August 23, 1888] 



NATURE 



405 



to the displacement of the centre relatively to the external ether. 

 Let x v x 2 , . . . xj, be the absolute displacement of the/shells, 

 and £ the displacement of the ether ; and let c x , c.,, . . . cj, be 

 the magnitudes of the elastic forces. We then have the following 

 equations : — 



JV1 -t (t JC-t / w \ / \ 



. . .(1) 



4^ 



Let the point £ have a periodic motion given by 



I 



(2) 



Then this motion will gradually be communicated to the centres 

 of Ihe shells in a manner which has been fully worked out by 

 Thomson. The value of T will vary, and after a certain 

 interval a steady condition will be arrived at in which all the 

 points will have periodic motions, so that 



Xi = a t - cos 



awt 

 T 



(3) 



where T is now arbitrary. 



Writing a x = M,/T 2 - a -c* + i, equations (1) give 



- C A « «. - c «; 



_ y 



which may be written in the form — 



c£ m x I K x 2 -T 2 



K 2 2 R 2 , 

 K 2 -T 2 



K/Ry 



" J ,1 . (4) 

 K,- 2 - T 2 / w 



The constant R t represents the ratio of the energy of the 

 shell mi to the total energy of the system. The quantity Kj is 

 determined by the condition that when T = Kj the ether 

 remains at rest, or £ = o ; and it may be called a critical period 

 of the molecule, which will accordingly have / critical periods, 

 and the molecule may undergo vibrations corresponding to any 

 or all of them simultaneously without affecting the external 

 ether. 



Instead of this somewhat artificial structure, the molecule 

 may be regarded as consisting of a sphere filled with continuous 

 matter of density varying with the radius, the density having 

 different values for each of/' assigned values of the radius, but 

 though this would be a simpler physical representation, it would 

 lead to great difficulties in the mathematical treatment, though 

 the results would necessarily be of a similar nature to those 

 obtained for the discrete molecule, and it is therefore preferable 

 to retain this representation. 



To apply the theory to transparent media let M z /4ir 2 represent 

 the thickness instead of the mass of a shell, and let p/4* 2 and 

 //4tt' ; be the density and elasticity respectively of the ether. 



The vibrations of the ether will then be given by the 

 equation 



y dt- dx 1 



(5) 



And the vibrations of the outermost shell will, in virtue of the 

 assumptions which have been made, be connected with those of 

 the neighbouring ether particle £ by an equation of the form 



a*/**;^-©^ 



df- 



dx 



(6) 



in which c x only differs from its former value by an unimportant 

 factor. The axis of x is here supposed to be perpendicular to 

 the line of centres, or diameter, of the molecule. 



Suppose a light -wave in a direction perpendicular to this axis, 

 and given by the equation 



\ 



-<z-t) 



(7) 



to strike the molecule ; then on the assumption that within a 

 definite interval only one wave strikes the molecule, or that the 

 diameter of the molecule is small in comparison with the wave- 



M = 



length, where /* is the index of refraction of the medium, and 

 v the velocity of the wave in it, equation (6) gives the equation 



expressing the index of refraction as a function of the period of 

 vibration of the ray. For waves of period equal to one of the 

 critical periods of the molecule, jj. becomes infinite, so that the 

 medium is opaque for such waves, which are entirely absorbed 

 in increasing the energy of the internal vibrations of the mole- 

 cules. The critical periods of the molecule are therefore the 

 vibration-periods of the dark lines of its absorption spectrum. 



§ 2. — The Index of Refraction as a Function of the 

 Wave-Length. 



As a preliminary to the more general investigation, it will be 

 advisable to trace the dependence of the index of refraction upon 

 the period of vibration in the simple cases / = 1 and/ = 2. 



For/ = 1 the molecule will consist of a core and a single 

 shell, and equation (8) will reduce to 



t - f il! - c *Jt K2 R 



/ / " ~lm x K 2 - T 2 *" 



F 



(9) 



Writing 



P - 

 I 



- ii- 

 / 



ft - 



lm x 



T 2 



(10) 

 (11) 



this may be written in the form 



j(K 2 - x) = (a + $x) (K 2 - x) +yx 2 



the equation of a hyperbola having the asymptotes 



x = K x 2 , y = (0 - i)x + a - 7K 2 



The former represents the single critical period, and the latter 

 practically determines by its direction whether the index of 

 refraction increases or diminishes as T, the period of vibration, 

 increases, and this the more exactly the more nearly the curve 

 coincides with its asymptotes — that is, the more nearly the value 

 of its determinant, which reduces to - 7K 2 /4 approaches the 

 value zero. 



There will therefore be three cases to consider — 



(a) |8 - y > o, /u increases as T increases. 



(d) - 7 = o, fi approximately constant. 



(c) j8 - 7 ■< o, /j. diminishes as T increases. 



There will be two expansions for fi- in powers of T, viz. : 

 For T < K, 



'L^T-'-fl - I' 

 lm x \ K 2 



+ 



K* 



+ &c 



= 1 -S X T 2 



I I 

 For T > K, 



/x 2 = o + fix 



T* 



&c 



} 



■} ■ (12) 



yx 



K- 



1 + - - + 



k : + 



&c, 



_ p , cy-KHl _ fj 



f + - 



//«! 



+ 



U' 



^ 2 K 8 R 



<-iK 2 R\ Ti 

 m, I 



I + 



K 2 



+ &c. 



(12a) 



The coefficient of T 2 must be very small in order that the 

 formulae may be in accordance with experimental results. 



Both the equations (12) and (12a) give, as a first approxi- 

 mation to the relation between wave-length and period of 

 vibration in the medium considered — . 



= VK^7 M 



But A is approximately proportional to T, so that 



f = A + B * 2 + *£b' 



where \ is the wave-length corresponding to the period T = K. 

 This agrees with the results of Helmholtz's theory, and with 

 experiment. 1 

 For values of T not in the neighbourhood of K, the hyperbola 



1 x Wullner's " Experimental-Physik," vol. ii. p. 161, fourth edition. 



