58o 



NATURE 



[Oct. ii, 1888 



The constant M, according to (28), depends on the two current 

 elements, and measures the electrical energy of the medium 

 between them. 



In (30) 6 = o, 77 = o ; in (31) e = o, 93 = 1 ; in (32) e = 1, 

 T) = o; in (33) e = 1, 7j = i. 1 



Substituting for p its value from (29), and neglecting the 

 second and higher powers, we find for the electro-dynamic 

 potential of the two current elements — 



2M 

 dV — , cos 6 cos 0' dsds' (34) 



C" 



which gives for the potential of two closed circuits — 



V 



-*// 



M cos cos 6' dsds' . 



•(35) 



where M is an electrostatic constant and c the velocity of light. 



In the case of closed circuits we know that the value of V 

 remains unchanged if cos cos 0' is replaced by cos (as, ds'), 

 and therefore we arrive at Neumann's expression for the mutual 

 potential of two closed circuits, namely — 



V = A / /m cos (ds.ds') dsds' (36) 



These expressions for V have been obtained by neglecting the 

 second and higher powers of p/c, \jc . ds/dt, and i/c . ds'/dt ; 

 moreover, the dependence of the energy on the wave-length was 

 only expressed in terms of a mean value, \' ; so that the ex- 

 pressions are only to be considered as approximately true. It 

 is evident that they cannot hold good if either of the quantities 



P> y > or become equal to or greater than the velocity of light — 

 dt dt 



that is to say, both the relative and absolute velocities of the 



particles must be less than that of light ; and it will be shown in 



what follows that this limitation is of the utmost importance. - 



§ 14. Weber s Fundamental Law. 

 von Helmholtz has investigated the mutual potential of two 

 current elements on the assumption that it is of the form — 



— --J (1 + k) cos (ds,ds') + (1 - k) cos 6 cos 6' Ydsds'. 



Putting k = - 1, this expression agrees with Weber's law and 

 also with (34), showing that the author's theory leads to Weber's 

 law. In fact, putting 6 = o, A' = ir, and ds = ds' — dr/2, and 

 taking the sum 3 of the electrostatic and electro-dynamic poten- 

 tials, we arrive at Weber's expression for the potential of the 

 two particles, namely — 



and the author's expression for d V leads to Weber's expression 

 for the repulsion between two particles, namely — 



r* I 2c 1 \dt J c* df J ' 

 von Helmholtz's objections against Weber's law must now be 

 considered, and his own examples may be taken. 4 



1 All the electric rays proceeding from 2 will not be absorbed by 1' unless 

 (§ 12) the two conduc tors are of the same material ; if they are of different 

 material, e and 11 can only approximately assume the value unity, and there- 

 fore the expression (35) will only give an approximate value of the mutual 

 potential. From a physical point of view, it would perhaps be more reason- 

 able to assume that the particles in the elements ds and ds' respectively, 

 instead of being, one strongly electrified and one unelectrified, are distri- 

 buted in an approximately regular manner throughout all the intermediate 

 stages. In this case the sum of the four expressions (3o)-(33) will have to 

 be replaced by a double integral, of which this sum will be the mean value. 



* These conditions are known experimentally to be fulfilled, for while the 

 velocity of light is about 300,000 kilometres a second, that of electricity in 

 wires is, according to Fizeau, Gounelle, Frohlich, and W. Siemens, from 

 100,000 to 260,000 kilometres a second, See Sir W. Thomson, " Mathe- 

 matical and Physical Papers,". vol. ii. p. 131, and Wullner's " Experimental 

 Physik," vol. iv. p. 403, 4th edition. According to the author's theory, 

 the propagation of electric waves in vacuo must take place with the velocity 

 of light ; but the theory would not be affected if the velocity in air were 

 found to be different. See vonHelmholtz, " WissenschaftlicheAbhandlungen," 

 vol. ii. p. 629 et seq. In fact. Hertz has found this velocity to be distinctly 

 greater than that of \\gh.t(Sitziingsberickte der Berliner Akademie, Febru- 

 ary 1888). The increase may be due to the electrical excitation of the air 

 particles, and their consequent repulsive action on one another. With 

 respect to electro-dynamic determinations of the constant c, see Himstedt, 

 Wiedemann's Annalen, vols, xxviii. and xxix. 



3 See Riemann, " Schwere, Electricitat, und Magnetismus," §§ 96 and 97. 

 It should be noted that Riemann uses c to denote the velocity of light mul- 

 tiplied by a/2. It may also be noted that the author uses ds/dt and ds'ldt to 

 denote the velocity of propagation of an electrical disturbance, and not 

 directly that of a molecule. 



* " Wissenschaftliche Abhandlungen," vol. ii. p. 636 et seq. The two equa- 

 tions which follow may be interpreted as meaning that the quantity of 

 electricity in motion depends on r, which is in agreement with § 12. 



Suppose a ponderable electrified particle of mass /j. to be re- 

 pelled by a stationary quantity of electricity at the origin, in the 

 direction of the joining line r. Let a force R of the ordinary 

 kind act on the m.iss p so as to diminish r, then the differential 

 equation of motion of the electrified particle will be — 



d-r M I I fdr\" , 



or, putting M 



£*» 



PV.C-- 



fx{\ 



rjdt- . r- I 2c\dt) ) 



+ R. 



Choosing the initial circumstances, so that t = o, when the 

 velocity and the work done by R are both zero, and supposing 

 that r then has the value r, the principle of conservation of 

 energy gives — 



■ K*-|)(£)'= M (K) + * 



where 



m 



Jo dt 



If, now, R;- 2 < - M, von Helmholtz points out that the moving 

 particle must always approach the stationary one ; its velocity 

 meanwhile increases without limit until, for a distance r = p 

 (the so-called critical distance, see § 12), it becomes infinite, so 

 that a finite force can give an infinitely great velocity to a 

 mass ft by a finite expenditure of work. This impossible result 

 is not, however, a consequence of the author's theory, owing 



to the limitations stated at the end of § 13. For if the velocity . — 



at 



increases without limit, it must exceed that of the velocity of 



light, and then Weber's law ceases to hold good. 



It would be easy, by expanding the four previously-considered 

 j partial potential expressions, in terms of c/p, c/ds/dt, and 

 c I ds'jdt, to obtain a law for the further motion ; but there is no 

 object in doing so, as it will be seen from what follows that this 

 new law would again only hold up to a certain limit not far 

 removed from the first. 



In the first place, it is doubtful whether, when moving so 

 rapidly, the ponderable molecules could traverse the ether 

 without resistance. In the second place, the electrical energy 

 transferred from the fixed origin to the moving particle has been 

 assumed to be inversely proportional to the wave-length, and 

 the latter has been regarded as varying gradually within the 

 given limits. This was allowable for good conductors, since 

 their molecules must be specially sensitive to electrical dis- 

 turbance, and therefore have a very large number of very small 

 critical periods. With the very great velocity assumed, the 

 wave-lengths of the disturbances proceeding from the origin will 

 be greatly shortened before acting on the mass /*. It will follow, 

 therefore, that only such vibrations will cause electrical excita- 

 tion which already have so great a wave-length that they will 

 really appear as light, or ultra-violet, vibrations, and not as 

 electrical vibrations. Now, in the case of all known substances, 

 these critical wave-lengths do not come together in great num- 

 bers, and therefore cannot be treated as forming a continuous 

 series. 



If such rays are emitted from the origin, they can only give 

 rise to electrical excitation by separate impulses, and will there- 

 fore only cause a slight temporary variation in the acceleration 

 of the particle fj. due to the steady action of the force R. 



We may therefore conclude that a particle easily susceptible 

 of electric excitation will be electrified if it is made to approach 

 a source of light with very great velocity, and this the more 

 readily, the higher the refrangibility of the light from the source. 

 The requisite velocity must exceed that of light by a definite 

 amount. 



The author is not aware that this conclusion has as yet been 

 directly verified by any experimental evidence, unless Hertz's 

 observations of the effect of light on the electric spark x may be 

 explained in this way, but it is indirectly supported by the 

 phenomena observed in Geissler tubes, as will be shown below. 

 Consider, moreover, the motion of the particle n away from the 

 origin at an equally great velocity, then electrical waves proceed- 

 ing from the origin will be lengthened, and act on the particle 

 as light waves, causing it to glow. This electric glow will first 

 appear of a blue colour, gradually passing through the various 

 colours of the spectrum towards the red, as the velocity further 



1 Sitznngsberichtc der BerlinerAkademie, 1887, pp. 487 and 895. 



