r 



Nov. IS, I 



NATURE 



65 



practical purposes we may assume T^ = To, which will give as 

 the angle of rotation — 





The assumption Tj = Tj is not in exact agreement with the 

 foregoing equations for (41) and (42) give respectively — 



.12=-?- 



A, 2rr 



Tj ^ B / o B-i ~ 



Aj 2/fs + \^ -"^i "^ 4/.Tj'''' 



(45) 



where yu, and /x, represent the indices of refraction of the medium 

 for vibrations of the periods Tj and Tg respectively. 

 For Ti == T,, and therefore fi^ = ^2, it follows that — 



' * = <.-i) = 'p <^^> 



that is to say, the angle of rotation is inversely proportional to 

 the square of the period of vibration, a result which is in 

 approximate agreement with observation. 



We may bring this result into still closer agreement with 

 experiment by observing that B itself is a function of T. Now B 

 determines the inductive action of the solenoid on the small 

 circle, and must therefore be the differential of a potential, and 

 therefore inversely proportional to the square of the velocity of 

 light. The latter is, however, not constant in the medium, as it 

 is the reciprocal of the index of refraction, and therefore a known 

 function of T. 



We may therefore put — 



B = M-C, ,^ = TT ^, M- (47) 



where C is a constant. The function jx- can be deteimined as in 

 § 2 (August 23, p. 404). If we take Cauchy's approximate 

 formula — 



we obtain an expression for ^ of exactly the same character 

 as that which Boltzmann deduced from his experimental results. 

 The direction of the rotation depends upon the relative values 

 of Aj and Aj. For Tj = T,, (41) and (42) give i/Ai>i/A2, and 

 therefore y\i>o. If, however, Tj > T,, and 



A*.-'=-^^ + '; ('i'(Ti)- 1), 



then /Zj2<;u./ for media of normal dispersive power, while for 

 media of anomalous dispersive power ix^^>ix<f, and vice versA 

 for Tj< Tj. Now, it was found that the velocity of propagation, 

 z/j, of the first wave was diminished, while that of the second 

 wave, V.2, was increased, so that v<^>z\. 



The equations (41) and (42) may be written — 



B 



/Ai' 



_L z= " _ ^ 



v.^ '^'' Ik; 



Therefore when B is small enough we shall always have fi{>fi.2, 

 and therefore Tj >T2 in the case of anomalous dispersion, and 

 Tj<T2 in the case of normal dispersion. 

 In the latter case — 



Mi" 



Ti^-^A.- 



M2« 



T./ 



But in the former case it may be the reverse. The rotation of 

 the plane of polarization will therefore be generally in the 

 positive direction, being negative only in media of anomalous 

 dispersive power, always supposing that B does not exceed a 

 certain fixed limit. This result is confirmed by the observed 

 fact that the loevo-rotatory substances, chloride of iron, and 

 chromic acid, give anomalous dispersion. According to Kundt,^ 

 iron is dextro-rotatory in spite of its anomalous di-ipersion, 

 which may be explained by a large value of B, and, as a matter 

 of fact, the angle of rotation is exceptionally large. Maxwell's 

 theory also led to the result that the angle of rotation is approxi- 



' SitzuHgsberichle tier Berliner Akadeinie, February 1888. 



mately inversely proportional to A", but it gave no explanation of 

 the decomposition of the ray into two circularly-polarized rays. 

 Maxwell starts with the assumption of a circularly-polarized ray, 

 and his A appears to represent the wave-length of this ray, and 

 is therefore diflferent from the A of the author. His theory rests 

 on the assumption of the existence of molecular vortices, and 

 therefore his differential equations are not the same as 

 Lindemann's (40) and (40^)- 



It has been suggested that the electric current may possibly pro- 

 duce a structure in the medium, similar to that already existing in 

 crystals which rotate the plane of polarization. Hitherto, however, 

 this hypothesis has not been of much use in explaining the pheno- 

 menon, as no explanation has been given of the manner in which 

 the ray is decomposed into two circularly-polarized rays such as 

 occur, for example, in a crystal of quartz. The author endeavours, 

 on the other hand, to explain the action of quartz on the hypothesis 

 of an electro-magnetic effect of the ray of light passing through 

 the crystal. There is, however, a difference between the two 

 phenomena which has to be accounted for— namely, in the 

 crystal the rotation is reversed when the direction of the ray is 

 reversed, so that if the ray passes through the crystal, and re- 

 turns along its own path, the total rotation is zero, while in the 

 case of the solenoid the effect is to double the angle of rotation. 

 The molecules of quartz must therefore be oppositely related to 

 opposite directions, which seems to suggest that the arrangement 

 may be due to the passage of the ray itself causing electric ex- 

 citation in the quartz, and this is confirmed by the observed fact 

 that quartz can be electrified by the action of light. Now suppose 

 that the molecules of quartz are unequally susceptible to the 

 electrical action of the ether in different directions, and suppose 

 further that the molecules most sensitive to a ray in the direction 

 of the axis are arranged in a spiral, having for its axis the 

 principal axis of the crystal. Then a ray traversing the crystal 

 in the direction of the axis will successively produce an electrical 

 excitation at every point of this spiral, which will therefore act 

 exactly as if the spiral were a conductor carrying a current. The 

 effect on the ray will therefore be to decompose it into a pair of 

 circularly-polarized rays differing in wave-length and in rate of 

 propagation, and the plane of polarization will therefore be 

 rotated. If the direction of the ray is reversed, the direction of 

 the current in the spiral will be reversed, causing a rotation in 

 the opposite direction. 



§ 18. Fai-amagtictism and Dianiagnetisvt. 



According to the theory of the rotation of the plane of polar- 

 ization which was developed in the last paragraph, an electric 

 current traversing a medium excites small molecular currents, 

 each one of which acts like a magnet. These currents as a 

 whole cannot give rise to any magnetic action, since alternate 

 currents flow in opposite directions, but this will not be the case 

 if one set of currents is absorbed by the medium, and not the 

 other, which will happen if one and only oneof the wave-lengths 

 Aj and A.2, which together give the wave of length A, is one of 

 the critical wave-lengths for the molecules of the medium, while 

 the other is not. 



The existence of Amperian currents in magnets can be ex- 

 plained in a similar manner. Here the currents cannot be 

 excited by the action of light, but it may be assumed that the 

 molecules, even of rigid bodies, continuously perform small 

 steady vibrations about their positions of equilibrium, and there- 

 fore come into collision with the neighbouring molecules on 

 every side, thereby exciting the internal critical vibrations, which 

 are visibly communicated to the ether when the substance attains 

 the temperature of redness. 



If such a substance is placed within a solenoid, the light- 

 vibrations in the direction of its axis will remain unaffected, 

 while the perpendicular ones will be decomposed into opposite 

 circular vibrations ; and the same thing will happen to the com- 

 ponents perpendicular to the axis of vibration in any other 

 direction, the components parallel to the axis remaining un- 

 changed. If one of these currents is absorbed, and the remaining 

 one is in the same direction as the current in the solenoid — that is, 

 with a right-handed rotation— the substance will be paramagnetic, 

 while if the rotation is left-handed the substance will be dia- 

 magnetic. In order that a sufficiently large number of vibrations 

 should be absorbed, the substance must have a large number of 

 critical periods, and therefore its spectrum must contain a large 

 number of lines, a result which agrees with the fact that the 

 spectrum of iron contains a larger number of lines than that of 

 any other known element. 



