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XI. The 1 "i heory of Anomalous Dispersion. 

 By Lord Rayleigh, F.R.S.* 



I HAVE lately discovered that Maxwell, earlier than 

 Sellmeier or any other writer, had considered this 

 question. His results are given in the Mathematical Tripos 

 Examination for 1869 (see 'Cambridge Calendar' for that 

 year). In the paper for Jan. 21, l^ h -4 h , Question IX. is: — 



" Shew from dynamical principles that if the elasticity of a 

 medium be such that a tangential displacement rj (in the 

 direction of y) of one surface of a sfratum of thickness a 

 calls into action a force of restitution equal to Kv/a per unit of 

 area, then the equation of propagation of .such displacements is 



d 2 y _ ^ d 2 r) 

 P dt 2 ~ dx 2 ' 



" Suppose that every part of this medium is connected with 

 an atom of other matter by an attractive force varying as 

 distance, and that there is also a force of resistance between 

 the medium and the atoms varying as their relative velocity, 

 the atoms being independent of each other; show that the 

 equations of propagation of waves in this compound medium 

 are 



d 2 V vd'v I rf?\ Jd 2 l ,d%\ 



Pd¥ ~ B d? =a \ 1J ^dtJ^-^KW + 3?> 



where p and a are the quantities of the medium and of the 

 atoms respectively in unit of volume, rj is the displacement of 

 the medium, and v + £ that of the atoms, ap 2 £ is the attraction, 

 and aRdt/dt is the resistance to the relative motion per unit 

 of volume. 



" If one term of the value of n be Ge~ x/l cos n(t — x/v), shew 

 that 



1 1 _ p 4- o" or? p" — n 2 



tf + ZV ~ ~E~ + "E" (2> 8 -n 8 ) 2 + RW 



2 _an 2 Rn 



vln~ E {f-?i 2 ) 2 + R 2 n 2 ' 



" If a be very small, one of the values of v 2 will be less than 

 E/p, and if R be very small v will diminish as n increases, 

 except when n is nearly equal to p, and in the last case I will 



* Communicated bv the Author. 



