 force is cyclic around the wire and could not be cyclic 

 if the front escaped into free space. In fact, the value of 

 the magnetic force above given obviously satisfies this 

 necessary condition, its circulation 27rr sin 0.A. tani£> 

 being equal to 47rAr along the wire and equal to zero 

 along its prolongation ; the current in the wire near the 

 end is thus A>. We have, therefore, only to show that 

 the characteristic equation of a magnetic field disposed 

 in circles around the wire is satisfied ; and this is so, for 

 by the Amperean relation it leads to a longitudinal 

 component Z of electric force proportional to r~\ which 

 is of the right form, being near the end practically 

 e" r jr, which satisfies the equation V -Z + k 2 Z = o. The 

 transverse component of the electric force s similarly 



found to be proportional to 



Hanifl thus the 



resultant force is in the direction bisecting the angle 

 between r and the direction of the wire produced, and 

 is therefore tangential to parabolic wave-fronts as above 

 stated, being wholly transverse close to the wire. 

 There is some temptation to imagine the wings of the 

 parabolic part as advancing towards each other and 

 forming a narrow neck which is finally nipped through, 

 the main part of the front then going off as a plane 

 sheet of radiation, while the other part retreats back 

 into the wire and gives rise to a reflected wave, some- 

 what in the manner described by Hertz (" Electric 

 Waves,'' p. 144) for the case of an oscillating doublet. 1 

 For free oscillations on a wire with two ends, the 

 radiation is, however, sideways. 



The circumstance that the general features of some of 

 the author's conclusions can be traced by simple reason- 

 ing, as he himself indicates, does not, of course, detract 

 from their value or novelty ; it rather tends to confirm the 

 validity of the powerful mathematical analysis to the 

 results of which they are a first approximation, and should 

 stimulate similar inquiry as regards the other part of his 

 results. That this type of analysis is yet destined to 

 point the way into the heart of other important problems 

 in mathematical physics there can be no doubt ; now 

 that spherical and ellipsoidal forms have received such 



1 Mr. Macdonald informs me that this view is supported by the graph of 

 his second approximation in § 77. 



